Subgroups — C₂×C₃₁×#2228 #21490381

Order 7936.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2228	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×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#125	—	2	1	no	—
11	C₂×C₄×D₄×C₃₁	—	2	1	no	—
12	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#499	—	2	2	no	—
15	C₃₁×#499	C₂²	1	4	yes	complemented
16	C₄×D₈×C₃₁	—	2	2	no	—
17	C₄×D₈×C₃₁	C₂²	1	4	yes	—
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#106	—	2	1	no	—
22	C₂×C₃₁×#102	—	2	1	no	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#367	C₂²	1	4	yes	complemented
25	C₃₁×#366	C₂²	1	4	yes	complemented
26	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	complemented
27	C₃₁×#357	C₂²	1	4	yes	—
28	C₃₁×#357	C₂²	1	4	yes	—
29	C₂²×D₄×C₃₁	—	2	1	no	—
30	C₂³×C₄×C₃₁	—	4	1	no	—
31	C₂×D₈×C₃₁	—	2	1	no	—
32	C₂×D₈×C₃₁	—	2	2	no	—
33	C₂×D₈×C₃₁	Q₈	1	4	yes	—
34	C₂×D₈×C₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#125	—	2	4	no	—
38	C₃₁×#125	—	4	2	no	—
39	C₄×D₄×C₃₁	—	2	4	no	—
40	C₄×D₄×C₃₁	—	4	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
45	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
46	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₁₆×C₃₁	—	2	2	no	—
49	C₂×C₁₆×C₃₁	—	2	2	no	—
50	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
51	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
52	C₃₁×#107	—	2	1	no	—
53	C₃₁×#107	—	2	1	no	—
54	C₃₁×#107	—	2	1	no	—
55	C₃₁×#107	C₂³	1	2	yes	—
56	C₃₁×#107	C₂³	1	2	yes	—
57	C₃₁×#107	C₂³	1	2	yes	—
58	C₃₁×#106	—	2	4	no	—
59	C₃₁×#102	—	2	4	no	—
60	C₄×C₈×C₃₁	D₄	1	2	yes	—
61	C₄×C₈×C₃₁	C₂³	1	2	yes	—
62	C₂⁴×C₃₁	—	4	1	no	—
63	C₂×D₄×C₃₁	—	2	4	no	—
64	C₂×D₄×C₃₁	—	2	2	no	—
65	C₂×D₄×C₃₁	—	4	1	no	—
66	C₂×D₄×C₃₁	—	4	2	no	—
67	C₂²×C₄×C₃₁	—	4	1	no	—
68	C₂²×C₄×C₃₁	—	4	1	no	—
69	C₂²×C₄×C₃₁	—	4	1	no	—
70	C₂²×C₄×C₃₁	—	4	4	no	—
71	C₂²×C₄×C₃₁	—	8	4	no	—
72	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	D₈×C₃₁	—	2	4	no	—
76	D₈×C₃₁	—	2	2	no	—
77	D₈×C₃₁	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₃₁	#41	1	2	yes	—
83	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	4	1	no	—
88	C₄⋊C₄×C₃₁	—	4	1	no	—
89	C₄⋊C₄×C₃₁	—	4	1	no	—
90	C₄⋊C₄×C₃₁	—	4	4	no	—
91	C₄⋊C₄×C₃₁	—	4	4	no	—
92	C₂²⋊C₄×C₃₁	—	4	4	no	—
93	C₄²×C₃₁	D₈	1	2	yes	—
94	C₄²×C₃₁	C₂×D₄	1	2	yes	—
95	C₁₆×C₃₁	—	2	2	no	—
96	C₁₆×C₃₁	—	2	2	no	—
97	C₂×#2228	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	4	2	no	—
99	C₂³×C₃₁	—	4	4	no	—
100	C₂³×C₃₁	—	8	2	no	—
101	C₂³×C₃₁	—	8	2	no	—
102	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
103	D₄×C₃₁	—	2	4	no	—
104	D₄×C₃₁	—	4	2	no	—
105	D₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	8	8	no	—
112	C₂×C₄×C₃₁	—	8	2	no	—
113	C₂×C₄×C₃₁	—	8	2	no	—
114	C₂×C₄×C₃₁	—	8	2	no	—
115	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	#137	1	2	yes	—
120	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₃₁	—	2	2	no	—
122	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
123	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
125	C₂×#499	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
127	#2228	C₂×C₃₁	1	8	yes	direct product factor
128	C₂×#367	C₂×C₃₁	1	1	yes	characteristic, complemented
129	C₂×#366	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₂×#357	C₂×C₃₁	1	1	yes	characteristic
132	C₂×#357	C₂×C₃₁	1	1	yes	characteristic
133	C₂²×C₃₁	—	4	4	no	—
134	C₂²×C₃₁	—	8	2	no	—
135	C₂²×C₃₁	—	8	2	no	—
136	C₂²×C₃₁	—	8	8	no	—
137	C₂²×C₃₁	#474	1	2	yes	—
138	C₂²×C₃₁	#508	1	2	yes	—
139	C₂²×C₃₁	#509	1	2	yes	—
140	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₃₁	—	2	2	no	—
142	C₄×C₃₁	—	2	2	no	—
143	C₄×C₃₁	—	8	2	no	—
144	C₄×C₃₁	—	8	2	no	—
145	C₄×C₃₁	—	8	2	no	—
146	C₄×C₃₁	#474	1	2	yes	—
147	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
148	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
149	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×#125	—	2	1	no	—
151	C₂×C₄×D₄	—	2	1	no	—
152	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
154	#499	—	2	2	no	—
