Subgroups — C₃₁×#3330 #16843263

Order 3968.

420 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#484	C₂	1	2	yes	maximal, complemented
5	C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#468	C₂	1	2	yes	maximal, complemented
7	C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#449	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#417	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₃₁	—	2	1	no	—
13	C₂×#41×C₃₁	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
17	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₃₁×#123	—	2	2	no	—
20	C₃₁×#123	—	2	2	no	—
21	C₃₁×C₂²⋊Q₈	—	2	1	no	—
22	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₃₁×#121	—	2	1	no	—
24	C₃₁×#121	C₂²	1	2	yes	—
25	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#107	C₂²	1	1	yes	characteristic
31	C₃₁×#107	C₂²	1	2	yes	—
32	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃₁×#103	C₂²	1	2	yes	—
34	C₃₁×#103	C₂²	1	2	yes	—
35	C₃₁×#103	C₂²	1	2	yes	—
36	C₃₁×#102	C₂²	1	1	yes	characteristic
37	C₃₁×#102	C₂²	1	2	yes	—
38	C₃₁×#102	C₂²	1	1	yes	characteristic
39	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	#41×C₃₁	—	2	4	no	—
42	#41×C₃₁	—	4	2	no	—
43	#41×C₃₁	—	4	4	no	—
44	C₂×Q₈×C₃₁	—	2	1	no	—
45	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂×D₄×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	—	2	2	no	—
48	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	1	no	—
55	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
56	SD₁₆×C₃₁	—	4	2	no	—
57	D₈×C₃₁	—	2	2	no	—
58	M₁₆×C₃₁	—	4	2	no	—
59	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂³	1	2	yes	—
62	C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
76	C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂³×C₃₁	—	2	1	no	—
78	C₂³×C₃₁	—	2	1	no	—
79	C₂³×C₃₁	C₂×D₄	1	2	yes	—
80	Q₈×C₃₁	—	2	2	no	—
81	Q₈×C₃₁	—	2	2	no	—
82	Q₈×C₃₁	—	4	2	no	—
83	D₄×C₃₁	—	2	2	no	—
84	D₄×C₃₁	—	2	1	no	—
85	D₄×C₃₁	—	2	1	no	—
86	D₄×C₃₁	—	2	1	no	—
87	D₄×C₃₁	—	4	4	no	—
88	D₄×C₃₁	—	4	4	no	—
89	D₄×C₃₁	#41	1	2	yes	complemented
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	4	4	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
104	C₈×C₃₁	—	2	1	no	—
105	C₈×C₃₁	—	2	1	no	—
106	C₈×C₃₁	—	2	2	no	—
107	#3330	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	2	2	no	—
111	C₂²×C₃₁	—	2	2	no	—
112	C₂²×C₃₁	—	4	1	no	—
113	C₂²×C₃₁	—	4	1	no	—
114	C₂²×C₃₁	—	4	1	no	—
115	C₂²×C₃₁	—	4	1	no	—
116	C₂²×C₃₁	—	4	1	no	—
117	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₃₁	—	2	2	no	—
119	C₄×C₃₁	—	2	1	no	—
120	C₄×C₃₁	—	4	2	no	—
121	C₄×C₃₁	—	4	1	no	—
122	C₄×C₃₁	—	4	1	no	—
123	C₄×C₃₁	—	4	2	no	—
124	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
125	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
126	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
127	#547	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#547	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#487	C₂×C₃₁	1	1	yes	characteristic, complemented
130	#484	C₂×C₃₁	1	2	yes	complemented
131	#474	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#468	C₂×C₃₁	1	2	yes	complemented
133	#461	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#449	C₂×C₃₁	1	2	yes	complemented
135	C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#436	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#417	C₂×C₃₁	1	2	yes	complemented
138	C₂×C₃₁	—	4	2	no	—
139	C₂×C₃₁	—	4	2	no	—
140	C₂×C₃₁	—	8	1	no	—
