Subgroups — C₃₁×#21382 #21478141

Order 7936.

1256 conjugacy classes of subgroups.

The subgroup lattice has 626 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	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#117	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×#1871	C₂	1	4	yes	maximal, complemented
7	C₄×C₃₁×#104	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
10	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₃₁×#529	—	2	1	no	—
16	C₃₁×#529	—	2	1	no	—
17	C₃₁×#529	—	2	1	no	—
18	C₃₁×#529	—	2	1	no	—
19	C₃₁×#529	C₄	1	2	yes	complemented
20	C₃₁×#529	C₄	1	2	yes	complemented
21	C₃₁×#525	—	2	1	no	—
22	C₃₁×#525	—	2	1	no	—
23	C₃₁×#525	C₂²	1	2	yes	—
24	C₄×#41×C₃₁	—	2	1	no	—
25	C₄×#41×C₃₁	—	2	1	no	—
26	C₄×#41×C₃₁	C₂²	1	2	yes	—
27	C₂×C₃₁×#117	—	2	1	no	—
28	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#104	C₂²	1	8	yes	—
30	C₄×M₁₆×C₃₁	—	2	1	no	—
31	C₄×M₁₆×C₃₁	—	2	1	no	—
32	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
33	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
34	C₂×C₃₁×#97	—	2	1	no	—
35	C₂×C₄×C₈×C₃₁	—	2	1	no	—
36	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
39	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
40	C₃₁×#376	—	2	1	no	—
41	C₄³×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	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	8	no	—
51	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
52	C₂²×C₈×C₃₁	—	2	2	no	—
53	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
54	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
56	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
57	C₃₁×#126	—	2	2	no	—
58	C₃₁×#126	—	2	2	no	—
59	C₃₁×#125	—	2	1	no	—
60	C₃₁×#125	—	2	1	no	—
61	C₃₁×#125	C₂×C₄	1	2	yes	—
62	C₃₁×#124	—	2	1	no	—
63	C₃₁×#124	—	2	1	no	—
64	C₃₁×#124	C₂×C₄	1	2	yes	—
65	C₃₁×#123	—	2	2	no	—
66	C₃₁×#123	—	2	2	no	—
67	C₃₁×#123	—	2	2	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	—	2	1	no	—
71	C₃₁×C₂²⋊Q₈	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
73	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	—	2	1	no	—
77	C₃₁×#121	—	2	1	no	—
78	C₃₁×#121	C₂×C₄	1	2	yes	—
79	C₃₁×#121	C₂×C₄	1	2	yes	—
80	C₃₁×#120	—	2	2	no	—
81	C₄×Q₈×C₃₁	—	2	1	no	—
82	C₄×Q₈×C₃₁	—	2	1	no	—
83	C₄×Q₈×C₃₁	—	2	1	no	—
84	C₄×Q₈×C₃₁	—	2	1	no	—
85	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
86	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
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	2	no	—
91	C₄×D₄×C₃₁	—	2	2	no	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
98	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
99	C₃₁×#117	—	2	8	no	—
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₃₁×#117	—	2	2	no	—
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	—	2	2	no	—
105	C₃₁×#117	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
108	C₂×C₄²×C₃₁	—	2	2	no	—
109	C₂×C₄²×C₃₁	—	2	2	no	—
110	C₂×C₄²×C₃₁	—	2	2	no	—
111	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	C₂×C₄	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	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
124	C₃₁×#104	—	2	16	no	—
125	C₃₁×#97	—	2	2	no	—
126	C₃₁×#97	—	2	2	no	—
127	C₄×C₈×C₃₁	—	2	2	no	—
128	C₄×C₈×C₃₁	—	2	2	no	—
129	C₃₁×#95	—	2	2	no	—
130	C₃₁×#95	—	2	2	no	—
131	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
132	#41×C₃₁	—	2	8	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	4	no	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	4	2	no	—
151	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
169	M₁₆×C₃₁	—	2	8	no	—
170	M₁₆×C₃₁	—	2	4	no	—
171	M₁₆×C₃₁	#41	1	8	yes	complemented
172	C₂×C₈×C₃₁	—	2	8	no	—
173	C₂×C₈×C₃₁	—	2	4	no	—
