Subgroups — C₃₁×#24784 #21481543

Order 7936.

1040 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#469	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#435	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2414	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2033	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1944	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1875	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1809	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#121	C₂²	1	2	yes	—
16	C₂×C₃₁×#121	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#117	C₂²	1	2	yes	—
19	C₃₁×#469	—	2	4	no	—
20	C₃₁×#466	—	2	4	no	—
21	C₃₁×#435	—	2	4	no	—
22	C₃₁×#427	—	2	4	no	—
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#107	C₂²	1	2	yes	—
25	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₃₁×#394	—	2	2	no	—
35	C₃₁×#392	—	2	2	no	—
36	C₃₁×#390	—	2	2	no	—
37	C₃₁×#390	C₂²	1	2	yes	—
38	C₃₁×#389	C₂²	1	2	yes	—
39	C₃₁×#388	—	2	2	no	—
40	C₃₁×#380	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₂²×D₄×C₃₁	—	2	2	no	—
44	C₂²×D₄×C₃₁	—	2	2	no	—
45	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	D₄	1	4	yes	complemented
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#121	—	2	4	no	—
63	C₃₁×#121	—	2	4	no	—
64	C₃₁×#121	—	2	4	no	—
65	C₃₁×#121	—	4	2	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₃₁×#117	—	2	4	no	—
69	C₃₁×#117	—	4	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	—	2	2	no	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#107	—	2	2	no	—
86	C₃₁×#107	—	2	4	no	—
87	C₃₁×#106	—	2	2	no	—
88	C₃₁×#106	D₄	1	4	yes	complemented
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×#102	—	2	4	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	4	2	no	—
93	C₃₁×C₂²⋊C₈	—	2	2	no	—
94	C₃₁×C₂²⋊C₈	—	2	2	no	—
95	C₃₁×#97	—	2	2	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	—	2	2	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	4	2	no	—
109	C₂×D₄×C₃₁	—	4	4	no	—
110	C₂×D₄×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	8	2	no	—
113	C₂×D₄×C₃₁	—	8	2	no	—
114	C₂×D₄×C₃₁	—	8	2	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	#41	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
135	D₈×C₃₁	—	4	4	no	—
136	M₁₆×C₃₁	—	4	4	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	8	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
147	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
148	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₃₁	#41	1	2	yes	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	8	2	no	—
157	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
158	C₂²⋊C₄×C₃₁	—	2	2	no	—
159	C₂²⋊C₄×C₃₁	—	2	2	no	—
160	C₂²⋊C₄×C₃₁	—	4	4	no	—
161	C₂²⋊C₄×C₃₁	—	4	4	no	—
162	C₂²⋊C₄×C₃₁	—	4	4	no	—
163	C₂²⋊C₄×C₃₁	—	4	4	no	—
164	C₂²⋊C₄×C₃₁	—	8	2	no	—
165	C₂²⋊C₄×C₃₁	—	8	2	no	—
166	C₄²×C₃₁	—	2	1	no	—
167	C₄²×C₃₁	—	2	1	no	—
168	C₄²×C₃₁	—	4	4	no	—
169	#24784	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	2	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	4	2	no	—
185	C₂³×C₃₁	—	4	2	no	—
186	C₂³×C₃₁	#121	1	2	yes	—
187	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	D₄×C₃₁	—	4	4	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	4	2	no	—
193	D₄×C₃₁	—	8	8	no	—
194	D₄×C₃₁	—	8	8	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	2	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	#121	1	2	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
228	C₈×C₃₁	—	2	2	no	—
229	C₈×C₃₁	—	4	1	no	—
230	C₈×C₃₁	—	4	1	no	—
231	C₈×C₃₁	#121	1	4	yes	—
232	C₂×#469	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂×#466	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂×#435	C₂×C₃₁	1	1	yes	characteristic, complemented
235	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#2414	C₂×C₃₁	1	2	yes	complemented
237	#2067	C₂×C₃₁	1	2	yes	complemented
