Subgroups — C₃₁×#34956 #21491715

Order 7936.

3394 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2600	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2579	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2534	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2405	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2403	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2392	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×#123	C₂	1	4	yes	maximal, complemented
10	C₄×C₃₁×#120	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	complemented
15	C₂×C₃₁×#123	C₂²	1	4	yes	—
16	C₂×C₃₁×#120	C₂²	1	4	yes	—
17	C₂×C₄×D₄×C₃₁	—	2	16	no	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
21	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
22	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
23	C₂×C₃₁×#117	C₂²	1	2	yes	complemented
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
27	C₂²×C₄²×C₃₁	C₂²	1	2	yes	complemented
28	C₃₁×#398	C₂²	1	4	yes	complemented
29	C₃₁×#397	C₂²	1	4	yes	complemented
30	C₃₁×#396	C₂²	1	8	yes	complemented
31	C₃₁×#394	C₂²	1	8	yes	complemented
32	C₃₁×#394	C₂²	1	4	yes	complemented
33	C₃₁×#392	C₂²	1	4	yes	—
34	C₃₁×#388	C₂²	1	8	yes	complemented
35	C₃₁×#387	C₂²	1	8	yes	complemented
36	C₃₁×#386	C₂²	1	4	yes	—
37	C₃₁×#386	C₂²	1	4	yes	—
38	C₃₁×#386	C₂²	1	8	yes	complemented
39	C₃₁×#385	C₂²	1	8	yes	complemented
40	C₃₁×#385	C₂²	1	4	yes	complemented
41	C₃₁×#385	C₂²	1	4	yes	complemented
42	C₃₁×#381	C₂²	1	2	yes	complemented
43	C₃₁×#381	C₂²	1	4	yes	complemented
44	C₃₁×#380	C₂²	1	4	yes	complemented
45	C₃₁×#379	C₂²	1	1	yes	characteristic
46	C₃₁×#379	C₂²	1	1	yes	characteristic
47	C₃₁×#379	C₂²	1	4	yes	complemented
48	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
50	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
51	C₂⁵×C₃₁	C₂³	1	2	yes	—
52	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₃₁	—	2	8	no	—
54	C₂²×D₄×C₃₁	—	2	16	no	—
55	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
56	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
57	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
58	C₂³×C₄×C₃₁	—	2	8	no	—
59	C₂³×C₄×C₃₁	—	4	4	no	—
60	C₂³×C₄×C₃₁	—	4	8	no	—
61	C₂³×C₄×C₃₁	D₄	1	8	yes	complemented
62	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
69	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#123	—	2	16	no	—
72	C₃₁×#120	—	2	16	no	—
73	C₄×D₄×C₃₁	—	2	16	no	—
74	C₄×D₄×C₃₁	—	2	16	no	—
75	C₄×D₄×C₃₁	—	2	16	no	—
76	C₄×D₄×C₃₁	—	2	16	no	—
77	C₄×D₄×C₃₁	—	4	8	no	—
78	C₃₁×#117	—	4	4	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
87	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₂×C₄²×C₃₁	—	2	8	no	—
99	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
102	C₃₁×#95	C₂³	1	8	yes	—
103	C₃₁×#95	C₂³	1	4	yes	—
104	C₃₁×#95	C₂³	1	8	yes	—
105	C₃₁×#95	C₂³	1	4	yes	—
106	C₃₁×#95	C₂³	1	8	yes	—
107	C₂⁴×C₃₁	—	2	4	no	—
108	C₂⁴×C₃₁	—	2	8	no	—
109	C₂⁴×C₃₁	—	2	8	no	—
110	C₂⁴×C₃₁	—	2	8	no	—
111	C₂⁴×C₃₁	—	4	2	no	—
112	C₂⁴×C₃₁	—	4	4	no	—
113	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
114	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
115	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
116	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
117	C₂×D₄×C₃₁	—	2	64	no	—
118	C₂×D₄×C₃₁	—	2	8	no	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	—	2	8	no	—
121	C₂×D₄×C₃₁	—	2	8	no	—
122	C₂×D₄×C₃₁	—	2	16	no	—
