Subgroups — C₃₁×#21900 #21478659

Order 7936.

1106 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1986	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1980	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1897	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1806	C₂	1	4	yes	maximal, complemented
8	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
9	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
10	C₃₁×#533	C₂²	1	1	yes	characteristic
11	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#529	—	2	1	no	—
15	C₃₁×#529	—	2	1	no	—
16	C₃₁×#529	—	2	1	no	—
17	C₃₁×#529	—	2	1	no	—
18	C₃₁×#529	C₄	1	2	yes	complemented
19	C₃₁×#529	C₄	1	2	yes	complemented
20	C₃₁×#525	—	2	1	no	—
21	C₃₁×#525	—	2	1	no	—
22	C₃₁×#525	C₂²	1	2	yes	—
23	C₄×#41×C₃₁	—	2	1	no	—
24	C₄×#41×C₃₁	—	2	1	no	—
25	C₄×#41×C₃₁	C₂²	1	2	yes	—
26	C₂×C₃₁×#117	C₂²	1	2	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#432	—	2	1	no	—
29	C₃₁×#432	—	2	1	no	—
30	C₃₁×#431	—	2	1	no	—
31	C₃₁×#431	—	2	1	no	—
32	C₃₁×#423	C₂²	1	2	yes	—
33	C₂×C₃₁×#104	C₂²	1	8	yes	—
34	C₃₁×#407	C₂²	1	2	yes	—
35	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#384	—	2	1	no	—
37	C₃₁×#382	—	2	1	no	—
38	C₃₁×#380	—	2	1	no	—
39	C₃₁×#380	—	2	1	no	—
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
42	C₃₁×#339	C₂²	1	8	yes	—
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂×#41×C₃₁	—	2	1	no	—
45	C₂×#41×C₃₁	C₂³	1	2	yes	—
46	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
50	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
52	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
53	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
54	C₃₁×#126	—	2	2	no	—
55	C₃₁×#126	—	2	2	no	—
56	C₃₁×#125	—	2	1	no	—
57	C₃₁×#125	—	2	1	no	—
58	C₃₁×#125	C₂×C₄	1	2	yes	—
59	C₃₁×#124	—	2	1	no	—
60	C₃₁×#124	—	2	1	no	—
61	C₃₁×#124	C₂×C₄	1	2	yes	—
62	C₃₁×#123	—	2	2	no	—
63	C₃₁×#123	—	2	2	no	—
64	C₃₁×#123	—	2	2	no	—
65	C₃₁×C₂²⋊Q₈	—	2	1	no	—
66	C₃₁×C₂²⋊Q₈	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
70	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
71	C₃₁×#121	—	2	1	no	—
72	C₃₁×#121	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	C₂×C₄	1	2	yes	—
76	C₃₁×#121	C₂×C₄	1	2	yes	—
77	C₃₁×#120	—	2	2	no	—
78	C₄×Q₈×C₃₁	—	2	1	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×Q₈×C₃₁	—	2	1	no	—
81	C₄×Q₈×C₃₁	—	2	1	no	—
82	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
83	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
84	C₄×D₄×C₃₁	—	2	2	no	—
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	1	no	—
91	C₄×D₄×C₃₁	—	2	1	no	—
92	C₄×D₄×C₃₁	—	2	1	no	—
93	C₄×D₄×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
95	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
96	C₃₁×#117	—	2	1	no	—
97	C₃₁×#117	—	2	1	no	—
98	C₃₁×#117	—	2	2	no	—
99	C₃₁×#117	—	2	2	no	—
100	C₃₁×#117	—	2	2	no	—
101	C₃₁×#117	—	4	4	no	—
102	C₃₁×#117	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄²×C₃₁	—	2	2	no	—
108	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
119	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₃₁×C₄⋊C₈	—	2	2	no	—
122	C₃₁×C₄⋊C₈	—	2	2	no	—
123	C₃₁×#104	—	4	8	no	—
124	C₃₁×C₂²⋊C₈	—	2	2	no	—
125	C₃₁×C₂²⋊C₈	—	2	2	no	—
126	C₃₁×#97	—	2	2	no	—
127	C₄×C₈×C₃₁	—	2	2	no	—
128	C₃₁×#95	—	2	2	no	—
129	C₃₁×#95	—	2	2	no	—
130	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
131	#41×C₃₁	—	2	4	no	—
132	#41×C₃₁	—	2	4	no	—
133	#41×C₃₁	—	4	4	no	—
134	#41×C₃₁	—	4	4	no	—
135	C₂×Q₈×C₃₁	—	2	1	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₃₁	—	2	2	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	4	2	no	—
151	C₂²×C₄×C₃₁	—	4	2	no	—
152	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	M₁₆×C₃₁	—	2	4	no	—
