Subgroups — C₃₁×#36624 #21493383

Order 7936.

3902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#525	C₂	1	2	yes	maximal, complemented
3	C₂³×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2857	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2856	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2854	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2850	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2847	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×#121	C₂	1	8	yes	maximal, complemented
10	C₃₁×#2297	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#532	—	2	8	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
19	C₃₁×#525	—	2	16	no	—
20	C₂×C₃₁×#123	C₂²	1	4	yes	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
22	C₂×C₃₁×#121	C₂²	1	4	yes	—
23	C₂×C₃₁×#121	C₂²	1	8	yes	—
24	C₂×C₃₁×#121	C₂²	1	4	yes	—
25	C₂×C₃₁×#120	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
28	C₂×C₃₁×#117	C₂²	1	2	yes	—
29	C₂²×C₄²×C₃₁	—	2	8	no	—
30	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
31	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
33	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
35	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#391	C₂²	1	4	yes	complemented
37	C₃₁×#390	C₂²	1	4	yes	complemented
38	C₃₁×#390	C₂²	1	8	yes	complemented
39	C₃₁×#389	C₂²	1	8	yes	complemented
40	C₃₁×#388	C₂²	1	16	yes	complemented
41	C₃₁×#386	C₂²	1	16	yes	—
42	C₃₁×#386	C₂²	1	8	yes	—
43	C₃₁×#384	C₂²	1	2	yes	complemented
44	C₃₁×#382	C₂²	1	2	yes	complemented
45	C₃₁×#380	C₂²	1	4	yes	—
46	C₃₁×#380	C₂²	1	8	yes	—
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
48	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
49	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×#41×C₃₁	—	2	8	no	—
51	C₂×#41×C₃₁	—	2	8	no	—
52	C₂×#41×C₃₁	—	2	8	no	—
53	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
55	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
56	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
57	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	—	2	8	no	—
59	C₂³×C₄×C₃₁	—	2	4	no	—
60	C₂³×C₄×C₃₁	—	2	4	no	—
61	C₂³×C₄×C₃₁	—	2	16	no	—
62	C₂³×C₄×C₃₁	D₄	1	8	yes	complemented
63	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×C₄⋊Q₈	—	2	4	no	—
73	C₃₁×#127	—	2	4	no	—
74	C₃₁×#124	—	2	8	no	—
75	C₃₁×#123	—	2	16	no	—
76	C₃₁×C₂²⋊Q₈	—	2	16	no	—
77	C₃₁×#121	—	2	16	no	—
78	C₃₁×#121	—	2	16	no	—
79	C₃₁×#121	—	2	32	no	—
80	C₃₁×#120	—	2	16	no	—
81	C₄×D₄×C₃₁	—	2	32	no	—
82	C₄×D₄×C₃₁	—	2	16	no	—
83	C₃₁×#117	—	2	8	no	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
93	C₂×C₄²×C₃₁	—	2	8	no	—
94	C₂×C₄²×C₃₁	—	2	4	no	—
95	C₂×C₄²×C₃₁	—	2	16	no	—
96	C₂×C₄²×C₃₁	—	2	8	no	—
97	C₂×C₄²×C₃₁	—	2	8	no	—
98	C₂×C₄²×C₃₁	D₄	1	8	yes	complemented
99	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
102	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
107	C₃₁×#95	C₂³	1	16	yes	—
108	C₃₁×#95	C₂³	1	8	yes	—
109	C₂⁴×C₃₁	—	2	2	no	—
110	C₂⁴×C₃₁	—	2	8	no	—
111	C₂⁴×C₃₁	—	2	2	no	—
112	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
113	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
114	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
115	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
116	#41×C₃₁	—	4	32	no	—
117	C₂×Q₈×C₃₁	—	2	4	no	—
118	C₂×Q₈×C₃₁	—	2	4	no	—
119	C₂×Q₈×C₃₁	—	2	4	no	—
120	C₂×D₄×C₃₁	—	2	16	no	—
121	C₂×D₄×C₃₁	—	2	8	no	—
122	C₂×D₄×C₃₁	—	2	8	no	—
123	C₂×D₄×C₃₁	—	2	32	no	—
124	C₂×D₄×C₃₁	—	2	8	no	—
125	C₂×D₄×C₃₁	—	2	8	no	—
126	C₂×D₄×C₃₁	—	2	4	no	—
127	C₂×D₄×C₃₁	—	2	4	no	—
128	C₂×D₄×C₃₁	—	2	16	no	—
129	C₂×D₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	64	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	8	no	—
