Subgroups — C₃₁×#3046 #16842979

Order 3968.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#464	C₂	1	2	yes	maximal, complemented
6	C₃₁×#463	C₂	1	2	yes	maximal, complemented
7	C₃₁×#462	C₂	1	2	yes	maximal, complemented
8	C₃₁×#461	C₂	1	2	yes	maximal, complemented
9	C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#417	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#137	—	2	4	no	—
16	C₃₁×#135	—	2	4	no	—
17	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
21	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₃₁×#124	C₂²	1	2	yes	—
27	C₃₁×#123	—	2	2	no	—
28	C₃₁×C₂²⋊Q₈	—	2	1	no	—
29	C₃₁×C₂²⋊Q₈	—	2	1	no	—
30	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₄×D₄×C₃₁	C₂²	1	2	yes	—
33	C₃₁×#117	—	2	1	no	—
34	C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
38	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
39	C₃₁×#103	C₂²	1	2	yes	—
40	C₃₁×#103	C₂²	1	2	yes	—
41	C₃₁×#103	C₂²	1	2	yes	—
42	C₃₁×#102	C₂²	1	2	yes	—
43	#41×C₃₁	—	2	2	no	—
44	#41×C₃₁	—	2	2	no	—
45	#41×C₃₁	—	2	2	no	—
46	#41×C₃₁	D₄	1	4	yes	—
47	C₂×Q₈×C₃₁	—	2	1	no	—
48	C₂×Q₈×C₃₁	—	2	2	no	—
49	C₂×Q₈×C₃₁	—	2	2	no	—
50	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
51	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₃₁	—	2	1	no	—
54	C₂×D₄×C₃₁	—	2	1	no	—
55	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	1	no	—
58	C₂²×C₄×C₃₁	—	2	1	no	—
59	C₂²×C₄×C₃₁	—	2	1	no	—
60	C₂²×C₄×C₃₁	—	2	1	no	—
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	Q₁₆×C₃₁	—	2	2	no	—
64	Q₁₆×C₃₁	—	2	4	no	—
65	SD₁₆×C₃₁	—	2	2	no	—
66	SD₁₆×C₃₁	—	2	2	no	—
67	SD₁₆×C₃₁	—	2	4	no	—
68	D₈×C₃₁	—	2	2	no	—
69	M₁₆×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	C₂³	1	2	yes	—
72	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
81	C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂²⋊C₄×C₃₁	—	2	2	no	—
83	C₂²⋊C₄×C₃₁	—	2	2	no	—
84	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
85	C₄²×C₃₁	—	2	2	no	—
86	C₄²×C₃₁	C₂³	1	2	yes	—
87	C₂³×C₃₁	—	2	1	no	—
88	C₂³×C₃₁	—	2	1	no	—
89	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₃₁	—	2	1	no	—
91	Q₈×C₃₁	—	2	2	no	—
92	Q₈×C₃₁	—	2	1	no	—
93	Q₈×C₃₁	—	2	2	no	—
94	Q₈×C₃₁	—	2	1	no	—
95	Q₈×C₃₁	—	4	2	no	—
96	Q₈×C₃₁	C₂×D₄	1	2	yes	—
97	D₄×C₃₁	—	2	4	no	—
98	D₄×C₃₁	—	2	1	no	—
99	D₄×C₃₁	—	2	1	no	—
100	D₄×C₃₁	—	2	1	no	—
101	D₄×C₃₁	—	4	2	no	—
102	D₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	8	2	no	—
111	C₂×C₄×C₃₁	—	8	1	no	—
112	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₃₁	#41	1	2	yes	—
117	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
118	C₈×C₃₁	—	2	1	no	—
119	C₈×C₃₁	—	2	1	no	—
120	C₈×C₃₁	—	2	2	no	—
121	#3046	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	2	1	no	—
124	C₂²×C₃₁	—	2	2	no	—
125	C₂²×C₃₁	—	4	1	no	—
126	C₂²×C₃₁	—	4	1	no	—
127	C₂²×C₃₁	—	4	1	no	—
128	C₂²×C₃₁	—	4	1	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	#121	1	2	yes	—
131	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₄×C₃₁	—	2	2	no	—
133	C₄×C₃₁	—	2	2	no	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	4	1	no	—
136	C₄×C₃₁	—	4	2	no	—
137	C₄×C₃₁	—	4	2	no	—
138	C₄×C₃₁	#121	1	2	yes	—
139	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
141	C₂×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂×#135	C₂×C₃₁	1	1	yes	characteristic
143	#536	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#520	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#464	C₂×C₃₁	1	2	yes	complemented
146	#463	C₂×C₃₁	1	2	yes	complemented
147	#462	C₂×C₃₁	1	2	yes	complemented
148	#461	C₂×C₃₁	1	2	yes	complemented
149	#424	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#417	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂×C₃₁	—	2	2	no	—
153	C₂×C₃₁	—	4	2	no	—