171	M₁₆×C₃₁	—	2	4	no	—
172	C₂×C₈×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	—	4	4	no	—
184	C₄⋊C₄×C₃₁	—	4	4	no	—
185	C₄⋊C₄×C₃₁	—	4	4	no	—
186	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
187	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
188	C₂²⋊C₄×C₃₁	—	2	2	no	—
189	C₂²⋊C₄×C₃₁	—	2	2	no	—
190	C₂²⋊C₄×C₃₁	—	2	2	no	—
191	C₂²⋊C₄×C₃₁	—	2	2	no	—
192	C₂²⋊C₄×C₃₁	—	2	2	no	—
193	C₂²⋊C₄×C₃₁	—	2	2	no	—
194	C₂²⋊C₄×C₃₁	—	2	2	no	—
195	C₂²⋊C₄×C₃₁	—	2	2	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₂²⋊C₄×C₃₁	—	4	4	no	—
198	C₄²×C₃₁	—	2	2	no	—
199	C₄²×C₃₁	—	2	2	no	—
200	C₄²×C₃₁	—	2	2	no	—
201	C₄²×C₃₁	—	2	8	no	—
202	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
209	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
214	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
218	#21900	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₃₁	—	2	1	no	—
220	C₂³×C₃₁	—	2	1	no	—
221	C₂³×C₃₁	—	2	2	no	—
222	C₂³×C₃₁	—	4	1	no	—
223	C₂³×C₃₁	—	4	1	no	—
224	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
227	Q₈×C₃₁	—	2	4	no	—
228	Q₈×C₃₁	—	4	2	no	—
229	Q₈×C₃₁	—	4	2	no	—
230	D₄×C₃₁	—	2	4	no	—
231	D₄×C₃₁	—	4	2	no	—
232	D₄×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	4	4	no	—
234	D₄×C₃₁	—	4	2	no	—
235	D₄×C₃₁	—	4	2	no	—
236	D₄×C₃₁	—	4	2	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	4	1	no	—
253	C₂×C₄×C₃₁	—	4	1	no	—
254	C₂×C₄×C₃₁	—	4	1	no	—
255	C₂×C₄×C₃₁	—	4	1	no	—
256	C₂×C₄×C₃₁	—	4	1	no	—
257	C₂×C₄×C₃₁	—	4	1	no	—
258	C₂×C₄×C₃₁	—	4	4	no	—
259	C₂×C₄×C₃₁	—	4	4	no	—
260	C₂×C₄×C₃₁	—	4	4	no	—
261	C₂×C₄×C₃₁	—	4	4	no	—
262	C₂×C₄×C₃₁	—	4	4	no	—
263	C₂×C₄×C₃₁	—	4	4	no	—
264	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
275	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
277	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
278	C₈×C₃₁	—	4	4	no	—
279	#3444	C₂×C₃₁	1	1	yes	characteristic
280	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#1986	C₂×C₃₁	1	2	yes	complemented
283	#1980	C₂×C₃₁	1	2	yes	complemented
284	#1897	C₂×C₃₁	1	4	yes	complemented
285	#1806	C₂×C₃₁	1	4	yes	complemented
286	C₂²×C₃₁	—	2	1	no	—
287	C₂²×C₃₁	—	2	1	no	—
288	C₂²×C₃₁	—	2	1	no	—
289	C₂²×C₃₁	—	2	1	no	—
290	C₂²×C₃₁	—	2	1	no	—
291	C₂²×C₃₁	—	2	1	no	—
292	C₂²×C₃₁	—	2	1	no	—
293	C₂²×C₃₁	—	2	1	no	—
294	C₂²×C₃₁	—	4	2	no	—
295	C₂²×C₃₁	—	4	2	no	—
296	C₂²×C₃₁	—	4	2	no	—
297	C₂²×C₃₁	—	4	2	no	—
298	C₂²×C₃₁	—	4	2	no	—
299	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
300	C₂²×C₃₁	#535	1	1	yes	characteristic
301	C₂²×C₃₁	#536	1	1	yes	characteristic
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	2	1	no	—
304	C₄×C₃₁	—	2	1	no	—
305	C₄×C₃₁	—	2	1	no	—
306	C₄×C₃₁	—	2	1	no	—
307	C₄×C₃₁	—	2	1	no	—
308	C₄×C₃₁	—	2	1	no	—
309	C₄×C₃₁	—	2	1	no	—
310	C₄×C₃₁	—	4	2	no	—
311	C₄×C₃₁	—	4	2	no	—
312	C₄×C₃₁	—	4	2	no	—
313	C₄×C₃₁	—	8	4	no	—
314	C₄×C₃₁	#421	1	2	yes	—
315	C₄×C₃₁	#421	1	2	yes	—
316	C₄×C₃₁	#520	1	2	yes	—
317	C₄×C₃₁	#537	1	2	yes	—
318	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
319	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
320	#533	C₂²×C₃₁	1	1	yes	characteristic
321	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
322	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
323	#532	C₂²×C₃₁	1	1	yes	characteristic
324	#529	—	2	1	no	—
325	#529	—	2	1	no	—
326	#529	—	2	1	no	—
327	#529	—	2	1	no	—
328	#529	C₄×C₃₁	1	2	yes	complemented
329	#529	C₄×C₃₁	1	2	yes	complemented
330	#525	—	2	1	no	—
331	#525	—	2	1	no	—
332	#525	C₂²×C₃₁	1	2	yes	—
333	C₄×#41	—	2	1	no	—
334	C₄×#41	—	2	1	no	—
335	C₄×#41	C₂²×C₃₁	1	2	yes	—
336	C₂×#117	C₂²×C₃₁	1	2	yes	—
337	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
338	#432	—	2	1	no	—
