Subgroups — C₃₁×#33167 #21489926

Order 7936.

1162 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#464	C₂	1	3	yes	maximal, complemented
2	C₂×C₃₁×#460	C₂	1	3	yes	maximal, complemented
3	C₃₁×#2657	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2276	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1983	C₂	1	3	yes	maximal, complemented
6	C₃₁×#1982	C₂	1	3	yes	maximal, complemented
7	C₃₁×#1843	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₃₁	C₂²	1	3	yes	—
10	C₂×C₃₁×#127	C₂²	1	3	yes	—
11	C₂×C₃₁×#124	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
13	C₃₁×#464	—	2	12	no	—
14	C₃₁×#460	—	2	12	no	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₃₁×#103	C₂²	1	3	yes	—
18	C₂×C₃₁×#102	C₂²	1	3	yes	—
19	C₂×C₃₁×#102	C₂²	1	6	yes	—
20	C₃₁×#392	—	2	1	no	—
21	C₃₁×#392	—	2	3	no	—
22	C₄×C₄⋊C₄×C₃₁	—	2	3	no	—
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
24	C₃₁×#376	—	2	1	no	—
25	C₃₁×#376	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×D₄×C₃₁	—	2	3	no	—
29	C₂²×D₄×C₃₁	—	2	3	no	—
30	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
31	C₂×Q₁₆×C₃₁	—	2	6	no	—
32	C₂×SD₁₆×C₃₁	—	2	12	no	—
33	C₂×SD₁₆×C₃₁	—	2	6	no	—
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
36	C₃₁×#127	—	2	6	no	—
37	C₃₁×#127	—	2	6	no	—
38	C₃₁×#124	—	2	6	no	—
39	C₃₁×#124	—	2	6	no	—
40	C₄×Q₈×C₃₁	—	2	6	no	—
41	C₄×Q₈×C₃₁	D₄	1	12	yes	complemented
42	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
46	C₂×C₄²×C₃₁	—	2	3	no	—
47	C₂×C₄²×C₃₁	—	2	3	no	—
48	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
49	C₃₁×C₄⋊C₈	—	2	6	no	—
50	C₃₁×C₄⋊C₈	—	2	6	no	—
51	C₃₁×#103	—	2	6	no	—
52	C₃₁×#102	—	2	12	no	—
53	C₃₁×#102	—	2	6	no	—
54	C₃₁×#95	—	2	3	no	—
55	C₃₁×#95	D₄	1	1	yes	characteristic
56	C₃₁×#95	D₄	1	3	yes	—
57	C₂⁴×C₃₁	—	2	3	no	—
58	C₂×Q₈×C₃₁	—	2	3	no	—
59	C₂×Q₈×C₃₁	—	2	3	no	—
60	C₂×Q₈×C₃₁	—	2	3	no	—
61	C₂×Q₈×C₃₁	C₂×D₄	1	6	yes	—
62	C₂×D₄×C₃₁	—	2	3	no	—
63	C₂×D₄×C₃₁	—	2	6	no	—
64	C₂×D₄×C₃₁	—	2	3	no	—
65	C₂×D₄×C₃₁	—	2	6	no	—
66	C₂×D₄×C₃₁	—	4	6	no	—
67	C₂×D₄×C₃₁	—	4	6	no	—
68	C₂×D₄×C₃₁	—	8	3	no	—
69	C₂×D₄×C₃₁	—	8	6	no	—
70	C₂×D₄×C₃₁	—	8	3	no	—
71	C₂²×C₄×C₃₁	—	2	1	no	—
72	C₂²×C₄×C₃₁	—	2	3	no	—
73	C₂²×C₄×C₃₁	—	2	1	no	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	Q₁₆×C₃₁	—	4	4	no	—
78	SD₁₆×C₃₁	—	4	12	no	—
79	C₂×C₈×C₃₁	—	2	6	no	—
80	C₂×C₈×C₃₁	—	2	3	no	—
81	C₂×C₈×C₃₁	—	2	3	no	—
82	C₄⋊C₄×C₃₁	—	2	12	no	—
83	C₄⋊C₄×C₃₁	—	2	3	no	—
84	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
85	C₂²⋊C₄×C₃₁	—	4	6	no	—
86	C₂²⋊C₄×C₃₁	—	4	6	no	—
87	C₄²×C₃₁	—	2	12	no	—
88	C₄²×C₃₁	—	8	3	no	—
89	C₄²×C₃₁	C₂×D₄	1	6	yes	—
90	C₄²×C₃₁	#41	1	6	yes	—
91	#33167	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	4	3	no	—
93	C₂³×C₃₁	—	4	6	no	—
94	C₂³×C₃₁	—	4	3	no	—
95	C₂³×C₃₁	—	4	3	no	—
96	C₂³×C₃₁	—	4	6	no	—
97	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₃₁	—	2	6	no	—
99	Q₈×C₃₁	—	4	1	no	—
100	Q₈×C₃₁	—	4	3	no	—
101	Q₈×C₃₁	#127	1	4	yes	—
102	D₄×C₃₁	—	4	3	no	—
103	D₄×C₃₁	—	4	3	no	—