123	C₂×D₄×C₃₁	—	2	8	no	—
124	C₂×D₄×C₃₁	—	4	4	no	—
125	C₂×D₄×C₃₁	#41	1	16	yes	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	8	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	8	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	8	no	—
133	C₂²×C₄×C₃₁	—	2	8	no	—
134	C₂²×C₄×C₃₁	—	2	8	no	—
135	C₂²×C₄×C₃₁	—	2	16	no	—
136	C₂²×C₄×C₃₁	—	4	16	no	—
137	C₂²×C₄×C₃₁	—	4	4	no	—
138	C₂²×C₄×C₃₁	—	4	2	no	—
139	C₂²×C₄×C₃₁	—	4	8	no	—
140	C₂²×C₄×C₃₁	—	4	4	no	—
141	C₂²×C₄×C₃₁	—	4	32	no	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
154	C₄⋊C₄×C₃₁	—	2	8	no	—
155	C₄⋊C₄×C₃₁	—	2	8	no	—
156	C₄⋊C₄×C₃₁	—	2	16	no	—
157	C₄⋊C₄×C₃₁	—	4	4	no	—
158	C₄⋊C₄×C₃₁	—	4	4	no	—
159	C₂²⋊C₄×C₃₁	—	2	8	no	—
160	C₂²⋊C₄×C₃₁	—	2	8	no	—
161	C₂²⋊C₄×C₃₁	—	2	8	no	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	16	no	—
164	C₂²⋊C₄×C₃₁	—	2	8	no	—
165	C₂²⋊C₄×C₃₁	—	2	8	no	—
166	C₂²⋊C₄×C₃₁	—	2	16	no	—
167	C₂²⋊C₄×C₃₁	—	4	2	no	—
168	C₂²⋊C₄×C₃₁	—	4	2	no	—
169	C₂²⋊C₄×C₃₁	—	4	8	no	—
170	C₄²×C₃₁	—	2	8	no	—
171	C₄²×C₃₁	—	2	8	no	—
172	C₄²×C₃₁	—	2	8	no	—
173	#34956	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	4	no	—
175	C₂³×C₃₁	—	2	4	no	—
176	C₂³×C₃₁	—	2	16	no	—
177	C₂³×C₃₁	—	2	16	no	—
178	C₂³×C₃₁	—	2	4	no	—
179	C₂³×C₃₁	—	2	8	no	—
180	C₂³×C₃₁	—	2	4	no	—
181	C₂³×C₃₁	—	2	4	no	—
182	C₂³×C₃₁	—	2	2	no	—
183	C₂³×C₃₁	—	2	8	no	—
184	C₂³×C₃₁	—	4	4	no	—
185	C₂³×C₃₁	—	4	16	no	—
186	C₂³×C₃₁	—	4	32	no	—
187	C₂³×C₃₁	—	4	8	no	—
188	C₂³×C₃₁	—	4	8	no	—
189	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
190	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
191	C₂³×C₃₁	C₂×#41	1	8	yes	—
192	C₂³×C₃₁	C₂×#41	1	8	yes	—
193	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
194	D₄×C₃₁	—	2	32	no	—
195	D₄×C₃₁	—	2	32	no	—
196	D₄×C₃₁	—	4	16	no	—
197	C₂×C₄×C₃₁	—	2	16	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	16	no	—
203	C₂×C₄×C₃₁	—	2	8	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	8	no	—
207	C₂×C₄×C₃₁	—	2	8	no	—
208	C₂×C₄×C₃₁	—	2	8	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	8	no	—
212	C₂×C₄×C₃₁	—	4	4	no	—
213	C₂×C₄×C₃₁	—	4	8	no	—
214	C₂×C₄×C₃₁	—	4	16	no	—
215	C₂×C₄×C₃₁	—	4	8	no	—
216	C₂×C₄×C₃₁	—	4	8	no	—
217	C₂×C₄×C₃₁	—	4	16	no	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
222	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
223	C₂²×C₄×D₄	C₂×C₃₁	1	2	yes	complemented
224	#2600	C₂×C₃₁	1	4	yes	complemented
225	#2579	C₂×C₃₁	1	4	yes	complemented
226	#2534	C₂×C₃₁	1	4	yes	complemented
227	#2483	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#2405	C₂×C₃₁	1	2	yes	complemented
229	#2403	C₂×C₃₁	1	1	yes	characteristic
230	#2392	C₂×C₃₁	1	4	yes	complemented
231	C₄×#123	C₂×C₃₁	1	4	yes	complemented
232	C₄×#120	C₂×C₃₁	1	4	yes	complemented
233	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂²×C₃₁	—	2	4	no	—
235	C₂²×C₃₁	—	2	4	no	—
236	C₂²×C₃₁	—	2	16	no	—
237	C₂²×C₃₁	—	2	4	no	—
238	C₂²×C₃₁	—	2	8	no	—
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	2	16	no	—
241	C₂²×C₃₁	—	4	16	no	—
242	C₂²×C₃₁	—	4	32	no	—
243	C₂²×C₃₁	—	4	4	no	—
244	C₂²×C₃₁	—	4	4	no	—
245	C₂²×C₃₁	—	4	8	no	—
246	C₂²×C₃₁	—	4	4	no	—
247	C₂²×C₃₁	#525	1	8	yes	—
248	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
249	C₂²×C₃₁	C₂²×#41	1	4	yes	—
250	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
251	C₄×C₃₁	—	2	8	no	—
252	C₄×C₃₁	—	2	4	no	—
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	4	4	no	—
255	C₄×C₃₁	—	4	4	no	—
