Subgroups — C₃₁×#22507 #21479266

Order 7936.

1126 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3475	C₂	1	1	yes	maximal, characteristic
2	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1986	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1897	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1871	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1826	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×#104	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1733	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#556	C₄	1	1	yes	characteristic, complemented
14	C₃₁×#555	C₄	1	1	yes	characteristic, complemented
15	C₃₁×#554	C₄	1	1	yes	characteristic, complemented
16	C₃₁×#553	C₄	1	2	yes	complemented
17	C₃₁×#552	—	2	1	no	—
18	C₃₁×#552	C₄	1	2	yes	complemented
19	C₃₁×#551	—	2	2	no	—
20	C₃₁×#547	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#546	—	2	1	no	—
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₃₁×#529	—	2	2	no	—
24	C₃₁×#529	—	2	1	no	—
25	C₃₁×#525	C₂²	1	2	yes	—
26	C₄×#41×C₃₁	—	2	1	no	—
27	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#117	C₂²	1	2	yes	—
30	C₂×C₃₁×#117	C₂²	1	2	yes	—
31	C₃₁×#436	—	2	1	no	—
32	C₃₁×#435	—	2	2	no	—
33	C₈×D₄×C₃₁	—	2	1	no	—
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#104	C₂²	1	2	yes	—
38	C₃₁×#407	C₂²	1	2	yes	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#390	—	2	1	no	—
41	C₃₁×#389	—	2	1	no	—
42	C₃₁×#388	—	2	2	no	—
43	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
45	C₄³×C₃₁	C₂²	1	1	yes	characteristic
46	C₃₁×#339	C₂²	1	2	yes	—
47	C₃₁×#339	C₂²	1	1	yes	characteristic
48	C₃₁×#339	C₂²	1	2	yes	—
49	C₃₁×#339	C₂²	1	2	yes	—
50	C₃₁×#339	C₂²	1	2	yes	—
51	C₃₁×#339	C₂²	1	2	yes	—
52	C₃₁×#339	C₂²	1	1	yes	characteristic
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	2	4	no	—
59	C₂×M₁₆×C₃₁	—	4	2	no	—
60	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
66	C₃₁×#126	—	2	2	no	—
67	C₃₁×#126	—	2	2	no	—
68	C₃₁×#126	—	2	2	no	—
69	C₃₁×#125	—	2	2	no	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#125	C₂×C₄	1	2	yes	—
72	C₃₁×#124	—	2	1	no	—
73	C₃₁×#124	—	2	2	no	—
74	C₃₁×#123	—	2	2	no	—
75	C₃₁×#123	—	2	2	no	—
76	C₃₁×#123	—	2	2	no	—
77	C₃₁×#123	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	1	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
83	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
84	C₃₁×#121	—	2	2	no	—
85	C₃₁×#121	—	2	1	no	—
86	C₃₁×#121	C₂×C₄	1	2	yes	—
87	C₃₁×#121	C₂×C₄	1	2	yes	—
88	C₃₁×#120	—	2	2	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	—	2	1	no	—
91	C₄×Q₈×C₃₁	—	2	1	no	—
92	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	—	2	2	no	—
100	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
101	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
102	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
103	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
104	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
105	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
106	C₄×D₄×C₃₁	D₄	1	2	yes	—
107	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
108	C₃₁×#117	—	2	2	no	—
109	C₃₁×#117	—	2	2	no	—
110	C₃₁×#117	—	2	1	no	—
111	C₃₁×#117	—	2	2	no	—
112	C₃₁×#117	—	2	1	no	—
113	C₃₁×#117	—	2	4	no	—
114	C₃₁×#117	—	4	2	no	—