104	D₄×C₃₁	—	4	6	no	—
105	D₄×C₃₁	—	8	12	no	—
106	D₄×C₃₁	—	8	12	no	—
107	C₂×C₄×C₃₁	—	2	6	no	—
108	C₂×C₄×C₃₁	—	2	3	no	—
109	C₂×C₄×C₃₁	—	2	6	no	—
110	C₂×C₄×C₃₁	—	2	3	no	—
111	C₂×C₄×C₃₁	—	2	6	no	—
112	C₂×C₄×C₃₁	—	4	3	no	—
113	C₂×C₄×C₃₁	—	4	3	no	—
114	C₂×C₄×C₃₁	—	4	3	no	—
115	C₂×C₄×C₃₁	#121	1	6	yes	—
116	C₂×C₄×C₃₁	#121	1	6	yes	—
117	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
118	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
119	C₈×C₃₁	—	4	3	no	—
120	C₈×C₃₁	—	4	1	no	—
121	C₂×#464	C₂×C₃₁	1	3	yes	complemented
122	C₂×#460	C₂×C₃₁	1	3	yes	complemented
123	#2657	C₂×C₃₁	1	1	yes	characteristic
124	#2276	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#1983	C₂×C₃₁	1	3	yes	complemented
126	#1982	C₂×C₃₁	1	3	yes	complemented
127	#1843	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂²×C₃₁	—	8	3	no	—
129	C₂²×C₃₁	—	8	6	no	—
130	C₂²×C₃₁	—	8	3	no	—
131	C₂²×C₃₁	—	8	6	no	—
132	C₂²×C₃₁	—	8	3	no	—
133	C₂²×C₃₁	C₂×#121	1	3	yes	—
134	C₂²×C₃₁	#576	1	1	yes	characteristic
135	C₂²×C₃₁	#577	1	3	yes	—
136	C₄×C₃₁	—	2	4	no	—
137	C₄×C₃₁	—	2	6	no	—
138	C₄×C₃₁	—	2	6	no	—
139	C₄×C₃₁	—	4	6	no	—
140	C₄×C₃₁	—	8	1	no	—
141	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
142	C₄×C₃₁	#532	1	3	yes	—
143	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
144	C₂²×SD₁₆	C₂²×C₃₁	1	3	yes	—
145	C₂×#127	C₂²×C₃₁	1	3	yes	—
146	C₂×#124	C₂²×C₃₁	1	3	yes	—
147	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
148	#464	—	2	12	no	—
149	#460	—	2	12	no	—
150	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
151	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
152	C₂×#103	C₂²×C₃₁	1	3	yes	—
153	C₂×#102	C₂²×C₃₁	1	3	yes	—
154	C₂×#102	C₂²×C₃₁	1	6	yes	—
155	#392	—	2	1	no	—
156	#392	—	2	3	no	—
157	C₄×C₄⋊C₄	—	2	3	no	—
158	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
159	#376	—	2	1	no	—
160	#376	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×C₃₁	—	16	3	no	—
162	C₂×C₃₁	#2856	1	1	yes	characteristic
163	C₂×C₃₁	#3043	1	3	yes	—
164	C₂×C₃₁	#3045	1	3	yes	—
165	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
166	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
167	C₂²×D₄	—	2	3	no	—
168	C₂²×D₄	—	2	3	no	—
169	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
170	C₂×Q₁₆	—	2	6	no	—
171	C₂×SD₁₆	—	2	12	no	—
172	C₂×SD₁₆	—	2	6	no	—
173	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
174	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
175	#127	—	2	6	no	—
176	#127	—	2	6	no	—
177	#124	—	2	6	no	—
178	#124	—	2	6	no	—
179	C₄×Q₈	—	2	6	no	—
180	C₄×Q₈	D₄×C₃₁	1	12	yes	complemented
181	C₂×C₄⋊C₄	—	2	3	no	—
182	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
183	C₂×C₂²⋊C₄	—	2	3	no	—
184	C₂×C₂²⋊C₄	—	2	3	no	—
185	C₂×C₄²	—	2	3	no	—
186	C₂×C₄²	—	2	3	no	—
187	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
188	C₄⋊C₈	—	2	6	no	—
189	C₄⋊C₈	—	2	6	no	—
190	#103	—	2	6	no	—
191	#102	—	2	12	no	—
192	#102	—	2	6	no	—
193	#95	—	2	3	no	—
194	#95	D₄×C₃₁	1	1	yes	characteristic
