Subgroups — C₃₁×#34363 #21491122

Order 7936.

2954 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2377	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2363	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2344	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2342	C₂	1	4	yes	maximal, complemented
6	D₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2333	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2324	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2309	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2308	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#121	C₄	1	16	yes	complemented
17	C₂×C₃₁×#121	C₂²	1	2	yes	—
18	C₂×C₃₁×#121	C₂²	1	2	yes	—
19	C₂×C₃₁×#121	C₂²	1	2	yes	—
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
22	C₂×C₃₁×#117	C₂²	1	4	yes	—
23	C₂×C₃₁×#117	C₂²	1	2	yes	complemented
24	C₂×C₃₁×#117	C₂²	1	2	yes	complemented
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#390	C₂²	1	8	yes	complemented
29	C₃₁×#390	C₂²	1	8	yes	complemented
30	C₃₁×#390	C₂²	1	4	yes	complemented
31	C₃₁×#389	C₂²	1	4	yes	complemented
32	C₃₁×#388	C₂²	1	8	yes	complemented
33	C₃₁×#388	C₂²	1	8	yes	complemented
34	C₃₁×#387	C₂²	1	8	yes	complemented
35	C₃₁×#386	C₂²	1	8	yes	complemented
36	C₃₁×#386	C₂²	1	8	yes	complemented
37	C₃₁×#386	C₂²	1	8	yes	complemented
38	C₃₁×#385	C₂²	1	8	yes	complemented
39	C₃₁×#381	C₂²	1	4	yes	—
40	C₃₁×#381	C₂²	1	2	yes	complemented
41	C₃₁×#381	C₂²	1	2	yes	complemented
42	C₃₁×#380	C₂²	1	2	yes	—
43	C₃₁×#380	C₂²	1	2	yes	—
44	C₃₁×#380	C₂²	1	2	yes	complemented
45	C₃₁×#380	C₂²	1	2	yes	complemented
46	C₃₁×#379	C₂²	1	1	yes	characteristic
47	C₃₁×#379	C₂²	1	2	yes	—
48	C₃₁×#379	C₂²	1	1	yes	characteristic
49	C₃₁×#379	C₂²	1	4	yes	—
50	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
51	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
52	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
53	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
54	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
55	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
56	C₂⁵×C₃₁	C₂³	1	2	yes	—
57	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₃₁	—	4	4	no	—
59	C₂²×D₄×C₃₁	—	4	8	no	—
60	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
61	C₂²×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
62	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
63	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
64	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
65	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
66	C₂³×C₄×C₃₁	—	4	4	no	—
67	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
68	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#121	C₂×C₄	1	16	yes	complemented
77	C₃₁×#121	C₂×C₄	1	16	yes	complemented
78	C₃₁×#121	C₂×C₄	1	16	yes	complemented
79	C₃₁×#121	C₂×C₄	1	16	yes	complemented
80	C₄×D₄×C₃₁	—	4	8	no	—
81	C₃₁×#117	—	4	4	no	—
82	C₃₁×#117	—	4	8	no	—
83	C₃₁×#117	—	4	4	no	—
84	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
90	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
91	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	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	4	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
108	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
109	C₃₁×#95	C₂³	1	4	yes	—
110	C₃₁×#95	C₂³	1	4	yes	—
111	C₃₁×#95	C₂³	1	4	yes	—
112	C₃₁×#95	C₂³	1	4	yes	—
113	C₃₁×#95	C₂³	1	8	yes	—
114	C₃₁×#95	C₂³	1	8	yes	—
115	C₂⁴×C₃₁	—	2	4	no	—
116	C₂⁴×C₃₁	—	2	8	no	—
117	C₂⁴×C₃₁	—	4	2	no	—
118	C₂⁴×C₃₁	—	4	2	no	—
119	C₂⁴×C₃₁	—	4	4	no	—
120	C₂⁴×C₃₁	—	4	4	no	—
121	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
122	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
123	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
124	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
125	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
126	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
127	C₂×D₄×C₃₁	—	2	8	no	—