115	C₃₁×#117	C₂×C₄	1	2	yes	—
116	C₃₁×#117	C₂×C₄	1	2	yes	—
117	C₃₁×#117	C₂³	1	1	yes	characteristic
118	C₃₁×#117	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
122	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
123	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
124	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
125	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
126	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
127	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
128	C₂×C₄²×C₃₁	—	2	2	no	—
129	C₂×C₄²×C₃₁	—	2	1	no	—
130	C₂×C₄²×C₃₁	—	2	1	no	—
131	C₂×C₄²×C₃₁	—	2	1	no	—
132	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
136	C₃₁×C₄⋊C₈	—	2	1	no	—
137	C₃₁×C₄⋊C₈	—	2	1	no	—
138	C₃₁×#104	—	2	2	no	—
139	C₃₁×#104	—	2	2	no	—
140	C₃₁×#104	—	4	2	no	—
141	C₃₁×C₂²⋊C₈	—	2	2	no	—
142	C₃₁×C₂²⋊C₈	—	2	2	no	—
143	C₃₁×#97	—	2	1	no	—
144	C₄×C₈×C₃₁	—	2	1	no	—
145	C₃₁×#95	—	2	2	no	—
146	C₃₁×#95	—	2	1	no	—
147	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
148	#41×C₃₁	—	2	1	no	—
149	#41×C₃₁	—	2	1	no	—
150	#41×C₃₁	—	2	1	no	—
151	#41×C₃₁	—	2	1	no	—
152	#41×C₃₁	—	4	4	no	—
153	C₂×Q₈×C₃₁	—	2	1	no	—
154	C₂×Q₈×C₃₁	—	2	1	no	—
155	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	—	2	1	no	—
158	C₂×D₄×C₃₁	—	2	2	no	—
159	C₂×D₄×C₃₁	—	4	2	no	—
160	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
161	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
162	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
163	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	2	no	—
167	C₂²×C₄×C₃₁	—	2	2	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	1	no	—
170	C₂²×C₄×C₃₁	—	2	1	no	—
171	C₂²×C₄×C₃₁	—	2	2	no	—
172	C₂²×C₄×C₃₁	—	2	1	no	—
173	C₂²×C₄×C₃₁	—	2	1	no	—
174	C₂²×C₄×C₃₁	—	2	1	no	—
175	C₂²×C₄×C₃₁	—	2	2	no	—
176	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
184	M₁₆×C₃₁	—	2	1	no	—
185	M₁₆×C₃₁	—	2	1	no	—
186	M₁₆×C₃₁	—	2	2	no	—
187	M₁₆×C₃₁	—	2	2	no	—
188	M₁₆×C₃₁	—	2	1	no	—
189	M₁₆×C₃₁	—	4	4	no	—
190	M₁₆×C₃₁	#41	1	2	yes	complemented
191	C₂×C₈×C₃₁	—	2	1	no	—
192	C₂×C₈×C₃₁	—	2	1	no	—
193	C₂×C₈×C₃₁	—	2	2	no	—
194	C₂×C₈×C₃₁	—	4	4	no	—
195	C₄⋊C₄×C₃₁	—	2	2	no	—
196	C₄⋊C₄×C₃₁	—	2	2	no	—
197	C₄⋊C₄×C₃₁	—	2	1	no	—
198	C₄⋊C₄×C₃₁	—	2	2	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	—	2	1	no	—
201	C₄⋊C₄×C₃₁	—	2	1	no	—
202	C₄⋊C₄×C₃₁	—	2	2	no	—
203	C₄⋊C₄×C₃₁	—	2	2	no	—
204	C₄⋊C₄×C₃₁	—	2	1	no	—
205	C₄⋊C₄×C₃₁	—	4	4	no	—
206	C₄⋊C₄×C₃₁	—	4	2	no	—
207	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
208	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
209	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
210	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
211	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
212	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
213	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
214	C₂²⋊C₄×C₃₁	—	2	2	no	—
215	C₂²⋊C₄×C₃₁	—	2	2	no	—
216	C₂²⋊C₄×C₃₁	—	2	2	no	—
217	C₂²⋊C₄×C₃₁	—	2	2	no	—
218	C₂²⋊C₄×C₃₁	—	2	1	no	—
219	C₂²⋊C₄×C₃₁	—	2	2	no	—
220	C₂²⋊C₄×C₃₁	—	2	2	no	—
221	C₂²⋊C₄×C₃₁	—	2	1	no	—
222	C₂²⋊C₄×C₃₁	—	2	1	no	—
223	C₂²⋊C₄×C₃₁	—	2	2	no	—
224	C₂²⋊C₄×C₃₁	—	4	4	no	—
225	C₂²⋊C₄×C₃₁	—	4	4	no	—
226	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
227	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
