Subgroups — #16847741

Order 3968.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521231	C₂	1	3	yes	maximal
2	#14520848	C₂	1	3	yes	maximal, complemented
3	Q₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758564	C₂²	1	1	yes	characteristic
5	C₃₁×C₄⋊Q₈	—	2	3	no	—
6	C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
7	C₄×Q₈×C₃₁	—	2	3	no	—
8	#11758519	—	2	3	no	—
9	#11758424	C₂²	1	3	yes	—
10	#11758393	—	2	3	no	—
11	C₂×Q₈×C₃₁	—	2	1	no	—
12	C₂×Q₈×C₃₁	—	2	1	no	—
13	C₂×Q₈×C₃₁	D₄	1	3	yes	—
14	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
15	#92499	—	4	1	no	—
16	Q₈×D₃₁	—	4	1	no	—
17	#92495	—	2	3	no	—
18	C₄⋊C₄×C₃₁	—	2	3	no	—
19	C₄⋊C₄×C₃₁	—	2	6	no	—
20	C₄²×C₃₁	—	2	3	no	—
21	C₄²×C₃₁	D₄	1	3	yes	—
22	#92482	—	4	3	no	—
23	#92481	—	4	3	no	—
24	#92474	—	2	3	no	—
25	Q₈×C₃₁	—	2	4	no	—
26	Q₈×C₃₁	—	4	1	no	—
27	Q₈×C₃₁	—	4	3	no	—
28	Q₈×C₃₁	—	8	1	no	—
29	C₂×C₄×C₃₁	—	2	3	no	—
30	C₂×C₄×C₃₁	—	2	3	no	—
31	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
32	C₂²⋊D₃₁	—	4	1	no	—
33	D₁₂₄	—	2	3	no	—
34	C₄×D₃₁	—	4	3	no	—
35	Dic₆₂	—	2	3	no	—
36	C₃₁⋊C₈	—	4	3	no	—
37	#2205	—	31	1	no	maximal, Sylow
38	C₂²×C₃₁	#120	1	1	yes	characteristic
39	C₂×D₃₁	—	4	1	no	—
40	C₄×C₃₁	—	2	6	no	—
41	C₄×C₃₁	—	2	3	no	—
42	C₄×C₃₁	—	4	3	no	—
43	Dic₃₁	—	4	1	no	—
44	Q₈²	D₃₁	1	1	yes	characteristic, complemented
45	#456	—	31	3	no	—
46	#356	—	31	3	no	—
47	C₂×C₃₁	—	2	1	no	—
48	C₂×C₃₁	B(4,𝔽₂)	1	1	yes	characteristic
49	D₃₁	—	8	1	no	—
50	2₋¹⁺⁴	—	31	1	no	—
51	#137	—	62	3	no	—
52	C₄⋊Q₈	—	2	3	no	—
53	C₄⋊Q₈	C₂×D₃₁	1	3	yes	—
54	C₄×Q₈	—	2	3	no	—
55	#104	—	62	3	no	—
56	#101	—	31	3	no	—
57	C₃₁	#2205	1	1	yes	characteristic, complemented, Sylow
58	#41	—	62	3	no	—
59	#41	—	124	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	124	1	no	—
63	C₂×Q₈	C₂²⋊D₃₁	1	3	yes	—
64	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
65	Q₁₆	—	124	3	no	—
66	SD₁₆	—	124	3	no	—
67	M₁₆	—	62	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	C₂²⋊D₃₁	1	3	yes	—
72	Q₈	—	2	4	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	3	no	—
75	Q₈	—	8	1	no	—
76	Q₈	—	62	3	no	—
77	D₄	—	62	3	no	—
78	D₄	—	124	1	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	124	3	no	—
82	C₂×C₄	C₂×C₂²⋊D₃₁	1	3	yes	—
83	C₈	—	124	3	no	—
84	C₂²	—	124	1	no	—
85	C₂²	#11758530	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	124	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	248	1	no	—
92	C₂	#14521236	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14521236
commutator, cosocle	C₂×Q₈×C₃₁	C₂³
Frattini	C₂×Q₈	C₂²×D₃₁
Fitting	Q₈²×C₃₁	C₂
socle	C₂×C₃₁	B(4,𝔽₂)
prime radical	#11758564	C₂²
hypercenter	Q₈²	D₃₁
hypocenter	C₃₁	#2205
LCS	G ▷ C₂×Q₈×C₃₁ ▷ C₂²×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
2. #14520848	C₂	semidirect
3. Q₈²×C₃₁	C₂	semidirect
44. Q₈²	D₃₁	semidirect
57. C₃₁	#2205	semidirect

Maximal subgroups

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

subgroup	index
#14521231	2
#14520848	2
Q₈²×C₃₁	2
#2205	31