134	C₂²×C₄×C₃₁	—	2	2	no	—
135	C₂²×C₄×C₃₁	—	2	32	no	—
136	C₂²×C₄×C₃₁	—	2	8	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	8	no	—
140	C₂²×C₄×C₃₁	—	2	8	no	—
141	C₂²×C₄×C₃₁	—	2	8	no	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	—	2	4	no	—
144	C₂²×C₄×C₃₁	—	2	8	no	—
145	C₂²×C₄×C₃₁	—	2	8	no	—
146	C₂²×C₄×C₃₁	—	2	8	no	—
147	C₂²×C₄×C₃₁	—	2	8	no	—
148	C₂²×C₄×C₃₁	—	2	8	no	—
149	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
155	C₂²×C₄×C₃₁	#41	1	4	yes	—
156	C₂²×C₄×C₃₁	#41	1	16	yes	complemented
157	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
165	C₄⋊C₄×C₃₁	—	2	8	no	—
166	C₄⋊C₄×C₃₁	—	2	8	no	—
167	C₄⋊C₄×C₃₁	—	2	16	no	—
168	C₄⋊C₄×C₃₁	—	2	16	no	—
169	C₂²⋊C₄×C₃₁	—	2	8	no	—
170	C₂²⋊C₄×C₃₁	—	2	16	no	—
171	C₂²⋊C₄×C₃₁	—	2	16	no	—
172	C₂²⋊C₄×C₃₁	—	2	16	no	—
173	C₂²⋊C₄×C₃₁	—	2	32	no	—
174	C₄²×C₃₁	—	2	8	no	—
175	C₄²×C₃₁	—	2	4	no	—
176	C₄²×C₃₁	—	2	4	no	—
177	C₄²×C₃₁	—	2	8	no	—
178	C₄²×C₃₁	—	2	8	no	—
179	C₄²×C₃₁	C₂×D₄	1	8	yes	—
180	C₄²×C₃₁	C₂×D₄	1	4	yes	—
181	C₄²×C₃₁	C₂×D₄	1	4	yes	—
182	#36624	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	16	no	—
184	C₂³×C₃₁	—	2	16	no	—
185	C₂³×C₃₁	—	2	8	no	—
186	C₂³×C₃₁	—	2	16	no	—
187	C₂³×C₃₁	—	2	2	no	—
188	C₂³×C₃₁	—	2	2	no	—
189	C₂³×C₃₁	—	2	4	no	—
190	C₂³×C₃₁	—	2	4	no	—
191	C₂³×C₃₁	—	2	4	no	—
192	C₂³×C₃₁	—	2	4	no	—
193	C₂³×C₃₁	—	2	16	no	—
194	C₂³×C₃₁	—	2	4	no	—
195	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
196	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
197	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
198	C₂³×C₃₁	C₂×#41	1	2	yes	—
199	C₂³×C₃₁	C₂×#41	1	8	yes	—
200	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
201	Q₈×C₃₁	—	4	8	no	—
202	D₄×C₃₁	—	4	16	no	—
203	D₄×C₃₁	—	4	32	no	—
204	D₄×C₃₁	—	4	8	no	—
205	D₄×C₃₁	—	4	16	no	—
206	C₂×C₄×C₃₁	—	2	32	no	—
207	C₂×C₄×C₃₁	—	2	16	no	—
208	C₂×C₄×C₃₁	—	2	8	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	16	no	—
215	C₂×C₄×C₃₁	—	2	16	no	—
216	C₂×C₄×C₃₁	—	2	16	no	—
217	C₂×C₄×C₃₁	—	2	32	no	—
218	C₂×C₄×C₃₁	—	2	8	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	16	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	4	16	no	—
224	C₂×C₄×C₃₁	#121	1	16	yes	complemented
225	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
232	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
233	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
234	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
235	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
236	C₂×#532	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂×#525	C₂×C₃₁	1	2	yes	complemented
238	C₂³×C₄²	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#2857	C₂×C₃₁	1	2	yes	complemented
240	#2856	C₂×C₃₁	1	2	yes	complemented
241	#2854	C₂×C₃₁	1	4	yes	complemented
242	#2850	C₂×C₃₁	1	4	yes	complemented
243	#2847	C₂×C₃₁	1	4	yes	complemented
244	C₄×#121	C₂×C₃₁	1	8	yes	complemented
245	#2297	C₂×C₃₁	1	2	yes	complemented
246	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂²×C₃₁	—	2	32	no	—
248	C₂²×C₃₁	—	2	4	no	—
249	C₂²×C₃₁	—	2	4	no	—
250	C₂²×C₃₁	—	2	8	no	—
251	C₂²×C₃₁	—	2	4	no	—
252	C₂²×C₃₁	—	2	4	no	—
253	C₂²×C₃₁	—	2	4	no	—
254	C₂²×C₃₁	—	2	8	no	—
255	C₂²×C₃₁	—	4	4	no	—
256	C₂²×C₃₁	—	4	8	no	—
257	C₂²×C₃₁	—	4	8	no	—
258	C₂²×C₃₁	—	4	8	no	—
259	C₂²×C₃₁	C₂×#121	1	4	yes	—
260	C₂²×C₃₁	#532	1	8	yes	complemented
261	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
262	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
263	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
264	C₂²×C₃₁	C₂²×#41	1	4	yes	—
265	C₄×C₃₁	—	2	4	no	—
266	C₄×C₃₁	—	2	4	no	—
267	C₄×C₃₁	—	2	8	no	—