228	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
229	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
230	C₄²×C₃₁	—	2	2	no	—
231	C₄²×C₃₁	—	2	1	no	—
232	C₄²×C₃₁	—	2	2	no	—
233	C₄²×C₃₁	—	2	1	no	—
234	C₄²×C₃₁	—	2	1	no	—
235	C₄²×C₃₁	—	2	2	no	—
236	C₄²×C₃₁	—	2	1	no	—
237	C₄²×C₃₁	—	2	1	no	—
238	C₄²×C₃₁	—	2	2	no	—
239	C₄²×C₃₁	—	2	2	no	—
240	C₄²×C₃₁	—	2	1	no	—
241	C₄²×C₃₁	—	4	2	no	—
242	C₄²×C₃₁	—	4	2	no	—
243	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
244	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
245	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
246	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
247	C₄²×C₃₁	#41	1	2	yes	complemented
248	#22507	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂³×C₃₁	—	2	2	no	—
250	C₂³×C₃₁	—	2	2	no	—
251	C₂³×C₃₁	—	2	1	no	—
252	C₂³×C₃₁	—	4	2	no	—
253	C₂³×C₃₁	—	4	2	no	—
254	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
255	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
256	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
257	Q₈×C₃₁	—	2	1	no	—
258	Q₈×C₃₁	—	2	1	no	—
259	Q₈×C₃₁	—	4	2	no	—
260	Q₈×C₃₁	—	8	1	no	—
261	D₄×C₃₁	—	2	1	no	—
262	D₄×C₃₁	—	2	1	no	—
263	D₄×C₃₁	—	2	2	no	—
264	D₄×C₃₁	—	2	2	no	—
265	D₄×C₃₁	—	4	2	no	—
266	D₄×C₃₁	—	4	4	no	—
267	D₄×C₃₁	—	4	1	no	—
268	D₄×C₃₁	—	4	1	no	—
269	D₄×C₃₁	—	8	2	no	—
270	C₂×C₄×C₃₁	—	2	2	no	—
271	C₂×C₄×C₃₁	—	2	2	no	—
272	C₂×C₄×C₃₁	—	2	2	no	—
273	C₂×C₄×C₃₁	—	2	2	no	—
274	C₂×C₄×C₃₁	—	2	2	no	—
275	C₂×C₄×C₃₁	—	2	2	no	—
276	C₂×C₄×C₃₁	—	2	2	no	—
277	C₂×C₄×C₃₁	—	2	2	no	—
278	C₂×C₄×C₃₁	—	2	2	no	—
279	C₂×C₄×C₃₁	—	2	2	no	—
280	C₂×C₄×C₃₁	—	2	2	no	—
281	C₂×C₄×C₃₁	—	2	2	no	—
282	C₂×C₄×C₃₁	—	2	1	no	—
283	C₂×C₄×C₃₁	—	2	1	no	—
284	C₂×C₄×C₃₁	—	2	1	no	—
285	C₂×C₄×C₃₁	—	2	2	no	—
286	C₂×C₄×C₃₁	—	2	2	no	—
287	C₂×C₄×C₃₁	—	2	2	no	—
288	C₂×C₄×C₃₁	—	4	1	no	—
289	C₂×C₄×C₃₁	—	4	1	no	—
290	C₂×C₄×C₃₁	—	4	1	no	—
291	C₂×C₄×C₃₁	—	4	1	no	—
292	C₂×C₄×C₃₁	—	4	4	no	—
293	C₂×C₄×C₃₁	—	4	1	no	—
294	C₂×C₄×C₃₁	—	4	1	no	—
295	C₂×C₄×C₃₁	—	4	1	no	—
296	C₂×C₄×C₃₁	—	4	2	no	—
297	C₂×C₄×C₃₁	—	4	2	no	—
298	C₂×C₄×C₃₁	—	4	2	no	—
299	C₂×C₄×C₃₁	—	4	2	no	—
300	C₂×C₄×C₃₁	—	8	1	no	—
301	C₂×C₄×C₃₁	—	8	2	no	—
302	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
303	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
304	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
305	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
306	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
307	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
308	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
309	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
311	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
312	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
313	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
314	C₈×C₃₁	—	2	2	no	—
315	C₈×C₃₁	—	4	2	no	—
316	C₈×C₃₁	—	4	1	no	—
317	#3475	C₂×C₃₁	1	1	yes	characteristic
318	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
319	#2293	C₂×C₃₁	1	1	yes	characteristic, complemented
320	#1994	C₂×C₃₁	1	1	yes	characteristic, complemented
321	#1986	C₂×C₃₁	1	1	yes	characteristic, complemented
322	#1897	C₂×C₃₁	1	2	yes	complemented
323	#1871	C₂×C₃₁	1	2	yes	complemented
324	#1826	C₂×C₃₁	1	2	yes	complemented
325	#1806	C₂×C₃₁	1	1	yes	characteristic, complemented
