Subgroups — C₈×C₃₁×#137 #21474610

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#137	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1579	C₂	1	2	yes	maximal, complemented
5	C₃₁×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₈×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1477	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₄	1	1	yes	characteristic
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₃₁	—	2	1	no	—
16	C₈×Q₈×C₃₁	—	2	1	no	—
17	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#441	C₄	1	2	yes	—
22	C₃₁×#440	C₄	1	2	yes	—
23	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
25	C₈×D₄×C₃₁	—	2	1	no	—
26	C₈×D₄×C₃₁	—	2	1	no	—
27	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#431	—	2	1	no	—
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#428	C₄	1	1	yes	characteristic
33	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃₁×#419	C₄	1	1	yes	characteristic
36	C₃₁×#417	C₄	1	1	yes	characteristic
37	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₃₁	—	2	1	no	—
39	C₂×C₄×C₈×C₃₁	—	2	1	no	—
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	2	yes	—
42	C₃₁×#334	C₂²	1	2	yes	—
43	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
45	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
46	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
47	C₃₁×#322	C₂²	1	2	yes	—
48	C₈²×C₃₁	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₃₁×#137	C₈	1	8	yes	direct product factor
52	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
61	C₄×Q₈×C₃₁	—	2	1	no	—
62	C₄×Q₈×C₃₁	—	2	1	no	—
63	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
64	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
66	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	—	2	1	no	—
69	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#117	—	2	1	no	—
72	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	—	2	1	no	—
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#107	C₂×C₄	1	2	yes	—
79	C₃₁×#106	C₂×C₄	1	2	yes	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	—	2	1	no	—
84	C₃₁×C₄⋊C₈	—	2	1	no	—
85	C₃₁×C₄⋊C₈	—	2	1	no	—
86	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
89	C₃₁×#103	C₂×C₄	1	2	yes	—
90	C₃₁×#103	C₂×C₄	1	2	yes	—
91	C₃₁×#103	C₂×C₄	1	2	yes	—
92	C₃₁×#102	C₂×C₄	1	2	yes	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×C₂²⋊C₈	—	2	1	no	—
95	C₃₁×C₂²⋊C₈	—	2	1	no	—
96	C₃₁×C₂²⋊C₈	—	2	1	no	—
97	C₃₁×#97	C₂×C₄	1	2	yes	—
98	C₄×C₈×C₃₁	—	2	1	no	—
99	C₄×C₈×C₃₁	—	2	2	no	—
100	C₄×C₈×C₃₁	—	2	1	no	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	C₄×C₈×C₃₁	—	2	1	no	—
103	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
104	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
105	C₄×C₈×C₃₁	C₂³	1	2	yes	—
106	C₄×C₈×C₃₁	C₂³	1	2	yes	—
107	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
109	#41×C₃₁	—	2	2	no	—
110	#41×C₃₁	C₂×C₈	1	4	yes	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	2	no	—
113	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	complemented
114	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
115	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
126	Q₁₆×C₃₁	C₂×C₈	1	8	yes	complemented
127	SD₁₆×C₃₁	C₂×C₈	1	8	yes	complemented
128	M₁₆×C₃₁	—	2	2	no	—
129	M₁₆×C₃₁	C₂×C₈	1	4	yes	complemented
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	4	1	no	—
136	C₂×C₈×C₃₁	—	4	2	no	—
137	C₂×C₈×C₃₁	—	4	2	no	—
138	C₂×C₈×C₃₁	—	4	2	no	—
139	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
141	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
151	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂²⋊C₄×C₃₁	—	2	1	no	—
154	C₂²⋊C₄×C₃₁	—	2	1	no	—
155	C₂²⋊C₄×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	2	1	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	—	2	2	no	—
161	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₃₁	#41	1	1	yes	characteristic
164	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
165	C₈×#137	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₃₁	—	2	1	no	—
167	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
168	Q₈×C₃₁	—	2	2	no	—
169	Q₈×C₃₁	—	2	1	no	—
170	Q₈×C₃₁	—	2	4	no	—
171	Q₈×C₃₁	C₂²×C₈	1	4	yes	—
172	Q₈×C₃₁	C₂²×C₈	1	2	yes	—
173	D₄×C₃₁	—	2	4	no	—
174	D₄×C₃₁	—	2	1	no	—
175	D₄×C₃₁	C₂²×C₈	1	2	yes	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	4	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
194	C₂×C₄×C₃₁	#131	1	2	yes	—
195	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₈×C₃₁	—	2	2	no	—
