Subgroups — C₃₁×#33947 #21490706

Order 7936.

476 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3380	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2253	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2228	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2226	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
8	C₄×C₃₁×D₁₆	C₂	1	2	yes	maximal
9	C₃₁×#554	—	2	2	no	—
10	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#499	C₂²	1	4	yes	—
13	C₃₁×#478	—	2	4	no	—
14	C₃₁×#476	—	2	2	no	—
15	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×D₈×C₃₁	C₂²	1	2	yes	—
17	C₄×D₈×C₃₁	C₂²	1	4	yes	—
18	C₃₁×#367	C₂²	1	4	yes	—
19	C₃₁×#366	C₂²	1	2	yes	—
20	C₃₁×#366	C₂²	1	1	yes	characteristic
21	C₃₁×#366	C₂²	1	2	yes	—
22	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₃₁×#357	C₂²	1	4	yes	—
25	C₃₁×#357	C₂²	1	2	yes	—
26	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
28	C₂×D₈×C₃₁	C₂³	1	2	yes	—
29	C₃₁×C₄⋊Q₈	—	2	2	no	—
30	C₃₁×#125	—	2	4	no	—
31	C₃₁×C₂²⋊Q₈	—	4	2	no	—
32	C₃₁×C₂²⋊Q₈	—	4	4	no	—
33	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
34	C₄×D₄×C₃₁	—	2	2	no	—
35	C₄×D₄×C₃₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
37	C₃₁×D₁₆	Q₈	1	4	yes	complemented
38	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#107	—	2	2	no	—
42	C₃₁×#107	C₂³	1	2	yes	—
43	C₃₁×#107	C₂³	1	4	yes	—
44	C₃₁×#106	—	2	4	no	—
45	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
46	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
47	C₃₁×#102	—	2	4	no	—
48	C₃₁×#102	—	2	2	no	—
49	C₄×C₈×C₃₁	C₂³	1	2	yes	—
50	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₃₁	—	4	2	no	—
52	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
53	C₂×D₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	—	4	4	no	—
56	D₈×C₃₁	—	2	2	no	—
57	D₈×C₃₁	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₄⋊C₄×C₃₁	—	4	4	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₄⋊C₄×C₃₁	—	4	4	no	—
66	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
67	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
68	C₂²⋊C₄×C₃₁	—	4	2	no	—
69	C₂²⋊C₄×C₃₁	—	4	4	no	—
70	C₄²×C₃₁	C₂×D₄	1	2	yes	—
71	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₃₁	—	2	1	no	—
73	C₁₆×C₃₁	—	2	2	no	—
74	C₁₆×C₃₁	C₂×Q₈	1	2	yes	—
75	#33947	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	—	4	2	no	—
77	Q₈×C₃₁	—	2	2	no	—
78	Q₈×C₃₁	—	16	2	no	—
79	D₄×C₃₁	—	2	4	no	—
80	D₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	—	8	4	no	—
84	C₂×C₄×C₃₁	—	16	4	no	—
85	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
90	C₈×C₃₁	—	2	2	no	—
91	C₈×C₃₁	—	2	1	no	—
92	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
94	#3380	C₂×C₃₁	1	2	yes	complemented
95	#2253	C₂×C₃₁	1	2	yes	complemented
96	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
97	#2228	C₂×C₃₁	1	4	yes	complemented
98	#2226	C₂×C₃₁	1	2	yes	complemented
99	Q₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
100	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic
101	C₄×D₁₆	C₂×C₃₁	1	2	yes	—
102	C₂²×C₃₁	—	4	4	no	—
103	C₂²×C₃₁	—	8	2	no	—
104	C₂²×C₃₁	—	8	2	no	—
105	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₃₁	—	2	2	no	—
107	C₄×C₃₁	—	2	2	no	—
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	8	2	no	—
110	C₄×C₃₁	—	8	4	no	—
111	C₄×C₃₁	#508	1	2	yes	—
112	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
113	C₄×C₃₁	#578	1	1	yes	characteristic
114	#554	—	2	2	no	—
115	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
116	#500	C₂²×C₃₁	1	2	yes	—
117	#499	C₂²×C₃₁	1	4	yes	—
118	#478	—	2	4	no	—
119	#476	—	2	2	no	—
120	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
