Subgroups — C₃₁×#33501 #21490260

Order 7936.

444 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2183	C₂	1	8	yes	maximal, complemented
3	C₃₁×#2164	C₂	1	4	yes	maximal
4	C₃₁×#2163	C₂	1	2	yes	maximal
5	C₃₁×#533	C₄	1	1	yes	characteristic
6	C₃₁×#446	C₄	1	4	yes	—
7	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
8	C₃₁×#424	C₄	1	2	yes	—
9	C₃₁×#423	C₄	1	1	yes	characteristic
10	C₃₁×#405	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	—
13	C₃₁×C₄⋊C₁₆	C₂²	1	8	yes	—
14	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
15	C₃₁×#346	C₂²	1	4	yes	—
16	C₃₁×#346	C₂²	1	4	yes	—
17	C₄×C₁₆×C₃₁	C₂²	1	4	yes	—
18	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₃₁×C₄⋊Q₈	C₈	1	2	yes	—
22	C₃₁×#125	C₈	1	2	yes	—
23	C₃₁×C₂²⋊Q₈	C₈	1	4	yes	—
24	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
25	C₃₁×#117	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₃₁	C₂³	1	8	yes	—
28	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
30	C₃₁×#97	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₃₁	C₂³	1	2	yes	—
33	C₄×C₈×C₃₁	C₂³	1	4	yes	—
34	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₃₁	—	4	2	no	—
38	C₂×C₈×C₃₁	—	2	2	no	—
39	C₂×C₈×C₃₁	—	4	1	no	—
40	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
42	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
43	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
44	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
45	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
46	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
47	C₂²⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
48	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
51	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
52	C₁₆×C₃₁	—	2	8	no	—
53	#33501	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
55	Q₈×C₃₁	—	4	4	no	—
56	C₂×C₄×C₃₁	—	2	1	no	—
57	C₂×C₄×C₃₁	—	2	1	no	—
58	C₂×C₄×C₃₁	—	4	2	no	—
59	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
61	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₃₁	C₂²×C₈	1	8	yes	—
63	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
64	C₈×C₃₁	—	2	1	no	—
65	C₈×C₃₁	—	2	2	no	—
66	C₈×C₃₁	—	2	4	no	—
67	C₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
68	#2960	C₂×C₃₁	1	1	yes	characteristic
69	#2183	C₂×C₃₁	1	8	yes	complemented
70	#2164	C₂×C₃₁	1	4	yes	—
71	#2163	C₂×C₃₁	1	2	yes	—
72	C₂²×C₃₁	—	2	2	no	—
73	C₂²×C₃₁	—	2	1	no	—
74	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
75	C₄×C₃₁	—	2	2	no	—
76	C₄×C₃₁	—	2	2	no	—
77	C₄×C₃₁	—	2	8	no	—
78	C₄×C₃₁	—	2	1	no	—
79	C₄×C₃₁	#519	1	1	yes	characteristic
80	C₄×C₃₁	#568	1	1	yes	characteristic
81	#533	C₄×C₃₁	1	1	yes	characteristic
82	#446	C₄×C₃₁	1	4	yes	—
83	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
84	#424	C₄×C₃₁	1	2	yes	—
85	#423	C₄×C₃₁	1	1	yes	characteristic
86	#405	C₂²×C₃₁	1	2	yes	—
87	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
88	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	—
89	C₄⋊C₁₆	C₂²×C₃₁	1	8	yes	—
90	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	—
91	#346	C₂²×C₃₁	1	4	yes	—
92	#346	C₂²×C₃₁	1	4	yes	—
93	C₄×C₁₆	C₂²×C₃₁	1	4	yes	—
94	C₂×C₃₁	—	4	1	no	—
95	C₂×C₃₁	#2982	1	1	yes	characteristic
96	C₂×C₃₁	#3407	1	2	yes	—
97	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
98	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
99	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
100	C₄⋊Q₈	C₈×C₃₁	1	2	yes	—
101	#125	C₈×C₃₁	1	2	yes	—
102	C₂²⋊Q₈	C₈×C₃₁	1	4	yes	—
103	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
104	#117	C₂×C₄×C₃₁	1	2	yes	—
105	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
106	C₂×C₁₆	C₂³×C₃₁	1	8	yes	—
107	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
108	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
109	#97	C₂×C₄×C₃₁	1	4	yes	—
110	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
111	C₄×C₈	C₂³×C₃₁	1	4	yes	—
112	C₄×C₈	C₂³×C₃₁	1	2	yes	—
113	C₃₁	#33501	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	—
115	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
116	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
121	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
122	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
123	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
124	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
125	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
126	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
127	C₂²⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
128	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
129	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
130	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
131	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
132	C₁₆	—	2	8	no	—
133	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
140	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
141	C₂×C₄	C₂²×C₈×C₃₁	1	8	yes	—
142	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	4	no	—
146	C₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	8	no	—
153	C₄	—	2	1	no	—
154	C₄	C₃₁×#519	1	1	yes	characteristic
155	C₄	C₃₁×#568	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	C₃₁×#2982	1	1	yes	characteristic
158	C₂	C₃₁×#3407	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2183	C₂	semidirect
53. #33501	C₃₁	direct (N × H)
69. #2183	C₂×C₃₁	semidirect
113. C₃₁	#33501	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×#2960	2
C₃₁×#2183	2
C₃₁×#2164	2
C₃₁×#2163	2
#33501	31

Subgroups — C31×#33501 #21490260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33501 #21490260