Subgroups — C₃₁×#34593 #21491352

Order 7936.

2934 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2636	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2633	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2631	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2626	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2623	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2617	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2613	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2398	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2397	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2392	C₂	1	4	yes	maximal, complemented
13	C₄×C₃₁×#120	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#529	—	2	8	no	—
18	C₂×C₃₁×#126	C₂²	1	2	yes	complemented
19	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃₁×#123	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#120	C₂²	1	4	yes	—
25	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₄×D₄×C₃₁	—	2	8	no	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	complemented
34	C₂×C₃₁×#117	C₂²	1	2	yes	complemented
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#400	C₂²	1	4	yes	complemented
39	C₃₁×#399	C₂²	1	4	yes	complemented
40	C₃₁×#399	C₂²	1	4	yes	complemented
41	C₃₁×#398	C₂²	1	4	yes	complemented
42	C₃₁×#397	C₂²	1	4	yes	complemented
43	C₃₁×#397	C₂²	1	4	yes	complemented
44	C₃₁×#396	C₂²	1	8	yes	complemented
45	C₃₁×#394	C₂²	1	4	yes	—
46	C₃₁×#394	C₂²	1	4	yes	—
47	C₃₁×#394	C₂²	1	8	yes	complemented
48	C₃₁×#393	C₂²	1	4	yes	complemented
49	C₃₁×#392	C₂²	1	4	yes	—
50	C₃₁×#392	C₂²	1	4	yes	—
51	C₃₁×#392	C₂²	1	4	yes	complemented
52	C₃₁×#386	C₂²	1	4	yes	—
53	C₃₁×#386	C₂²	1	8	yes	complemented
54	C₃₁×#385	C₂²	1	4	yes	—
55	C₃₁×#385	C₂²	1	8	yes	complemented
56	C₃₁×#383	C₂²	1	2	yes	complemented
57	C₃₁×#382	C₂²	1	2	yes	complemented
58	C₃₁×#381	C₂²	1	2	yes	—
59	C₃₁×#380	C₂²	1	2	yes	—
60	C₃₁×#379	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
62	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
64	C₃₁×#376	C₂²	1	2	yes	complemented
65	C₂⁵×C₃₁	C₂³	1	2	yes	—
66	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₃₁	—	2	8	no	—
68	C₂²×D₄×C₃₁	—	2	4	no	—
69	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
70	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
71	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
72	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
73	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	—	2	8	no	—
75	C₂³×C₄×C₃₁	—	4	8	no	—
76	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
78	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
81	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
82	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#126	—	2	8	no	—
84	C₃₁×#125	—	2	4	no	—
85	C₃₁×#124	—	2	4	no	—
86	C₃₁×#123	—	2	8	no	—
87	C₃₁×C₂²⋊Q₈	—	2	4	no	—
88	C₃₁×#121	—	2	4	no	—
89	C₃₁×#120	—	2	16	no	—
90	C₄×Q₈×C₃₁	—	2	4	no	—
91	C₄×D₄×C₃₁	—	2	8	no	—
92	C₄×D₄×C₃₁	—	2	8	no	—
93	C₄×D₄×C₃₁	—	2	8	no	—
94	C₄×D₄×C₃₁	—	2	8	no	—
95	C₄×D₄×C₃₁	—	2	8	no	—
96	C₄×D₄×C₃₁	—	2	4	no	—
97	C₄×D₄×C₃₁	—	4	16	no	—
98	C₃₁×#117	—	2	8	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
110	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
122	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
123	C₂×C₄²×C₃₁	D₄	1	8	yes	complemented
124	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
130	C₃₁×#95	C₂³	1	8	yes	—
131	C₃₁×#95	C₂³	1	8	yes	—
132	C₃₁×#95	C₂³	1	4	yes	—
133	C₃₁×#95	C₂³	1	4	yes	—
134	C₃₁×#95	C₂³	1	4	yes	—
135	C₃₁×#95	C₂³	1	4	yes	—
136	C₂⁴×C₃₁	—	2	4	no	—
137	C₂⁴×C₃₁	—	2	4	no	—
138	C₂⁴×C₃₁	—	4	4	no	—
