Subgroups — C₂×C₃₁×#2117 #21490098

Order 7936.

1154 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2117	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#350	C₂	1	4	yes	maximal, complemented
6	C₂²×#41×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₃₁×#131	—	2	1	no	—
8	C₂×C₃₁×#131	—	2	2	no	—
9	C₂×C₃₁×#131	—	2	2	no	—
10	C₂×C₃₁×#131	—	2	1	no	—
11	C₂×C₃₁×#131	C₄	1	2	yes	—
12	C₂×C₃₁×#131	C₄	1	4	yes	—
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₂×C₃₁×#131	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₃₁	—	2	1	no	—
16	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
17	C₂³×C₈×C₃₁	—	2	1	no	—
18	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×M₃₂	—	2	2	no	—
20	C₂×C₃₁×M₃₂	—	2	1	no	—
21	C₂×C₃₁×M₃₂	—	2	1	no	—
22	C₂×C₃₁×M₃₂	—	2	2	no	—
23	C₂×C₃₁×M₃₂	—	2	1	no	—
24	C₂×C₃₁×M₃₂	C₂²	1	2	yes	complemented
25	C₂×C₃₁×M₃₂	C₂²	1	4	yes	complemented
26	C₂×C₃₁×M₃₂	C₂²	1	4	yes	complemented
27	C₂²×C₁₆×C₃₁	—	2	1	no	—
28	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	complemented
29	C₃₁×#350	C₂²	1	16	yes	complemented
30	C₂×#41×C₃₁	—	2	2	no	—
31	C₂×#41×C₃₁	—	2	1	no	—
32	C₂×#41×C₃₁	—	2	1	no	—
33	C₂×#41×C₃₁	—	2	2	no	—
34	C₂×#41×C₃₁	C₈	1	2	yes	—
35	C₂×#41×C₃₁	C₈	1	4	yes	—
36	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
38	C₂²×Q₈×C₃₁	C₈	1	1	yes	characteristic
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	C₈	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	—	2	1	no	—
42	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₃₁×#131	—	2	4	no	—
44	C₃₁×#131	—	2	4	no	—
45	C₃₁×#131	—	2	4	no	—
46	C₃₁×#131	—	2	2	no	—
47	C₃₁×#131	—	2	4	no	—
48	C₃₁×#131	—	2	4	no	—
49	C₃₁×#131	—	2	2	no	—
50	C₃₁×#131	C₂×C₄	1	4	yes	—
51	C₃₁×#131	C₂×C₄	1	8	yes	—
52	C₃₁×#131	C₂³	1	4	yes	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂×M₁₆×C₃₁	—	2	4	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	2	4	no	—
59	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
61	C₂²×C₈×C₃₁	—	2	1	no	—
62	C₂²×C₈×C₃₁	—	2	4	no	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	—	2	4	no	—
65	C₂²×C₈×C₃₁	—	2	2	no	—
66	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
68	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₃₁	D₄	1	2	yes	—
71	C₂²×C₈×C₃₁	D₄	1	2	yes	—
72	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×M₃₂	—	2	4	no	—
77	C₃₁×M₃₂	—	2	2	no	—
78	C₃₁×M₃₂	—	2	4	no	—
79	C₃₁×M₃₂	—	2	4	no	—
80	C₃₁×M₃₂	C₂³	1	4	yes	complemented
81	C₂×C₁₆×C₃₁	—	2	4	no	—
82	C₂×C₁₆×C₃₁	—	2	2	no	—
83	C₂×C₁₆×C₃₁	—	2	2	no	—
84	C₂×C₁₆×C₃₁	—	2	2	no	—
85	C₂×C₁₆×C₃₁	C₂³	1	4	yes	complemented
86	C₂⁴×C₃₁	—	2	1	no	—
87	C₂⁴×C₃₁	M₁₆	1	1	yes	characteristic
88	#41×C₃₁	—	2	4	no	—
89	#41×C₃₁	—	2	4	no	—
90	#41×C₃₁	—	2	4	no	—
91	#41×C₃₁	—	2	2	no	—
92	#41×C₃₁	—	2	4	no	—
93	#41×C₃₁	—	2	2	no	—
94	#41×C₃₁	—	2	4	no	—
95	#41×C₃₁	C₂×C₈	1	4	yes	—
96	#41×C₃₁	C₂×C₈	1	8	yes	—
97	#41×C₃₁	C₂²×C₄	1	4	yes	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	1	no	—
100	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	—
101	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	4	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	4	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	C₂×C₈	1	4	yes	—
109	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	4	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
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	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
121	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
123	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
125	M₁₆×C₃₁	—	2	2	no	—
126	M₁₆×C₃₁	—	2	4	no	—
127	M₁₆×C₃₁	—	2	4	no	—
128	M₁₆×C₃₁	—	2	8	no	—
129	M₁₆×C₃₁	—	2	4	no	—
130	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
131	C₂×C₈×C₃₁	—	2	4	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	8	no	—
134	C₂×C₈×C₃₁	—	2	4	no	—
135	C₂×C₈×C₃₁	—	2	4	no	—
136	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