154	C₂×C₃₁	—	8	1	no	—
155	C₂×C₃₁	C₂×#121	1	1	yes	characteristic
156	C₂×C₃₁	#577	1	1	yes	characteristic
157	C₂×C₃₁	#578	1	1	yes	characteristic
158	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
160	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
161	#137	—	2	4	no	—
162	#135	—	2	4	no	—
163	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
165	C₂×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
168	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
170	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
171	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
172	#124	C₂²×C₃₁	1	2	yes	—
173	#123	—	2	2	no	—
174	C₂²⋊Q₈	—	2	1	no	—
175	C₂²⋊Q₈	—	2	1	no	—
176	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	C₂²×C₃₁	1	2	yes	—
179	#117	—	2	1	no	—
180	#117	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
183	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
184	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
185	#103	C₂²×C₃₁	1	2	yes	—
186	#103	C₂²×C₃₁	1	2	yes	—
187	#103	C₂²×C₃₁	1	2	yes	—
188	#102	C₂²×C₃₁	1	2	yes	—
189	C₃₁	#3046	1	1	yes	characteristic, direct product factor, Sylow
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	D₄×C₃₁	1	4	yes	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
198	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
199	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
203	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
210	Q₁₆	—	2	2	no	—
211	Q₁₆	—	2	4	no	—
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	2	4	no	—
215	D₈	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	C₂³×C₃₁	1	2	yes	—
219	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
220	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
226	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
227	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	C₂³×C₃₁	1	2	yes	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	—	2	2	no	—
239	Q₈	—	2	1	no	—
240	Q₈	—	2	2	no	—
241	Q₈	—	2	1	no	—
242	Q₈	—	4	2	no	—
243	Q₈	C₂×D₄×C₃₁	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	1	no	—
248	D₄	—	4	2	no	—
249	D₄	C₂×D₄×C₃₁	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	1	no	—
259	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
261	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
263	C₂×C₄	#41×C₃₁	1	2	yes	—
264	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	C₃₁×#121	1	2	yes	—
277	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	C₃₁×#121	1	2	yes	—
285	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
286	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	8	1	no	—
290	C₂	C₂×C₃₁×#121	1	1	yes	characteristic
291	C₂	C₃₁×#577	1	1	yes	characteristic
292	C₂	C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#137	C₂	semidirect
3. C₃₁×#536	C₂	semidirect
4. C₃₁×#520	C₂	semidirect
5. C₃₁×#464	C₂	semidirect
6. C₃₁×#463	C₂	semidirect
7. C₃₁×#462	C₂	semidirect
8. C₃₁×#461	C₂	semidirect
9. C₃₁×#424	C₂	semidirect
10. C₃₁×#417	C₂	semidirect
11. C₂×C₃₁×#103	C₂	semidirect
121. #3046	C₃₁	direct (N × H)
141. C₂×#137	C₂×C₃₁	semidirect
143. #536	C₂×C₃₁	semidirect
144. #520	C₂×C₃₁	semidirect
145. #464	C₂×C₃₁	semidirect
146. #463	C₂×C₃₁	semidirect
147. #462	C₂×C₃₁	semidirect
148. #461	C₂×C₃₁	semidirect
149. #424	C₂×C₃₁	semidirect
150. #417	C₂×C₃₁	semidirect
151. C₂×#103	C₂×C₃₁	semidirect
189. C₃₁	#3046	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×#137	2
C₂×C₃₁×#135	2
C₃₁×#536	2
C₃₁×#520	2
C₃₁×#464	2
C₃₁×#463	2
C₃₁×#462	2
C₃₁×#461	2
C₃₁×#424	2
C₃₁×#417	2
C₂×C₃₁×#103	2
#3046	31

Subgroups — C31×#3046 #16842979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3046 #16842979