199	C₈×C₃₁	—	2	1	no	—
200	C₈×C₃₁	—	2	1	no	—
201	C₈×C₃₁	—	2	1	no	—
202	C₈×C₃₁	—	4	2	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	—	4	1	no	—
205	C₈×C₃₁	—	4	2	no	—
206	C₈×C₃₁	C₂²×C₈	1	4	yes	—
207	C₈×C₃₁	#137	1	2	yes	direct product factor
208	C₈×#41	C₂×C₃₁	1	1	yes	characteristic
209	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
210	C₄×#137	C₂×C₃₁	1	1	yes	characteristic
211	#1579	C₂×C₃₁	1	2	yes	complemented
212	SD₁₆⋊C₈	C₂×C₃₁	1	2	yes	complemented
213	C₈×Q₁₆	C₂×C₃₁	1	2	yes	complemented
214	C₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
215	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1479	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1477	C₂×C₃₁	1	1	yes	characteristic
218	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₂²×C₃₁	—	2	1	no	—
220	C₂²×C₃₁	—	2	1	no	—
221	C₂²×C₃₁	—	2	2	no	—
222	C₂²×C₃₁	—	4	1	no	—
223	C₂²×C₃₁	—	4	1	no	—
224	C₂²×C₃₁	C₈×D₄	1	2	yes	—
225	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
226	C₄×C₃₁	—	2	2	no	—
227	C₄×C₃₁	—	2	4	no	—
228	C₄×C₃₁	—	2	1	no	—
229	C₄×C₃₁	—	2	1	no	—
230	C₄×C₃₁	—	2	1	no	—
231	C₄×C₃₁	—	4	1	no	—
232	C₄×C₃₁	—	4	1	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	C₈×D₄	1	2	yes	—
235	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
236	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
237	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
238	C₄×C₃₁	#575	1	1	yes	characteristic
239	C₂×#137	C₄×C₃₁	1	1	yes	characteristic
240	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₈×Q₈	—	2	1	no	—
243	C₈×Q₈	—	2	1	no	—
244	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
245	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
247	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
248	#441	C₄×C₃₁	1	2	yes	—
249	#440	C₄×C₃₁	1	2	yes	—
250	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
251	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
252	C₈×D₄	—	2	1	no	—
253	C₈×D₄	—	2	1	no	—
254	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
255	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
256	#431	—	2	1	no	—
257	#431	C₂²×C₃₁	1	1	yes	characteristic
258	#431	C₂²×C₃₁	1	1	yes	characteristic
259	#428	C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄⋊C₈	—	2	1	no	—
261	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
262	#419	C₄×C₃₁	1	1	yes	characteristic
263	#417	C₄×C₃₁	1	1	yes	characteristic
264	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄×C₈	—	2	1	no	—
266	C₂×C₄×C₈	—	2	1	no	—
267	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
268	#335	C₂²×C₃₁	1	2	yes	—
269	#334	C₂²×C₃₁	1	2	yes	—
270	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
271	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
272	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
273	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
274	#322	C₂²×C₃₁	1	2	yes	—
275	C₈²	C₂²×C₃₁	1	2	yes	—
276	C₂×C₃₁	—	2	2	no	—
277	C₂×C₃₁	—	4	2	no	—
278	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
279	C₂×C₃₁	C₄×#137	1	1	yes	characteristic
280	C₂×C₃₁	#2955	1	1	yes	characteristic
281	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
282	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
283	#137	C₈×C₃₁	1	8	yes	direct product factor
284	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
285	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
286	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
287	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
288	C₂²×C₈	—	2	1	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	—	2	1	no	—
291	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
292	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
293	C₄×Q₈	—	2	1	no	—
294	C₄×Q₈	—	2	1	no	—
295	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
296	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
297	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
298	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
303	#117	—	2	1	no	—
304	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄²	—	2	1	no	—
308	C₂×C₄²	—	2	1	no	—
309	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
310	#107	C₂×C₄×C₃₁	1	2	yes	—
311	#106	C₂×C₄×C₃₁	1	2	yes	—
312	C₄⋊C₈	—	2	2	no	—
313	C₄⋊C₈	—	2	2	no	—
314	C₄⋊C₈	—	2	1	no	—
315	C₄⋊C₈	—	2	1	no	—
316	C₄⋊C₈	—	2	1	no	—
317	C₄⋊C₈	—	2	1	no	—
318	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
319	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