155	#499	C₂²×C₃₁	1	4	yes	complemented
156	C₄×D₈	—	2	2	no	—
157	C₄×D₈	C₂²×C₃₁	1	4	yes	—
158	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×#106	—	2	1	no	—
162	C₂×#102	—	2	1	no	—
163	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
164	#367	C₂²×C₃₁	1	4	yes	complemented
165	#366	C₂²×C₃₁	1	4	yes	complemented
166	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	complemented
167	#357	C₂²×C₃₁	1	4	yes	—
168	#357	C₂²×C₃₁	1	4	yes	—
169	C₂×C₃₁	—	8	4	no	—
170	C₂×C₃₁	#2228	1	4	yes	direct product factor
171	C₂×C₃₁	C₂×#474	1	1	yes	characteristic
172	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
173	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
174	C₂²×D₄	—	2	1	no	—
175	C₂³×C₄	—	4	1	no	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	—	2	2	no	—
178	C₂×D₈	Q₈×C₃₁	1	4	yes	—
179	C₂×D₈	C₂³×C₃₁	1	2	yes	—
180	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
182	#125	—	2	4	no	—
183	#125	—	4	2	no	—
184	C₄×D₄	—	2	4	no	—
185	C₄×D₄	—	4	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	4	1	no	—
190	C₂×C₄⋊C₄	—	4	1	no	—
191	C₂×C₂²⋊C₄	—	4	1	no	—
192	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
193	C₂×C₁₆	—	2	2	no	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
196	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
197	#107	—	2	1	no	—
198	#107	—	2	1	no	—
199	#107	—	2	1	no	—
200	#107	C₂³×C₃₁	1	2	yes	—
201	#107	C₂³×C₃₁	1	2	yes	—
202	#107	C₂³×C₃₁	1	2	yes	—
203	#106	—	2	4	no	—
204	#102	—	2	4	no	—
205	C₄×C₈	D₄×C₃₁	1	2	yes	—
206	C₄×C₈	C₂³×C₃₁	1	2	yes	—
207	C₃₁	C₂×#2228	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	—	4	1	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	—	8	4	no	—
218	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
221	D₈	—	2	4	no	—
222	D₈	—	2	2	no	—
223	D₈	C₂×Q₈×C₃₁	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
228	C₂×C₈	#41×C₃₁	1	2	yes	—
229	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	D₈×C₃₁	1	2	yes	—
240	C₄²	C₂×D₄×C₃₁	1	2	yes	—
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₂³	—	4	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
261	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
262	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
263	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₃₁×#137	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
268	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
269	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
270	C₂²	—	4	4	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	8	no	—
274	C₂²	C₃₁×#474	1	2	yes	—
275	C₂²	C₃₁×#508	1	2	yes	—
276	C₂²	C₃₁×#509	1	2	yes	—
277	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	C₃₁×#474	1	2	yes	—
284	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
285	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
286	C₂	—	8	4	no	—
287	C₂	C₃₁×#2228	1	4	yes	direct product factor
288	C₂	C₂×C₃₁×#474	1	1	yes	characteristic
289	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
290	C₂	C₂×C₃₁×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#499	C₂	semidirect
3. C₃₁×#2228	C₂	direct (N × H)
4. C₂×C₃₁×#367	C₂	semidirect
5. C₂×C₃₁×#366	C₂	semidirect
6. C₂×C₃₁×C₄⋊C₁₆	C₂	semidirect
15. C₃₁×#499	C₂²	semidirect
24. C₃₁×#367	C₂²	semidirect
25. C₃₁×#366	C₂²	semidirect
26. C₃₁×C₄⋊C₁₆	C₂²	semidirect
97. C₂×#2228	C₃₁	direct (N × H)
125. C₂×#499	C₂×C₃₁	semidirect
127. #2228	C₂×C₃₁	direct (N × H)
128. C₂×#367	C₂×C₃₁	semidirect
129. C₂×#366	C₂×C₃₁	semidirect
130. C₂×C₄⋊C₁₆	C₂×C₃₁	semidirect
155. #499	C₂²×C₃₁	semidirect
164. #367	C₂²×C₃₁	semidirect
165. #366	C₂²×C₃₁	semidirect
166. C₄⋊C₁₆	C₂²×C₃₁	semidirect
170. C₂×C₃₁	#2228	direct (N × H)
207. C₃₁	C₂×#2228	direct (N × H)
287. C₂	C₃₁×#2228	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×#499	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2228	2
C₂×C₃₁×#367	2
C₂×C₃₁×#366	2
C₂×C₃₁×C₄⋊C₁₆	2
C₂×C₃₁×#357	2
C₂×#2228	31

Subgroups — C2×C31×#2228 #21490381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2228 #21490381