Subgroups — #27187546

Order 12928.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266098	C₂	1	2	yes	maximal
2	C₄⋊#13108077	C₂	1	1	yes	maximal, characteristic
3	#20266063	C₂	1	2	yes	maximal
4	C₄×#13108077	C₂	1	1	yes	maximal, characteristic
5	#20265301	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108077	C₂²	1	1	yes	characteristic
7	C₂×#13108077	C₂²	1	2	yes	—
8	C₂×#13108077	C₂²	1	1	yes	characteristic
9	C₄×Dic₂₀₂	C₄	1	1	yes	characteristic
10	#15692184	C₄	1	2	yes	—
11	C₄×#11717579	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₀₁	C₂²	1	2	yes	—
13	C₄⋊C₈×C₁₀₁	C₄	1	1	yes	characteristic
14	#13108077	—	2	1	no	—
15	#13108077	—	2	2	no	—
16	#13108077	Q₈	1	2	yes	complemented
17	C₂×Dic₂₀₂	C₈	1	1	yes	characteristic
18	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
19	C₂×#11717579	C₂³	1	1	yes	characteristic
20	C₄⋊Dic₁₀₁	C₈	1	1	yes	characteristic
21	#13108053	C₈	1	2	yes	—
22	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
24	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₁₀₁	C₂×C₈	1	2	yes	—
29	C₂×Dic₁₀₁	C₂×C₈	1	2	yes	—
30	Dic₂₀₂	C₁₆	1	4	yes	complemented
31	C₈×C₁₀₁	—	2	2	no	—
32	#11717579	—	2	1	no	—
33	#11717579	C₂×Q₈	1	1	yes	characteristic
34	#11717579	#41	1	1	yes	characteristic
35	C₂²×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₀₁	—	2	1	no	—
37	C₄×C₁₀₁	C₂×C₁₆	1	2	yes	—
38	C₄×C₁₀₁	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₀₁	#131	1	1	yes	characteristic
40	Dic₁₀₁	—	2	2	no	—
41	Dic₁₀₁	C₂×C₁₆	1	4	yes	—
42	C₂×C₁₀₁	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₀₁	C₂²×C₁₆	1	1	yes	characteristic
44	C₂×C₁₀₁	#504	1	1	yes	characteristic
45	Q₈×C₁₆	—	101	1	no	maximal, Sylow
46	C₁₀₁	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	—	101	1	no	—
48	C₄⋊C₁₆	—	101	1	no	—
49	C₄⋊C₁₆	—	101	2	no	—
50	C₄×C₁₆	—	101	1	no	—
51	C₄×C₁₆	—	101	2	no	—
52	C₄×Q₈	—	101	1	no	—
53	C₂×C₁₆	—	101	1	no	—
54	C₂×C₁₆	—	101	1	no	—
55	C₂×C₁₆	—	101	2	no	—
56	C₄⋊C₈	—	101	2	no	—
57	C₄⋊C₈	#87938	1	1	yes	characteristic
58	C₄×C₈	—	101	2	no	—
59	C₄×C₈	—	101	1	no	—
60	C₂×Q₈	—	101	1	no	—
61	C₂×C₈	—	101	1	no	—
62	C₂×C₈	—	101	1	no	—
63	C₂×C₈	C₂×#87938	1	2	yes	—
64	C₄⋊C₄	—	101	2	no	—
65	C₄⋊C₄	—	101	1	no	—
66	C₄²	—	101	2	no	—
67	C₄²	C₂×#87938	1	1	yes	characteristic
68	C₁₆	—	101	2	no	—
69	C₁₆	—	202	1	no	—
70	C₁₆	—	202	2	no	—
71	Q₈	—	101	4	no	—
72	C₂×C₄	—	101	2	no	—
73	C₂×C₄	—	101	2	no	—
74	C₂×C₄	D₁₀₁⋊C₈	1	1	yes	characteristic
75	C₂×C₄	D₁₀₁⋊C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	101	1	no	—
79	C₈	—	101	1	no	—
80	C₈	—	202	1	no	—
81	C₂²	C₂×D₁₀₁⋊C₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	101	4	no	—
84	C₄	—	202	2	no	—
85	C₄	D₁₀₁⋊C₁₆	1	2	yes	—
86	C₄	Q₈×#87938	1	1	yes	characteristic
87	C₄	#15692342	1	1	yes	characteristic
88	C₂	#20266226	1	1	yes	characteristic
89	C₂	C₂×D₁₀₁⋊C₁₆	1	1	yes	characteristic
90	C₂	#20266236	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#87938
commutator	C₂×C₁₀₁	C₂²×C₁₆
Fitting	C₄⋊C₈×C₁₀₁	C₄
socle, prime radical	C₂²×C₁₀₁	C₂²×C₈
cosocle	C₂×#11717579	C₂³
hypercenter	C₄⋊C₈	#87938
hypocenter	C₁₀₁	Q₈×C₁₆
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
16. #13108077	Q₈	semidirect
30. Dic₂₀₂	C₁₆	semidirect
46. C₁₀₁	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20266098	2
C₄⋊#13108077	2
#20266063	2
C₄×#13108077	2
#20265301	2
Q₈×C₁₆	101