238	#2033	C₂×C₃₁	1	2	yes	complemented
239	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#1944	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1875	C₂×C₃₁	1	2	yes	complemented
242	#1809	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	2	2	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	—	4	1	no	—
250	C₂²×C₃₁	—	4	2	no	—
251	C₂²×C₃₁	—	4	1	no	—
252	C₂²×C₃₁	—	4	1	no	—
253	C₂²×C₃₁	—	8	2	no	—
254	C₂²×C₃₁	—	8	2	no	—
255	C₂²×C₃₁	—	8	2	no	—
256	C₂²×C₃₁	—	8	2	no	—
257	C₂²×C₃₁	—	8	2	no	—
258	C₂²×C₃₁	—	8	2	no	—
259	C₂²×C₃₁	—	8	2	no	—
260	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
261	C₂²×C₃₁	D₄²	1	1	yes	characteristic
262	C₂²×C₃₁	#547	1	1	yes	characteristic
263	C₂²×C₃₁	#575	1	2	yes	—
264	C₂²×C₃₁	#576	1	2	yes	—
265	C₄×C₃₁	—	2	2	no	—
266	C₄×C₃₁	—	4	1	no	—
267	C₄×C₃₁	—	4	4	no	—
268	C₄×C₃₁	—	4	1	no	—
269	C₄×C₃₁	—	4	1	no	—
270	C₄×C₃₁	—	8	2	no	—
271	C₄×C₃₁	—	8	2	no	—
272	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
273	C₄×C₃₁	#537	1	1	yes	characteristic
274	C₄×C₃₁	#548	1	1	yes	characteristic
275	C₄×C₃₁	#553	1	1	yes	characteristic
276	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
277	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
278	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#121	C₂²×C₃₁	1	2	yes	—
280	C₂×#121	C₂²×C₃₁	1	2	yes	—
281	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×#117	C₂²×C₃₁	1	2	yes	—
283	#469	—	2	4	no	—
284	#466	—	2	4	no	—
285	#435	—	2	4	no	—
286	#427	—	2	4	no	—
287	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#107	C₂²×C₃₁	1	2	yes	—
289	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#102	C₂²×C₃₁	1	2	yes	—
293	C₂×#102	C₂²×C₃₁	1	2	yes	—
294	C₂×#102	C₂²×C₃₁	1	2	yes	—
295	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
298	#394	—	2	2	no	—
299	#392	—	2	2	no	—
300	#390	—	2	2	no	—
301	#390	C₂²×C₃₁	1	2	yes	—
302	#389	C₂²×C₃₁	1	2	yes	—
303	#388	—	2	2	no	—
304	#380	C₂²×C₃₁	1	2	yes	—
305	#340	C₂²×C₃₁	1	2	yes	—
306	#340	C₂²×C₃₁	1	2	yes	—
307	C₂×C₃₁	—	4	2	no	—
308	C₂×C₃₁	—	8	1	no	—
309	C₂×C₃₁	—	16	2	no	—
310	C₂×C₃₁	#2416	1	1	yes	characteristic
311	C₂×C₃₁	#3054	1	1	yes	characteristic
312	C₂×C₃₁	#3058	1	1	yes	characteristic
313	C₂×C₃₁	#3268	1	2	yes	—
314	C₂×C₃₁	#3347	1	2	yes	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	—	2	2	no	—
317	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
318	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
319	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
321	C₂×D₈	—	2	2	no	—
322	C₂×D₈	—	2	2	no	—
323	C₂×D₈	—	2	2	no	—
324	C₂×M₁₆	—	2	2	no	—
325	C₂×M₁₆	—	2	2	no	—
326	C₂×M₁₆	—	2	2	no	—
327	C₂²×C₈	—	2	2	no	—
328	C₂²×C₈	—	2	2	no	—
329	C₂²×C₈	D₄×C₃₁	1	4	yes	complemented
330	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
334	#121	—	2	4	no	—
335	#121	—	2	4	no	—
336	#121	—	2	4	no	—
337	#121	—	4	2	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	#117	—	2	4	no	—
341	#117	—	4	2	no	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
346	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₂²⋊C₄	—	2	2	no	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	2	no	—
353	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
354	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
355	C₂×C₄²	—	2	2	no	—
356	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
357	#107	—	2	2	no	—
358	#107	—	2	4	no	—
359	#106	—	2	2	no	—
360	#106	D₄×C₃₁	1	4	yes	complemented
361	C₄⋊C₈	—	2	2	no	—
362	#102	—	2	4	no	—
363	#102	—	2	4	no	—
364	#102	—	4	2	no	—
365	C₂²⋊C₈	—	2	2	no	—
366	C₂²⋊C₈	—	2	2	no	—
367	#97	—	2	2	no	—
368	#95	—	2	2	no	—
369	#95	—	2	2	no	—
370	C₃₁	#24784	1	1	yes	characteristic, direct product factor, Sylow
371	C₂⁴	—	2	2	no	—
372	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	—	4	4	no	—
383	C₂×D₄	—	4	4	no	—
384	C₂×D₄	—	4	4	no	—
385	C₂×D₄	—	8	2	no	—
386	C₂×D₄	—	8	2	no	—
387	C₂×D₄	—	8	2	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
