Subgroups — C₂×C₃₁×#1695 #21475895

Order 7936.

694 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#476	C₂	1	2	yes	maximal
3	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1695	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#333	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₃₁	—	2	1	no	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#500	—	2	2	no	—
11	C₃₁×#500	C₂²	1	4	yes	complemented
12	C₃₁×#476	—	2	4	no	—
13	C₃₁×#476	C₂²	1	8	yes	—
14	C₃₁×#436	—	2	2	no	—
15	C₃₁×#436	C₂²	1	4	yes	—
16	C₂×C₃₁×#107	—	2	1	no	—
17	C₂×C₃₁×#107	—	2	1	no	—
18	C₂×C₃₁×#106	—	2	2	no	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	—	2	2	no	—
22	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#334	C₂²	1	4	yes	complemented
25	C₃₁×#333	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	—	2	1	no	—
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	—	4	4	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
35	C₃₁×C₄⋊Q₈	—	2	2	no	—
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
39	C₄×D₄×C₃₁	—	2	1	no	—
40	C₄×D₄×C₃₁	—	2	1	no	—
41	C₄×D₄×C₃₁	D₄	1	2	yes	—
42	C₄×D₄×C₃₁	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#107	—	2	4	no	—
49	C₃₁×#107	—	2	4	no	—
50	C₃₁×#106	—	2	8	no	—
51	C₃₁×C₄⋊C₈	—	2	1	no	—
52	C₃₁×C₄⋊C₈	—	2	2	no	—
53	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
54	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
55	C₃₁×#102	—	2	8	no	—
56	C₃₁×C₂²⋊C₈	—	2	4	no	—
57	C₄×C₈×C₃₁	D₄	1	2	yes	—
58	C₄×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂⁴×C₃₁	—	2	1	no	—
60	C₂×Q₈×C₃₁	—	2	4	no	—
61	C₂×Q₈×C₃₁	—	4	2	no	—
62	C₂×Q₈×C₃₁	—	8	4	no	—
63	C₂×D₄×C₃₁	—	2	1	no	—
64	C₂×D₄×C₃₁	—	2	4	no	—
65	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	4	2	no	—
70	C₂²×C₄×C₃₁	—	4	2	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
75	M₁₆×C₃₁	—	4	8	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	—	4	4	no	—
81	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
82	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₃₁	—	2	1	no	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	4	2	no	—
88	C₄⋊C₄×C₃₁	—	4	2	no	—
89	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
90	C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₃₁	#41	1	1	yes	characteristic
95	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₂×#1695	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	2	2	no	—
98	C₂³×C₃₁	—	4	2	no	—
99	C₂³×C₃₁	—	4	2	no	—
100	C₂³×C₃₁	—	4	2	no	—
101	C₂³×C₃₁	#121	1	1	yes	characteristic
102	Q₈×C₃₁	—	8	4	no	—
103	Q₈×C₃₁	—	8	4	no	—
104	D₄×C₃₁	—	2	4	no	—
105	D₄×C₃₁	—	4	2	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	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	8	4	no	—
114	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₃₁	#137	1	2	yes	—
119	C₂×C₄×C₃₁	#137	1	4	yes	—
120	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
123	C₈×C₃₁	—	2	2	no	—
124	C₈×C₃₁	—	4	2	no	—
125	C₈×C₃₁	—	4	4	no	—
126	C₈×C₃₁	C₂×SD₁₆	1	4	yes	—
127	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂×#476	C₂×C₃₁	1	2	yes	—
129	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
130	#1695	C₂×C₃₁	1	8	yes	direct product factor
131	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
132	C₂×#333	C₂×C₃₁	1	2	yes	complemented
133	C₂²×C₃₁	—	4	2	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	—	8	4	no	—
137	C₂²×C₃₁	#461	1	2	yes	—
138	C₂²×C₃₁	#470	1	2	yes	—
139	C₂²×C₃₁	#472	1	2	yes	—
140	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	4	2	no	—
144	C₄×C₃₁	—	8	4	no	—
145	C₄×C₃₁	#461	1	2	yes	—
146	C₄×C₃₁	#470	1	2	yes	—
147	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
148	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
149	C₄×C₃₁	C₂×#137	1	2	yes	—
150	C₂²×M₁₆	—	2	1	no	—
151	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