128	C₂×D₄×C₃₁	—	4	16	no	—
129	C₂×D₄×C₃₁	—	4	32	no	—
130	C₂×D₄×C₃₁	—	4	4	no	—
131	C₂×D₄×C₃₁	—	4	4	no	—
132	C₂×D₄×C₃₁	—	4	4	no	—
133	C₂×D₄×C₃₁	—	4	4	no	—
134	C₂×D₄×C₃₁	—	4	4	no	—
135	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
136	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
137	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
138	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
139	C₂²×C₄×C₃₁	—	2	8	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	—	4	16	no	—
144	C₂²×C₄×C₃₁	—	4	2	no	—
145	C₂²×C₄×C₃₁	—	4	2	no	—
146	C₂²×C₄×C₃₁	—	4	2	no	—
147	C₂²×C₄×C₃₁	—	4	8	no	—
148	C₂²×C₄×C₃₁	—	4	8	no	—
149	C₂²×C₄×C₃₁	—	4	8	no	—
150	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
151	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
155	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
165	C₄⋊C₄×C₃₁	—	2	8	no	—
166	C₄⋊C₄×C₃₁	—	4	4	no	—
167	C₄⋊C₄×C₃₁	—	4	8	no	—
168	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
169	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	—	2	8	no	—
172	C₂²⋊C₄×C₃₁	—	2	4	no	—
173	C₂²⋊C₄×C₃₁	—	4	8	no	—
174	C₂²⋊C₄×C₃₁	—	4	2	no	—
175	C₂²⋊C₄×C₃₁	—	4	8	no	—
176	C₂²⋊C₄×C₃₁	—	4	4	no	—
177	C₂²⋊C₄×C₃₁	—	4	2	no	—
178	C₂²⋊C₄×C₃₁	—	4	2	no	—
179	C₂²⋊C₄×C₃₁	—	4	2	no	—
180	C₂²⋊C₄×C₃₁	—	4	4	no	—
181	C₂²⋊C₄×C₃₁	—	4	8	no	—
182	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
183	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
184	C₄²×C₃₁	—	2	8	no	—
185	C₄²×C₃₁	—	4	4	no	—
186	C₄²×C₃₁	—	4	8	no	—
187	#34363	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂³×C₃₁	—	2	2	no	—
189	C₂³×C₃₁	—	2	2	no	—
190	C₂³×C₃₁	—	2	8	no	—
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	4	no	—
194	C₂³×C₃₁	—	2	8	no	—
195	C₂³×C₃₁	—	2	4	no	—
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	2	no	—
200	C₂³×C₃₁	—	2	8	no	—
201	C₂³×C₃₁	—	2	16	no	—
202	C₂³×C₃₁	—	4	4	no	—
203	C₂³×C₃₁	—	4	4	no	—
204	C₂³×C₃₁	—	4	8	no	—
205	C₂³×C₃₁	—	4	8	no	—
206	C₂³×C₃₁	—	4	16	no	—
207	C₂³×C₃₁	—	8	16	no	—
208	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
209	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
210	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
211	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
212	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
213	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
214	D₄×C₃₁	—	2	16	no	—
215	D₄×C₃₁	—	4	8	no	—
216	D₄×C₃₁	—	4	16	no	—
217	D₄×C₃₁	—	4	16	no	—
218	C₂×C₄×C₃₁	—	2	16	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	8	no	—
225	C₂×C₄×C₃₁	—	2	8	no	—
226	C₂×C₄×C₃₁	—	2	8	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	8	no	—
231	C₂×C₄×C₃₁	—	2	8	no	—
232	C₂×C₄×C₃₁	—	2	4	no	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	2	8	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	4	4	no	—
237	C₂×C₄×C₃₁	—	4	4	no	—
238	C₂×C₄×C₃₁	—	4	4	no	—
239	C₂×C₄×C₃₁	—	8	16	no	—
240	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
243	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
244	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
245	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
246	C₂²×#121	C₂×C₃₁	1	1	yes	characteristic
247	#2377	C₂×C₃₁	1	4	yes	complemented
248	#2363	C₂×C₃₁	1	4	yes	complemented
249	#2344	C₂×C₃₁	1	2	yes	complemented
250	#2342	C₂×C₃₁	1	4	yes	complemented
251	D₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
252	#2333	C₂×C₃₁	1	4	yes	complemented
253	#2324	C₂×C₃₁	1	4	yes	complemented
254	#2309	C₂×C₃₁	1	2	yes	complemented
255	#2308	C₂×C₃₁	1	2	yes	complemented
256	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
258	C₂²×C₃₁	—	2	2	no	—
259	C₂²×C₃₁	—	2	2	no	—
260	C₂²×C₃₁	—	2	8	no	—
261	C₂²×C₃₁	—	2	2	no	—
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	2	8	no	—
264	C₂²×C₃₁	—	2	2	no	—
265	C₂²×C₃₁	—	2	2	no	—
266	C₂²×C₃₁	—	2	8	no	—
267	C₂²×C₃₁	—	2	2	no	—
268	C₂²×C₃₁	—	2	2	no	—
269	C₂²×C₃₁	—	2	8	no	—
270	C₂²×C₃₁	—	4	16	no	—
271	C₂²×C₃₁	—	4	8	no	—
272	C₂²×C₃₁	—	4	4	no	—