339	#432	—	2	1	no	—
340	#431	—	2	1	no	—
341	#431	—	2	1	no	—
342	#423	C₂²×C₃₁	1	2	yes	—
343	C₂×#104	C₂²×C₃₁	1	8	yes	—
344	#407	C₂²×C₃₁	1	2	yes	—
345	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
346	#384	—	2	1	no	—
347	#382	—	2	1	no	—
348	#380	—	2	1	no	—
349	#380	—	2	1	no	—
350	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
351	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
352	#339	C₂²×C₃₁	1	8	yes	—
353	C₂×C₃₁	—	2	1	no	—
354	C₂×C₃₁	—	2	1	no	—
355	C₂×C₃₁	—	4	1	no	—
356	C₂×C₃₁	—	4	1	no	—
357	C₂×C₃₁	—	8	2	no	—
358	C₂×C₃₁	#2311	1	1	yes	characteristic
359	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
360	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
361	C₂×#41	—	2	1	no	—
362	C₂×#41	—	2	1	no	—
363	C₂×#41	C₂³×C₃₁	1	2	yes	—
364	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
365	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
366	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
368	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
369	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
370	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
371	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
372	#126	—	2	2	no	—
373	#126	—	2	2	no	—
374	#125	—	2	1	no	—
375	#125	—	2	1	no	—
376	#125	C₂×C₄×C₃₁	1	2	yes	—
377	#124	—	2	1	no	—
378	#124	—	2	1	no	—
379	#124	C₂×C₄×C₃₁	1	2	yes	—
380	#123	—	2	2	no	—
381	#123	—	2	2	no	—
382	#123	—	2	2	no	—
383	C₂²⋊Q₈	—	2	1	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	C₂²⋊Q₈	—	2	1	no	—
386	C₂²⋊Q₈	—	2	1	no	—
387	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
388	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
389	#121	—	2	1	no	—
390	#121	—	2	1	no	—
391	#121	—	2	1	no	—
392	#121	—	2	1	no	—
393	#121	C₂×C₄×C₃₁	1	2	yes	—
394	#121	C₂×C₄×C₃₁	1	2	yes	—
395	#120	—	2	2	no	—
396	C₄×Q₈	—	2	1	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
401	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	1	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
413	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
414	#117	—	2	1	no	—
415	#117	—	2	1	no	—
416	#117	—	2	2	no	—
417	#117	—	2	2	no	—
418	#117	—	2	2	no	—
419	#117	—	4	4	no	—
420	#117	C₂³×C₃₁	1	2	yes	—
421	C₂×C₄⋊C₄	—	2	2	no	—
422	C₂×C₄⋊C₄	—	2	2	no	—
423	C₂×C₄⋊C₄	—	2	2	no	—
424	C₂×C₂²⋊C₄	—	2	2	no	—
425	C₂×C₄²	—	2	2	no	—
426	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
428	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
435	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
437	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
439	C₄⋊C₈	—	2	2	no	—
440	C₄⋊C₈	—	2	2	no	—
441	#104	—	4	8	no	—
442	C₂²⋊C₈	—	2	2	no	—
443	C₂²⋊C₈	—	2	2	no	—
444	#97	—	2	2	no	—
445	C₄×C₈	—	2	2	no	—
446	#95	—	2	2	no	—
447	#95	—	2	2	no	—
448	C₃₁	#21900	1	1	yes	characteristic, direct product factor, Sylow
449	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
450	#41	—	2	4	no	—
451	#41	—	2	4	no	—
452	#41	—	4	4	no	—
453	#41	—	4	4	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
479	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
489	M₁₆	—	2	4	no	—
490	M₁₆	—	2	4	no	—
491	C₂×C₈	—	2	4	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	4	4	no	—
503	C₄⋊C₄	—	4	4	no	—
504	C₄⋊C₄	—	4	4	no	—
505	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
506	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	4	4	no	—
517	C₄²	—	2	2	no	—
518	C₄²	—	2	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	8	no	—
521	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
523	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
524	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
