Subgroups — C₃₁×#34575 #21491334

Order 7936.

2794 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2663	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2661	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2659	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2656	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2639	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2617	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2395	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2388	C₂	1	4	yes	maximal, complemented
10	C₄×C₃₁×#121	C₂	1	4	yes	maximal, complemented
11	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#525	—	2	16	no	—
15	C₂×C₃₁×#123	C₂²	1	4	yes	—
16	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₃₁×#121	C₂²	1	4	yes	—
19	C₂×C₃₁×#121	C₂²	1	2	yes	—
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₂×C₃₁×#120	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
25	C₂×C₃₁×#117	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#400	C₂²	1	4	yes	complemented
29	C₃₁×#399	C₂²	1	4	yes	complemented
30	C₃₁×#398	C₂²	1	4	yes	complemented
31	C₃₁×#397	C₂²	1	8	yes	—
32	C₃₁×#397	C₂²	1	4	yes	complemented
33	C₃₁×#395	C₂²	1	4	yes	complemented
34	C₃₁×#394	C₂²	1	8	yes	—
35	C₃₁×#394	C₂²	1	8	yes	—
36	C₃₁×#394	C₂²	1	4	yes	complemented
37	C₃₁×#393	C₂²	1	4	yes	complemented
38	C₃₁×#392	C₂²	1	8	yes	—
39	C₃₁×#392	C₂²	1	4	yes	complemented
40	C₃₁×#390	C₂²	1	4	yes	complemented
41	C₃₁×#389	C₂²	1	4	yes	complemented
42	C₃₁×#388	C₂²	1	8	yes	complemented
43	C₃₁×#386	C₂²	1	8	yes	—
44	C₃₁×#380	C₂²	1	4	yes	—
45	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
46	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
47	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
48	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
49	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
51	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×#41×C₃₁	—	2	8	no	—
53	C₂×#41×C₃₁	—	4	8	no	—
54	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
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	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
59	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
60	C₂³×C₄×C₃₁	—	4	4	no	—
61	C₂³×C₄×C₃₁	D₄	1	8	yes	complemented
62	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	C₂³	1	2	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	2	yes	—
69	C₃₁×#123	—	2	16	no	—
70	C₃₁×C₂²⋊Q₈	—	2	8	no	—
71	C₃₁×C₂²⋊Q₈	—	2	8	no	—
72	C₃₁×#121	—	2	8	no	—
73	C₃₁×#121	—	2	8	no	—
74	C₃₁×#121	—	2	16	no	—
75	C₃₁×#120	—	2	16	no	—
76	C₄×D₄×C₃₁	—	2	16	no	—
77	C₄×D₄×C₃₁	—	2	16	no	—
78	C₄×D₄×C₃₁	—	4	8	no	—
79	C₃₁×#117	—	2	8	no	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
97	C₂×C₄²×C₃₁	—	4	4	no	—
98	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
101	C₃₁×#95	C₂³	1	8	yes	—
102	C₃₁×#95	C₂³	1	4	yes	—
103	C₃₁×#95	C₂³	1	8	yes	—
104	C₃₁×#95	C₂³	1	8	yes	—
105	C₃₁×#95	C₂³	1	4	yes	—
106	C₂⁴×C₃₁	—	2	4	no	—
107	C₂⁴×C₃₁	—	4	2	no	—
108	C₂⁴×C₃₁	—	4	1	no	—
109	C₂⁴×C₃₁	—	4	1	no	—
110	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
111	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
112	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
113	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
114	#41×C₃₁	—	4	32	no	—
115	C₂×Q₈×C₃₁	—	2	4	no	—
116	C₂×Q₈×C₃₁	—	4	4	no	—
117	C₂×D₄×C₃₁	—	2	8	no	—