256	C₄×C₃₁	—	4	4	no	—
257	C₄×C₃₁	—	4	4	no	—
258	C₄×C₃₁	#534	1	4	yes	—
259	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
260	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	complemented
262	C₂×#123	C₂²×C₃₁	1	4	yes	—
263	C₂×#120	C₂²×C₃₁	1	4	yes	—
264	C₂×C₄×D₄	—	2	16	no	—
265	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
266	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
267	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
268	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
269	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
270	C₂×#117	C₂²×C₃₁	1	2	yes	complemented
271	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
272	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
273	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
274	C₂²×C₄²	C₂²×C₃₁	1	2	yes	complemented
275	#398	C₂²×C₃₁	1	4	yes	complemented
276	#397	C₂²×C₃₁	1	4	yes	complemented
277	#396	C₂²×C₃₁	1	8	yes	complemented
278	#394	C₂²×C₃₁	1	8	yes	complemented
279	#394	C₂²×C₃₁	1	4	yes	complemented
280	#392	C₂²×C₃₁	1	4	yes	—
281	#388	C₂²×C₃₁	1	8	yes	complemented
282	#387	C₂²×C₃₁	1	8	yes	complemented
283	#386	C₂²×C₃₁	1	4	yes	—
284	#386	C₂²×C₃₁	1	4	yes	—
285	#386	C₂²×C₃₁	1	8	yes	complemented
286	#385	C₂²×C₃₁	1	8	yes	complemented
287	#385	C₂²×C₃₁	1	4	yes	complemented
288	#385	C₂²×C₃₁	1	4	yes	complemented
289	#381	C₂²×C₃₁	1	2	yes	complemented
290	#381	C₂²×C₃₁	1	4	yes	complemented
291	#380	C₂²×C₃₁	1	4	yes	complemented
292	#379	C₂²×C₃₁	1	1	yes	characteristic
293	#379	C₂²×C₃₁	1	1	yes	characteristic
294	#379	C₂²×C₃₁	1	4	yes	complemented
295	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
296	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
297	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
298	C₂×C₃₁	—	2	8	no	—
299	C₂×C₃₁	—	4	2	no	—
300	C₂×C₃₁	—	4	8	no	—
301	C₂×C₃₁	C₂×#525	1	2	yes	—
302	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
303	C₂×C₃₁	#3481	1	4	yes	—
304	C₂⁵	C₂³×C₃₁	1	2	yes	—
305	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
306	C₂²×D₄	—	2	8	no	—
307	C₂²×D₄	—	2	16	no	—
308	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
309	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
310	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
311	C₂³×C₄	—	2	8	no	—
312	C₂³×C₄	—	4	4	no	—
313	C₂³×C₄	—	4	8	no	—
314	C₂³×C₄	D₄×C₃₁	1	8	yes	complemented
315	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
316	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
317	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
318	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
320	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
321	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
322	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
323	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
324	#123	—	2	16	no	—
325	#120	—	2	16	no	—
326	C₄×D₄	—	2	16	no	—
327	C₄×D₄	—	2	16	no	—
328	C₄×D₄	—	2	16	no	—
329	C₄×D₄	—	2	16	no	—
330	C₄×D₄	—	4	8	no	—
331	#117	—	4	4	no	—
332	C₂×C₄⋊C₄	—	2	8	no	—
333	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
334	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
338	C₂×C₂²⋊C₄	—	2	8	no	—
339	C₂×C₂²⋊C₄	—	2	8	no	—
340	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
341	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
342	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
