Subgroups — #27169095

Order 12928.

98 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	C₄×#13108049	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₀₄	C₄	1	1	yes	characteristic
5	C₄×Dic₂₀₂	—	2	1	no	—
6	C₂×#13108049	—	2	1	no	—
7	#15692189	C₄	1	1	yes	characteristic
8	#15692180	—	2	1	no	—
9	C₄⋊C₈×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	C₄⋊Dic₁₀₁	—	2	1	no	—
15	#13108053	—	4	1	no	—
16	C₄×Dic₁₀₁	—	4	1	no	—
17	#13108049	—	2	2	no	—
18	C₂×C₈×C₁₀₁	—	4	1	no	—
19	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
21	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
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	—	8	1	no	—
31	C₈×C₁₀₁	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₀₁	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₀₁	#102	1	1	yes	characteristic
34	C₄×C₁₀₁	—	2	1	no	—
35	C₄×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₁₀₁	#104	1	1	yes	characteristic
37	C₄×C₁₀₁	SD₃₂	1	1	yes	characteristic
38	C₄×C₁₀₁	Q₃₂	1	1	yes	characteristic
39	Dic₁₀₁	—	4	2	no	—
40	Dic₁₀₁	—	8	1	no	—
41	C₂×C₁₀₁	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₁₀₁	#358	1	1	yes	characteristic
43	C₂×C₁₀₁	#359	1	1	yes	characteristic
44	#1332	—	101	1	no	maximal, Sylow
45	C₁₀₁	#1332	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	101	1	no	—
47	C₄×C₁₆	—	101	1	no	—
48	#335	D₁₀₁	1	1	yes	characteristic
49	C₂×Q₁₆	—	101	1	no	—
50	C₄×Q₈	—	202	1	no	—
51	C₂×C₁₆	—	202	1	no	—
52	#107	—	101	1	no	—
53	C₄⋊C₈	—	2	1	no	—
54	#103	—	202	1	no	—
55	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
56	C₂×Q₈	—	202	1	no	—
57	Q₁₆	—	101	2	no	—
58	Q₁₆	—	202	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	C₄×D₁₀₁	1	1	yes	characteristic
61	C₂×C₈	D₄₀₄	1	1	yes	characteristic
62	C₄⋊C₄	—	404	1	no	—
63	C₄⋊C₄	—	202	1	no	—
64	C₄²	—	404	1	no	—
65	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
66	C₁₆	—	202	2	no	—
67	Q₈	—	404	1	no	—
68	Q₈	—	202	2	no	—
69	C₂×C₄	—	404	1	no	—
70	C₂×C₄	—	404	1	no	—
71	C₂×C₄	#13108051	1	1	yes	characteristic
72	C₂×C₄	#13108059	1	1	yes	characteristic
73	C₂×C₄	#13108061	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	8	1	no	—
76	C₈	C₈×D₁₀₁	1	1	yes	characteristic
77	C₈	#13108057	1	1	yes	characteristic
78	C₂²	#15692175	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	808	1	no	—
81	C₄	—	404	2	no	—
82	C₄	#15692167	1	1	yes	characteristic
83	C₄	#15692181	1	1	yes	characteristic
84	C₄	#15692199	1	1	yes	characteristic
85	C₄	#15692201	1	1	yes	characteristic
86	C₂	#20264904	1	1	yes	characteristic
87	C₂	#20264980	1	1	yes	characteristic
88	C₂	#20264983	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13108059
commutator	C₈×C₁₀₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	#335×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#102
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	#335	D₁₀₁
hypocenter	C₁₀₁	#1332
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
13. Dic₄₀₄	C₈	semidirect
45. C₁₀₁	#1332	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
C₄×#13108049	2
#335×C₁₀₁	2
#1332	101