152	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
153	#500	—	2	2	no	—
154	#500	C₂²×C₃₁	1	4	yes	complemented
155	#476	—	2	4	no	—
156	#476	C₂²×C₃₁	1	8	yes	—
157	#436	—	2	2	no	—
158	#436	C₂²×C₃₁	1	4	yes	—
159	C₂×#107	—	2	1	no	—
160	C₂×#107	—	2	1	no	—
161	C₂×#106	—	2	2	no	—
162	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
164	C₂×#102	—	2	2	no	—
165	C₂×C₂²⋊C₈	—	2	1	no	—
166	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
167	#334	C₂²×C₃₁	1	4	yes	complemented
168	#333	C₂²×C₃₁	1	8	yes	complemented
169	C₂×C₃₁	—	8	2	no	—
170	C₂×C₃₁	#1695	1	4	yes	direct product factor
171	C₂×C₃₁	C₂×#461	1	1	yes	characteristic
172	C₂×C₃₁	C₂×#470	1	1	yes	characteristic
173	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
174	C₂²×Q₈	—	2	2	no	—
175	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
176	C₂³×C₄	—	2	1	no	—
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	—	4	4	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
182	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	D₄×C₃₁	1	2	yes	—
190	C₄×D₄	C₂³×C₃₁	1	2	yes	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	—	2	1	no	—
195	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
196	#107	—	2	4	no	—
197	#107	—	2	4	no	—
198	#106	—	2	8	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
202	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
203	#102	—	2	8	no	—
204	C₂²⋊C₈	—	2	4	no	—
205	C₄×C₈	D₄×C₃₁	1	2	yes	—
206	C₄×C₈	C₂³×C₃₁	1	2	yes	—
207	C₃₁	C₂×#1695	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	—	2	1	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	8	4	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
224	M₁₆	—	4	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
231	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₄²	#41×C₃₁	1	1	yes	characteristic
244	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
245	C₂³	—	2	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	C₃₁×#121	1	1	yes	characteristic
250	Q₈	—	8	4	no	—
251	Q₈	—	8	4	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
263	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
264	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
265	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
266	C₂×C₄	C₃₁×#137	1	2	yes	—
267	C₂×C₄	C₃₁×#137	1	4	yes	—
268	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	4	no	—
274	C₈	C₂×SD₁₆×C₃₁	1	4	yes	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	8	4	no	—
279	C₂²	C₃₁×#461	1	2	yes	—
280	C₂²	C₃₁×#470	1	2	yes	—
281	C₂²	C₃₁×#472	1	2	yes	—
282	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	—	8	4	no	—
287	C₄	C₃₁×#461	1	2	yes	—
288	C₄	C₃₁×#470	1	2	yes	—
289	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
290	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
291	C₄	C₂×C₃₁×#137	1	2	yes	—
292	C₂	—	8	2	no	—
293	C₂	C₃₁×#1695	1	4	yes	direct product factor
294	C₂	C₂×C₃₁×#461	1	1	yes	characteristic
295	C₂	C₂×C₃₁×#470	1	1	yes	characteristic
296	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
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
1. C₂×C₃₁×#500	C₂	semidirect
4. C₃₁×#1695	C₂	direct (N × H)
5. C₂×C₃₁×#334	C₂	semidirect
6. C₂×C₃₁×#333	C₂	semidirect
11. C₃₁×#500	C₂²	semidirect
24. C₃₁×#334	C₂²	semidirect
25. C₃₁×#333	C₂²	semidirect
96. C₂×#1695	C₃₁	direct (N × H)
127. C₂×#500	C₂×C₃₁	semidirect
130. #1695	C₂×C₃₁	direct (N × H)
131. C₂×#334	C₂×C₃₁	semidirect
132. C₂×#333	C₂×C₃₁	semidirect
154. #500	C₂²×C₃₁	semidirect
167. #334	C₂²×C₃₁	semidirect
168. #333	C₂²×C₃₁	semidirect
170. C₂×C₃₁	#1695	direct (N × H)
207. C₃₁	C₂×#1695	direct (N × H)
293. C₂	C₃₁×#1695	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×#500	2
C₂×C₃₁×#476	2
C₂×C₃₁×#436	2
C₃₁×#1695	2
C₂×C₃₁×#334	2
C₂×C₃₁×#333	2
C₂×#1695	31

Subgroups — C2×C31×#1695 #21475895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1695 #21475895