Subgroups — #27173162

Order 12928.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265359	C₂	1	3	yes	maximal
2	#20264976	C₂	1	3	yes	maximal, complemented
3	Q₈²×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692303	C₂²	1	1	yes	characteristic
5	#15692286	—	2	3	no	—
6	#15692183	—	2	3	no	—
7	#15692174	C₂²	1	3	yes	—
8	C₄⋊Q₈×C₁₀₁	—	2	3	no	—
9	C₄⋊Q₈×C₁₀₁	C₂²	1	3	yes	—
10	C₄×Q₈×C₁₀₁	—	2	3	no	—
11	#13108075	—	4	1	no	—
12	#13108074	—	2	3	no	—
13	Q₈×D₁₀₁	—	4	1	no	—
14	#13108064	—	4	3	no	—
15	#13108061	—	4	3	no	—
16	#13108058	—	2	3	no	—
17	C₂×Q₈×C₁₀₁	—	2	1	no	—
18	C₂×Q₈×C₁₀₁	—	2	1	no	—
19	C₂×Q₈×C₁₀₁	D₄	1	3	yes	—
20	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₁	—	2	3	no	—
22	C₄⋊C₄×C₁₀₁	—	2	6	no	—
23	C₄²×C₁₀₁	—	2	3	no	—
24	C₄²×C₁₀₁	D₄	1	3	yes	—
25	Q₈×C₁₀₁	—	2	4	no	—
26	Q₈×C₁₀₁	—	4	1	no	—
27	Q₈×C₁₀₁	—	4	3	no	—
28	Q₈×C₁₀₁	—	8	1	no	—
29	C₂×C₄×C₁₀₁	—	2	3	no	—
30	C₂×C₄×C₁₀₁	—	2	3	no	—
31	C₂×C₄×C₁₀₁	C₂×D₄	1	3	yes	—
32	C₂²⋊D₁₀₁	—	4	1	no	—
33	D₄₀₄	—	2	3	no	—
34	C₄×D₁₀₁	—	4	3	no	—
35	Dic₂₀₂	—	2	3	no	—
36	#11717579	—	4	3	no	—
37	C₂²×C₁₀₁	#120	1	1	yes	characteristic
38	C₂×D₁₀₁	—	4	1	no	—
39	C₄×C₁₀₁	—	2	6	no	—
40	C₄×C₁₀₁	—	2	3	no	—
41	C₄×C₁₀₁	—	4	3	no	—
42	Dic₁₀₁	—	4	1	no	—
43	C₂×C₁₀₁	—	2	1	no	—
44	C₂×C₁₀₁	B(4,𝔽₂)	1	1	yes	characteristic
45	D₁₀₁	—	8	1	no	—
46	#2205	—	101	1	no	maximal, Sylow
47	C₁₀₁	#2205	1	1	yes	characteristic, complemented, Sylow
48	Q₈²	D₁₀₁	1	1	yes	characteristic, complemented
49	#456	—	101	3	no	—
50	#356	—	101	3	no	—
51	2₋¹⁺⁴	—	101	1	no	—
52	#137	—	202	3	no	—
53	C₄⋊Q₈	—	2	3	no	—
54	C₄⋊Q₈	C₂×D₁₀₁	1	3	yes	—
55	C₄×Q₈	—	2	3	no	—
56	#104	—	202	3	no	—
57	#101	—	101	3	no	—
58	#41	—	404	1	no	—
59	#41	—	202	3	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	404	1	no	—
63	C₂×Q₈	C₂²⋊D₁₀₁	1	3	yes	—
64	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
65	Q₁₆	—	404	3	no	—
66	SD₁₆	—	404	3	no	—
67	M₁₆	—	202	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	C₂²⋊D₁₀₁	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	—	202	3	no	—
77	D₄	—	404	1	no	—
78	D₄	—	202	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	404	3	no	—
82	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	3	yes	—
83	C₈	—	404	3	no	—
84	C₂²	—	404	1	no	—
85	C₂²	#15692224	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	404	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	808	1	no	—
92	C₂	#20265364	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20265364
commutator, cosocle	C₂×Q₈×C₁₀₁	C₂³
Frattini	C₂×Q₈	C₂²×D₁₀₁
Fitting	Q₈²×C₁₀₁	C₂
socle	C₂×C₁₀₁	B(4,𝔽₂)
prime radical	#15692303	C₂²
hypercenter	Q₈²	D₁₀₁
hypocenter	C₁₀₁	#2205
LCS	G ▷ C₂×Q₈×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
2. #20264976	C₂	semidirect
3. Q₈²×C₁₀₁	C₂	semidirect
47. C₁₀₁	#2205	semidirect
48. Q₈²	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265359	2
#20264976	2
Q₈²×C₁₀₁	2
#2205	101