195	#95	D₄×C₃₁	1	3	yes	—
196	C₃₁	#33167	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	—	2	3	no	—
198	C₂×Q₈	—	2	3	no	—
199	C₂×Q₈	—	2	3	no	—
200	C₂×Q₈	—	2	3	no	—
201	C₂×Q₈	C₂×D₄×C₃₁	1	6	yes	—
202	C₂×D₄	—	2	3	no	—
203	C₂×D₄	—	2	6	no	—
204	C₂×D₄	—	2	3	no	—
205	C₂×D₄	—	2	6	no	—
206	C₂×D₄	—	4	6	no	—
207	C₂×D₄	—	4	6	no	—
208	C₂×D₄	—	8	3	no	—
209	C₂×D₄	—	8	6	no	—
210	C₂×D₄	—	8	3	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	3	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
215	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
216	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
217	Q₁₆	—	4	4	no	—
218	SD₁₆	—	4	12	no	—
219	C₂×C₈	—	2	6	no	—
220	C₂×C₈	—	2	3	no	—
221	C₂×C₈	—	2	3	no	—
222	C₄⋊C₄	—	2	12	no	—
223	C₄⋊C₄	—	2	3	no	—
224	C₄⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
225	C₂²⋊C₄	—	4	6	no	—
226	C₂²⋊C₄	—	4	6	no	—
227	C₄²	—	2	12	no	—
228	C₄²	—	8	3	no	—
229	C₄²	C₂×D₄×C₃₁	1	6	yes	—
230	C₄²	#41×C₃₁	1	6	yes	—
231	C₂³	—	4	3	no	—
232	C₂³	—	4	6	no	—
233	C₂³	—	4	3	no	—
234	C₂³	—	4	3	no	—
235	C₂³	—	4	6	no	—
236	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
237	Q₈	—	2	6	no	—
238	Q₈	—	4	1	no	—
239	Q₈	—	4	3	no	—
240	Q₈	C₃₁×#127	1	4	yes	—
241	D₄	—	4	3	no	—
242	D₄	—	4	3	no	—
243	D₄	—	4	6	no	—
244	D₄	—	8	12	no	—
245	D₄	—	8	12	no	—
246	C₂×C₄	—	2	6	no	—
247	C₂×C₄	—	2	3	no	—
248	C₂×C₄	—	2	6	no	—
249	C₂×C₄	—	2	3	no	—
250	C₂×C₄	—	2	6	no	—
251	C₂×C₄	—	4	3	no	—
252	C₂×C₄	—	4	3	no	—
253	C₂×C₄	—	4	3	no	—
254	C₂×C₄	C₃₁×#121	1	6	yes	—
255	C₂×C₄	C₃₁×#121	1	6	yes	—
256	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
257	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
258	C₈	—	4	3	no	—
259	C₈	—	4	1	no	—
260	C₂²	—	8	3	no	—
261	C₂²	—	8	6	no	—
262	C₂²	—	8	3	no	—
263	C₂²	—	8	6	no	—
264	C₂²	—	8	3	no	—
265	C₂²	C₂×C₃₁×#121	1	3	yes	—
266	C₂²	C₃₁×#576	1	1	yes	characteristic
267	C₂²	C₃₁×#577	1	3	yes	—
268	C₄	—	2	4	no	—
269	C₄	—	2	6	no	—
270	C₄	—	2	6	no	—
271	C₄	—	4	6	no	—
272	C₄	—	8	1	no	—
273	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
274	C₄	C₃₁×#532	1	3	yes	—
275	C₂	—	16	3	no	—
276	C₂	C₃₁×#2856	1	1	yes	characteristic
277	C₂	C₃₁×#3043	1	3	yes	—
278	C₂	C₃₁×#3045	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2657	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#464	C₂	semidirect
2. C₂×C₃₁×#460	C₂	semidirect
4. C₃₁×#2276	C₂	semidirect
5. C₃₁×#1983	C₂	semidirect
6. C₃₁×#1982	C₂	semidirect
7. C₃₁×#1843	C₂	semidirect
41. C₄×Q₈×C₃₁	D₄	semidirect
91. #33167	C₃₁	direct (N × H)
121. C₂×#464	C₂×C₃₁	semidirect
122. C₂×#460	C₂×C₃₁	semidirect
124. #2276	C₂×C₃₁	semidirect
125. #1983	C₂×C₃₁	semidirect
126. #1982	C₂×C₃₁	semidirect
127. #1843	C₂×C₃₁	semidirect
180. C₄×Q₈	D₄×C₃₁	semidirect
196. C₃₁	#33167	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×#464	2
C₂×C₃₁×#460	2
C₃₁×#2657	2
C₃₁×#2276	2
C₃₁×#1983	2
C₃₁×#1982	2
C₃₁×#1843	2
#33167	31

Subgroups — C31×#33167 #21489926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33167 #21489926