326	C₄×#104	C₂×C₃₁	1	1	yes	characteristic, complemented
327	#1733	C₂×C₃₁	1	2	yes	complemented
328	C₂²×C₃₁	—	2	1	no	—
329	C₂²×C₃₁	—	2	1	no	—
330	C₂²×C₃₁	—	2	2	no	—
331	C₂²×C₃₁	—	2	2	no	—
332	C₂²×C₃₁	—	2	2	no	—
333	C₂²×C₃₁	—	2	2	no	—
334	C₂²×C₃₁	—	2	2	no	—
335	C₂²×C₃₁	—	2	2	no	—
336	C₂²×C₃₁	—	4	1	no	—
337	C₂²×C₃₁	—	4	1	no	—
338	C₂²×C₃₁	—	4	1	no	—
339	C₂²×C₃₁	—	8	2	no	—
340	C₂²×C₃₁	—	8	2	no	—
341	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
342	C₂²×C₃₁	#546	1	1	yes	characteristic
343	C₂²×C₃₁	#547	1	1	yes	characteristic
344	C₄×C₃₁	—	2	2	no	—
345	C₄×C₃₁	—	2	2	no	—
346	C₄×C₃₁	—	2	1	no	—
347	C₄×C₃₁	—	2	1	no	—
348	C₄×C₃₁	—	2	1	no	—
349	C₄×C₃₁	—	4	2	no	—
350	C₄×C₃₁	—	4	2	no	—
351	C₄×C₃₁	—	4	1	no	—
352	C₄×C₃₁	—	4	1	no	—
353	C₄×C₃₁	—	4	2	no	—
354	C₄×C₃₁	—	4	1	no	—
355	C₄×C₃₁	—	4	2	no	—
356	C₄×C₃₁	—	8	2	no	—
357	C₄×C₃₁	#421	1	2	yes	—
358	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
359	C₄×C₃₁	#520	1	1	yes	characteristic
360	C₄×C₃₁	#546	1	1	yes	characteristic
361	C₄×C₃₁	#547	1	1	yes	characteristic
362	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
363	#556	C₄×C₃₁	1	1	yes	characteristic, complemented
364	#555	C₄×C₃₁	1	1	yes	characteristic, complemented
365	#554	C₄×C₃₁	1	1	yes	characteristic, complemented
366	#553	C₄×C₃₁	1	2	yes	complemented
367	#552	—	2	1	no	—
368	#552	C₄×C₃₁	1	2	yes	complemented
369	#551	—	2	2	no	—
370	#547	C₄×C₃₁	1	1	yes	characteristic, complemented
371	#546	—	2	1	no	—
372	#533	C₂²×C₃₁	1	1	yes	characteristic
373	#529	—	2	2	no	—
374	#529	—	2	1	no	—
375	#525	C₂²×C₃₁	1	2	yes	—
376	C₄×#41	—	2	1	no	—
377	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
378	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
379	C₂×#117	C₂²×C₃₁	1	2	yes	—
380	C₂×#117	C₂²×C₃₁	1	2	yes	—
381	#436	—	2	1	no	—
382	#435	—	2	2	no	—
383	C₈×D₄	—	2	1	no	—
384	#423	C₂²×C₃₁	1	1	yes	characteristic
385	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
386	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
387	C₂×#104	C₂²×C₃₁	1	2	yes	—
388	#407	C₂²×C₃₁	1	2	yes	—
389	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
390	#390	—	2	1	no	—
391	#389	—	2	1	no	—
392	#388	—	2	2	no	—
393	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
394	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
395	C₄³	C₂²×C₃₁	1	1	yes	characteristic
396	#339	C₂²×C₃₁	1	2	yes	—
397	#339	C₂²×C₃₁	1	1	yes	characteristic
398	#339	C₂²×C₃₁	1	2	yes	—
399	#339	C₂²×C₃₁	1	2	yes	—
400	#339	C₂²×C₃₁	1	2	yes	—
401	#339	C₂²×C₃₁	1	2	yes	—
402	#339	C₂²×C₃₁	1	1	yes	characteristic
403	C₂×C₃₁	—	2	1	no	—
404	C₂×C₃₁	—	2	1	no	—
405	C₂×C₃₁	—	4	2	no	—
406	C₂×C₃₁	—	4	2	no	—
407	C₂×C₃₁	—	8	1	no	—
408	C₂×C₃₁	#2339	1	1	yes	characteristic
409	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
410	C₂×C₃₁	#2901	1	1	yes	characteristic
411	C₂×#41	—	2	1	no	—
412	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
413	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
414	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
415	C₂×M₁₆	—	2	2	no	—
416	C₂×M₁₆	—	2	4	no	—
417	C₂×M₁₆	—	4	2	no	—
418	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
419	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
420	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
421	C₂²×C₈	—	2	2	no	—