139	C₂⁴×C₃₁	—	4	4	no	—
140	C₂⁴×C₃₁	#41	1	8	yes	—
141	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
142	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
143	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
144	C₂×Q₈×C₃₁	—	2	2	no	—
145	C₂×Q₈×C₃₁	—	4	2	no	—
146	C₂×D₄×C₃₁	—	2	32	no	—
147	C₂×D₄×C₃₁	—	2	8	no	—
148	C₂×D₄×C₃₁	—	2	8	no	—
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	16	no	—
151	C₂×D₄×C₃₁	—	2	8	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	4	no	—
154	C₂×D₄×C₃₁	—	2	2	no	—
155	C₂×D₄×C₃₁	—	4	8	no	—
156	C₂×D₄×C₃₁	—	4	2	no	—
157	C₂×D₄×C₃₁	—	4	8	no	—
158	C₂²×C₄×C₃₁	—	2	4	no	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	4	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	16	no	—
166	C₂²×C₄×C₃₁	—	4	8	no	—
167	C₂²×C₄×C₃₁	—	4	4	no	—
168	C₂²×C₄×C₃₁	—	4	2	no	—
169	C₂²×C₄×C₃₁	—	4	32	no	—
170	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
184	C₄⋊C₄×C₃₁	—	2	4	no	—
185	C₄⋊C₄×C₃₁	—	2	4	no	—
186	C₄⋊C₄×C₃₁	—	2	4	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	4	4	no	—
193	C₄⋊C₄×C₃₁	—	4	8	no	—
194	C₂²⋊C₄×C₃₁	—	2	4	no	—
195	C₂²⋊C₄×C₃₁	—	2	16	no	—
196	C₂²⋊C₄×C₃₁	—	2	8	no	—
197	C₂²⋊C₄×C₃₁	—	2	8	no	—
198	C₂²⋊C₄×C₃₁	—	2	8	no	—
199	C₂²⋊C₄×C₃₁	—	2	4	no	—
200	C₂²⋊C₄×C₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	—	2	4	no	—
203	C₂²⋊C₄×C₃₁	—	4	8	no	—
204	C₂²⋊C₄×C₃₁	—	4	8	no	—
205	C₂²⋊C₄×C₃₁	—	4	8	no	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	—	2	8	no	—
208	C₄²×C₃₁	C₂×D₄	1	4	yes	—
209	C₄²×C₃₁	C₂×D₄	1	4	yes	—
210	C₄²×C₃₁	C₂×D₄	1	4	yes	—
211	C₄²×C₃₁	C₂×D₄	1	4	yes	—
212	#34593	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₃₁	—	2	2	no	—
214	C₂³×C₃₁	—	2	2	no	—
215	C₂³×C₃₁	—	2	16	no	—
216	C₂³×C₃₁	—	2	4	no	—
217	C₂³×C₃₁	—	2	4	no	—
218	C₂³×C₃₁	—	2	8	no	—
219	C₂³×C₃₁	—	2	8	no	—
220	C₂³×C₃₁	—	2	2	no	—
221	C₂³×C₃₁	—	2	4	no	—
222	C₂³×C₃₁	—	2	4	no	—
223	C₂³×C₃₁	—	2	4	no	—
224	C₂³×C₃₁	—	4	16	no	—
225	C₂³×C₃₁	—	4	8	no	—
226	C₂³×C₃₁	—	4	8	no	—
227	C₂³×C₃₁	—	4	16	no	—
228	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
229	C₂³×C₃₁	C₂×#41	1	8	yes	—
230	C₂³×C₃₁	C₂×#41	1	4	yes	—
231	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
232	Q₈×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	2	16	no	—
234	D₄×C₃₁	—	2	16	no	—
235	D₄×C₃₁	—	4	4	no	—
236	D₄×C₃₁	—	4	32	no	—
237	D₄×C₃₁	—	4	8	no	—
238	C₂×C₄×C₃₁	—	2	8	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	8	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	2	8	no	—
251	C₂×C₄×C₃₁	—	2	8	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	8	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	8	no	—
256	C₂×C₄×C₃₁	—	4	4	no	—
257	C₂×C₄×C₃₁	—	4	16	no	—
258	C₂×C₄×C₃₁	—	4	8	no	—
259	C₂×C₄×C₃₁	—	4	8	no	—
260	C₂×C₄×C₃₁	—	4	16	no	—
261	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
268	C₂×#529	C₂×C₃₁	1	1	yes	characteristic, complemented
269	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#2636	C₂×C₃₁	1	4	yes	complemented
271	#2633	C₂×C₃₁	1	2	yes	complemented
272	#2631	C₂×C₃₁	1	2	yes	complemented
273	#2626	C₂×C₃₁	1	2	yes	complemented
274	#2623	C₂×C₃₁	1	2	yes	complemented
275	#2617	C₂×C₃₁	1	2	yes	complemented
276	#2613	C₂×C₃₁	1	2	yes	complemented
277	#2398	C₂×C₃₁	1	2	yes	complemented
278	#2397	C₂×C₃₁	1	2	yes	complemented
279	#2392	C₂×C₃₁	1	4	yes	complemented
280	C₄×#120	C₂×C₃₁	1	4	yes	complemented
281	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
282	C₂²×C₃₁	—	2	2	no	—
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	2	8	no	—
285	C₂²×C₃₁	—	2	8	no	—
286	C₂²×C₃₁	—	2	16	no	—
287	C₂²×C₃₁	—	4	32	no	—
288	C₂²×C₃₁	—	4	4	no	—
289	C₂²×C₃₁	—	4	4	no	—
290	C₂²×C₃₁	—	4	8	no	—
291	C₂²×C₃₁	—	4	2	no	—
