Subgroups — C₃₁×#22136 #21478895

Order 7936.

896 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2392	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1955	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1955	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1924	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1869	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1850	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1798	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₈×D₄×C₃₁	—	2	4	no	—
17	C₃₁×#433	—	2	4	no	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#97	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#398	C₄	1	1	yes	characteristic
33	C₃₁×#397	C₄	1	1	yes	characteristic
34	C₃₁×#396	C₄	1	2	yes	—
35	C₃₁×#394	C₄	1	1	yes	characteristic
36	C₃₁×#394	C₄	1	2	yes	—
37	C₃₁×#392	C₄	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	—	2	4	no	—
48	C₂²×C₈×C₃₁	—	4	2	no	—
49	C₂²×C₈×C₃₁	—	4	2	no	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	4	2	no	—
59	C₄×D₄×C₃₁	—	4	2	no	—
60	C₃₁×#117	—	2	2	no	—
61	C₃₁×#117	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
73	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
74	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	4	2	no	—
82	C₃₁×C₂²⋊C₈	—	2	2	no	—
83	C₃₁×C₂²⋊C₈	—	2	2	no	—
84	C₃₁×C₂²⋊C₈	—	2	4	no	—
85	C₃₁×C₂²⋊C₈	—	4	2	no	—
86	C₃₁×C₂²⋊C₈	—	4	2	no	—
87	C₃₁×C₂²⋊C₈	—	4	2	no	—
88	C₃₁×#97	—	2	2	no	—
89	C₃₁×#97	—	4	2	no	—
90	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
91	C₃₁×#95	C₂×C₄	1	2	yes	—
92	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
93	C₃₁×#95	C₂×C₄	1	2	yes	—
94	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
95	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	2	4	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	4	2	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	4	8	no	—
140	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	4	2	no	—
146	C₄⋊C₄×C₃₁	—	4	2	no	—
147	C₂²⋊C₄×C₃₁	—	2	4	no	—
148	C₂²⋊C₄×C₃₁	—	2	2	no	—
149	C₂²⋊C₄×C₃₁	—	2	2	no	—
150	C₂²⋊C₄×C₃₁	—	2	4	no	—
151	C₂²⋊C₄×C₃₁	—	4	2	no	—
152	C₂²⋊C₄×C₃₁	—	4	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	2	no	—
154	C₄²×C₃₁	—	2	1	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	—	2	2	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	C₂×D₄	1	2	yes	—
159	C₄²×C₃₁	#41	1	2	yes	—
160	#22136	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	2	2	no	—
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	2	2	no	—
167	C₂³×C₃₁	—	2	1	no	—
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	#131	1	4	yes	—
173	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
174	D₄×C₃₁	—	2	8	no	—
175	D₄×C₃₁	—	4	8	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
203	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
204	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
209	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
210	C₈×C₃₁	—	2	2	no	—
211	C₈×C₃₁	—	2	2	no	—
212	C₈×C₃₁	—	4	1	no	—
213	C₈×C₃₁	—	4	2	no	—
214	C₈×C₃₁	—	4	1	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
217	C₂×#433	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#2392	C₂×C₃₁	1	1	yes	characteristic
219	#1955	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1955	C₂×C₃₁	1	2	yes	complemented
221	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#1924	C₂×C₃₁	1	2	yes	complemented
223	#1869	C₂×C₃₁	1	2	yes	complemented
224	#1850	C₂×C₃₁	1	2	yes	complemented
225	#1798	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂²×C₃₁	—	2	4	no	—
228	C₂²×C₃₁	—	2	4	no	—
229	C₂²×C₃₁	—	2	4	no	—
230	C₂²×C₃₁	—	4	2	no	—
231	C₂²×C₃₁	—	4	2	no	—
232	C₂²×C₃₁	—	4	2	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	1	no	—
235	C₂²×C₃₁	—	4	1	no	—
236	C₂²×C₃₁	—	4	1	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
240	C₂²×C₃₁	#518	1	1	yes	characteristic
241	C₂²×C₃₁	#520	1	1	yes	characteristic
242	C₂²×C₃₁	C₂×#131	1	2	yes	—
243	C₂²×C₃₁	#568	1	2	yes	—
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	4	1	no	—
248	C₄×C₃₁	—	4	1	no	—
249	C₄×C₃₁	—	4	1	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	#535	1	1	yes	characteristic
253	C₄×C₃₁	#537	1	1	yes	characteristic
254	C₄×C₃₁	#540	1	1	yes	characteristic
255	C₄×C₃₁	#542	1	1	yes	characteristic
256	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
257	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
259	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
260	C₈×D₄	—	2	4	no	—
261	#433	—	2	4	no	—
262	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
267	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
270	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
271	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
272	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
273	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#97	C₂²×C₃₁	1	2	yes	—
275	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
276	#398	C₄×C₃₁	1	1	yes	characteristic
277	#397	C₄×C₃₁	1	1	yes	characteristic
278	#396	C₄×C₃₁	1	2	yes	—
279	#394	C₄×C₃₁	1	2	yes	—
280	#394	C₄×C₃₁	1	1	yes	characteristic
281	#392	C₄×C₃₁	1	1	yes	characteristic
282	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
283	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
284	#336	C₂²×C₃₁	1	2	yes	—
285	#336	C₂²×C₃₁	1	2	yes	—
286	#336	C₂²×C₃₁	1	2	yes	—
287	C₂×C₃₁	—	4	4	no	—
288	C₂×C₃₁	—	8	1	no	—
289	C₂×C₃₁	#2324	1	1	yes	characteristic
290	C₂×C₃₁	#2929	1	1	yes	characteristic