268	C₄×C₃₁	—	4	4	no	—
269	C₄×C₃₁	—	4	8	no	—
270	C₄×C₃₁	C₂×#121	1	4	yes	—
271	C₄×C₃₁	C₂×#121	1	4	yes	—
272	C₄×C₃₁	#525	1	8	yes	complemented
273	C₂²×#41	C₂²×C₃₁	1	2	yes	—
274	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
275	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
276	#532	—	2	8	no	—
277	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
278	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
279	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
280	#525	—	2	16	no	—
281	C₂×#123	C₂²×C₃₁	1	4	yes	—
282	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
283	C₂×#121	C₂²×C₃₁	1	4	yes	—
284	C₂×#121	C₂²×C₃₁	1	8	yes	—
285	C₂×#121	C₂²×C₃₁	1	4	yes	—
286	C₂×#120	C₂²×C₃₁	1	4	yes	—
287	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
288	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
289	C₂×#117	C₂²×C₃₁	1	2	yes	—
290	C₂²×C₄²	—	2	8	no	—
291	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
292	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
293	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
294	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
295	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
296	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
297	#391	C₂²×C₃₁	1	4	yes	complemented
298	#390	C₂²×C₃₁	1	4	yes	complemented
299	#390	C₂²×C₃₁	1	8	yes	complemented
300	#389	C₂²×C₃₁	1	8	yes	complemented
301	#388	C₂²×C₃₁	1	16	yes	complemented
302	#386	C₂²×C₃₁	1	16	yes	—
303	#386	C₂²×C₃₁	1	8	yes	—
304	#384	C₂²×C₃₁	1	2	yes	complemented
305	#382	C₂²×C₃₁	1	2	yes	complemented
306	#380	C₂²×C₃₁	1	4	yes	—
307	#380	C₂²×C₃₁	1	8	yes	—
308	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
309	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
310	C₂×C₃₁	—	2	4	no	—
311	C₂×C₃₁	—	2	8	no	—
312	C₂×C₃₁	—	8	4	no	—
313	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
314	C₂×C₃₁	C₂×#525	1	2	yes	—
315	C₂×C₃₁	C₂×#532	1	2	yes	—
316	C₂×C₃₁	#3444	1	2	yes	—
317	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
318	C₂×#41	—	2	8	no	—
319	C₂×#41	—	2	8	no	—
320	C₂×#41	—	2	8	no	—
321	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
322	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
323	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
324	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
325	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
326	C₂³×C₄	—	2	8	no	—
327	C₂³×C₄	—	2	4	no	—
328	C₂³×C₄	—	2	4	no	—
329	C₂³×C₄	—	2	16	no	—
330	C₂³×C₄	D₄×C₃₁	1	8	yes	complemented
331	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
332	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
333	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
334	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
335	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
336	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
337	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
339	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₄⋊Q₈	—	2	4	no	—
341	#127	—	2	4	no	—
342	#124	—	2	8	no	—
343	#123	—	2	16	no	—
344	C₂²⋊Q₈	—	2	16	no	—
345	#121	—	2	16	no	—
346	#121	—	2	16	no	—
347	#121	—	2	32	no	—
348	#120	—	2	16	no	—
349	C₄×D₄	—	2	32	no	—
350	C₄×D₄	—	2	16	no	—
351	#117	—	2	8	no	—
352	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
354	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
355	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
361	C₂×C₄²	—	2	8	no	—
362	C₂×C₄²	—	2	4	no	—
363	C₂×C₄²	—	2	16	no	—
364	C₂×C₄²	—	2	8	no	—
365	C₂×C₄²	—	2	8	no	—
366	C₂×C₄²	D₄×C₃₁	1	8	yes	complemented
367	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