273	C₂²×C₃₁	—	4	4	no	—
274	C₂²×C₃₁	—	4	8	no	—
275	C₂²×C₃₁	—	8	16	no	—
276	C₂²×C₃₁	#518	1	4	yes	—
277	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
278	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
279	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
280	C₂²×C₃₁	#585	1	2	yes	—
281	C₄×C₃₁	—	2	2	no	—
282	C₄×C₃₁	—	2	2	no	—
283	C₄×C₃₁	—	2	2	no	—
284	C₄×C₃₁	—	2	8	no	—
285	C₄×C₃₁	—	4	4	no	—
286	C₄×C₃₁	—	4	8	no	—
287	C₄×C₃₁	—	4	4	no	—
288	C₄×C₃₁	#518	1	4	yes	—
289	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
290	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
291	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#121	C₄×C₃₁	1	16	yes	complemented
293	C₂×#121	C₂²×C₃₁	1	2	yes	—
294	C₂×#121	C₂²×C₃₁	1	2	yes	—
295	C₂×#121	C₂²×C₃₁	1	2	yes	—
296	C₂×#121	C₂²×C₃₁	1	2	yes	—
297	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
298	C₂×#117	C₂²×C₃₁	1	4	yes	—
299	C₂×#117	C₂²×C₃₁	1	2	yes	complemented
300	C₂×#117	C₂²×C₃₁	1	2	yes	complemented
301	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
302	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
303	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
304	#390	C₂²×C₃₁	1	8	yes	complemented
305	#390	C₂²×C₃₁	1	8	yes	complemented
306	#390	C₂²×C₃₁	1	4	yes	complemented
307	#389	C₂²×C₃₁	1	4	yes	complemented
308	#388	C₂²×C₃₁	1	8	yes	complemented
309	#388	C₂²×C₃₁	1	8	yes	complemented
310	#387	C₂²×C₃₁	1	8	yes	complemented
311	#386	C₂²×C₃₁	1	8	yes	complemented
312	#386	C₂²×C₃₁	1	8	yes	complemented
313	#386	C₂²×C₃₁	1	8	yes	complemented
314	#385	C₂²×C₃₁	1	8	yes	complemented
315	#381	C₂²×C₃₁	1	4	yes	—
316	#381	C₂²×C₃₁	1	2	yes	complemented
317	#381	C₂²×C₃₁	1	2	yes	complemented
318	#380	C₂²×C₃₁	1	2	yes	—
319	#380	C₂²×C₃₁	1	2	yes	—
320	#380	C₂²×C₃₁	1	2	yes	complemented
321	#380	C₂²×C₃₁	1	2	yes	complemented
322	#379	C₂²×C₃₁	1	1	yes	characteristic
323	#379	C₂²×C₃₁	1	2	yes	—
324	#379	C₂²×C₃₁	1	1	yes	characteristic
325	#379	C₂²×C₃₁	1	4	yes	—
326	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
327	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
328	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
329	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
330	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
331	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
332	C₂×C₃₁	—	2	4	no	—
333	C₂×C₃₁	—	4	2	no	—
334	C₂×C₃₁	—	4	2	no	—
335	C₂×C₃₁	—	4	8	no	—
336	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
337	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
338	C₂×C₃₁	#3428	1	1	yes	characteristic
339	C₂×C₃₁	#3542	1	4	yes	—
340	C₂⁵	C₂³×C₃₁	1	2	yes	—
341	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
342	C₂²×D₄	—	4	4	no	—
343	C₂²×D₄	—	4	8	no	—
344	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
345	C₂²×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
346	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
347	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
348	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
349	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
350	C₂³×C₄	—	4	4	no	—
351	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	complemented
352	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
353	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
357	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
359	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
360	#121	C₂×C₄×C₃₁	1	16	yes	complemented
361	#121	C₂×C₄×C₃₁	1	16	yes	complemented
362	#121	C₂×C₄×C₃₁	1	16	yes	complemented
363	#121	C₂×C₄×C₃₁	1	16	yes	complemented
364	C₄×D₄	—	4	8	no	—
365	#117	—	4	4	no	—
366	#117	—	4	8	no	—
367	#117	—	4	4	no	—
368	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
374	C₂×C₂²⋊C₄	—	4	8	no	—
375	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
376	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
390	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
391	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
