Subgroups — C₂×C₃₁×#2237 #21490401

Order 7936.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2237	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
12	C₂×C₃₁×#117	—	2	1	no	—
13	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#467	—	2	2	no	—
16	C₃₁×#467	C₂²	1	4	yes	—
17	C₃₁×#427	—	2	2	no	—
18	C₃₁×#427	C₂²	1	4	yes	—
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#106	—	2	1	no	—
23	C₂×C₃₁×#103	—	2	1	no	—
24	C₃₁×#367	C₂²	1	4	yes	complemented
25	C₃₁×#366	C₂²	1	4	yes	complemented
26	C₃₁×#358	C₂²	1	4	yes	complemented
27	C₃₁×#358	C₂²	1	4	yes	complemented
28	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
29	C₂²×Q₈×C₃₁	—	2	1	no	—
30	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
31	C₂×Q₁₆×C₃₁	—	2	1	no	—
32	C₂×Q₁₆×C₃₁	—	2	2	no	—
33	C₂×Q₁₆×C₃₁	—	2	2	no	—
34	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	D₄	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₂²⋊Q₈	—	2	4	no	—
42	C₃₁×C₂²⋊Q₈	—	4	2	no	—
43	C₃₁×#117	—	2	4	no	—
44	C₃₁×#117	—	4	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
51	C₂×C₄²×C₃₁	—	4	1	no	—
52	C₂×C₁₆×C₃₁	—	2	2	no	—
53	C₂×C₁₆×C₃₁	—	2	2	no	—
54	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
55	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
56	C₃₁×#107	—	2	1	no	—
57	C₃₁×#107	—	2	1	no	—
58	C₃₁×#107	—	2	1	no	—
59	C₃₁×#107	C₂³	1	2	yes	—
60	C₃₁×#107	C₂³	1	2	yes	—
61	C₃₁×#107	C₂³	1	2	yes	—
62	C₃₁×#106	—	2	4	no	—
63	C₃₁×#103	—	2	4	no	—
64	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
65	C₂×Q₈×C₃₁	—	2	2	no	—
66	C₂×Q₈×C₃₁	—	4	1	no	—
67	C₂×Q₈×C₃₁	—	4	2	no	—
68	C₂×Q₈×C₃₁	—	4	2	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	4	1	no	—
72	C₂²×C₄×C₃₁	—	4	1	no	—
73	C₂²×C₄×C₃₁	—	4	1	no	—
74	C₂²×C₄×C₃₁	—	4	1	no	—
75	C₂²×C₄×C₃₁	D₈	1	2	yes	—
76	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₃₁	—	2	2	no	—
80	Q₁₆×C₃₁	—	2	2	no	—
81	Q₁₆×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	4	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	#41	1	2	yes	—
89	C₂×C₈×C₃₁	#41	1	2	yes	—
90	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	4	1	no	—
95	C₄⋊C₄×C₃₁	—	4	1	no	—
96	C₄⋊C₄×C₃₁	—	4	1	no	—
97	C₄⋊C₄×C₃₁	—	4	4	no	—
98	C₂²⋊C₄×C₃₁	—	4	4	no	—
99	C₂²⋊C₄×C₃₁	—	4	4	no	—
100	C₄²×C₃₁	—	4	4	no	—
101	C₁₆×C₃₁	—	2	2	no	—
102	C₁₆×C₃₁	—	2	2	no	—
103	C₂×#2237	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	—	2	2	no	—
105	C₂³×C₃₁	—	2	2	no	—
106	C₂³×C₃₁	—	2	2	no	—
107	C₂³×C₃₁	C₂×D₈	1	2	yes	—
108	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
109	Q₈×C₃₁	—	4	2	no	—
110	Q₈×C₃₁	—	4	2	no	—
111	Q₈×C₃₁	—	8	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	4	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	8	2	no	—
121	C₂×C₄×C₃₁	—	8	2	no	—
122	C₂×C₄×C₃₁	—	8	2	no	—
123	C₂×C₄×C₃₁	—	8	2	no	—
124	C₂×C₄×C₃₁	#123	1	2	yes	—
125	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#137	1	2	yes	—
128	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₃₁	—	2	2	no	—
130	C₈×C₃₁	#123	1	2	yes	—
131	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
132	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
133	C₂×#467	C₂×C₃₁	1	1	yes	characteristic
134	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
135	#2237	C₂×C₃₁	1	8	yes	direct product factor
136	C₂×#367	C₂×C₃₁	1	1	yes	characteristic, complemented
137	C₂×#366	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₂×#358	C₂×C₃₁	1	1	yes	characteristic, complemented
139	C₂×#358	C₂×C₃₁	1	1	yes	characteristic, complemented
140	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	—	2	2	no	—
143	C₂²×C₃₁	—	2	2	no	—
144	C₂²×C₃₁	—	4	4	no	—
145	C₂²×C₃₁	#480	1	2	yes	—
146	C₂²×C₃₁	#508	1	2	yes	—
147	C₂²×C₃₁	#510	1	2	yes	—
148	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	8	2	no	—
151	C₄×C₃₁	—	8	2	no	—
152	C₄×C₃₁	—	8	2	no	—
153	C₄×C₃₁	—	8	2	no	—
154	C₄×C₃₁	#480	1	2	yes	—
155	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
156	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