138	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
139	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
141	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
142	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
143	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
148	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
150	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
151	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
154	C₁₆×C₃₁	—	2	4	no	—
155	C₁₆×C₃₁	—	2	4	no	—
156	C₂×#2117	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	2	4	no	—
158	C₂³×C₃₁	—	2	2	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	4	no	—
161	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
162	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
163	C₂³×C₃₁	C₂²⋊C₈	1	2	yes	—
164	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
165	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
166	Q₈×C₃₁	—	2	2	no	—
167	Q₈×C₃₁	—	2	4	no	—
168	Q₈×C₃₁	C₂²×C₈	1	4	yes	—
169	D₄×C₃₁	—	2	4	no	—
170	D₄×C₃₁	—	2	8	no	—
171	D₄×C₃₁	—	2	4	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	2	4	no	—
174	D₄×C₃₁	C₂²×C₈	1	4	yes	—
175	C₂×C₄×C₃₁	—	2	4	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	8	no	—
179	C₂×C₄×C₃₁	—	2	4	no	—
180	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
181	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
183	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
185	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
186	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
187	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
188	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
190	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
192	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
194	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
196	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
198	C₈×C₃₁	—	2	4	no	—
199	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
200	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
203	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic
206	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#2117	C₂×C₃₁	1	8	yes	direct product factor
209	C₂×#350	C₂×C₃₁	1	4	yes	complemented
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	2	2	no	—
213	C₂²×C₃₁	—	2	4	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	8	no	—
217	C₂²×C₃₁	—	4	4	no	—
218	C₂²×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
219	C₂²×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
220	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
221	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
222	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
223	C₄×C₃₁	—	2	4	no	—
224	C₄×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
225	C₄×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
226	C₄×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
227	C₄×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
228	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
229	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
230	C₂²×#41	C₄×C₃₁	1	1	yes	characteristic
231	C₂×#131	—	2	1	no	—
232	C₂×#131	—	2	2	no	—
233	C₂×#131	—	2	1	no	—
234	C₂×#131	—	2	2	no	—
235	C₂×#131	C₄×C₃₁	1	2	yes	—
236	C₂×#131	C₄×C₃₁	1	4	yes	—
237	C₂×#131	C₂²×C₃₁	1	2	yes	—
238	C₂×#131	C₂²×C₃₁	1	4	yes	—
239	C₂²×M₁₆	—	2	1	no	—
240	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
241	C₂³×C₈	—	2	1	no	—
242	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×M₃₂	—	2	1	no	—
244	C₂×M₃₂	—	2	2	no	—
245	C₂×M₃₂	—	2	1	no	—
246	C₂×M₃₂	—	2	1	no	—
247	C₂×M₃₂	—	2	2	no	—
248	C₂×M₃₂	C₂²×C₃₁	1	2	yes	complemented
249	C₂×M₃₂	C₂²×C₃₁	1	4	yes	complemented
250	C₂×M₃₂	C₂²×C₃₁	1	4	yes	complemented
251	C₂²×C₁₆	—	2	1	no	—