174	C₄⋊C₄×C₃₁	—	2	8	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	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	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
186	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
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	C₄²×C₃₁	—	2	8	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	—	4	4	no	—
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	C₂²×C₄	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	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
219	#21382	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₃₁	—	2	2	no	—
221	C₂³×C₃₁	—	2	2	no	—
222	C₂³×C₃₁	—	2	2	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	4	no	—
235	D₄×C₃₁	—	4	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	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	8	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	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	4	no	—
257	C₂×C₄×C₃₁	—	4	4	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	#104	1	4	yes	complemented
262	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
273	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
275	C₈×C₃₁	—	2	8	no	—
276	#3444	C₂×C₃₁	1	1	yes	characteristic
277	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
278	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1986	C₂×C₃₁	1	2	yes	complemented
280	#1980	C₂×C₃₁	1	2	yes	complemented
281	#1871	C₂×C₃₁	1	4	yes	complemented
282	C₄×#104	C₂×C₃₁	1	4	yes	complemented
283	C₂²×C₃₁	—	2	1	no	—
284	C₂²×C₃₁	—	2	1	no	—
285	C₂²×C₃₁	—	2	1	no	—
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	—	4	4	no	—
291	C₂²×C₃₁	—	4	2	no	—
292	C₂²×C₃₁	—	4	2	no	—
293	C₂²×C₃₁	—	4	2	no	—
294	C₂²×C₃₁	C₂×#104	1	2	yes	—
295	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
296	C₂²×C₃₁	#532	1	2	yes	—
297	C₄×C₃₁	—	2	1	no	—
298	C₄×C₃₁	—	2	1	no	—
299	C₄×C₃₁	—	2	1	no	—
300	C₄×C₃₁	—	2	1	no	—
301	C₄×C₃₁	—	2	1	no	—
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	2	1	no	—
304	C₄×C₃₁	—	4	8	no	—
305	C₄×C₃₁	—	4	2	no	—
306	C₄×C₃₁	—	4	2	no	—
307	C₄×C₃₁	—	4	2	no	—
308	C₄×C₃₁	C₂×#104	1	2	yes	—
309	C₄×C₃₁	C₂×#104	1	2	yes	—
310	C₄×C₃₁	#421	1	2	yes	—
311	C₄×C₃₁	C₄×#41	1	2	yes	—
312	C₄×C₃₁	#525	1	2	yes	—
313	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
314	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
315	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
316	#533	C₂²×C₃₁	1	1	yes	characteristic
317	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
318	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
319	#532	C₂²×C₃₁	1	1	yes	characteristic
320	#529	—	2	1	no	—
321	#529	—	2	1	no	—
322	#529	—	2	1	no	—
323	#529	—	2	1	no	—
324	#529	C₄×C₃₁	1	2	yes	complemented
325	#529	C₄×C₃₁	1	2	yes	complemented
326	#525	—	2	1	no	—
327	#525	—	2	1	no	—
328	#525	C₂²×C₃₁	1	2	yes	—
329	C₄×#41	—	2	1	no	—
330	C₄×#41	—	2	1	no	—
331	C₄×#41	C₂²×C₃₁	1	2	yes	—
332	C₂×#117	—	2	1	no	—
333	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×#104	C₂²×C₃₁	1	8	yes	—
335	C₄×M₁₆	—	2	1	no	—
336	C₄×M₁₆	—	2	1	no	—
337	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
338	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
339	C₂×#97	—	2	1	no	—
340	C₂×C₄×C₈	—	2	1	no	—
341	C₄×C₄⋊C₄	—	2	1	no	—
342	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
343	C₄×C₂²⋊C₄	—	2	1	no	—
344	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
345	#376	—	2	1	no	—
346	C₄³	C₂²×C₃₁	1	2	yes	—
347	#339	C₂²×C₃₁	1	8	yes	—
348	C₂×C₃₁	—	2	2	no	—
349	C₂×C₃₁	—	2	2	no	—
350	C₂×C₃₁	—	4	1	no	—
351	C₂×C₃₁	—	8	2	no	—
352	C₂×C₃₁	#2297	1	1	yes	characteristic
353	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
354	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
355	C₂×#41	—	2	1	no	—
356	C₂×#41	—	2	1	no	—