368	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
369	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
370	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
371	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
372	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
374	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
375	#95	C₂³×C₃₁	1	16	yes	—
376	#95	C₂³×C₃₁	1	8	yes	—
377	C₃₁	#36624	1	1	yes	characteristic, direct product factor, Sylow
378	C₂⁴	—	2	2	no	—
379	C₂⁴	—	2	8	no	—
380	C₂⁴	—	2	2	no	—
381	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
382	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
383	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
384	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
385	#41	—	4	32	no	—
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	4	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×D₄	—	2	16	no	—
390	C₂×D₄	—	2	8	no	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	2	32	no	—
393	C₂×D₄	—	2	8	no	—
394	C₂×D₄	—	2	8	no	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	2	16	no	—
398	C₂×D₄	—	2	4	no	—
399	C₂²×C₄	—	2	64	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	8	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	32	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	8	no	—
414	C₂²×C₄	—	2	8	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	2	8	no	—
418	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
421	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
422	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
423	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
424	C₂²×C₄	#41×C₃₁	1	4	yes	—
425	C₂²×C₄	#41×C₃₁	1	16	yes	complemented
426	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
427	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
428	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
429	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
430	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
431	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
434	C₄⋊C₄	—	2	8	no	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	2	16	no	—
437	C₄⋊C₄	—	2	16	no	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₂²⋊C₄	—	2	16	no	—
440	C₂²⋊C₄	—	2	16	no	—
441	C₂²⋊C₄	—	2	16	no	—
442	C₂²⋊C₄	—	2	32	no	—
443	C₄²	—	2	8	no	—
444	C₄²	—	2	4	no	—
445	C₄²	—	2	4	no	—
446	C₄²	—	2	8	no	—
447	C₄²	—	2	8	no	—
448	C₄²	C₂×D₄×C₃₁	1	8	yes	—
449	C₄²	C₂×D₄×C₃₁	1	4	yes	—
450	C₄²	C₂×D₄×C₃₁	1	4	yes	—
451	C₂³	—	2	16	no	—
452	C₂³	—	2	16	no	—
453	C₂³	—	2	8	no	—
454	C₂³	—	2	16	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	16	no	—
462	C₂³	—	2	4	no	—
463	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
464	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
465	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
466	C₂³	C₂×#41×C₃₁	1	2	yes	—
467	C₂³	C₂×#41×C₃₁	1	8	yes	—
468	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
469	Q₈	—	4	8	no	—
470	D₄	—	4	16	no	—
471	D₄	—	4	32	no	—
472	D₄	—	4	8	no	—
473	D₄	—	4	16	no	—
474	C₂×C₄	—	2	32	no	—
475	C₂×C₄	—	2	16	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	16	no	—
483	C₂×C₄	—	2	16	no	—
484	C₂×C₄	—	2	16	no	—
485	C₂×C₄	—	2	32	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	16	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	4	16	no	—
492	C₂×C₄	C₃₁×#121	1	16	yes	complemented
493	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
494	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
495	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
496	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
497	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