118	C₂×D₄×C₃₁	—	2	8	no	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	—	2	16	no	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	—	2	8	no	—
123	C₂×D₄×C₃₁	—	4	4	no	—
124	C₂×D₄×C₃₁	—	4	4	no	—
125	C₂×D₄×C₃₁	—	4	4	no	—
126	C₂×D₄×C₃₁	—	4	4	no	—
127	C₂×D₄×C₃₁	—	4	4	no	—
128	C₂×D₄×C₃₁	—	4	8	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	8	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	8	no	—
133	C₂²×C₄×C₃₁	—	2	16	no	—
134	C₂²×C₄×C₃₁	—	4	16	no	—
135	C₂²×C₄×C₃₁	—	4	2	no	—
136	C₂²×C₄×C₃₁	—	4	2	no	—
137	C₂²×C₄×C₃₁	—	4	4	no	—
138	C₂²×C₄×C₃₁	—	4	2	no	—
139	C₂²×C₄×C₃₁	—	4	2	no	—
140	C₂²×C₄×C₃₁	—	4	2	no	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
145	C₂²×C₄×C₃₁	#41	1	8	yes	complemented
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	4	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	8	no	—
157	C₄⋊C₄×C₃₁	—	2	8	no	—
158	C₄⋊C₄×C₃₁	—	2	8	no	—
159	C₄⋊C₄×C₃₁	—	4	4	no	—
160	C₄⋊C₄×C₃₁	—	4	4	no	—
161	C₂²⋊C₄×C₃₁	—	2	8	no	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	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	—	2	16	no	—
168	C₂²⋊C₄×C₃₁	—	4	8	no	—
169	C₂²⋊C₄×C₃₁	—	4	8	no	—
170	C₄²×C₃₁	—	2	8	no	—
171	C₄²×C₃₁	—	2	8	no	—
172	C₄²×C₃₁	—	4	4	no	—
173	#34575	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	8	no	—
178	C₂³×C₃₁	—	2	4	no	—
179	C₂³×C₃₁	—	2	4	no	—
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	2	2	no	—
182	C₂³×C₃₁	—	2	4	no	—
183	C₂³×C₃₁	—	2	8	no	—
184	C₂³×C₃₁	—	2	8	no	—
185	C₂³×C₃₁	—	2	4	no	—
186	C₂³×C₃₁	—	2	4	no	—
187	C₂³×C₃₁	—	4	4	no	—
188	C₂³×C₃₁	—	4	4	no	—
189	C₂³×C₃₁	—	4	8	no	—
190	C₂³×C₃₁	—	4	8	no	—
191	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
192	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
193	C₂³×C₃₁	C₂×#41	1	4	yes	—
194	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
195	Q₈×C₃₁	—	4	8	no	—
196	D₄×C₃₁	—	4	16	no	—
197	D₄×C₃₁	—	4	8	no	—
198	D₄×C₃₁	—	4	16	no	—
199	D₄×C₃₁	—	4	16	no	—
200	D₄×C₃₁	—	4	16	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	8	no	—
206	C₂×C₄×C₃₁	—	2	16	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	4	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	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	8	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	4	8	no	—
218	C₂×C₄×C₃₁	—	4	4	no	—
219	C₂×C₄×C₃₁	—	4	8	no	—
220	C₂×C₄×C₃₁	—	4	8	no	—
221	C₂×C₄×C₃₁	—	4	16	no	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
226	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
227	C₂×#525	C₂×C₃₁	1	2	yes	complemented
228	#2663	C₂×C₃₁	1	4	yes	complemented
229	#2661	C₂×C₃₁	1	4	yes	complemented
230	#2659	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#2656	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#2639	C₂×C₃₁	1	2	yes	complemented
233	#2617	C₂×C₃₁	1	4	yes	complemented
234	#2395	C₂×C₃₁	1	4	yes	complemented
235	#2388	C₂×C₃₁	1	4	yes	complemented
236	C₄×#121	C₂×C₃₁	1	4	yes	complemented
237	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×C₃₁	—	2	2	no	—
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	2	4	no	—
241	C₂²×C₃₁	—	2	4	no	—
242	C₂²×C₃₁	—	2	4	no	—
243	C₂²×C₃₁	—	2	4	no	—
244	C₂²×C₃₁	—	2	4	no	—
245	C₂²×C₃₁	—	4	16	no	—
246	C₂²×C₃₁	—	4	4	no	—
247	C₂²×C₃₁	—	4	4	no	—
248	C₂²×C₃₁	—	4	4	no	—
249	C₂²×C₃₁	—	4	4	no	—
250	C₂²×C₃₁	—	4	4	no	—
251	C₂²×C₃₁	—	4	8	no	—
252	C₂²×C₃₁	—	4	8	no	—
253	C₂²×C₃₁	#532	1	4	yes	—
254	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
255	C₂²×C₃₁	C₂²×#41	1	2	yes	—