357	C₂×#41	C₂³×C₃₁	1	2	yes	—
358	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
359	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
360	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
361	C₂×M₁₆	—	2	2	no	—
362	C₂×M₁₆	—	2	8	no	—
363	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
364	C₂²×C₈	—	2	2	no	—
365	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
366	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
367	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
368	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
369	#126	—	2	2	no	—
370	#126	—	2	2	no	—
371	#125	—	2	1	no	—
372	#125	—	2	1	no	—
373	#125	C₂×C₄×C₃₁	1	2	yes	—
374	#124	—	2	1	no	—
375	#124	—	2	1	no	—
376	#124	C₂×C₄×C₃₁	1	2	yes	—
377	#123	—	2	2	no	—
378	#123	—	2	2	no	—
379	#123	—	2	2	no	—
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	—	2	1	no	—
382	C₂²⋊Q₈	—	2	1	no	—
383	C₂²⋊Q₈	—	2	1	no	—
384	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
385	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
386	#121	—	2	1	no	—
387	#121	—	2	1	no	—
388	#121	—	2	1	no	—
389	#121	—	2	1	no	—
390	#121	C₂×C₄×C₃₁	1	2	yes	—
391	#121	C₂×C₄×C₃₁	1	2	yes	—
392	#120	—	2	2	no	—
393	C₄×Q₈	—	2	1	no	—
394	C₄×Q₈	—	2	1	no	—
395	C₄×Q₈	—	2	1	no	—
396	C₄×Q₈	—	2	1	no	—
397	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
398	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	—	2	1	no	—
408	C₄×D₄	—	2	1	no	—
409	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
410	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
411	#117	—	2	8	no	—
412	#117	—	2	1	no	—
413	#117	—	2	1	no	—
414	#117	—	2	2	no	—
415	#117	—	2	2	no	—
416	#117	—	2	2	no	—
417	#117	C₂³×C₃₁	1	2	yes	—
418	C₂×C₄⋊C₄	—	2	2	no	—
419	C₂×C₂²⋊C₄	—	2	2	no	—
420	C₂×C₄²	—	2	2	no	—
421	C₂×C₄²	—	2	2	no	—
422	C₂×C₄²	—	2	2	no	—
423	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
424	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
425	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
426	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
428	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄²	D₄×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	C₂³×C₃₁	1	4	yes	—
434	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
436	#104	—	2	16	no	—
437	#97	—	2	2	no	—
438	#97	—	2	2	no	—
439	C₄×C₈	—	2	2	no	—
440	C₄×C₈	—	2	2	no	—
441	#95	—	2	2	no	—
442	#95	—	2	2	no	—
443	C₃₁	#21382	1	1	yes	characteristic, direct product factor, Sylow
444	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
445	#41	—	2	8	no	—
446	#41	—	4	4	no	—
447	#41	—	4	4	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	4	2	no	—
464	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
467	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
468	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
469	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
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	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
482	M₁₆	—	2	8	no	—
483	M₁₆	—	2	4	no	—
484	M₁₆	#41×C₃₁	1	8	yes	complemented
485	C₂×C₈	—	2	8	no	—
486	C₂×C₈	—	2	4	no	—
487	C₄⋊C₄	—	2	8	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
499	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₄²	—	2	8	no	—
511	C₄²	—	2	2	no	—
512	C₄²	—	2	2	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	8	no	—
515	C₄²	—	4	4	no	—
516	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
517	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
518	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
519	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
520	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
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	C₂×D₄×C₃₁	1	1	yes	characteristic