292	C₂²×C₃₁	—	4	2	no	—
293	C₂²×C₃₁	—	4	2	no	—
294	C₂²×C₃₁	—	4	8	no	—
295	C₂²×C₃₁	#525	1	4	yes	—
296	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
297	C₂²×C₃₁	C₂²×#41	1	2	yes	—
298	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
299	C₂²×C₃₁	#585	1	2	yes	—
300	C₄×C₃₁	—	2	2	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	2	no	—
303	C₄×C₃₁	—	2	2	no	—
304	C₄×C₃₁	—	2	2	no	—
305	C₄×C₃₁	—	4	2	no	—
306	C₄×C₃₁	—	4	2	no	—
307	C₄×C₃₁	—	4	4	no	—
308	C₄×C₃₁	—	4	2	no	—
309	C₄×C₃₁	—	4	8	no	—
310	C₄×C₃₁	#535	1	4	yes	—
311	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
312	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
313	#529	—	2	8	no	—
314	C₂×#126	C₂²×C₃₁	1	2	yes	complemented
315	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
316	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
317	C₂×#123	C₂²×C₃₁	1	2	yes	—
318	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×#120	C₂²×C₃₁	1	4	yes	—
321	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
322	C₂×C₄×D₄	—	2	8	no	—
323	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
324	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
325	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
326	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
327	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
328	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
329	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	complemented
330	C₂×#117	C₂²×C₃₁	1	2	yes	complemented
331	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
332	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
333	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
334	#400	C₂²×C₃₁	1	4	yes	complemented
335	#399	C₂²×C₃₁	1	4	yes	complemented
336	#399	C₂²×C₃₁	1	4	yes	complemented
337	#398	C₂²×C₃₁	1	4	yes	complemented
338	#397	C₂²×C₃₁	1	4	yes	complemented
339	#397	C₂²×C₃₁	1	4	yes	complemented
340	#396	C₂²×C₃₁	1	8	yes	complemented
341	#394	C₂²×C₃₁	1	4	yes	—
342	#394	C₂²×C₃₁	1	4	yes	—
343	#394	C₂²×C₃₁	1	8	yes	complemented
344	#393	C₂²×C₃₁	1	4	yes	complemented
345	#392	C₂²×C₃₁	1	4	yes	—
346	#392	C₂²×C₃₁	1	4	yes	—
347	#392	C₂²×C₃₁	1	4	yes	complemented
348	#386	C₂²×C₃₁	1	4	yes	—
349	#386	C₂²×C₃₁	1	8	yes	complemented
350	#385	C₂²×C₃₁	1	4	yes	—
351	#385	C₂²×C₃₁	1	8	yes	complemented
352	#383	C₂²×C₃₁	1	2	yes	complemented
353	#382	C₂²×C₃₁	1	2	yes	complemented
354	#381	C₂²×C₃₁	1	2	yes	—
355	#380	C₂²×C₃₁	1	2	yes	—
356	#379	C₂²×C₃₁	1	4	yes	—
357	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
358	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
359	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
360	#376	C₂²×C₃₁	1	2	yes	complemented
361	C₂×C₃₁	—	2	4	no	—
362	C₂×C₃₁	—	4	8	no	—
363	C₂×C₃₁	—	8	2	no	—
364	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
365	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
366	C₂×C₃₁	#3449	1	1	yes	characteristic
367	C₂×C₃₁	#3459	1	4	yes	—
368	C₂⁵	C₂³×C₃₁	1	2	yes	—
369	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
370	C₂²×D₄	—	2	8	no	—
371	C₂²×D₄	—	2	4	no	—
372	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
373	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
374	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
375	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
376	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
377	C₂³×C₄	—	2	8	no	—
378	C₂³×C₄	—	4	8	no	—
379	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
380	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
381	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
382	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
