Subgroups — #27169403

Order 12928.

736 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692162	C₂	1	6	yes	maximal
2	C₂³×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂³×Dic₁₀₁	C₂²	1	4	yes	—
4	#15692162	—	2	12	no	—
5	C₂²×Q₈×C₁₀₁	—	2	12	no	—
6	C₂²×Q₈×C₁₀₁	C₄	1	4	yes	complemented
7	C₂³×C₄×C₁₀₁	C₂²	1	3	yes	—
8	C₂²×Dic₁₀₁	—	2	12	no	—
9	C₂²×Dic₁₀₁	—	2	12	no	—
10	C₂²×Dic₁₀₁	Q₈	1	8	yes	complemented
11	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
12	C₂×Q₈×C₁₀₁	—	2	32	no	—
13	C₂×Q₈×C₁₀₁	—	2	24	no	—
14	C₂²×C₄×C₁₀₁	—	2	12	no	—
15	C₂²×C₄×C₁₀₁	C₂×C₄	1	6	yes	—
16	C₂²×C₄×C₁₀₁	D₄	1	12	yes	—
17	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
18	C₂³×C₁₀₁	C₂×D₄	1	6	yes	—
19	C₂³×C₁₀₁	C₂×Q₈	1	4	yes	—
20	C₂³×C₁₀₁	#41	1	4	yes	—
21	Q₈×C₁₀₁	—	2	32	no	—
22	C₂×C₄×C₁₀₁	—	2	12	no	—
23	C₂×C₄×C₁₀₁	—	2	24	no	—
24	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	12	yes	—
25	C₂×Dic₁₀₁	—	2	24	no	—
26	C₂×Dic₁₀₁	—	4	4	no	—
27	C₂×Dic₁₀₁	—	4	12	no	—
28	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	3	yes	—
29	C₂²×C₁₀₁	C₄×Q₈	1	4	yes	—
30	C₂²×C₁₀₁	#120	1	4	yes	—
31	C₂²×C₁₀₁	C₂²⋊Q₈	1	12	yes	—
32	C₂²×C₁₀₁	#124	1	6	yes	—
33	C₂²×C₁₀₁	C₄⋊Q₈	1	6	yes	—
34	C₄×C₁₀₁	—	2	12	no	—
35	Dic₁₀₁	—	4	8	no	—
36	C₂×C₁₀₁	#379	1	1	yes	characteristic
37	C₂×C₁₀₁	#391	1	6	yes	—
38	C₂×C₁₀₁	#393	1	4	yes	—
39	C₂×C₁₀₁	#395	1	4	yes	—
40	#1446	—	101	1	no	maximal, Sylow
41	C₁₀₁	#1446	1	1	yes	characteristic, complemented, Sylow
42	C₂³×Q₈	D₁₀₁	1	1	yes	characteristic
43	C₂×#95	—	101	6	no	—
44	C₂²×Q₈	—	2	12	no	—
45	C₂²×Q₈	Dic₁₀₁	1	4	yes	complemented
46	C₂³×C₄	—	101	4	no	—
47	C₂³×C₄	C₂×D₁₀₁	1	3	yes	—
48	#95	—	202	12	no	—
49	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
50	C₂×Q₈	—	2	32	no	—
51	C₂×Q₈	—	2	24	no	—
52	C₂²×C₄	—	2	12	no	—
53	C₂²×C₄	—	101	8	no	—
54	C₂²×C₄	—	202	12	no	—
55	C₂²×C₄	—	202	12	no	—
56	C₂²×C₄	C₂×Dic₁₀₁	1	6	yes	—
57	C₂²×C₄	C₂²⋊D₁₀₁	1	12	yes	—
58	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
59	C₂³	C₂×C₂²⋊D₁₀₁	1	6	yes	—
60	C₂³	Q₈×D₁₀₁	1	4	yes	—
61	C₂³	#13108075	1	4	yes	—
62	Q₈	—	2	32	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	—	2	24	no	—
65	C₂×C₄	—	404	4	no	—
66	C₂×C₄	—	404	12	no	—
67	C₂×C₄	—	202	24	no	—
68	C₂×C₄	C₂²⋊Dic₁₀₁	1	12	yes	—
69	C₂²	C₂×C₂²⋊Dic₁₀₁	1	3	yes	—
70	C₂²	Q₈×Dic₁₀₁	1	4	yes	—
71	C₂²	#15692224	1	4	yes	—
72	C₂²	#15692231	1	12	yes	—
73	C₂²	#15692244	1	6	yes	—
74	C₂²	#15692253	1	6	yes	—
75	C₄	—	2	12	no	—
76	C₄	—	404	8	no	—
77	C₂	#20265033	1	1	yes	characteristic
78	C₂	#20265084	1	6	yes	—
79	C₂	#20265088	1	4	yes	—
80	C₂	#20265096	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₀₁
commutator	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₂³×Q₈×C₁₀₁	C₂
socle, prime radical, cosocle	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂³×Q₈	D₁₀₁
hypocenter	C₁₀₁	#1446
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
6. C₂²×Q₈×C₁₀₁	C₄	semidirect
10. C₂²×Dic₁₀₁	Q₈	semidirect
41. C₁₀₁	#1446	semidirect
45. C₂²×Q₈	Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692162	2
C₂³×Q₈×C₁₀₁	2
#1446	101