343	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
344	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
351	C₂×C₄²	—	2	8	no	—
352	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
353	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
354	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
355	#95	C₂³×C₃₁	1	8	yes	—
356	#95	C₂³×C₃₁	1	8	yes	—
357	#95	C₂³×C₃₁	1	8	yes	—
358	#95	C₂³×C₃₁	1	4	yes	—
359	#95	C₂³×C₃₁	1	4	yes	—
360	C₃₁	#34956	1	1	yes	characteristic, direct product factor, Sylow
361	C₂⁴	—	2	4	no	—
362	C₂⁴	—	2	8	no	—
363	C₂⁴	—	2	8	no	—
364	C₂⁴	—	2	8	no	—
365	C₂⁴	—	4	2	no	—
366	C₂⁴	—	4	4	no	—
367	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
368	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
369	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
370	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
371	C₂×D₄	—	2	64	no	—
372	C₂×D₄	—	2	8	no	—
373	C₂×D₄	—	2	8	no	—
374	C₂×D₄	—	2	8	no	—
375	C₂×D₄	—	2	8	no	—
376	C₂×D₄	—	2	16	no	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	4	4	no	—
379	C₂×D₄	#41×C₃₁	1	16	yes	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	8	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	16	no	—
390	C₂²×C₄	—	4	16	no	—
391	C₂²×C₄	—	4	4	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	8	no	—
394	C₂²×C₄	—	4	4	no	—
395	C₂²×C₄	—	4	32	no	—
396	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
397	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
400	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
401	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	16	no	—
411	C₄⋊C₄	—	4	4	no	—
412	C₄⋊C₄	—	4	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	16	no	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	—	2	16	no	—
421	C₂²⋊C₄	—	4	2	no	—
422	C₂²⋊C₄	—	4	2	no	—
423	C₂²⋊C₄	—	4	8	no	—
424	C₄²	—	2	8	no	—
425	C₄²	—	2	8	no	—
426	C₄²	—	2	8	no	—
427	C₂³	—	2	4	no	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	16	no	—
430	C₂³	—	2	16	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	2	8	no	—
433	C₂³	—	2	4	no	—
434	C₂³	—	2	4	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	8	no	—
437	C₂³	—	4	4	no	—
438	C₂³	—	4	16	no	—
439	C₂³	—	4	32	no	—
440	C₂³	—	4	8	no	—
441	C₂³	—	4	8	no	—
442	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
443	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
444	C₂³	C₂×#41×C₃₁	1	8	yes	—
445	C₂³	C₂×#41×C₃₁	1	8	yes	—
446	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
447	D₄	—	2	32	no	—
448	D₄	—	2	32	no	—
449	D₄	—	4	16	no	—
450	C₂×C₄	—	2	16	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	16	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	8	no	—
460	C₂×C₄	—	2	8	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	8	no	—
465	C₂×C₄	—	4	4	no	—
466	C₂×C₄	—	4	8	no	—
467	C₂×C₄	—	4	16	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	—	4	16	no	—
471	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
472	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
474	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
475	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
476	C₂²	—	2	4	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	16	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	8	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	16	no	—
483	C₂²	—	4	16	no	—
484	C₂²	—	4	32	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	8	no	—
488	C₂²	—	4	4	no	—
