Subgroups — C₃₁×#3502 #16843435

Order 3968.

964 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₃₁	C₂	1	4	yes	maximal, complemented
2	C₃₁×#557	C₂	1	8	yes	maximal, complemented
3	C₃₁×#557	C₂	1	4	yes	maximal, complemented
4	C₃₁×#544	C₂	1	6	yes	maximal
5	C₃₁×#533	C₂	1	6	yes	maximal
6	C₃₁×#519	C₂	1	2	yes	maximal
7	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×C₄⋊Q₈	C₂²	1	6	yes	—
10	C₃₁×C₄⋊Q₈	C₂²	1	6	yes	—
11	C₃₁×C₄⋊Q₈	C₂²	1	24	yes	—
12	C₃₁×#125	C₂²	1	12	yes	—
13	C₃₁×#125	C₂²	1	24	yes	—
14	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
15	C₃₁×C₂²⋊Q₈	C₂²	1	12	yes	—
16	C₄×Q₈×C₃₁	C₂²	1	4	yes	—
17	C₄×Q₈×C₃₁	C₂²	1	4	yes	—
18	C₄×Q₈×C₃₁	C₂²	1	8	yes	—
19	C₄×Q₈×C₃₁	C₂²	1	8	yes	—
20	C₄×Q₈×C₃₁	C₂²	1	12	yes	—
21	C₄×Q₈×C₃₁	C₂²	1	12	yes	—
22	C₃₁×#117	C₂²	1	12	yes	—
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
24	C₂×C₄²×C₃₁	C₂²	1	3	yes	—
25	C₂×Q₈×C₃₁	Q₈	1	8	yes	complemented
26	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
27	C₂×Q₈×C₃₁	C₂³	1	12	yes	—
28	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
31	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
32	C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
33	C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
34	C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
35	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
36	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
37	C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
38	C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
39	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
40	C₄²×C₃₁	C₂³	1	6	yes	—
41	C₄²×C₃₁	C₂³	1	6	yes	—
42	C₄²×C₃₁	C₂³	1	24	yes	—
43	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
44	Q₈×C₃₁	—	2	8	no	—
45	Q₈×C₃₁	—	4	12	no	—
46	Q₈×C₃₁	C₂×Q₈	1	16	yes	—
47	C₂×C₄×C₃₁	—	2	2	no	—
48	C₂×C₄×C₃₁	—	4	3	no	—
49	C₂×C₄×C₃₁	C₂×Q₈	1	12	yes	—
50	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
52	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
53	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
54	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
55	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
56	#3502	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₃₁	—	2	1	no	—
58	C₂²×C₃₁	—	2	1	no	—
59	C₂²×C₃₁	C₂²×Q₈	1	2	yes	—
60	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
61	C₄×C₃₁	—	2	1	no	—
62	C₄×C₃₁	—	2	8	no	—
63	C₄×C₃₁	—	2	6	no	—
64	C₄×C₃₁	—	2	8	no	—
65	C₄×C₃₁	C₂²×Q₈	1	12	yes	—
66	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
67	Q₈²	C₂×C₃₁	1	4	yes	complemented
68	#557	C₂×C₃₁	1	8	yes	complemented
69	#557	C₂×C₃₁	1	4	yes	complemented
70	#544	C₂×C₃₁	1	6	yes	—
71	#533	C₂×C₃₁	1	6	yes	—
72	#519	C₂×C₃₁	1	2	yes	—
73	C₂×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
74	C₂×C₃₁	—	2	2	no	—
75	C₂×C₃₁	C₂³×Q₈	1	1	yes	characteristic
76	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
77	C₂×C₃₁	#585	1	1	yes	characteristic
78	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
79	C₄⋊Q₈	C₂²×C₃₁	1	6	yes	—
80	C₄⋊Q₈	C₂²×C₃₁	1	6	yes	—
81	C₄⋊Q₈	C₂²×C₃₁	1	24	yes	—
82	#125	C₂²×C₃₁	1	12	yes	—
83	#125	C₂²×C₃₁	1	24	yes	—
84	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
85	C₂²⋊Q₈	C₂²×C₃₁	1	12	yes	—
86	C₄×Q₈	C₂²×C₃₁	1	4	yes	—
87	C₄×Q₈	C₂²×C₃₁	1	4	yes	—
88	C₄×Q₈	C₂²×C₃₁	1	8	yes	—
89	C₄×Q₈	C₂²×C₃₁	1	8	yes	—
90	C₄×Q₈	C₂²×C₃₁	1	12	yes	—
91	C₄×Q₈	C₂²×C₃₁	1	12	yes	—
92	#117	C₂²×C₃₁	1	12	yes	—
93	C₂×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
94	C₂×C₄²	C₂²×C₃₁	1	3	yes	—
95	C₃₁	#3502	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×Q₈	Q₈×C₃₁	1	8	yes	complemented
97	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
98	C₂×Q₈	C₂³×C₃₁	1	12	yes	—
99	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
100	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
101	C₂²×C₄	C₂³×C₃₁	1	3	yes	—
102	C₂²×C₄	C₂³×C₃₁	1	3	yes	—
103	C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
104	C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
105	C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
106	C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
107	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
108	C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
109	C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
110	C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
111	C₄²	C₂³×C₃₁	1	6	yes	—
112	C₄²	C₂³×C₃₁	1	6	yes	—
113	C₄²	C₂³×C₃₁	1	24	yes	—
114	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
115	Q₈	—	2	8	no	—
116	Q₈	—	4	12	no	—
117	Q₈	C₂×Q₈×C₃₁	1	16	yes	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	3	no	—
120	C₂×C₄	C₂×Q₈×C₃₁	1	12	yes	—
121	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
122	C₂×C₄	C₂⁴×C₃₁	1	8	yes	—
123	C₂×C₄	C₂⁴×C₃₁	1	8	yes	—
124	C₂×C₄	C₂⁴×C₃₁	1	6	yes	—
125	C₂×C₄	C₂⁴×C₃₁	1	6	yes	—
126	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂²×Q₈×C₃₁	1	2	yes	—
130	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	8	no	—
133	C₄	—	2	6	no	—
134	C₄	—	2	8	no	—
135	C₄	C₂²×Q₈×C₃₁	1	12	yes	—
136	C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂³×Q₈×C₃₁	1	1	yes	characteristic
139	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
140	C₂	C₃₁×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
prime radical	C₂³×C₃₁	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₃₁	C₂	semidirect
2. C₃₁×#557	C₂	semidirect
3. C₃₁×#557	C₂	semidirect
25. C₂×Q₈×C₃₁	Q₈	semidirect
56. #3502	C₃₁	direct (N × H)
67. Q₈²	C₂×C₃₁	semidirect
68. #557	C₂×C₃₁	semidirect
69. #557	C₂×C₃₁	semidirect
95. C₃₁	#3502	direct (N × H)
96. C₂×Q₈	Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈²×C₃₁	2
C₃₁×#557	2
C₃₁×#544	2
C₃₁×#533	2
C₃₁×#519	2
C₂×C₄×Q₈×C₃₁	2
#3502	31

Subgroups — C31×#3502 #16843435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3502 #16843435