Subgroups — #27169103

Order 12928.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₄₀₄	C₂	1	1	yes	maximal, characteristic
2	#20264928	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₀₄	C₄	1	1	yes	characteristic
5	C₄×Dic₂₀₂	—	2	1	no	—
6	#15692189	C₄	1	1	yes	characteristic
7	C₄⋊#11717579	—	2	1	no	—
8	#15692180	—	2	1	no	—
9	C₂×C₁₆×C₁₀₁	—	2	1	no	—
10	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₀₂	—	2	1	no	—
12	Dic₄₀₄	—	2	1	no	—
13	Dic₄₀₄	C₈	1	2	yes	complemented
14	C₂×#11717579	—	4	1	no	—
15	C₄⋊Dic₁₀₁	—	2	1	no	—
16	#13108053	—	4	1	no	—
17	C₄×Dic₁₀₁	—	4	1	no	—
18	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
20	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
21	C₁₆×C₁₀₁	—	2	2	no	—
22	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₀₁	SD₁₆	1	1	yes	characteristic
25	C₂×Dic₁₀₁	—	4	1	no	—
26	C₂×Dic₁₀₁	—	4	1	no	—
27	Dic₂₀₂	—	2	2	no	—
28	Dic₂₀₂	—	4	1	no	—
29	C₈×C₁₀₁	—	2	1	no	—
30	C₈×C₁₀₁	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₀₁	M₁₆	1	1	yes	characteristic
32	#11717579	—	8	1	no	—
33	C₂²×C₁₀₁	#102	1	1	yes	characteristic
34	C₄×C₁₀₁	—	2	1	no	—
35	C₄×C₁₀₁	—	2	1	no	—
36	C₄×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₁₀₁	#104	1	1	yes	characteristic
38	Dic₁₀₁	—	4	2	no	—
39	Dic₁₀₁	—	8	1	no	—
40	C₂×C₁₀₁	D₄⋊C₈	1	1	yes	characteristic
41	C₂×C₁₀₁	#360	1	1	yes	characteristic
42	C₂×C₁₀₁	#362	1	1	yes	characteristic
43	#1334	—	101	1	no	maximal, Sylow
44	C₁₀₁	#1334	1	1	yes	characteristic, complemented, Sylow
45	C₄×Q₁₆	—	101	1	no	—
46	#346	D₁₀₁	1	1	yes	characteristic
47	#334	—	101	1	no	—
48	C₂×Q₁₆	—	101	1	no	—
49	C₄×Q₈	—	202	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	#107	—	101	1	no	—
52	C₄⋊C₈	—	202	1	no	—
53	#103	—	202	1	no	—
54	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
55	C₂×Q₈	—	202	1	no	—
56	Q₁₆	—	101	2	no	—
57	Q₁₆	—	202	1	no	—
58	C₂×C₈	—	404	1	no	—
59	C₂×C₈	C₄×D₁₀₁	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
61	C₄⋊C₄	—	404	1	no	—
62	C₄⋊C₄	—	202	1	no	—
63	C₄²	—	404	1	no	—
64	C₄²	D₄₀₄	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	Q₈	—	404	1	no	—
67	Q₈	—	202	2	no	—
68	C₂×C₄	—	404	1	no	—
69	C₂×C₄	—	404	1	no	—
70	C₂×C₄	#13108051	1	1	yes	characteristic
71	C₂×C₄	D₈₀₈	1	1	yes	characteristic
72	C₂×C₄	#13108063	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	808	1	no	—
75	C₈	C₈×D₁₀₁	1	1	yes	characteristic
76	C₈	#13108057	1	1	yes	characteristic
77	C₂²	#15692177	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	808	1	no	—
81	C₄	—	404	2	no	—
82	C₄	#15692167	1	1	yes	characteristic
83	C₄	#15692183	1	1	yes	characteristic
84	C₂	#20264906	1	1	yes	characteristic
85	C₂	#20264988	1	1	yes	characteristic
86	C₂	#20264994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692177
commutator	C₈×C₁₀₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	#346×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#102
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	#346	D₁₀₁
hypocenter	C₁₀₁	#1334
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. Dic₄₀₄	C₈	semidirect
44. C₁₀₁	#1334	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₄₀₄	2
#20264928	2
#346×C₁₀₁	2
#1334	101