121	C₄×D₈	C₂²×C₃₁	1	2	yes	—
122	C₄×D₈	C₂²×C₃₁	1	4	yes	—
123	#367	C₂²×C₃₁	1	4	yes	—
124	#366	C₂²×C₃₁	1	2	yes	—
125	#366	C₂²×C₃₁	1	1	yes	characteristic
126	#366	C₂²×C₃₁	1	2	yes	—
127	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
128	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
129	#357	C₂²×C₃₁	1	4	yes	—
130	#357	C₂²×C₃₁	1	2	yes	—
131	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
132	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
133	C₂×C₃₁	—	8	4	no	—
134	C₂×C₃₁	Q₈×D₈	1	1	yes	characteristic
135	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
136	C₂×C₃₁	#3415	1	1	yes	characteristic
137	C₂×D₈	C₂³×C₃₁	1	2	yes	—
138	C₄⋊Q₈	—	2	2	no	—
139	#125	—	2	4	no	—
140	C₂²⋊Q₈	—	4	2	no	—
141	C₂²⋊Q₈	—	4	4	no	—
142	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
143	C₄×D₄	—	2	2	no	—
144	C₄×D₄	—	2	4	no	—
145	C₂×C₄⋊C₄	—	4	2	no	—
146	D₁₆	Q₈×C₃₁	1	4	yes	complemented
147	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
148	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
149	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
150	#107	—	2	2	no	—
151	#107	C₂³×C₃₁	1	4	yes	—
152	#107	C₂³×C₃₁	1	2	yes	—
153	#106	—	2	4	no	—
154	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
155	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
156	#102	—	2	4	no	—
157	#102	—	2	2	no	—
158	C₄×C₈	C₂³×C₃₁	1	2	yes	—
159	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
160	C₃₁	#33947	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
163	C₂×D₄	—	2	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	4	no	—
166	D₈	—	2	2	no	—
167	D₈	C₂×Q₈×C₃₁	1	4	yes	—
168	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
169	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
170	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
177	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	4	no	—
180	C₄²	C₂×D₄×C₃₁	1	2	yes	—
181	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
182	C₁₆	—	2	1	no	—
183	C₁₆	—	2	2	no	—
184	C₁₆	C₂×Q₈×C₃₁	1	2	yes	—
185	C₂³	—	4	2	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	16	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	—	16	4	no	—
194	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
195	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
196	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
197	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	1	no	—
201	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
202	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
203	C₂²	—	4	4	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	4	no	—
212	C₄	C₃₁×#508	1	2	yes	—
213	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
214	C₄	C₃₁×#578	1	1	yes	characteristic
215	C₂	—	8	4	no	—
216	C₂	Q₈×D₈×C₃₁	1	1	yes	characteristic
217	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
218	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3380	C₂	semidirect
2. C₃₁×#2253	C₂	semidirect
3. C₃₁×#2252	C₂	semidirect
4. C₃₁×#2228	C₂	semidirect
5. C₃₁×#2226	C₂	semidirect
6. Q₈×C₁₆×C₃₁	C₂	semidirect
37. C₃₁×D₁₆	Q₈	semidirect
75. #33947	C₃₁	direct (N × H)
94. #3380	C₂×C₃₁	semidirect
95. #2253	C₂×C₃₁	semidirect
96. #2252	C₂×C₃₁	semidirect
97. #2228	C₂×C₃₁	semidirect
98. #2226	C₂×C₃₁	semidirect
99. Q₈×C₁₆	C₂×C₃₁	semidirect
146. D₁₆	Q₈×C₃₁	semidirect
160. C₃₁	#33947	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3380	2
C₃₁×#2253	2
C₃₁×#2252	2
C₃₁×#2228	2
C₃₁×#2226	2
Q₈×C₁₆×C₃₁	2
C₄×C₃₁×D₁₆	2
#33947	31

Subgroups — C31×#33947 #21490706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33947 #21490706