422	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
423	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
424	#126	—	2	2	no	—
425	#126	—	2	2	no	—
426	#126	—	2	2	no	—
427	#125	—	2	2	no	—
428	#125	—	2	1	no	—
429	#125	C₂×C₄×C₃₁	1	2	yes	—
430	#124	—	2	1	no	—
431	#124	—	2	2	no	—
432	#123	—	2	2	no	—
433	#123	—	2	2	no	—
434	#123	—	2	2	no	—
435	#123	—	2	2	no	—
436	C₂²⋊Q₈	—	2	1	no	—
437	C₂²⋊Q₈	—	2	1	no	—
438	C₂²⋊Q₈	—	2	1	no	—
439	C₂²⋊Q₈	—	2	2	no	—
440	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
441	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
442	#121	—	2	2	no	—
443	#121	—	2	1	no	—
444	#121	C₂×C₄×C₃₁	1	2	yes	—
445	#121	C₂×C₄×C₃₁	1	2	yes	—
446	#120	—	2	2	no	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	—	2	1	no	—
449	C₄×Q₈	—	2	1	no	—
450	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
451	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
452	C₄×D₄	—	2	1	no	—
453	C₄×D₄	—	2	2	no	—
454	C₄×D₄	—	2	1	no	—
455	C₄×D₄	—	2	1	no	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	2	no	—
458	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
459	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
460	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
461	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
462	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
463	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
464	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
465	C₄×D₄	D₄×C₃₁	1	2	yes	—
466	#117	—	2	2	no	—
467	#117	—	2	2	no	—
468	#117	—	2	1	no	—
469	#117	—	2	2	no	—
470	#117	—	2	1	no	—
471	#117	—	2	4	no	—
472	#117	—	4	2	no	—
473	#117	C₂×C₄×C₃₁	1	2	yes	—
474	#117	C₂×C₄×C₃₁	1	2	yes	—
475	#117	C₂³×C₃₁	1	1	yes	characteristic
476	#117	C₂³×C₃₁	1	2	yes	—
477	C₂×C₄⋊C₄	—	2	2	no	—
478	C₂×C₄⋊C₄	—	2	1	no	—
479	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
480	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
481	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₂²⋊C₄	—	2	2	no	—
483	C₂×C₂²⋊C₄	—	2	2	no	—
484	C₂×C₂²⋊C₄	—	2	2	no	—
485	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
486	C₂×C₄²	—	2	2	no	—
487	C₂×C₄²	—	2	1	no	—
488	C₂×C₄²	—	2	1	no	—
489	C₂×C₄²	—	2	1	no	—
490	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
491	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
492	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
493	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
494	C₄⋊C₈	—	2	1	no	—
495	C₄⋊C₈	—	2	1	no	—
496	#104	—	2	2	no	—
497	#104	—	2	2	no	—
498	#104	—	4	2	no	—
499	C₂²⋊C₈	—	2	2	no	—
500	C₂²⋊C₈	—	2	2	no	—
501	#97	—	2	1	no	—
502	C₄×C₈	—	2	1	no	—
503	#95	—	2	2	no	—
504	#95	—	2	1	no	—
505	C₃₁	#22507	1	1	yes	characteristic, direct product factor, Sylow
506	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
507	#41	—	2	1	no	—
508	#41	—	2	1	no	—
509	#41	—	2	1	no	—
510	#41	—	2	1	no	—
511	#41	—	4	4	no	—
512	C₂×Q₈	—	2	1	no	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
515	C₂×D₄	—	2	2	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	2	no	—
518	C₂×D₄	—	4	2	no	—
519	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
520	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
521	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
522	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	2	no	—