256	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
257	C₂²×C₃₁	#585	1	1	yes	characteristic
258	C₂²×C₃₁	#585	1	1	yes	characteristic
259	C₄×C₃₁	—	2	2	no	—
260	C₄×C₃₁	—	2	4	no	—
261	C₄×C₃₁	—	2	8	no	—
262	C₄×C₃₁	—	4	4	no	—
263	C₄×C₃₁	—	4	4	no	—
264	C₄×C₃₁	—	4	4	no	—
265	C₄×C₃₁	—	4	4	no	—
266	C₄×C₃₁	#534	1	4	yes	—
267	C₂²×#41	C₂²×C₃₁	1	2	yes	—
268	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
269	#525	—	2	16	no	—
270	C₂×#123	C₂²×C₃₁	1	4	yes	—
271	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
272	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
273	C₂×#121	C₂²×C₃₁	1	4	yes	—
274	C₂×#121	C₂²×C₃₁	1	2	yes	—
275	C₂×#121	C₂²×C₃₁	1	2	yes	—
276	C₂×#120	C₂²×C₃₁	1	4	yes	—
277	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
278	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
279	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
280	C₂×#117	C₂²×C₃₁	1	2	yes	—
281	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
282	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
283	#400	C₂²×C₃₁	1	4	yes	complemented
284	#399	C₂²×C₃₁	1	4	yes	complemented
285	#398	C₂²×C₃₁	1	4	yes	complemented
286	#397	C₂²×C₃₁	1	8	yes	—
287	#397	C₂²×C₃₁	1	4	yes	complemented
288	#395	C₂²×C₃₁	1	4	yes	complemented
289	#394	C₂²×C₃₁	1	8	yes	—
290	#394	C₂²×C₃₁	1	8	yes	—
291	#394	C₂²×C₃₁	1	4	yes	complemented
292	#393	C₂²×C₃₁	1	4	yes	complemented
293	#392	C₂²×C₃₁	1	8	yes	—
294	#392	C₂²×C₃₁	1	4	yes	complemented
295	#390	C₂²×C₃₁	1	4	yes	complemented
296	#389	C₂²×C₃₁	1	4	yes	complemented
297	#388	C₂²×C₃₁	1	8	yes	complemented
298	#386	C₂²×C₃₁	1	8	yes	—
299	#380	C₂²×C₃₁	1	4	yes	—
300	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
301	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
302	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
303	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
304	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×C₃₁	—	2	4	no	—
307	C₂×C₃₁	—	4	4	no	—
308	C₂×C₃₁	—	8	4	no	—
309	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
310	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
311	C₂×C₃₁	#3449	1	1	yes	characteristic
312	C₂×C₃₁	#3459	1	2	yes	—
313	C₂×C₃₁	#3460	1	2	yes	—
314	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×#41	—	2	8	no	—
316	C₂×#41	—	4	8	no	—
317	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
318	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
319	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
320	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
321	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
322	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
323	C₂³×C₄	—	4	4	no	—
324	C₂³×C₄	D₄×C₃₁	1	8	yes	complemented
325	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
326	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
327	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
328	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
329	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
330	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
331	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
332	#123	—	2	16	no	—
333	C₂²⋊Q₈	—	2	8	no	—
334	C₂²⋊Q₈	—	2	8	no	—
335	#121	—	2	8	no	—
336	#121	—	2	8	no	—
337	#121	—	2	16	no	—
338	#120	—	2	16	no	—
339	C₄×D₄	—	2	16	no	—
340	C₄×D₄	—	2	16	no	—
341	C₄×D₄	—	4	8	no	—
342	#117	—	2	8	no	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