489	C₂²	C₃₁×#525	1	8	yes	—
490	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
491	C₂²	C₂²×#41×C₃₁	1	4	yes	—
492	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
493	C₄	—	2	8	no	—
494	C₄	—	2	4	no	—
495	C₄	—	2	2	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	4	no	—
498	C₄	—	4	4	no	—
499	C₄	—	4	4	no	—
500	C₄	C₃₁×#534	1	4	yes	—
501	C₂	—	2	8	no	—
502	C₂	—	4	2	no	—
503	C₂	—	4	8	no	—
504	C₂	C₂×C₃₁×#525	1	2	yes	—
505	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
506	C₂	C₃₁×#3481	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₃₁×#2403	C₂
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. C₂²×C₄×D₄×C₃₁	C₂	semidirect
2. C₃₁×#2600	C₂	semidirect
3. C₃₁×#2579	C₂	semidirect
4. C₃₁×#2534	C₂	semidirect
5. C₃₁×#2483	C₂	semidirect
6. C₃₁×#2405	C₂	semidirect
8. C₃₁×#2392	C₂	semidirect
9. C₄×C₃₁×#123	C₂	semidirect
10. C₄×C₃₁×#120	C₂	semidirect
11. C₃₁×#2280	C₂	semidirect
14. C₂⁴×C₄×C₃₁	C₂²	semidirect
20. C₂×C₄×D₄×C₃₁	C₂²	semidirect
21. C₂×C₄×D₄×C₃₁	C₂²	semidirect
22. C₂×C₄×D₄×C₃₁	C₂²	semidirect
23. C₂×C₃₁×#117	C₂²	semidirect
24. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
26. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
27. C₂²×C₄²×C₃₁	C₂²	semidirect
28. C₃₁×#398	C₂²	semidirect
29. C₃₁×#397	C₂²	semidirect
30. C₃₁×#396	C₂²	semidirect
31. C₃₁×#394	C₂²	semidirect
32. C₃₁×#394	C₂²	semidirect
34. C₃₁×#388	C₂²	semidirect
35. C₃₁×#387	C₂²	semidirect
38. C₃₁×#386	C₂²	semidirect
39. C₃₁×#385	C₂²	semidirect
40. C₃₁×#385	C₂²	semidirect
41. C₃₁×#385	C₂²	semidirect
42. C₃₁×#381	C₂²	semidirect
43. C₃₁×#381	C₂²	semidirect
44. C₃₁×#380	C₂²	semidirect
47. C₃₁×#379	C₂²	semidirect
48. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
49. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
61. C₂³×C₄×C₃₁	D₄	semidirect
173. #34956	C₃₁	direct (N × H)
223. C₂²×C₄×D₄	C₂×C₃₁	semidirect
224. #2600	C₂×C₃₁	semidirect
225. #2579	C₂×C₃₁	semidirect
226. #2534	C₂×C₃₁	semidirect
227. #2483	C₂×C₃₁	semidirect
228. #2405	C₂×C₃₁	semidirect
230. #2392	C₂×C₃₁	semidirect
231. C₄×#123	C₂×C₃₁	semidirect
232. C₄×#120	C₂×C₃₁	semidirect
233. #2280	C₂×C₃₁	semidirect
261. C₂⁴×C₄	C₂²×C₃₁	semidirect
267. C₂×C₄×D₄	C₂²×C₃₁	semidirect
268. C₂×C₄×D₄	C₂²×C₃₁	semidirect
269. C₂×C₄×D₄	C₂²×C₃₁	semidirect
270. C₂×#117	C₂²×C₃₁	semidirect
271. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
273. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
274. C₂²×C₄²	C₂²×C₃₁	semidirect
275. #398	C₂²×C₃₁	semidirect
276. #397	C₂²×C₃₁	semidirect
277. #396	C₂²×C₃₁	semidirect
278. #394	C₂²×C₃₁	semidirect
279. #394	C₂²×C₃₁	semidirect
281. #388	C₂²×C₃₁	semidirect
282. #387	C₂²×C₃₁	semidirect
285. #386	C₂²×C₃₁	semidirect
286. #385	C₂²×C₃₁	semidirect
287. #385	C₂²×C₃₁	semidirect
288. #385	C₂²×C₃₁	semidirect
289. #381	C₂²×C₃₁	semidirect
290. #381	C₂²×C₃₁	semidirect
291. #380	C₂²×C₃₁	semidirect
294. #379	C₂²×C₃₁	semidirect
295. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
296. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
297. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
314. C₂³×C₄	D₄×C₃₁	semidirect
360. C₃₁	#34956	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2600	2
C₃₁×#2579	2
C₃₁×#2534	2
C₃₁×#2483	2
C₃₁×#2405	2
C₃₁×#2403	2
C₃₁×#2392	2
C₄×C₃₁×#123	2
C₄×C₃₁×#120	2
C₃₁×#2280	2
#34956	31

Subgroups — C31×#34956 #21491715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34956 #21491715