392	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
393	#95	C₂³×C₃₁	1	4	yes	—
394	#95	C₂³×C₃₁	1	4	yes	—
395	#95	C₂³×C₃₁	1	4	yes	—
396	#95	C₂³×C₃₁	1	4	yes	—
397	#95	C₂³×C₃₁	1	8	yes	—
398	#95	C₂³×C₃₁	1	8	yes	—
399	C₃₁	#34363	1	1	yes	characteristic, direct product factor, Sylow
400	C₂⁴	—	2	4	no	—
401	C₂⁴	—	2	8	no	—
402	C₂⁴	—	4	2	no	—
403	C₂⁴	—	4	2	no	—
404	C₂⁴	—	4	4	no	—
405	C₂⁴	—	4	4	no	—
406	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
407	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
408	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
409	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
410	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
411	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
412	C₂×D₄	—	2	8	no	—
413	C₂×D₄	—	4	16	no	—
414	C₂×D₄	—	4	32	no	—
415	C₂×D₄	—	4	4	no	—
416	C₂×D₄	—	4	4	no	—
417	C₂×D₄	—	4	4	no	—
418	C₂×D₄	—	4	4	no	—
419	C₂×D₄	—	4	4	no	—
420	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
421	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
422	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
423	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	4	16	no	—
429	C₂²×C₄	—	4	2	no	—
430	C₂²×C₄	—	4	2	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	4	8	no	—
433	C₂²×C₄	—	4	8	no	—
434	C₂²×C₄	—	4	8	no	—
435	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
436	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
437	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
438	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
440	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
441	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
444	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
445	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
447	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	4	4	no	—
452	C₄⋊C₄	—	4	8	no	—
453	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
454	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
455	C₂²⋊C₄	—	2	4	no	—
456	C₂²⋊C₄	—	2	8	no	—
457	C₂²⋊C₄	—	2	4	no	—
458	C₂²⋊C₄	—	4	8	no	—
459	C₂²⋊C₄	—	4	2	no	—
460	C₂²⋊C₄	—	4	8	no	—
461	C₂²⋊C₄	—	4	4	no	—
462	C₂²⋊C₄	—	4	2	no	—
463	C₂²⋊C₄	—	4	2	no	—
464	C₂²⋊C₄	—	4	2	no	—
465	C₂²⋊C₄	—	4	4	no	—
466	C₂²⋊C₄	—	4	8	no	—
467	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
468	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
469	C₄²	—	2	8	no	—
470	C₄²	—	4	4	no	—
471	C₄²	—	4	8	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	8	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	8	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	2	8	no	—
485	C₂³	—	2	16	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	4	no	—
488	C₂³	—	4	8	no	—
489	C₂³	—	4	8	no	—
490	C₂³	—	4	16	no	—
491	C₂³	—	8	16	no	—
492	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
493	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
494	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
495	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
496	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
497	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
498	D₄	—	2	16	no	—
499	D₄	—	4	8	no	—
500	D₄	—	4	16	no	—
501	D₄	—	4	16	no	—
502	C₂×C₄	—	2	16	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	8	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	8	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	8	16	no	—
524	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
525	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
526	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
527	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
528	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
529	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	8	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	8	no	—
542	C₂²	—	4	16	no	—
543	C₂²	—	4	8	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	8	no	—
547	C₂²	—	8	16	no	—