384	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
385	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
386	#126	—	2	8	no	—
387	#125	—	2	4	no	—
388	#124	—	2	4	no	—
389	#123	—	2	8	no	—
390	C₂²⋊Q₈	—	2	4	no	—
391	#121	—	2	4	no	—
392	#120	—	2	16	no	—
393	C₄×Q₈	—	2	4	no	—
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	—	2	8	no	—
396	C₄×D₄	—	2	8	no	—
397	C₄×D₄	—	2	8	no	—
398	C₄×D₄	—	2	8	no	—
399	C₄×D₄	—	2	4	no	—
400	C₄×D₄	—	4	16	no	—
401	#117	—	2	8	no	—
402	C₂×C₄⋊C₄	—	2	4	no	—
403	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
412	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
413	C₂×C₂²⋊C₄	—	2	8	no	—
414	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
426	C₂×C₄²	D₄×C₃₁	1	8	yes	complemented
427	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
428	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
429	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
433	#95	C₂³×C₃₁	1	4	yes	—
434	#95	C₂³×C₃₁	1	4	yes	—
435	#95	C₂³×C₃₁	1	4	yes	—
436	#95	C₂³×C₃₁	1	8	yes	—
437	#95	C₂³×C₃₁	1	8	yes	—
438	#95	C₂³×C₃₁	1	4	yes	—
439	C₃₁	#34593	1	1	yes	characteristic, direct product factor, Sylow
440	C₂⁴	—	2	4	no	—
441	C₂⁴	—	2	4	no	—
442	C₂⁴	—	4	4	no	—
443	C₂⁴	—	4	4	no	—
444	C₂⁴	#41×C₃₁	1	8	yes	—
445	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
446	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
447	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	4	2	no	—
450	C₂×D₄	—	2	32	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	4	no	—
454	C₂×D₄	—	2	16	no	—
455	C₂×D₄	—	2	8	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	4	8	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	—	4	8	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	2	8	no	—
468	C₂²×C₄	—	2	8	no	—
469	C₂²×C₄	—	2	16	no	—
470	C₂²×C₄	—	4	8	no	—
471	C₂²×C₄	—	4	4	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	4	32	no	—
474	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
475	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
476	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
478	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	4	4	no	—
497	C₄⋊C₄	—	4	8	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	16	no	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	—	2	8	no	—
502	C₂²⋊C₄	—	2	8	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	4	8	no	—
508	C₂²⋊C₄	—	4	8	no	—
509	C₂²⋊C₄	—	4	8	no	—
510	C₄²	—	2	4	no	—
511	C₄²	—	2	8	no	—
512	C₄²	C₂×D₄×C₃₁	1	4	yes	—
513	C₄²	C₂×D₄×C₃₁	1	4	yes	—
514	C₄²	C₂×D₄×C₃₁	1	4	yes	—
515	C₄²	C₂×D₄×C₃₁	1	4	yes	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	16	no	—
519	C₂³	—	2	4	no	—
520	C₂³	—	2	4	no	—
521	C₂³	—	2	8	no	—
522	C₂³	—	2	8	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	4	16	no	—
528	C₂³	—	4	8	no	—
529	C₂³	—	4	8	no	—
530	C₂³	—	4	16	no	—
531	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
532	C₂³	C₂×#41×C₃₁	1	8	yes	—
533	C₂³	C₂×#41×C₃₁	1	4	yes	—
534	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
535	Q₈	—	4	4	no	—
536	D₄	—	2	16	no	—
537	D₄	—	2	16	no	—
538	D₄	—	4	4	no	—
539	D₄	—	4	32	no	—
540	D₄	—	4	8	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	8	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	16	no	—
561	C₂×C₄	—	4	8	no	—
562	C₂×C₄	—	4	8	no	—
563	C₂×C₄	—	4	16	no	—
564	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
567	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
569	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
570	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	8	no	—
574	C₂²	—	2	8	no	—
575	C₂²	—	2	16	no	—