525	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
526	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
527	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
528	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
529	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
530	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
531	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
532	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
533	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
534	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
535	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
536	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
537	C₂³	—	2	1	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	1	no	—
542	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
543	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
544	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
545	Q₈	—	2	4	no	—
546	Q₈	—	4	2	no	—
547	Q₈	—	4	2	no	—
548	D₄	—	2	4	no	—
549	D₄	—	4	2	no	—
550	D₄	—	4	4	no	—
551	D₄	—	4	4	no	—
552	D₄	—	4	2	no	—
553	D₄	—	4	2	no	—
554	D₄	—	4	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	4	4	no	—
579	C₂×C₄	—	4	4	no	—
580	C₂×C₄	—	4	4	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
583	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
584	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
585	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
586	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
587	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
588	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
590	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
592	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
593	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
594	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
595	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
596	C₈	—	4	4	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	1	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
611	C₂²	C₃₁×#535	1	1	yes	characteristic
612	C₂²	C₃₁×#536	1	1	yes	characteristic
613	C₄	—	2	1	no	—
614	C₄	—	2	1	no	—
615	C₄	—	2	1	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	2	1	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	8	4	no	—
625	C₄	C₃₁×#421	1	2	yes	—
626	C₄	C₃₁×#421	1	2	yes	—
627	C₄	C₃₁×#520	1	2	yes	—
628	C₄	C₃₁×#537	1	2	yes	—
629	C₂	—	2	1	no	—
630	C₂	—	2	1	no	—
631	C₂	—	4	1	no	—
632	C₂	—	4	1	no	—
633	C₂	—	8	2	no	—
634	C₂	C₃₁×#2311	1	1	yes	characteristic
635	C₂	C₂×C₃₁×#421	1	1	yes	characteristic
636	C₂	C₂×C₃₁×#421	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×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2920	C₂	semidirect
3. C₃₁×#2285	C₂	semidirect
4. C₃₁×#1986	C₂	semidirect
5. C₃₁×#1980	C₂	semidirect
6. C₃₁×#1897	C₂	semidirect
7. C₃₁×#1806	C₂	semidirect
8. C₃₁×#533	C₄	semidirect
9. C₃₁×#533	C₄	semidirect
11. C₃₁×#532	C₄	semidirect
12. C₃₁×#532	C₄	semidirect
18. C₃₁×#529	C₄	semidirect
19. C₃₁×#529	C₄	semidirect
218. #21900	C₃₁	direct (N × H)
280. #2920	C₂×C₃₁	semidirect
281. #2285	C₂×C₃₁	semidirect
282. #1986	C₂×C₃₁	semidirect
283. #1980	C₂×C₃₁	semidirect
284. #1897	C₂×C₃₁	semidirect
285. #1806	C₂×C₃₁	semidirect
318. #533	C₄×C₃₁	semidirect
319. #533	C₄×C₃₁	semidirect
321. #532	C₄×C₃₁	semidirect
322. #532	C₄×C₃₁	semidirect
328. #529	C₄×C₃₁	semidirect
329. #529	C₄×C₃₁	semidirect
448. C₃₁	#21900	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3444	2
C₃₁×#2920	2
C₃₁×#2285	2
C₃₁×#1986	2
C₃₁×#1980	2
C₃₁×#1897	2
C₃₁×#1806	2
#21900	31

Subgroups — C31×#21900 #21478659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21900 #21478659