406	C₂²×C₄	#41×C₃₁	1	2	yes	—
407	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
408	D₈	—	4	4	no	—
409	M₁₆	—	4	4	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	8	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
420	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
421	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
422	C₂×C₈	#41×C₃₁	1	2	yes	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	8	2	no	—
430	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	2	2	no	—
433	C₂²⋊C₄	—	4	4	no	—
434	C₂²⋊C₄	—	4	4	no	—
435	C₂²⋊C₄	—	4	4	no	—
436	C₂²⋊C₄	—	4	4	no	—
437	C₂²⋊C₄	—	8	2	no	—
438	C₂²⋊C₄	—	8	2	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	4	4	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	1	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	1	no	—
447	C₂³	—	2	1	no	—
448	C₂³	—	2	1	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	4	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	—	4	2	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	4	2	no	—
457	C₂³	—	4	2	no	—
458	C₂³	C₃₁×#121	1	2	yes	—
459	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
460	D₄	—	4	4	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	2	no	—
464	D₄	—	4	2	no	—
465	D₄	—	8	8	no	—
466	D₄	—	8	8	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	C₃₁×#121	1	2	yes	—
494	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
496	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
497	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
498	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
499	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
500	C₈	—	2	2	no	—
501	C₈	—	4	1	no	—
502	C₈	—	4	1	no	—
503	C₈	C₃₁×#121	1	4	yes	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	1	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	1	no	—
514	C₂²	—	8	2	no	—
515	C₂²	—	8	2	no	—
516	C₂²	—	8	2	no	—
517	C₂²	—	8	2	no	—
518	C₂²	—	8	2	no	—
519	C₂²	—	8	2	no	—
520	C₂²	—	8	2	no	—
521	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
522	C₂²	D₄²×C₃₁	1	1	yes	characteristic
523	C₂²	C₃₁×#547	1	1	yes	characteristic
524	C₂²	C₃₁×#575	1	2	yes	—
525	C₂²	C₃₁×#576	1	2	yes	—
526	C₄	—	2	2	no	—
527	C₄	—	4	1	no	—
528	C₄	—	4	4	no	—
529	C₄	—	4	1	no	—
530	C₄	—	4	1	no	—
531	C₄	—	8	2	no	—
532	C₄	—	8	2	no	—
533	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
534	C₄	C₃₁×#537	1	1	yes	characteristic
535	C₄	C₃₁×#548	1	1	yes	characteristic
536	C₄	C₃₁×#553	1	1	yes	characteristic
537	C₂	—	4	2	no	—
538	C₂	—	8	1	no	—
539	C₂	—	16	2	no	—
540	C₂	C₃₁×#2416	1	1	yes	characteristic
541	C₂	C₃₁×#3054	1	1	yes	characteristic
542	C₂	C₃₁×#3058	1	1	yes	characteristic
543	C₂	C₃₁×#3268	1	2	yes	—
544	C₂	C₃₁×#3347	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#469	C₂	semidirect
2. C₂×C₃₁×#466	C₂	semidirect
3. C₂×C₃₁×#435	C₂	semidirect
4. C₂×C₃₁×#427	C₂	semidirect
5. C₃₁×#2414	C₂	semidirect
6. C₃₁×#2067	C₂	semidirect
7. C₃₁×#2033	C₂	semidirect
8. C₃₁×#1972	C₂	semidirect
9. C₃₁×#1944	C₂	semidirect
10. C₃₁×#1875	C₂	semidirect
11. C₃₁×#1809	C₂	semidirect
57. C₂²×C₈×C₃₁	D₄	semidirect
88. C₃₁×#106	D₄	semidirect
169. #24784	C₃₁	direct (N × H)
232. C₂×#469	C₂×C₃₁	semidirect
233. C₂×#466	C₂×C₃₁	semidirect
234. C₂×#435	C₂×C₃₁	semidirect
235. C₂×#427	C₂×C₃₁	semidirect
236. #2414	C₂×C₃₁	semidirect
237. #2067	C₂×C₃₁	semidirect
238. #2033	C₂×C₃₁	semidirect
239. #1972	C₂×C₃₁	semidirect
240. #1944	C₂×C₃₁	semidirect
241. #1875	C₂×C₃₁	semidirect
242. #1809	C₂×C₃₁	semidirect
329. C₂²×C₈	D₄×C₃₁	semidirect
360. #106	D₄×C₃₁	semidirect
370. C₃₁	#24784	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#469	2
C₂×C₃₁×#466	2
C₂×C₃₁×#435	2
C₂×C₃₁×#427	2
C₃₁×#2414	2
C₃₁×#2067	2
C₃₁×#2033	2
C₃₁×#1972	2
C₃₁×#1944	2
C₃₁×#1875	2
C₃₁×#1809	2
#24784	31

Subgroups — C31×#24784 #21481543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24784 #21481543