321	#103	C₂×C₄×C₃₁	1	2	yes	—
322	#103	C₂×C₄×C₃₁	1	2	yes	—
323	#103	C₂×C₄×C₃₁	1	2	yes	—
324	#102	C₂×C₄×C₃₁	1	2	yes	—
325	C₂²⋊C₈	—	2	1	no	—
326	C₂²⋊C₈	—	2	1	no	—
327	C₂²⋊C₈	—	2	1	no	—
328	C₂²⋊C₈	—	2	1	no	—
329	#97	C₂×C₄×C₃₁	1	2	yes	—
330	C₄×C₈	—	2	1	no	—
331	C₄×C₈	—	2	1	no	—
332	C₄×C₈	—	2	2	no	—
333	C₄×C₈	—	2	2	no	—
334	C₄×C₈	—	2	1	no	—
335	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
336	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
337	C₄×C₈	C₂³×C₃₁	1	2	yes	—
338	C₄×C₈	C₂³×C₃₁	1	2	yes	—
339	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
340	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
341	C₃₁	C₈×#137	1	1	yes	characteristic, direct product factor, Sylow
342	#41	—	2	2	no	—
343	#41	C₂×C₈×C₃₁	1	4	yes	—
344	C₂×Q₈	—	2	1	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	complemented
347	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
348	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
359	Q₁₆	C₂×C₈×C₃₁	1	8	yes	complemented
360	SD₁₆	C₂×C₈×C₃₁	1	8	yes	complemented
361	M₁₆	—	2	2	no	—
362	M₁₆	C₂×C₈×C₃₁	1	4	yes	complemented
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	4	1	no	—
369	C₂×C₈	—	4	2	no	—
370	C₂×C₈	—	4	2	no	—
371	C₂×C₈	—	4	2	no	—
372	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
373	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
374	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
384	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
385	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
386	C₂²⋊C₄	—	2	1	no	—
387	C₂²⋊C₄	—	2	1	no	—
388	C₂²⋊C₄	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	1	no	—
393	C₄²	—	2	2	no	—
394	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
395	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₄²	#41×C₃₁	1	1	yes	characteristic
397	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
398	C₂³	—	2	1	no	—
399	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
400	Q₈	—	2	4	no	—
401	Q₈	—	2	2	no	—
402	Q₈	—	2	1	no	—
403	Q₈	C₂²×C₈×C₃₁	1	4	yes	—
404	Q₈	C₂²×C₈×C₃₁	1	2	yes	—
405	D₄	—	2	4	no	—
406	D₄	—	2	1	no	—
407	D₄	C₂²×C₈×C₃₁	1	2	yes	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
426	C₂×C₄	C₃₁×#131	1	2	yes	—
427	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
430	C₈	—	2	2	no	—
431	C₈	—	2	1	no	—
432	C₈	—	2	1	no	—
433	C₈	—	2	1	no	—
434	C₈	—	4	2	no	—
435	C₈	—	4	1	no	—
436	C₈	—	4	1	no	—
437	C₈	—	4	2	no	—
438	C₈	C₂²×C₈×C₃₁	1	4	yes	—
439	C₈	C₃₁×#137	1	2	yes	direct product factor
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	4	1	no	—
445	C₂²	C₈×D₄×C₃₁	1	2	yes	—
446	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
447	C₄	—	2	2	no	—
448	C₄	—	2	4	no	—
449	C₄	—	2	1	no	—
450	C₄	—	2	1	no	—
451	C₄	—	2	1	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	1	no	—
454	C₄	—	4	2	no	—
455	C₄	C₈×D₄×C₃₁	1	2	yes	—
456	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
457	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
458	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
459	C₄	C₃₁×#575	1	1	yes	characteristic
460	C₂	—	2	2	no	—
461	C₂	—	4	2	no	—
462	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic
463	C₂	C₄×C₃₁×#137	1	1	yes	characteristic
464	C₂	C₃₁×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#1579	C₂	semidirect
5. C₃₁×SD₁₆⋊C₈	C₂	semidirect
6. C₈×Q₁₆×C₃₁	C₂	semidirect
7. C₈×SD₁₆×C₃₁	C₂	semidirect
8. C₃₁×#1565	C₂	semidirect
9. C₃₁×#1479	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
51. C₃₁×#137	C₈	direct (N × H)
113. C₂×Q₈×C₃₁	C₂×C₈	semidirect
126. Q₁₆×C₃₁	C₂×C₈	semidirect
127. SD₁₆×C₃₁	C₂×C₈	semidirect
129. M₁₆×C₃₁	C₂×C₈	semidirect
165. C₈×#137	C₃₁	direct (N × H)
207. C₈×C₃₁	#137	direct (N × H)
209. C₂×C₈×Q₈	C₂×C₃₁	semidirect
211. #1579	C₂×C₃₁	semidirect
212. SD₁₆⋊C₈	C₂×C₃₁	semidirect
213. C₈×Q₁₆	C₂×C₃₁	semidirect
214. C₈×SD₁₆	C₂×C₃₁	semidirect
215. #1565	C₂×C₃₁	semidirect
216. #1479	C₂×C₃₁	semidirect
218. C₈×M₁₆	C₂×C₃₁	semidirect
283. #137	C₈×C₃₁	direct (N × H)
341. C₃₁	C₈×#137	direct (N × H)
346. C₂×Q₈	C₂×C₈×C₃₁	semidirect
359. Q₁₆	C₂×C₈×C₃₁	semidirect
360. SD₁₆	C₂×C₈×C₃₁	semidirect
362. M₁₆	C₂×C₈×C₃₁	semidirect
439. C₈	C₃₁×#137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#41×C₃₁	2
C₂×C₈×Q₈×C₃₁	2
C₄×C₃₁×#137	2
C₃₁×#1579	2
C₃₁×SD₁₆⋊C₈	2
C₈×Q₁₆×C₃₁	2
C₈×SD₁₆×C₃₁	2
C₃₁×#1565	2
C₃₁×#1479	2
C₃₁×#1477	2
C₈×M₁₆×C₃₁	2
C₈×#137	31

Subgroups — C8×C31×#137 #21474610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₃₁×#137 #21474610