576	C₂²	—	4	32	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	8	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	8	no	—
584	C₂²	C₃₁×#525	1	4	yes	—
585	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
586	C₂²	C₂²×#41×C₃₁	1	2	yes	—
587	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
588	C₂²	C₃₁×#585	1	2	yes	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	4	no	—
596	C₄	—	4	2	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	8	no	—
599	C₄	C₃₁×#535	1	4	yes	—
600	C₂	—	2	4	no	—
601	C₂	—	4	8	no	—
602	C₂	—	8	2	no	—
603	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
604	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
605	C₂	C₃₁×#3449	1	1	yes	characteristic
606	C₂	C₃₁×#3459	1	4	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×#529	C₂	semidirect
2. C₂²×C₄×D₄×C₃₁	C₂	semidirect
3. C₃₁×#2636	C₂	semidirect
4. C₃₁×#2633	C₂	semidirect
5. C₃₁×#2631	C₂	semidirect
6. C₃₁×#2626	C₂	semidirect
7. C₃₁×#2623	C₂	semidirect
8. C₃₁×#2617	C₂	semidirect
9. C₃₁×#2613	C₂	semidirect
10. C₃₁×#2398	C₂	semidirect
11. C₃₁×#2397	C₂	semidirect
12. C₃₁×#2392	C₂	semidirect
13. C₄×C₃₁×#120	C₂	semidirect
14. C₃₁×#2283	C₂	semidirect
18. C₂×C₃₁×#126	C₂²	semidirect
19. C₂×C₃₁×#125	C₂²	semidirect
20. C₂×C₃₁×#124	C₂²	semidirect
25. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
32. C₂×C₄×D₄×C₃₁	C₂²	semidirect
33. C₂×C₄×D₄×C₃₁	C₂²	semidirect
34. C₂×C₃₁×#117	C₂²	semidirect
38. C₃₁×#400	C₂²	semidirect
39. C₃₁×#399	C₂²	semidirect
40. C₃₁×#399	C₂²	semidirect
41. C₃₁×#398	C₂²	semidirect
42. C₃₁×#397	C₂²	semidirect
43. C₃₁×#397	C₂²	semidirect
44. C₃₁×#396	C₂²	semidirect
47. C₃₁×#394	C₂²	semidirect
48. C₃₁×#393	C₂²	semidirect
51. C₃₁×#392	C₂²	semidirect
53. C₃₁×#386	C₂²	semidirect
55. C₃₁×#385	C₂²	semidirect
56. C₃₁×#383	C₂²	semidirect
57. C₃₁×#382	C₂²	semidirect
61. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
63. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
64. C₃₁×#376	C₂²	semidirect
123. C₂×C₄²×C₃₁	D₄	semidirect
212. #34593	C₃₁	direct (N × H)
268. C₂×#529	C₂×C₃₁	semidirect
269. C₂²×C₄×D₄	C₂×C₃₁	semidirect
270. #2636	C₂×C₃₁	semidirect
271. #2633	C₂×C₃₁	semidirect
272. #2631	C₂×C₃₁	semidirect
273. #2626	C₂×C₃₁	semidirect
274. #2623	C₂×C₃₁	semidirect
275. #2617	C₂×C₃₁	semidirect
276. #2613	C₂×C₃₁	semidirect
277. #2398	C₂×C₃₁	semidirect
278. #2397	C₂×C₃₁	semidirect
279. #2392	C₂×C₃₁	semidirect
280. C₄×#120	C₂×C₃₁	semidirect
281. #2283	C₂×C₃₁	semidirect
314. C₂×#126	C₂²×C₃₁	semidirect
315. C₂×#125	C₂²×C₃₁	semidirect
316. C₂×#124	C₂²×C₃₁	semidirect
321. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
328. C₂×C₄×D₄	C₂²×C₃₁	semidirect
329. C₂×C₄×D₄	C₂²×C₃₁	semidirect
330. C₂×#117	C₂²×C₃₁	semidirect
334. #400	C₂²×C₃₁	semidirect
335. #399	C₂²×C₃₁	semidirect
336. #399	C₂²×C₃₁	semidirect
337. #398	C₂²×C₃₁	semidirect
338. #397	C₂²×C₃₁	semidirect
339. #397	C₂²×C₃₁	semidirect
340. #396	C₂²×C₃₁	semidirect
343. #394	C₂²×C₃₁	semidirect
344. #393	C₂²×C₃₁	semidirect
347. #392	C₂²×C₃₁	semidirect
349. #386	C₂²×C₃₁	semidirect
351. #385	C₂²×C₃₁	semidirect
352. #383	C₂²×C₃₁	semidirect
353. #382	C₂²×C₃₁	semidirect
357. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
359. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
360. #376	C₂²×C₃₁	semidirect
426. C₂×C₄²	D₄×C₃₁	semidirect
439. C₃₁	#34593	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×#529	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2636	2
C₃₁×#2633	2
C₃₁×#2631	2
C₃₁×#2626	2
C₃₁×#2623	2
C₃₁×#2617	2
C₃₁×#2613	2
C₃₁×#2398	2
C₃₁×#2397	2
C₃₁×#2392	2
C₄×C₃₁×#120	2
C₃₁×#2283	2
#34593	31

Subgroups — C31×#34593 #21491352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34593 #21491352