Subgroups — C₂×C₃₁×#1630 #21475238

Order 7936.

1074 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×#559	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#477	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1630	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆×C₃₁	—	2	2	no	—
8	C₃₁×#559	—	2	2	no	—
9	C₃₁×#559	—	2	4	no	—
10	C₃₁×#559	C₂²	1	4	yes	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#124	—	2	2	no	—
14	C₂×C₃₁×#123	—	2	2	no	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₃₁	—	2	2	no	—
17	C₃₁×#492	—	2	2	no	—
18	C₃₁×#492	C₂²	1	4	yes	—
19	C₃₁×#477	—	2	4	no	—
20	C₃₁×#477	C₂²	1	8	yes	complemented
21	C₂×C₃₁×#107	—	2	2	no	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	—	2	2	no	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#331	C₂²	1	4	yes	—
26	C₃₁×Q₈⋊C₈	C₂²	1	8	yes	complemented
27	C₂²×Q₈×C₃₁	—	2	1	no	—
28	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
29	C₂²×D₄×C₃₁	—	2	2	no	—
30	C₂²×D₄×C₃₁	D₄	1	2	yes	—
31	C₂³×C₄×C₃₁	—	2	2	no	—
32	C₂×SD₁₆×C₃₁	—	2	8	no	—
33	C₂×SD₁₆×C₃₁	—	4	8	no	—
34	C₂²×C₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	—	2	2	no	—
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	—	2	1	no	—
38	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
39	C₃₁×#127	—	2	2	no	—
40	C₃₁×#127	D₄	1	2	yes	—
41	C₃₁×#127	C₂³	1	2	yes	—
42	C₃₁×#124	—	2	8	no	—
43	C₃₁×#124	—	2	8	no	—
44	C₃₁×#123	—	2	8	no	—
45	C₃₁×#123	—	2	8	no	—
46	C₄×Q₈×C₃₁	—	2	2	no	—
47	C₄×Q₈×C₃₁	—	2	2	no	—
48	C₄×Q₈×C₃₁	D₄	1	4	yes	—
49	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
50	C₄×D₄×C₃₁	—	2	8	no	—
51	C₄×D₄×C₃₁	—	2	8	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
56	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄²×C₃₁	—	2	2	no	—
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#107	—	2	8	no	—
60	C₃₁×C₄⋊C₈	—	2	2	no	—
61	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
62	C₃₁×#103	—	2	8	no	—
63	C₄×C₈×C₃₁	—	2	1	no	—
64	C₄×C₈×C₃₁	C₂³	1	2	yes	—
65	C₂⁴×C₃₁	—	2	2	no	—
66	C₂×Q₈×C₃₁	—	2	2	no	—
67	C₂×Q₈×C₃₁	—	2	4	no	—
68	C₂×Q₈×C₃₁	—	2	4	no	—
69	C₂×Q₈×C₃₁	—	2	2	no	—
70	C₂×Q₈×C₃₁	—	4	4	no	—
71	C₂×Q₈×C₃₁	—	4	1	no	—
72	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
73	C₂×D₄×C₃₁	—	2	4	no	—
74	C₂×D₄×C₃₁	—	2	2	no	—
75	C₂×D₄×C₃₁	—	2	4	no	—
76	C₂×D₄×C₃₁	—	4	4	no	—
77	C₂×D₄×C₃₁	—	4	4	no	—
78	C₂×D₄×C₃₁	—	4	4	no	—
79	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	8	no	—
83	C₂²×C₄×C₃₁	—	4	1	no	—
84	C₂²×C₄×C₃₁	—	8	4	no	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
87	SD₁₆×C₃₁	—	4	16	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	—	4	4	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
98	C₂²⋊C₄×C₃₁	—	2	16	no	—
99	C₂²⋊C₄×C₃₁	—	2	8	no	—
100	C₄²×C₃₁	—	2	4	no	—
101	C₄²×C₃₁	—	2	4	no	—
102	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₃₁	C₂×D₄	1	2	yes	—
104	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
105	C₂×#1630	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	2	4	no	—
107	C₂³×C₃₁	—	4	8	no	—
108	C₂³×C₃₁	—	4	4	no	—
109	C₂³×C₃₁	#120	1	1	yes	characteristic
110	Q₈×C₃₁	—	2	4	no	—
111	Q₈×C₃₁	—	2	4	no	—
112	Q₈×C₃₁	—	4	4	no	—
113	Q₈×C₃₁	—	4	4	no	—
114	Q₈×C₃₁	—	8	2	no	—
115	D₄×C₃₁	—	4	4	no	—
116	D₄×C₃₁	—	4	4	no	—
117	D₄×C₃₁	—	4	4	no	—
118	D₄×C₃₁	—	4	8	no	—
119	D₄×C₃₁	—	8	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	2	4	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	8	8	no	—
127	C₂×C₄×C₃₁	—	8	2	no	—
128	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#120	1	2	yes	—
130	C₂×C₄×C₃₁	#135	1	4	yes	—
131	C₂×C₄×C₃₁	#137	1	4	yes	—
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
134	C₈×C₃₁	—	2	4	no	—
135	C₈×C₃₁	—	4	4	no	—
136	C₂×#559	C₂×C₃₁	1	1	yes	characteristic
137	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
138	C₂×#477	C₂×C₃₁	1	2	yes	complemented
139	#1630	C₂×C₃₁	1	8	yes	direct product factor
140	C₂×#331	C₂×C₃₁	1	1	yes	characteristic
141	C₂×Q₈⋊C₈	C₂×C₃₁	1	2	yes	complemented
142	C₂²×C₃₁	—	4	8	no	—
143	C₂²×C₃₁	—	4	4	no	—
144	C₂²×C₃₁	—	8	8	no	—
145	C₂²×C₃₁	#452	1	4	yes	—
146	C₂²×C₃₁	#453	1	2	yes	—