346	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
347	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
348	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
349	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
350	C₂×C₂²⋊C₄	—	2	8	no	—
351	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	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×C₄²	—	4	4	no	—
361	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
362	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
363	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
364	#95	C₂³×C₃₁	1	4	yes	—
365	#95	C₂³×C₃₁	1	4	yes	—
366	#95	C₂³×C₃₁	1	8	yes	—
367	#95	C₂³×C₃₁	1	8	yes	—
368	#95	C₂³×C₃₁	1	8	yes	—
369	C₃₁	#34575	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁴	—	2	4	no	—
371	C₂⁴	—	4	2	no	—
372	C₂⁴	—	4	1	no	—
373	C₂⁴	—	4	1	no	—
374	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
375	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
376	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
377	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
378	#41	—	4	32	no	—
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	4	4	no	—
381	C₂×D₄	—	2	8	no	—
382	C₂×D₄	—	2	8	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	16	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	8	no	—
387	C₂×D₄	—	4	4	no	—
388	C₂×D₄	—	4	4	no	—
389	C₂×D₄	—	4	4	no	—
390	C₂×D₄	—	4	4	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	8	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	8	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	16	no	—
398	C₂²×C₄	—	4	16	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	2	no	—
401	C₂²×C₄	—	4	4	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	—	4	2	no	—
404	C₂²×C₄	—	4	2	no	—
405	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
406	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
407	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
408	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
409	C₂²×C₄	#41×C₃₁	1	8	yes	complemented
410	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
412	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
413	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
414	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
417	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	8	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	4	4	no	—
424	C₄⋊C₄	—	4	4	no	—
425	C₂²⋊C₄	—	2	8	no	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	16	no	—
431	C₂²⋊C₄	—	2	16	no	—
432	C₂²⋊C₄	—	4	8	no	—
433	C₂²⋊C₄	—	4	8	no	—
434	C₄²	—	2	8	no	—
435	C₄²	—	2	8	no	—
436	C₄²	—	4	4	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	8	no	—
441	C₂³	—	2	4	no	—
442	C₂³	—	2	4	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	8	no	—
447	C₂³	—	2	8	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	4	4	no	—
451	C₂³	—	4	4	no	—
452	C₂³	—	4	8	no	—
453	C₂³	—	4	8	no	—
454	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
455	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
456	C₂³	C₂×#41×C₃₁	1	4	yes	—
457	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
458	Q₈	—	4	8	no	—
459	D₄	—	4	16	no	—
460	D₄	—	4	8	no	—
461	D₄	—	4	16	no	—
462	D₄	—	4	16	no	—
463	D₄	—	4	16	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	8	no	—
469	C₂×C₄	—	2	16	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	4	4	no	—