498	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
499	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
500	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
501	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
502	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
503	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
504	C₂²	—	2	32	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	2	4	no	—
507	C₂²	—	2	8	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	2	8	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	4	8	no	—
514	C₂²	—	4	8	no	—
515	C₂²	—	4	8	no	—
516	C₂²	C₂×C₃₁×#121	1	4	yes	—
517	C₂²	C₃₁×#532	1	8	yes	complemented
518	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
519	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
520	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
521	C₂²	C₂²×#41×C₃₁	1	4	yes	—
522	C₄	—	2	4	no	—
523	C₄	—	2	4	no	—
524	C₄	—	2	8	no	—
525	C₄	—	4	4	no	—
526	C₄	—	4	8	no	—
527	C₄	C₂×C₃₁×#121	1	4	yes	—
528	C₄	C₂×C₃₁×#121	1	4	yes	—
529	C₄	C₃₁×#525	1	8	yes	complemented
530	C₂	—	2	4	no	—
531	C₂	—	2	8	no	—
532	C₂	—	8	4	no	—
533	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
534	C₂	C₂×C₃₁×#525	1	2	yes	—
535	C₂	C₂×C₃₁×#532	1	2	yes	—
536	C₂	C₃₁×#3444	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#532	C₂	semidirect
2. C₂×C₃₁×#525	C₂	semidirect
3. C₂³×C₄²×C₃₁	C₂	semidirect
4. C₃₁×#2857	C₂	semidirect
5. C₃₁×#2856	C₂	semidirect
6. C₃₁×#2854	C₂	semidirect
7. C₃₁×#2850	C₂	semidirect
8. C₃₁×#2847	C₂	semidirect
9. C₄×C₃₁×#121	C₂	semidirect
10. C₃₁×#2297	C₂	semidirect
11. C₃₁×#2285	C₂	semidirect
16. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
17. C₂×C₃₁×#127	C₂²	semidirect
18. C₂×C₃₁×#124	C₂²	semidirect
27. C₂×C₄×D₄×C₃₁	C₂²	semidirect
36. C₃₁×#391	C₂²	semidirect
37. C₃₁×#390	C₂²	semidirect
38. C₃₁×#390	C₂²	semidirect
39. C₃₁×#389	C₂²	semidirect
40. C₃₁×#388	C₂²	semidirect
43. C₃₁×#384	C₂²	semidirect
44. C₃₁×#382	C₂²	semidirect
47. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
48. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
62. C₂³×C₄×C₃₁	D₄	semidirect
98. C₂×C₄²×C₃₁	D₄	semidirect
156. C₂²×C₄×C₃₁	#41	semidirect
182. #36624	C₃₁	direct (N × H)
224. C₂×C₄×C₃₁	#121	semidirect
236. C₂×#532	C₂×C₃₁	semidirect
237. C₂×#525	C₂×C₃₁	semidirect
238. C₂³×C₄²	C₂×C₃₁	semidirect
239. #2857	C₂×C₃₁	semidirect
240. #2856	C₂×C₃₁	semidirect
241. #2854	C₂×C₃₁	semidirect
242. #2850	C₂×C₃₁	semidirect
243. #2847	C₂×C₃₁	semidirect
244. C₄×#121	C₂×C₃₁	semidirect
245. #2297	C₂×C₃₁	semidirect
246. #2285	C₂×C₃₁	semidirect
260. C₂²×C₃₁	#532	semidirect
272. C₄×C₃₁	#525	semidirect
277. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
278. C₂×#127	C₂²×C₃₁	semidirect
279. C₂×#124	C₂²×C₃₁	semidirect
288. C₂×C₄×D₄	C₂²×C₃₁	semidirect
297. #391	C₂²×C₃₁	semidirect
298. #390	C₂²×C₃₁	semidirect
299. #390	C₂²×C₃₁	semidirect
300. #389	C₂²×C₃₁	semidirect
301. #388	C₂²×C₃₁	semidirect
304. #384	C₂²×C₃₁	semidirect
305. #382	C₂²×C₃₁	semidirect
308. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
309. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
330. C₂³×C₄	D₄×C₃₁	semidirect
366. C₂×C₄²	D₄×C₃₁	semidirect
377. C₃₁	#36624	direct (N × H)
425. C₂²×C₄	#41×C₃₁	semidirect
492. C₂×C₄	C₃₁×#121	semidirect
517. C₂²	C₃₁×#532	semidirect
529. C₄	C₃₁×#525	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#532	2
C₂×C₃₁×#525	2
C₂³×C₄²×C₃₁	2
C₃₁×#2857	2
C₃₁×#2856	2
C₃₁×#2854	2
C₃₁×#2850	2
C₃₁×#2847	2
C₄×C₃₁×#121	2
C₃₁×#2297	2
C₃₁×#2285	2
#36624	31

Subgroups — C31×#36624 #21493383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36624 #21493383