147	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
148	C₄×C₃₁	—	2	1	no	—
149	C₄×C₃₁	—	2	1	no	—
150	C₄×C₃₁	—	4	4	no	—
151	C₄×C₃₁	—	4	4	no	—
152	C₄×C₃₁	—	8	2	no	—
153	C₄×C₃₁	#452	1	4	yes	—
154	C₄×C₃₁	C₂×#135	1	2	yes	—
155	C₄×C₃₁	C₂×#137	1	2	yes	—
156	C₂²×SD₁₆	—	2	2	no	—
157	#559	—	2	2	no	—
158	#559	—	2	4	no	—
159	#559	C₂²×C₃₁	1	4	yes	—
160	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×#124	—	2	2	no	—
163	C₂×#123	—	2	2	no	—
164	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
165	C₂×C₄×D₄	—	2	2	no	—
166	#492	—	2	2	no	—
167	#492	C₂²×C₃₁	1	4	yes	—
168	#477	—	2	4	no	—
169	#477	C₂²×C₃₁	1	8	yes	complemented
170	C₂×#107	—	2	2	no	—
171	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
172	C₂×#103	—	2	2	no	—
173	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
174	#331	C₂²×C₃₁	1	4	yes	—
175	Q₈⋊C₈	C₂²×C₃₁	1	8	yes	complemented
176	C₂×C₃₁	—	8	4	no	—
177	C₂×C₃₁	#1630	1	4	yes	direct product factor
178	C₂×C₃₁	C₂×#452	1	2	yes	—
179	C₂×C₃₁	C₂×#453	1	1	yes	characteristic
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
182	C₂²×D₄	—	2	2	no	—
183	C₂²×D₄	D₄×C₃₁	1	2	yes	—
184	C₂³×C₄	—	2	2	no	—
185	C₂×SD₁₆	—	2	8	no	—
186	C₂×SD₁₆	—	4	8	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
192	#127	—	2	2	no	—
193	#127	D₄×C₃₁	1	2	yes	—
194	#127	C₂³×C₃₁	1	2	yes	—
195	#124	—	2	8	no	—
196	#124	—	2	8	no	—
197	#123	—	2	8	no	—
198	#123	—	2	8	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×Q₈	D₄×C₃₁	1	4	yes	—
202	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
203	C₄×D₄	—	2	8	no	—
204	C₄×D₄	—	2	8	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	4	no	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
212	#107	—	2	8	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
215	#103	—	2	8	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	C₂³×C₃₁	1	2	yes	—
218	C₃₁	C₂×#1630	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	—	2	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	8	4	no	—
239	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
241	SD₁₆	—	4	16	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
252	C₂²⋊C₄	—	2	16	no	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₃₁	1	2	yes	—
258	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
259	C₂³	—	2	4	no	—
260	C₂³	—	4	8	no	—
261	C₂³	—	4	4	no	—
262	C₂³	C₃₁×#120	1	1	yes	characteristic
263	Q₈	—	2	4	no	—
264	Q₈	—	2	4	no	—
265	Q₈	—	4	4	no	—
266	Q₈	—	4	4	no	—
267	Q₈	—	8	2	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	8	no	—
270	D₄	—	4	4	no	—
271	D₄	—	4	4	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	8	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#120	1	2	yes	—
283	C₂×C₄	C₃₁×#135	1	4	yes	—
284	C₂×C₄	C₃₁×#137	1	4	yes	—
285	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
287	C₈	—	2	4	no	—
288	C₈	—	4	4	no	—
289	C₂²	—	4	8	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	8	8	no	—
292	C₂²	C₃₁×#452	1	4	yes	—
293	C₂²	C₃₁×#453	1	2	yes	—
294	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	4	no	—
298	C₄	—	4	4	no	—
299	C₄	—	8	2	no	—
300	C₄	C₃₁×#452	1	4	yes	—
301	C₄	C₂×C₃₁×#135	1	2	yes	—
302	C₄	C₂×C₃₁×#137	1	2	yes	—
303	C₂	—	8	4	no	—
304	C₂	C₃₁×#1630	1	4	yes	direct product factor
305	C₂	C₂×C₃₁×#452	1	2	yes	—
306	C₂	C₂×C₃₁×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#477	C₂	semidirect
4. C₃₁×#1630	C₂	direct (N × H)
6. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
20. C₃₁×#477	C₂²	semidirect
26. C₃₁×Q₈⋊C₈	C₂²	semidirect
105. C₂×#1630	C₃₁	direct (N × H)
138. C₂×#477	C₂×C₃₁	semidirect
139. #1630	C₂×C₃₁	direct (N × H)
141. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
169. #477	C₂²×C₃₁	semidirect
175. Q₈⋊C₈	C₂²×C₃₁	semidirect
177. C₂×C₃₁	#1630	direct (N × H)
218. C₃₁	C₂×#1630	direct (N × H)
304. C₂	C₃₁×#1630	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×#559	2
C₂×C₃₁×#492	2
C₂×C₃₁×#477	2
C₃₁×#1630	2
C₂×C₃₁×#331	2
C₂×C₃₁×Q₈⋊C₈	2
C₂×#1630	31

Subgroups — C2×C31×#1630 #21475238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1630 #21475238