548	C₂²	C₃₁×#518	1	4	yes	—
549	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
550	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
551	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
552	C₂²	C₃₁×#585	1	2	yes	—
553	C₄	—	2	2	no	—
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	8	no	—
557	C₄	—	4	4	no	—
558	C₄	—	4	8	no	—
559	C₄	—	4	4	no	—
560	C₄	C₃₁×#518	1	4	yes	—
561	C₂	—	2	4	no	—
562	C₂	—	4	2	no	—
563	C₂	—	4	2	no	—
564	C₂	—	4	8	no	—
565	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
566	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
567	C₂	C₃₁×#3428	1	1	yes	characteristic
568	C₂	C₃₁×#3542	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂²×C₃₁×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2377	C₂	semidirect
3. C₃₁×#2363	C₂	semidirect
4. C₃₁×#2344	C₂	semidirect
5. C₃₁×#2342	C₂	semidirect
6. D₄×C₂²⋊C₄×C₃₁	C₂	semidirect
7. C₃₁×#2333	C₂	semidirect
8. C₃₁×#2324	C₂	semidirect
9. C₃₁×#2309	C₂	semidirect
10. C₃₁×#2308	C₂	semidirect
11. C₃₁×#2280	C₂	semidirect
12. C₃₁×#2280	C₂	semidirect
16. C₂×C₃₁×#121	C₄	semidirect
21. C₂×C₄×D₄×C₃₁	C₂²	semidirect
23. C₂×C₃₁×#117	C₂²	semidirect
24. C₂×C₃₁×#117	C₂²	semidirect
28. C₃₁×#390	C₂²	semidirect
29. C₃₁×#390	C₂²	semidirect
30. C₃₁×#390	C₂²	semidirect
31. C₃₁×#389	C₂²	semidirect
32. C₃₁×#388	C₂²	semidirect
33. C₃₁×#388	C₂²	semidirect
34. C₃₁×#387	C₂²	semidirect
35. C₃₁×#386	C₂²	semidirect
36. C₃₁×#386	C₂²	semidirect
37. C₃₁×#386	C₂²	semidirect
38. C₃₁×#385	C₂²	semidirect
40. C₃₁×#381	C₂²	semidirect
41. C₃₁×#381	C₂²	semidirect
44. C₃₁×#380	C₂²	semidirect
45. C₃₁×#380	C₂²	semidirect
50. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
53. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
54. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
55. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
61. C₂²×D₄×C₃₁	C₂×C₄	semidirect
67. C₂³×C₄×C₃₁	C₂×C₄	semidirect
76. C₃₁×#121	C₂×C₄	semidirect
77. C₃₁×#121	C₂×C₄	semidirect
78. C₃₁×#121	C₂×C₄	semidirect
79. C₃₁×#121	C₂×C₄	semidirect
84. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
91. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
187. #34363	C₃₁	direct (N × H)
247. #2377	C₂×C₃₁	semidirect
248. #2363	C₂×C₃₁	semidirect
249. #2344	C₂×C₃₁	semidirect
250. #2342	C₂×C₃₁	semidirect
251. D₄×C₂²⋊C₄	C₂×C₃₁	semidirect
252. #2333	C₂×C₃₁	semidirect
253. #2324	C₂×C₃₁	semidirect
254. #2309	C₂×C₃₁	semidirect
255. #2308	C₂×C₃₁	semidirect
256. #2280	C₂×C₃₁	semidirect
257. #2280	C₂×C₃₁	semidirect
292. C₂×#121	C₄×C₃₁	semidirect
297. C₂×C₄×D₄	C₂²×C₃₁	semidirect
299. C₂×#117	C₂²×C₃₁	semidirect
300. C₂×#117	C₂²×C₃₁	semidirect
304. #390	C₂²×C₃₁	semidirect
305. #390	C₂²×C₃₁	semidirect
306. #390	C₂²×C₃₁	semidirect
307. #389	C₂²×C₃₁	semidirect
308. #388	C₂²×C₃₁	semidirect
309. #388	C₂²×C₃₁	semidirect
310. #387	C₂²×C₃₁	semidirect
311. #386	C₂²×C₃₁	semidirect
312. #386	C₂²×C₃₁	semidirect
313. #386	C₂²×C₃₁	semidirect
314. #385	C₂²×C₃₁	semidirect
316. #381	C₂²×C₃₁	semidirect
317. #381	C₂²×C₃₁	semidirect
320. #380	C₂²×C₃₁	semidirect
321. #380	C₂²×C₃₁	semidirect
326. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
329. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
330. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
331. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
345. C₂²×D₄	C₂×C₄×C₃₁	semidirect
351. C₂³×C₄	C₂×C₄×C₃₁	semidirect
360. #121	C₂×C₄×C₃₁	semidirect
361. #121	C₂×C₄×C₃₁	semidirect
362. #121	C₂×C₄×C₃₁	semidirect
363. #121	C₂×C₄×C₃₁	semidirect
368. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
375. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
399. C₃₁	#34363	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×C₃₁×#121	2
C₃₁×#2377	2
C₃₁×#2363	2
C₃₁×#2344	2
C₃₁×#2342	2
D₄×C₂²⋊C₄×C₃₁	2
C₃₁×#2333	2
C₃₁×#2324	2
C₃₁×#2309	2
C₃₁×#2308	2
C₃₁×#2280	2
#34363	31

Subgroups — C31×#34363 #21491122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34363 #21491122