141	C₂×C₃₁	#546	1	1	yes	characteristic
142	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₃₁	#578	1	1	yes	characteristic
144	C₂×#41	—	2	1	no	—
145	C₂×#41	C₂²×C₃₁	1	2	yes	—
146	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
147	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×M₁₆	C₂²×C₃₁	1	2	yes	—
149	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
151	#123	—	2	2	no	—
152	#123	—	2	2	no	—
153	C₂²⋊Q₈	—	2	1	no	—
154	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
155	#121	—	2	1	no	—
156	#121	C₂²×C₃₁	1	2	yes	—
157	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
158	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
159	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
162	#107	C₂²×C₃₁	1	1	yes	characteristic
163	#107	C₂²×C₃₁	1	2	yes	—
164	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
165	#103	C₂²×C₃₁	1	2	yes	—
166	#103	C₂²×C₃₁	1	2	yes	—
167	#103	C₂²×C₃₁	1	2	yes	—
168	#102	C₂²×C₃₁	1	1	yes	characteristic
169	#102	C₂²×C₃₁	1	2	yes	—
170	#102	C₂²×C₃₁	1	1	yes	characteristic
171	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
172	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₃₁	#3330	1	1	yes	characteristic, direct product factor, Sylow
174	#41	—	2	4	no	—
175	#41	—	4	2	no	—
176	#41	—	4	4	no	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
183	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
189	SD₁₆	—	4	2	no	—
190	D₈	—	2	2	no	—
191	M₁₆	—	4	2	no	—
192	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
194	C₂×C₈	C₂³×C₃₁	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
203	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
209	C₄²	C₂³×C₃₁	1	1	yes	characteristic
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₃₁	1	2	yes	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	4	2	no	—
216	D₄	—	2	2	no	—
217	D₄	—	2	1	no	—
218	D₄	—	2	1	no	—
219	D₄	—	2	1	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	4	no	—
222	D₄	#41×C₃₁	1	2	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	1	no	—
256	C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
257	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
258	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂	—	4	2	no	—
260	C₂	—	4	2	no	—
261	C₂	—	8	1	no	—
262	C₂	C₃₁×#546	1	1	yes	characteristic
263	C₂	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
264	C₂	C₃₁×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#547	C₂	semidirect
2. C₃₁×#547	C₂	semidirect
3. C₃₁×#487	C₂	semidirect
4. C₃₁×#484	C₂	semidirect
5. C₃₁×#474	C₂	semidirect
6. C₃₁×#468	C₂	semidirect
7. C₃₁×#461	C₂	semidirect
8. C₃₁×#449	C₂	semidirect
9. C₄×D₈×C₃₁	C₂	semidirect
10. C₃₁×#436	C₂	semidirect
11. C₃₁×#417	C₂	semidirect
89. D₄×C₃₁	#41	semidirect
107. #3330	C₃₁	direct (N × H)
127. #547	C₂×C₃₁	semidirect
128. #547	C₂×C₃₁	semidirect
129. #487	C₂×C₃₁	semidirect
130. #484	C₂×C₃₁	semidirect
131. #474	C₂×C₃₁	semidirect
132. #468	C₂×C₃₁	semidirect
133. #461	C₂×C₃₁	semidirect
134. #449	C₂×C₃₁	semidirect
135. C₄×D₈	C₂×C₃₁	semidirect
136. #436	C₂×C₃₁	semidirect
137. #417	C₂×C₃₁	semidirect
173. C₃₁	#3330	direct (N × H)
222. D₄	#41×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#547	2
C₃₁×#487	2
C₃₁×#484	2
C₃₁×#474	2
C₃₁×#468	2
C₃₁×#461	2
C₃₁×#449	2
C₄×D₈×C₃₁	2
C₃₁×#436	2
C₃₁×#417	2
#3330	31

Subgroups — C31×#3330 #16843263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3330 #16843263