525	C₂²×C₄	—	2	2	no	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	2	1	no	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
536	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
537	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
538	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
539	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
540	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
541	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
542	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
543	M₁₆	—	2	1	no	—
544	M₁₆	—	2	1	no	—
545	M₁₆	—	2	2	no	—
546	M₁₆	—	2	2	no	—
547	M₁₆	—	2	1	no	—
548	M₁₆	—	4	4	no	—
549	M₁₆	#41×C₃₁	1	2	yes	complemented
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	2	no	—
553	C₂×C₈	—	4	4	no	—
554	C₄⋊C₄	—	2	2	no	—
555	C₄⋊C₄	—	2	2	no	—
556	C₄⋊C₄	—	2	1	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	—	2	2	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	2	1	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	1	no	—
564	C₄⋊C₄	—	4	4	no	—
565	C₄⋊C₄	—	4	2	no	—
566	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
567	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
568	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
569	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
570	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
571	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
572	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	2	no	—
575	C₂²⋊C₄	—	2	2	no	—
576	C₂²⋊C₄	—	2	2	no	—
577	C₂²⋊C₄	—	2	1	no	—
578	C₂²⋊C₄	—	2	2	no	—
579	C₂²⋊C₄	—	2	2	no	—
580	C₂²⋊C₄	—	2	1	no	—
581	C₂²⋊C₄	—	2	1	no	—
582	C₂²⋊C₄	—	2	2	no	—
583	C₂²⋊C₄	—	4	4	no	—
584	C₂²⋊C₄	—	4	4	no	—
585	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
586	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
587	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
588	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
589	C₄²	—	2	2	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	2	no	—
592	C₄²	—	2	1	no	—
593	C₄²	—	2	1	no	—
594	C₄²	—	2	2	no	—
595	C₄²	—	2	1	no	—
596	C₄²	—	2	1	no	—
597	C₄²	—	2	2	no	—
598	C₄²	—	2	2	no	—
599	C₄²	—	2	1	no	—
600	C₄²	—	4	2	no	—
601	C₄²	—	4	2	no	—
602	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
603	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
604	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
605	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
606	C₄²	#41×C₃₁	1	2	yes	complemented
607	C₂³	—	2	2	no	—
608	C₂³	—	2	2	no	—
609	C₂³	—	2	1	no	—
610	C₂³	—	4	2	no	—
611	C₂³	—	4	2	no	—
612	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
613	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
614	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
615	Q₈	—	2	1	no	—
616	Q₈	—	2	1	no	—
617	Q₈	—	4	2	no	—
618	Q₈	—	8	1	no	—
619	D₄	—	2	2	no	—
620	D₄	—	2	2	no	—
621	D₄	—	2	1	no	—
622	D₄	—	2	1	no	—
623	D₄	—	4	2	no	—
624	D₄	—	4	4	no	—
625	D₄	—	4	1	no	—
626	D₄	—	4	1	no	—
627	D₄	—	8	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	4	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	2	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	2	no	—
656	C₂×C₄	—	4	2	no	—
657	C₂×C₄	—	4	2	no	—
658	C₂×C₄	—	8	1	no	—
659	C₂×C₄	—	8	2	no	—
