Subgroups — #27186957

Order 12928.

612 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692312	C₂	1	6	yes	maximal, complemented
2	C₂²×#13108075	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11717581	C₂²	1	4	yes	—
4	#15692312	—	2	12	no	—
5	C₂×#13108075	—	2	12	no	—
6	C₂²×D₄₀₄	C₄	1	3	yes	—
7	C₂²×C₄×D₁₀₁	C₂²	1	3	yes	—
8	C₂²×Q₈×C₁₀₁	C₄	1	1	yes	characteristic
9	C₂×#11717581	—	2	12	no	—
10	C₂³×D₁₀₁	C₂×C₄	1	3	yes	—
11	#13108075	—	2	32	no	—
12	C₂×D₄₀₄	—	2	12	no	—
13	C₂×D₄₀₄	—	2	6	no	—
14	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
15	C₂×C₄×D₁₀₁	—	2	12	no	—
16	C₂×C₄×D₁₀₁	D₄	1	12	yes	—
17	C₂×Q₈×C₁₀₁	—	2	6	no	—
18	C₂²×C₄×C₁₀₁	C₂×C₄	1	3	yes	—
19	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
20	C₂²×D₁₀₁	—	2	6	no	—
21	C₂²×D₁₀₁	—	2	12	no	—
22	C₂²×D₁₀₁	C₂²⋊C₄	1	6	yes	—
23	Q₈×C₁₀₁	—	2	8	no	—
24	C₂×C₄×C₁₀₁	—	2	6	no	—
25	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	6	yes	—
26	C₂×Dic₁₀₁	C₂×D₄	1	6	yes	—
27	D₄₀₄	—	2	24	no	—
28	C₄×D₁₀₁	—	2	24	no	—
29	#11717581	—	4	4	no	—
30	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	3	yes	—
31	C₂²×C₁₀₁	#131	1	4	yes	—
32	C₂×D₁₀₁	—	2	6	no	—
33	C₂×D₁₀₁	—	2	12	no	—
34	C₂×D₁₀₁	—	4	12	no	—
35	C₄×C₁₀₁	—	2	6	no	—
36	Dic₁₀₁	#120	1	4	yes	—
37	C₂×C₁₀₁	#379	1	1	yes	characteristic
38	C₂×C₁₀₁	#408	1	6	yes	—
39	D₁₀₁	—	4	6	no	—
40	#1780	—	101	1	no	maximal, Sylow
41	C₁₀₁	#1780	1	1	yes	characteristic, complemented, Sylow
42	C₂²×#41	—	101	1	no	—
43	C₂×C₂²⋊C₈	—	101	6	no	—
44	C₂×#41	—	202	12	no	—
45	C₂²×Q₈	#87938	1	1	yes	characteristic
46	C₂²×D₄	—	101	3	no	—
47	C₂³×C₄	—	101	3	no	—
48	C₂²×C₈	—	101	4	no	—
49	C₂²⋊C₈	—	202	12	no	—
50	C₂⁴	—	101	3	no	—
51	#41	—	202	32	no	—
52	C₂×Q₈	—	2	6	no	—
53	C₂×D₄	—	202	12	no	—
54	C₂×D₄	—	202	6	no	—
55	C₂²×C₄	—	101	1	no	—
56	C₂²×C₄	—	101	12	no	—
57	C₂²×C₄	—	202	12	no	—
58	C₂²×C₄	C₂×#87938	1	3	yes	—
59	C₂×C₈	—	202	12	no	—
60	C₂³	—	101	6	no	—
61	C₂³	—	202	6	no	—
62	C₂³	—	202	12	no	—
63	C₂³	C₂²×#87938	1	1	yes	characteristic
64	Q₈	—	2	8	no	—
65	D₄	—	202	24	no	—
66	C₂×C₄	—	2	6	no	—
67	C₂×C₄	—	101	6	no	—
68	C₂×C₄	—	202	24	no	—
69	C₂×C₄	C₂²⋊#87938	1	6	yes	—
70	C₈	—	404	4	no	—
71	C₂²	—	404	12	no	—
72	C₂²	—	202	6	no	—
73	C₂²	—	202	12	no	—
74	C₂²	C₂×C₂²⋊#87938	1	3	yes	—
75	C₂²	#15692343	1	4	yes	—
76	C₄	—	2	6	no	—
77	C₄	—	101	4	no	—
78	C₂	—	404	6	no	—
79	C₂	#20266118	1	1	yes	characteristic
80	C₂	#20266152	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87938
commutator, socle	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₂²×Q₈×C₁₀₁	C₄
prime radical	C₂²×#13108075	C₂
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	C₂²×Q₈	#87938
hypocenter	C₁₀₁	#1780
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15692312	C₂	semidirect
41. C₁₀₁	#1780	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692312	2
C₂²×#13108075	2
#1780	101