Subgroups — C₃₁×#33726 #21490485

Order 7936.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	2	yes	maximal
2	C₃₁×#2254	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2253	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2252	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2158	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#533	—	2	2	no	—
8	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#500	C₂²	1	4	yes	—
10	C₃₁×#499	—	2	2	no	—
11	C₃₁×#499	C₂²	1	4	yes	—
12	C₃₁×#498	—	2	2	no	—
13	C₃₁×#427	—	2	2	no	—
14	C₃₁×#427	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#367	C₂²	1	8	yes	—
17	C₃₁×#366	C₂²	1	8	yes	—
18	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
20	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
23	C₃₁×C₄⋊Q₈	—	2	4	no	—
24	C₃₁×#125	—	2	4	no	—
25	C₃₁×C₂²⋊Q₈	—	4	4	no	—
26	C₄×Q₈×C₃₁	—	4	4	no	—
27	C₃₁×#117	—	2	4	no	—
28	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₃₁	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
31	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
32	C₃₁×#107	—	2	4	no	—
33	C₃₁×#107	C₂³	1	8	yes	—
34	C₃₁×#106	—	2	4	no	—
35	C₃₁×#106	—	2	4	no	—
36	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×Q₈×C₃₁	—	4	4	no	—
41	C₂²×C₄×C₃₁	D₈	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₃₁	Q₁₆	1	4	yes	—
44	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
48	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₃₁	—	2	8	no	—
50	C₄⋊C₄×C₃₁	—	4	8	no	—
51	C₄⋊C₄×C₃₁	—	4	4	no	—
52	C₂²⋊C₄×C₃₁	—	4	4	no	—
53	C₄²×C₃₁	—	4	4	no	—
54	C₄²×C₃₁	D₈	1	2	yes	—
55	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₃₁	C₂×Q₈	1	4	yes	—
58	C₁₆×C₃₁	C₂×Q₈	1	4	yes	—
59	#33726	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
61	Q₈×C₃₁	—	8	8	no	—
62	C₂×C₄×C₃₁	—	2	4	no	—
63	C₂×C₄×C₃₁	—	4	8	no	—
64	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
69	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
70	C₈×C₃₁	C₄⋊Q₈	1	2	yes	—
71	C₈×C₃₁	C₂×Q₁₆	1	2	yes	—
72	C₈×C₃₁	C₂×Q₁₆	1	2	yes	—
73	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
74	#3160	C₂×C₃₁	1	2	yes	—
75	#2254	C₂×C₃₁	1	2	yes	complemented
76	#2253	C₂×C₃₁	1	4	yes	complemented
77	#2252	C₂×C₃₁	1	2	yes	complemented
78	#2158	C₂×C₃₁	1	4	yes	—
79	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
80	C₂²×C₃₁	—	2	2	no	—
81	C₂²×C₃₁	#500	1	2	yes	—
82	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
83	C₄×C₃₁	—	2	2	no	—
84	C₄×C₃₁	—	8	8	no	—
85	C₄×C₃₁	#500	1	2	yes	—
86	C₄×C₃₁	#508	1	4	yes	—
87	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
89	#533	—	2	2	no	—
90	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
91	#500	C₂²×C₃₁	1	4	yes	—
92	#499	—	2	2	no	—
93	#499	C₂²×C₃₁	1	4	yes	—
94	#498	—	2	2	no	—
95	#427	—	2	2	no	—
96	#427	C₂²×C₃₁	1	4	yes	—
97	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
98	#367	C₂²×C₃₁	1	8	yes	—
99	#366	C₂²×C₃₁	1	8	yes	—
100	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
101	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
102	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
103	C₂×C₃₁	—	2	2	no	—
104	C₂×C₃₁	C₂×#500	1	1	yes	characteristic
105	C₂×C₃₁	C₂×#508	1	2	yes	—
106	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
108	C₄⋊Q₈	—	2	4	no	—
109	#125	—	2	4	no	—
110	C₂²⋊Q₈	—	4	4	no	—
111	C₄×Q₈	—	4	4	no	—
112	#117	—	2	4	no	—
113	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
114	C₂×C₁₆	Q₈×C₃₁	1	8	yes	complemented
115	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
116	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
117	#107	—	2	4	no	—
118	#107	C₂³×C₃₁	1	8	yes	—
119	#106	—	2	4	no	—
120	#106	—	2	4	no	—
121	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
122	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
123	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
125	C₃₁	#33726	1	1	yes	characteristic, direct product factor, Sylow
126	C₂×Q₈	—	4	4	no	—
127	C₂²×C₄	D₈×C₃₁	1	2	yes	—
128	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
129	C₂×C₈	Q₁₆×C₃₁	1	4	yes	—
130	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
131	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
132	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
133	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
134	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	4	8	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₂²⋊C₄	—	4	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	D₈×C₃₁	1	2	yes	—
141	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
142	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
143	C₁₆	C₂×Q₈×C₃₁	1	4	yes	—
144	C₁₆	C₂×Q₈×C₃₁	1	4	yes	—
145	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
146	Q₈	—	8	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
150	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
151	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
152	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
153	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
154	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
155	C₈	C₃₁×C₄⋊Q₈	1	2	yes	—
156	C₈	C₂×Q₁₆×C₃₁	1	2	yes	—
157	C₈	C₂×Q₁₆×C₃₁	1	2	yes	—
158	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	C₃₁×#500	1	2	yes	—
161	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	8	8	no	—
164	C₄	C₃₁×#500	1	2	yes	—
165	C₄	C₃₁×#508	1	4	yes	—
166	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
167	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×C₃₁×#500	1	1	yes	characteristic
170	C₂	C₂×C₃₁×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₈
prime radical	C₂³×C₃₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2254	C₂	semidirect
3. C₃₁×#2253	C₂	semidirect
4. C₃₁×#2252	C₂	semidirect
29. C₂×C₁₆×C₃₁	Q₈	semidirect
59. #33726	C₃₁	direct (N × H)
75. #2254	C₂×C₃₁	semidirect
76. #2253	C₂×C₃₁	semidirect
77. #2252	C₂×C₃₁	semidirect
114. C₂×C₁₆	Q₈×C₃₁	semidirect
125. C₃₁	#33726	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×#3160	2
C₃₁×#2254	2
C₃₁×#2253	2
C₃₁×#2252	2
C₃₁×#2158	2
C₂×C₄×C₁₆×C₃₁	2
#33726	31

Subgroups — C31×#33726 #21490485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33726 #21490485