660	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
661	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
662	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
663	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
664	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
665	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
666	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
667	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
668	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
669	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
670	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
671	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
672	C₈	—	2	2	no	—
673	C₈	—	4	2	no	—
674	C₈	—	4	1	no	—
675	C₂²	—	2	1	no	—
676	C₂²	—	2	1	no	—
677	C₂²	—	2	2	no	—
678	C₂²	—	2	2	no	—
679	C₂²	—	2	2	no	—
680	C₂²	—	2	2	no	—
681	C₂²	—	2	2	no	—
682	C₂²	—	2	2	no	—
683	C₂²	—	4	1	no	—
684	C₂²	—	4	1	no	—
685	C₂²	—	4	1	no	—
686	C₂²	—	8	2	no	—
687	C₂²	—	8	2	no	—
688	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
689	C₂²	C₃₁×#546	1	1	yes	characteristic
690	C₂²	C₃₁×#547	1	1	yes	characteristic
691	C₄	—	2	2	no	—
692	C₄	—	2	2	no	—
693	C₄	—	2	1	no	—
694	C₄	—	2	1	no	—
695	C₄	—	2	1	no	—
696	C₄	—	4	2	no	—
697	C₄	—	4	2	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	1	no	—
700	C₄	—	4	2	no	—
701	C₄	—	4	1	no	—
702	C₄	—	4	2	no	—
703	C₄	—	8	2	no	—
704	C₄	C₃₁×#421	1	2	yes	—
705	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
706	C₄	C₃₁×#520	1	1	yes	characteristic
707	C₄	C₃₁×#546	1	1	yes	characteristic
708	C₄	C₃₁×#547	1	1	yes	characteristic
709	C₂	—	2	1	no	—
710	C₂	—	2	1	no	—
711	C₂	—	4	2	no	—
712	C₂	—	4	2	no	—
713	C₂	—	8	1	no	—
714	C₂	C₃₁×#2339	1	1	yes	characteristic
715	C₂	C₂×C₃₁×#421	1	1	yes	characteristic
716	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3475	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. D₄×M₁₆×C₃₁	C₂	semidirect
3. C₃₁×#2293	C₂	semidirect
4. C₃₁×#1994	C₂	semidirect
5. C₃₁×#1986	C₂	semidirect
6. C₃₁×#1897	C₂	semidirect
7. C₃₁×#1871	C₂	semidirect
8. C₃₁×#1826	C₂	semidirect
9. C₃₁×#1806	C₂	semidirect
10. C₄×C₃₁×#104	C₂	semidirect
11. C₃₁×#1733	C₂	semidirect
13. C₃₁×#556	C₄	semidirect
14. C₃₁×#555	C₄	semidirect
15. C₃₁×#554	C₄	semidirect
16. C₃₁×#553	C₄	semidirect
18. C₃₁×#552	C₄	semidirect
20. C₃₁×#547	C₄	semidirect
190. M₁₆×C₃₁	#41	semidirect
247. C₄²×C₃₁	#41	semidirect
248. #22507	C₃₁	direct (N × H)
318. D₄×M₁₆	C₂×C₃₁	semidirect
319. #2293	C₂×C₃₁	semidirect
320. #1994	C₂×C₃₁	semidirect
321. #1986	C₂×C₃₁	semidirect
322. #1897	C₂×C₃₁	semidirect
323. #1871	C₂×C₃₁	semidirect
324. #1826	C₂×C₃₁	semidirect
325. #1806	C₂×C₃₁	semidirect
326. C₄×#104	C₂×C₃₁	semidirect
327. #1733	C₂×C₃₁	semidirect
363. #556	C₄×C₃₁	semidirect
364. #555	C₄×C₃₁	semidirect
365. #554	C₄×C₃₁	semidirect
366. #553	C₄×C₃₁	semidirect
368. #552	C₄×C₃₁	semidirect
370. #547	C₄×C₃₁	semidirect
505. C₃₁	#22507	direct (N × H)
549. M₁₆	#41×C₃₁	semidirect
606. C₄²	#41×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3475	2
D₄×M₁₆×C₃₁	2
C₃₁×#2293	2
C₃₁×#1994	2
C₃₁×#1986	2
C₃₁×#1897	2
C₃₁×#1871	2
C₃₁×#1826	2
C₃₁×#1806	2
C₄×C₃₁×#104	2
C₃₁×#1733	2
#22507	31

Subgroups — C31×#22507 #21479266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22507 #21479266