529	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
530	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
531	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
532	C₂³	—	2	2	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	4	1	no	—
536	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
537	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
538	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
539	Q₈	—	2	4	no	—
540	Q₈	—	4	2	no	—
541	Q₈	—	4	2	no	—
542	D₄	—	2	4	no	—
543	D₄	—	4	2	no	—
544	D₄	—	4	4	no	—
545	D₄	—	4	4	no	—
546	D₄	—	4	4	no	—
547	D₄	—	4	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	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	8	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	4	no	—
569	C₂×C₄	—	4	4	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	C₃₁×#104	1	4	yes	complemented
574	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
575	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
576	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
577	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
578	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
579	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
580	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
581	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
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	C₂³×C₄×C₃₁	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
586	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
587	C₈	—	2	8	no	—
588	C₂²	—	2	1	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	2	1	no	—
592	C₂²	—	2	1	no	—
593	C₂²	—	2	1	no	—
594	C₂²	—	2	1	no	—
595	C₂²	—	4	4	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	4	2	no	—
599	C₂²	C₂×C₃₁×#104	1	2	yes	—
600	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
601	C₂²	C₃₁×#532	1	2	yes	—
602	C₄	—	2	1	no	—
603	C₄	—	2	1	no	—
604	C₄	—	2	1	no	—
605	C₄	—	2	1	no	—
606	C₄	—	2	1	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	4	8	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	2	no	—
612	C₄	—	4	2	no	—
613	C₄	C₂×C₃₁×#104	1	2	yes	—
614	C₄	C₂×C₃₁×#104	1	2	yes	—
615	C₄	C₃₁×#421	1	2	yes	—
616	C₄	C₄×#41×C₃₁	1	2	yes	—
617	C₄	C₃₁×#525	1	2	yes	—
618	C₂	—	2	2	no	—
619	C₂	—	2	2	no	—
620	C₂	—	4	1	no	—
621	C₂	—	8	2	no	—
622	C₂	C₃₁×#2297	1	1	yes	characteristic
623	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic
624	C₂	C₂×C₃₁×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₃₁	C₂×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 ◁ C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
3. C₄×C₃₁×#117	C₂	semidirect
4. C₃₁×#1986	C₂	semidirect
5. C₃₁×#1980	C₂	semidirect
6. C₃₁×#1871	C₂	semidirect
7. C₄×C₃₁×#104	C₂	semidirect
9. C₃₁×#533	C₄	semidirect
10. C₃₁×#533	C₄	semidirect
12. C₃₁×#532	C₄	semidirect
13. C₃₁×#532	C₄	semidirect
19. C₃₁×#529	C₄	semidirect
20. C₃₁×#529	C₄	semidirect
171. M₁₆×C₃₁	#41	semidirect
219. #21382	C₃₁	direct (N × H)
261. C₂×C₄×C₃₁	#104	semidirect
277. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
278. C₄×#117	C₂×C₃₁	semidirect
279. #1986	C₂×C₃₁	semidirect
280. #1980	C₂×C₃₁	semidirect
281. #1871	C₂×C₃₁	semidirect
282. C₄×#104	C₂×C₃₁	semidirect
314. #533	C₄×C₃₁	semidirect
315. #533	C₄×C₃₁	semidirect
317. #532	C₄×C₃₁	semidirect
318. #532	C₄×C₃₁	semidirect
324. #529	C₄×C₃₁	semidirect
325. #529	C₄×C₃₁	semidirect
443. C₃₁	#21382	direct (N × H)
484. M₁₆	#41×C₃₁	semidirect
573. C₂×C₄	C₃₁×#104	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3444	2
C₂×C₄×M₁₆×C₃₁	2
C₄×C₃₁×#117	2
C₃₁×#1986	2
C₃₁×#1980	2
C₃₁×#1871	2
C₄×C₃₁×#104	2
#21382	31

Subgroups — C31×#21382 #21478141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21382 #21478141