Subgroups — #27186261

Order 12928.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266086	C₂	1	1	yes	maximal, characteristic
2	#20266084	C₂	1	1	yes	maximal, characteristic
3	#20265569	C₂	1	1	yes	maximal, characteristic
4	#15692316	—	2	1	no	—
5	#15692314	—	2	1	no	—
6	C₂×#13108064	—	2	1	no	—
7	C₂×#13108061	—	2	1	no	—
8	#15692243	C₂²	1	1	yes	characteristic
9	#15692166	C₄	1	1	yes	characteristic, complemented
10	C₄⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic, complemented
11	#13108083	—	4	1	no	—
12	C₂²⋊#87938	—	4	1	no	—
13	C₂×Dic₂₀₂	D₄	1	1	yes	characteristic
14	C₂×D₄₀₄	D₄	1	1	yes	characteristic
15	#13108064	—	4	2	no	—
16	#13108061	—	4	2	no	—
17	C₂×#11717579	—	2	1	no	—
18	#13108051	—	4	1	no	—
19	C₂×Q₈×C₁₀₁	—	2	1	no	—
20	C₄⋊C₄×C₁₀₁	—	4	1	no	—
21	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₀₁	—	2	1	no	—
23	C₂×#87938	—	8	1	no	—
24	Q₈×C₁₀₁	—	4	1	no	—
25	Q₈×C₁₀₁	—	4	1	no	—
26	C₂×C₄×C₁₀₁	—	2	1	no	—
27	C₂×C₄×C₁₀₁	—	4	1	no	—
28	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×Dic₁₀₁	—	2	1	no	—
30	D₄₀₄	—	4	1	no	—
31	Dic₂₀₂	—	4	1	no	—
32	#11717581	—	8	1	no	—
33	#11717579	—	2	2	no	—
34	C₂²×C₁₀₁	#99	1	1	yes	characteristic
35	C₂×D₁₀₁	—	4	1	no	—
36	C₂×D₁₀₁	—	8	1	no	—
37	#87938	—	16	1	no	—
38	C₄×C₁₀₁	—	2	1	no	—
39	C₄×C₁₀₁	—	4	1	no	—
40	C₄×C₁₀₁	—	8	1	no	—
41	Dic₁₀₁	—	4	1	no	—
42	C₂×C₁₀₁	—	2	1	no	—
43	C₂×C₁₀₁	#353	1	1	yes	characteristic
44	D₁₀₁	—	8	1	no	—
45	#1407	—	101	1	no	maximal, Sylow
46	C₁₀₁	#1407	1	1	yes	characteristic, complemented, Sylow
47	#497	—	101	1	no	—
48	#355	—	101	1	no	—
49	#354	—	101	1	no	—
50	C₂×Q₁₆	—	202	1	no	—
51	C₂×SD₁₆	—	202	1	no	—
52	C₄⋊Q₈	#87938	1	1	yes	characteristic, complemented
53	#124	—	101	1	no	—
54	#101	—	202	1	no	—
55	#99	—	202	1	no	—
56	#97	—	101	1	no	—
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	—	101	1	no	—
59	C₂×D₄	—	101	1	no	—
60	Q₁₆	—	404	2	no	—
61	SD₁₆	—	404	2	no	—
62	M₁₆	—	404	1	no	—
63	C₂×C₈	—	202	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₂²⋊C₄	—	404	1	no	—
66	C₂²⋊C₄	—	404	1	no	—
67	C₄²	C₂×#87938	1	1	yes	characteristic
68	C₂³	—	202	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	404	1	no	—
72	D₄	—	404	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	808	1	no	—
76	C₂×C₄	—	202	1	no	—
77	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
78	C₈	—	808	1	no	—
79	C₈	—	202	2	no	—
80	C₂²	—	808	1	no	—
81	C₂²	—	404	1	no	—
82	C₂²	#15692313	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	1616	1	no	—
87	C₄	—	404	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	808	1	no	—
90	C₂	#20266081	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20266081
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	C₄²	C₂×#87938
Fitting	C₄⋊Q₈×C₁₀₁	C₄
socle	C₂×C₁₀₁	#353
prime radical	C₂×D₄₀₄	D₄
cosocle	#15692243	C₂²
hypercenter	C₄⋊Q₈	#87938
hypocenter	C₁₀₁	#1407
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
9. #15692166	C₄	semidirect
10. C₄⋊Q₈×C₁₀₁	C₄	semidirect
46. C₁₀₁	#1407	semidirect
52. C₄⋊Q₈	#87938	semidirect

Maximal subgroups

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

subgroup	index
#20266086	2
#20266084	2
#20265569	2
#1407	101