157	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
158	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×C₂²⋊Q₈	—	2	1	no	—
160	C₂×#117	—	2	1	no	—
161	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
162	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
163	#467	—	2	2	no	—
164	#467	C₂²×C₃₁	1	4	yes	—
165	#427	—	2	2	no	—
166	#427	C₂²×C₃₁	1	4	yes	—
167	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
168	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×#106	—	2	1	no	—
171	C₂×#103	—	2	1	no	—
172	#367	C₂²×C₃₁	1	4	yes	complemented
173	#366	C₂²×C₃₁	1	4	yes	complemented
174	#358	C₂²×C₃₁	1	4	yes	complemented
175	#358	C₂²×C₃₁	1	4	yes	complemented
176	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	—
177	C₂×C₃₁	—	4	2	no	—
178	C₂×C₃₁	#2237	1	4	yes	direct product factor
179	C₂×C₃₁	C₂×#480	1	1	yes	characteristic
180	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
181	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
182	C₂²×Q₈	—	2	1	no	—
183	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₈	D₄×C₃₁	1	2	yes	—
192	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
193	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
194	C₂²⋊Q₈	—	2	4	no	—
195	C₂²⋊Q₈	—	4	2	no	—
196	#117	—	2	4	no	—
197	#117	—	4	2	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	4	1	no	—
202	C₂×C₂²⋊C₄	—	4	1	no	—
203	C₂×C₂²⋊C₄	—	4	1	no	—
204	C₂×C₄²	—	4	1	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
208	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
209	#107	—	2	1	no	—
210	#107	—	2	1	no	—
211	#107	—	2	1	no	—
212	#107	C₂³×C₃₁	1	2	yes	—
213	#107	C₂³×C₃₁	1	2	yes	—
214	#107	C₂³×C₃₁	1	2	yes	—
215	#106	—	2	4	no	—
216	#103	—	2	4	no	—
217	C₃₁	C₂×#2237	1	1	yes	characteristic, direct product factor, Sylow
218	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	D₈×C₃₁	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂×C₈	#41×C₃₁	1	2	yes	—
243	C₂×C₈	#41×C₃₁	1	2	yes	—
244	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₂²⋊C₄	—	4	4	no	—
254	C₄²	—	4	4	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×D₈×C₃₁	1	2	yes	—
261	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	C₃₁×#123	1	2	yes	—
278	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
279	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₃₁×#137	1	2	yes	—
281	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	C₃₁×#123	1	2	yes	—
284	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
285	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	C₃₁×#480	1	2	yes	—
291	C₂²	C₃₁×#508	1	2	yes	—
292	C₂²	C₃₁×#510	1	2	yes	—
293	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	C₃₁×#480	1	2	yes	—
300	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
301	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	C₃₁×#2237	1	4	yes	direct product factor
304	C₂	C₂×C₃₁×#480	1	1	yes	characteristic
305	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
306	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2237	C₂	direct (N × H)
4. C₂×C₃₁×#367	C₂	semidirect
5. C₂×C₃₁×#366	C₂	semidirect
6. C₂×C₃₁×#358	C₂	semidirect
7. C₂×C₃₁×#358	C₂	semidirect
24. C₃₁×#367	C₂²	semidirect
25. C₃₁×#366	C₂²	semidirect
26. C₃₁×#358	C₂²	semidirect
27. C₃₁×#358	C₂²	semidirect
103. C₂×#2237	C₃₁	direct (N × H)
135. #2237	C₂×C₃₁	direct (N × H)
136. C₂×#367	C₂×C₃₁	semidirect
137. C₂×#366	C₂×C₃₁	semidirect
138. C₂×#358	C₂×C₃₁	semidirect
139. C₂×#358	C₂×C₃₁	semidirect
172. #367	C₂²×C₃₁	semidirect
173. #366	C₂²×C₃₁	semidirect
174. #358	C₂²×C₃₁	semidirect
175. #358	C₂²×C₃₁	semidirect
178. C₂×C₃₁	#2237	direct (N × H)
217. C₃₁	C₂×#2237	direct (N × H)
303. C₂	C₃₁×#2237	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×#467	2
C₂×C₃₁×#427	2
C₃₁×#2237	2
C₂×C₃₁×#367	2
C₂×C₃₁×#366	2
C₂×C₃₁×#358	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₂×#2237	31

Subgroups — C2×C31×#2237 #21490401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2237 #21490401