291	C₂×C₃₁	#2932	1	1	yes	characteristic
292	C₂×C₃₁	#2976	1	2	yes	—
293	C₂×C₃₁	#2992	1	2	yes	—
294	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
295	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
296	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
298	C₂²×C₈	—	2	4	no	—
299	C₂²×C₈	—	4	2	no	—
300	C₂²×C₈	—	4	2	no	—
301	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
302	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
304	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
307	C₄×D₄	—	2	2	no	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	—	4	2	no	—
310	C₄×D₄	—	4	2	no	—
311	#117	—	2	2	no	—
312	#117	—	2	2	no	—
313	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
316	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
317	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
318	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
319	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
320	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
321	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
322	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
323	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
324	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
325	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
327	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
328	C₄⋊C₈	—	2	2	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	2	2	no	—
332	C₄⋊C₈	—	4	2	no	—
333	C₂²⋊C₈	—	2	2	no	—
334	C₂²⋊C₈	—	2	2	no	—
335	C₂²⋊C₈	—	2	4	no	—
336	C₂²⋊C₈	—	4	2	no	—
337	C₂²⋊C₈	—	4	2	no	—
338	C₂²⋊C₈	—	4	2	no	—
339	#97	—	2	2	no	—
340	#97	—	4	2	no	—
341	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
342	#95	C₂×C₄×C₃₁	1	2	yes	—
343	#95	C₂×C₄×C₃₁	1	2	yes	—
344	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
345	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
346	C₃₁	#22136	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
348	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	2	4	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	4	2	no	—
353	C₂×D₄	—	4	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
368	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
369	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
370	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
371	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	4	8	no	—
392	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
393	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	4	2	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₂²⋊C₄	—	2	4	no	—
400	C₂²⋊C₄	—	2	2	no	—
401	C₂²⋊C₄	—	2	2	no	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	4	2	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₂²⋊C₄	—	4	2	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	2	2	no	—
410	C₄²	C₂×D₄×C₃₁	1	2	yes	—
411	C₄²	#41×C₃₁	1	2	yes	—
412	C₂³	—	2	2	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	C₃₁×#131	1	4	yes	—
424	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
425	D₄	—	2	8	no	—
426	D₄	—	4	8	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
454	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
455	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
456	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
458	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
459	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
460	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
461	C₈	—	2	2	no	—
462	C₈	—	2	2	no	—
463	C₈	—	4	1	no	—
464	C₈	—	4	2	no	—
465	C₈	—	4	1	no	—
466	C₈	—	4	2	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	2	4	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	1	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
480	C₂²	C₃₁×#518	1	1	yes	characteristic
481	C₂²	C₃₁×#520	1	1	yes	characteristic
482	C₂²	C₂×C₃₁×#131	1	2	yes	—
483	C₂²	C₃₁×#568	1	2	yes	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	1	no	—
488	C₄	—	4	1	no	—
489	C₄	—	4	1	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	2	no	—
492	C₄	C₃₁×#535	1	1	yes	characteristic
493	C₄	C₃₁×#537	1	1	yes	characteristic
494	C₄	C₃₁×#540	1	1	yes	characteristic
495	C₄	C₃₁×#542	1	1	yes	characteristic
496	C₂	—	4	4	no	—
497	C₂	—	8	1	no	—
498	C₂	C₃₁×#2324	1	1	yes	characteristic
499	C₂	C₃₁×#2929	1	1	yes	characteristic
500	C₂	C₃₁×#2932	1	1	yes	characteristic
501	C₂	C₃₁×#2976	1	2	yes	—
502	C₂	C₃₁×#2992	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₃₁×#392	C₄
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×C₈×D₄×C₃₁	C₂	semidirect
2. C₂×C₃₁×#433	C₂	semidirect
4. C₃₁×#1955	C₂	semidirect
5. C₃₁×#1955	C₂	semidirect
6. C₃₁×#1940	C₂	semidirect
7. C₃₁×#1924	C₂	semidirect
8. C₃₁×#1869	C₂	semidirect
9. C₃₁×#1850	C₂	semidirect
10. C₃₁×#1798	C₂	semidirect
11. C₃₁×#1770	C₂	semidirect
90. C₄×C₈×C₃₁	D₄	semidirect
160. #22136	C₃₁	direct (N × H)
216. C₂×C₈×D₄	C₂×C₃₁	semidirect
217. C₂×#433	C₂×C₃₁	semidirect
219. #1955	C₂×C₃₁	semidirect
220. #1955	C₂×C₃₁	semidirect
221. #1940	C₂×C₃₁	semidirect
222. #1924	C₂×C₃₁	semidirect
223. #1869	C₂×C₃₁	semidirect
224. #1850	C₂×C₃₁	semidirect
225. #1798	C₂×C₃₁	semidirect
226. #1770	C₂×C₃₁	semidirect
341. C₄×C₈	D₄×C₃₁	semidirect
346. C₃₁	#22136	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×C₈×D₄×C₃₁	2
C₂×C₃₁×#433	2
C₃₁×#2392	2
C₃₁×#1955	2
C₃₁×#1940	2
C₃₁×#1924	2
C₃₁×#1869	2
C₃₁×#1850	2
C₃₁×#1798	2
C₃₁×#1770	2
#22136	31

Subgroups — C31×#22136 #21478895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22136 #21478895