252	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	complemented
253	#350	C₂²×C₃₁	1	16	yes	complemented
254	C₂×C₃₁	—	2	2	no	—
255	C₂×C₃₁	—	2	1	no	—
256	C₂×C₃₁	—	2	2	no	—
257	C₂×C₃₁	—	2	1	no	—
258	C₂×C₃₁	—	4	4	no	—
259	C₂×C₃₁	#2117	1	2	yes	direct product factor
260	C₂×C₃₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
261	C₂×#41	—	2	1	no	—
262	C₂×#41	—	2	2	no	—
263	C₂×#41	—	2	1	no	—
264	C₂×#41	—	2	2	no	—
265	C₂×#41	C₈×C₃₁	1	2	yes	—
266	C₂×#41	C₈×C₃₁	1	4	yes	—
267	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
268	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
269	C₂²×Q₈	C₈×C₃₁	1	1	yes	characteristic
270	C₂²×D₄	—	2	1	no	—
271	C₂²×D₄	C₈×C₃₁	1	1	yes	characteristic
272	C₂³×C₄	—	2	1	no	—
273	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
274	#131	—	2	4	no	—
275	#131	—	2	4	no	—
276	#131	—	2	4	no	—
277	#131	—	2	2	no	—
278	#131	—	2	4	no	—
279	#131	—	2	4	no	—
280	#131	—	2	2	no	—
281	#131	C₂×C₄×C₃₁	1	4	yes	—
282	#131	C₂×C₄×C₃₁	1	8	yes	—
283	#131	C₂³×C₃₁	1	4	yes	—
284	C₂×M₁₆	—	2	1	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	—	2	2	no	—
287	C₂×M₁₆	—	2	4	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	—	2	4	no	—
290	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
291	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
292	C₂²×C₈	—	2	1	no	—
293	C₂²×C₈	—	2	4	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	—	2	4	no	—
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
298	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
299	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
301	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
302	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
303	C₂²×C₈	D₄×C₃₁	1	2	yes	—
304	C₂²×C₈	D₄×C₃₁	1	2	yes	—
305	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
307	M₃₂	—	2	4	no	—
308	M₃₂	—	2	2	no	—
309	M₃₂	—	2	4	no	—
310	M₃₂	—	2	4	no	—
311	M₃₂	C₂³×C₃₁	1	4	yes	complemented
312	C₂×C₁₆	—	2	4	no	—
313	C₂×C₁₆	—	2	2	no	—
314	C₂×C₁₆	—	2	2	no	—
315	C₂×C₁₆	—	2	2	no	—
316	C₂×C₁₆	C₂³×C₃₁	1	4	yes	complemented
317	C₃₁	C₂×#2117	1	1	yes	characteristic, direct product factor, Sylow
318	C₂⁴	—	2	1	no	—
319	C₂⁴	M₁₆×C₃₁	1	1	yes	characteristic
320	#41	—	2	4	no	—
321	#41	—	2	4	no	—
322	#41	—	2	4	no	—
323	#41	—	2	2	no	—
324	#41	—	2	4	no	—
325	#41	—	2	2	no	—
326	#41	—	2	4	no	—
327	#41	C₂×C₈×C₃₁	1	4	yes	—
328	#41	C₂×C₈×C₃₁	1	8	yes	—
329	#41	C₂²×C₄×C₃₁	1	4	yes	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	—
333	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	4	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	C₂×C₈×C₃₁	1	4	yes	—
341	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
353	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
355	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
357	M₁₆	—	2	2	no	—
358	M₁₆	—	2	4	no	—
359	M₁₆	—	2	4	no	—
360	M₁₆	—	2	8	no	—
361	M₁₆	—	2	4	no	—
362	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
363	C₂×C₈	—	2	4	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	8	no	—
366	C₂×C₈	—	2	4	no	—
367	C₂×C₈	—	2	4	no	—
368	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
369	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
370	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
371	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
372	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
373	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
374	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
375	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
376	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
377	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
380	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
381	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
382	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
383	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