481	C₂×C₄	—	4	8	no	—
482	C₂×C₄	—	4	8	no	—
483	C₂×C₄	—	4	8	no	—
484	C₂×C₄	—	4	16	no	—
485	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
487	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
488	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
489	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	2	4	no	—
495	C₂²	—	2	4	no	—
496	C₂²	—	2	4	no	—
497	C₂²	—	4	16	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	4	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	4	8	no	—
504	C₂²	—	4	8	no	—
505	C₂²	C₃₁×#532	1	4	yes	—
506	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
507	C₂²	C₂²×#41×C₃₁	1	2	yes	—
508	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
509	C₂²	C₃₁×#585	1	1	yes	characteristic
510	C₂²	C₃₁×#585	1	1	yes	characteristic
511	C₄	—	2	4	no	—
512	C₄	—	2	2	no	—
513	C₄	—	2	8	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	4	no	—
516	C₄	—	4	4	no	—
517	C₄	—	4	4	no	—
518	C₄	C₃₁×#534	1	4	yes	—
519	C₂	—	2	4	no	—
520	C₂	—	4	4	no	—
521	C₂	—	8	4	no	—
522	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
523	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
524	C₂	C₃₁×#3449	1	1	yes	characteristic
525	C₂	C₃₁×#3459	1	2	yes	—
526	C₂	C₃₁×#3460	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×#525	C₂	semidirect
2. C₃₁×#2663	C₂	semidirect
3. C₃₁×#2661	C₂	semidirect
4. C₃₁×#2659	C₂	semidirect
5. C₃₁×#2656	C₂	semidirect
6. C₃₁×#2639	C₂	semidirect
7. C₃₁×#2617	C₂	semidirect
8. C₃₁×#2395	C₂	semidirect
9. C₃₁×#2388	C₂	semidirect
10. C₄×C₃₁×#121	C₂	semidirect
11. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
24. C₂×C₄×D₄×C₃₁	C₂²	semidirect
28. C₃₁×#400	C₂²	semidirect
29. C₃₁×#399	C₂²	semidirect
30. C₃₁×#398	C₂²	semidirect
32. C₃₁×#397	C₂²	semidirect
33. C₃₁×#395	C₂²	semidirect
36. C₃₁×#394	C₂²	semidirect
37. C₃₁×#393	C₂²	semidirect
39. C₃₁×#392	C₂²	semidirect
40. C₃₁×#390	C₂²	semidirect
41. C₃₁×#389	C₂²	semidirect
42. C₃₁×#388	C₂²	semidirect
45. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
48. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
61. C₂³×C₄×C₃₁	D₄	semidirect
145. C₂²×C₄×C₃₁	#41	semidirect
173. #34575	C₃₁	direct (N × H)
227. C₂×#525	C₂×C₃₁	semidirect
228. #2663	C₂×C₃₁	semidirect
229. #2661	C₂×C₃₁	semidirect
230. #2659	C₂×C₃₁	semidirect
231. #2656	C₂×C₃₁	semidirect
232. #2639	C₂×C₃₁	semidirect
233. #2617	C₂×C₃₁	semidirect
234. #2395	C₂×C₃₁	semidirect
235. #2388	C₂×C₃₁	semidirect
236. C₄×#121	C₂×C₃₁	semidirect
237. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
279. C₂×C₄×D₄	C₂²×C₃₁	semidirect
283. #400	C₂²×C₃₁	semidirect
284. #399	C₂²×C₃₁	semidirect
285. #398	C₂²×C₃₁	semidirect
287. #397	C₂²×C₃₁	semidirect
288. #395	C₂²×C₃₁	semidirect
291. #394	C₂²×C₃₁	semidirect
292. #393	C₂²×C₃₁	semidirect
294. #392	C₂²×C₃₁	semidirect
295. #390	C₂²×C₃₁	semidirect
296. #389	C₂²×C₃₁	semidirect
297. #388	C₂²×C₃₁	semidirect
300. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
303. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
324. C₂³×C₄	D₄×C₃₁	semidirect
369. C₃₁	#34575	direct (N × H)
409. C₂²×C₄	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#525	2
C₃₁×#2663	2
C₃₁×#2661	2
C₃₁×#2659	2
C₃₁×#2656	2
C₃₁×#2639	2
C₃₁×#2617	2
C₃₁×#2395	2
C₃₁×#2388	2
C₄×C₃₁×#121	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
#34575	31

Subgroups — C31×#34575 #21491334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34575 #21491334