386	C₁₆	—	2	4	no	—
387	C₁₆	—	2	4	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	4	no	—
392	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
393	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
394	C₂³	C₃₁×C₂²⋊C₈	1	2	yes	—
395	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
396	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
397	Q₈	—	2	2	no	—
398	Q₈	—	2	4	no	—
399	Q₈	C₂²×C₈×C₃₁	1	4	yes	—
400	D₄	—	2	8	no	—
401	D₄	—	2	4	no	—
402	D₄	—	2	4	no	—
403	D₄	—	2	2	no	—
404	D₄	—	2	4	no	—
405	D₄	C₂²×C₈×C₃₁	1	4	yes	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	8	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
412	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
413	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
414	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
415	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
416	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
417	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
418	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
419	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
421	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
423	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
424	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
425	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
426	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
427	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
429	C₈	—	2	4	no	—
430	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
431	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
432	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
433	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
434	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
435	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	2	4	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	8	no	—
443	C₂²	—	4	4	no	—
444	C₂²	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
445	C₂²	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
446	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
447	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
448	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
449	C₄	—	2	4	no	—
450	C₄	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
451	C₄	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
452	C₄	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
453	C₄	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
454	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
455	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
456	C₂	—	2	2	no	—
457	C₂	—	2	2	no	—
458	C₂	—	2	1	no	—
459	C₂	—	2	1	no	—
460	C₂	—	4	4	no	—
461	C₂	C₃₁×#2117	1	2	yes	direct product factor
462	C₂	C₂²×C₃₁×C₂²⋊C₈	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×M₃₂	C₂	semidirect
3. C₂²×C₃₁×M₃₂	C₂	semidirect
4. C₃₁×#2117	C₂	direct (N × H)
5. C₂×C₃₁×#350	C₂	semidirect
24. C₂×C₃₁×M₃₂	C₂²	semidirect
25. C₂×C₃₁×M₃₂	C₂²	semidirect
26. C₂×C₃₁×M₃₂	C₂²	semidirect
28. C₂²×C₁₆×C₃₁	C₂²	semidirect
29. C₃₁×#350	C₂²	semidirect
80. C₃₁×M₃₂	C₂³	semidirect
85. C₂×C₁₆×C₃₁	C₂³	semidirect
156. C₂×#2117	C₃₁	direct (N × H)
206. C₂²×M₃₂	C₂×C₃₁	semidirect
207. C₂²×M₃₂	C₂×C₃₁	semidirect
208. #2117	C₂×C₃₁	direct (N × H)
209. C₂×#350	C₂×C₃₁	semidirect
248. C₂×M₃₂	C₂²×C₃₁	semidirect
249. C₂×M₃₂	C₂²×C₃₁	semidirect
250. C₂×M₃₂	C₂²×C₃₁	semidirect
252. C₂²×C₁₆	C₂²×C₃₁	semidirect
253. #350	C₂²×C₃₁	semidirect
259. C₂×C₃₁	#2117	direct (N × H)
311. M₃₂	C₂³×C₃₁	semidirect
316. C₂×C₁₆	C₂³×C₃₁	semidirect
317. C₃₁	C₂×#2117	direct (N × H)
461. C₂	C₃₁×#2117	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×#131	2
C₂²×C₃₁×M₃₂	2
C₃₁×#2117	2
C₂×C₃₁×#350	2
C₂×#2117	31

Subgroups — C2×C31×#2117 #21490098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2117 #21490098