Subgroups — #27186322

Order 12928.

736 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×#87938)⋊C₄	C₂	1	6	yes	maximal
2	C₂²×Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂³×#87938	C₂²	1	4	yes	—
4	(C₂×#87938)⋊C₄	—	2	12	no	—
5	C₂²×Dic₂₀₂	C₄	1	3	yes	complemented
6	C₂×Q₈×D₁₀₁	—	2	12	no	—
7	C₂²×C₄×D₁₀₁	C₂²	1	3	yes	—
8	C₂²×Q₈×C₁₀₁	C₄	1	1	yes	characteristic, complemented
9	C₂²×#87938	—	2	12	no	—
10	C₂²×#87938	—	2	12	no	—
11	C₂²×#87938	Q₈	1	8	yes	complemented
12	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
13	C₂×Dic₂₀₂	—	2	6	no	—
14	C₂×Dic₂₀₂	—	2	12	no	—
15	Q₈×D₁₀₁	—	2	32	no	—
16	C₂²×Dic₁₀₁	C₂×C₄	1	3	yes	—
17	C₂×C₄×D₁₀₁	—	2	12	no	—
18	C₂×C₄×D₁₀₁	D₄	1	12	yes	—
19	C₂×Q₈×C₁₀₁	—	2	6	no	—
20	C₂²×C₄×C₁₀₁	C₂×C₄	1	3	yes	—
21	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
22	C₂²×D₁₀₁	C₂×D₄	1	6	yes	—
23	C₂²×D₁₀₁	C₂×Q₈	1	4	yes	—
24	C₂²×D₁₀₁	#41	1	4	yes	—
25	C₂×#87938	—	2	24	no	—
26	C₂×#87938	—	4	4	no	—
27	C₂×#87938	—	4	12	no	—
28	Q₈×C₁₀₁	—	2	8	no	—
29	C₂×C₄×C₁₀₁	—	2	6	no	—
30	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	6	yes	—
31	C₂×Dic₁₀₁	—	2	6	no	—
32	C₂×Dic₁₀₁	C₂²⋊C₄	1	6	yes	—
33	C₄×D₁₀₁	—	2	24	no	—
34	Dic₂₀₂	—	2	24	no	—
35	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	3	yes	—
36	C₂²×C₁₀₁	C₄×Q₈	1	4	yes	—
37	C₂×D₁₀₁	#120	1	4	yes	—
38	C₂×D₁₀₁	C₂²⋊Q₈	1	12	yes	—
39	C₂×D₁₀₁	#124	1	6	yes	—
40	C₂×D₁₀₁	C₄⋊Q₈	1	6	yes	—
41	#87938	—	4	8	no	—
42	C₄×C₁₀₁	—	2	6	no	—
43	Dic₁₀₁	—	2	6	no	—
44	C₂×C₁₀₁	#379	1	1	yes	characteristic
45	C₂×C₁₀₁	#391	1	6	yes	—
46	D₁₀₁	#393	1	4	yes	—
47	D₁₀₁	#395	1	4	yes	—
48	#1446	—	101	1	no	maximal, Sylow
49	C₁₀₁	#1446	1	1	yes	characteristic, complemented, Sylow
50	C₂³×Q₈	—	101	1	no	—
51	C₂×#95	—	101	6	no	—
52	C₂²×Q₈	—	101	3	no	—
53	C₂²×Q₈	—	202	12	no	—
54	C₂²×Q₈	#87938	1	1	yes	characteristic, complemented
55	C₂³×C₄	—	101	3	no	—
56	C₂³×C₄	—	101	4	no	—
57	#95	—	202	12	no	—
58	C₂⁴	—	101	1	no	—
59	C₂×Q₈	—	2	6	no	—
60	C₂×Q₈	—	202	12	no	—
61	C₂×Q₈	—	202	32	no	—
62	C₂×Q₈	—	202	6	no	—
63	C₂²×C₄	—	101	3	no	—
64	C₂²×C₄	—	101	12	no	—
65	C₂²×C₄	—	101	8	no	—
66	C₂²×C₄	—	202	12	no	—
67	C₂²×C₄	—	202	12	no	—
68	C₂²×C₄	—	202	12	no	—
69	C₂²×C₄	C₂×#87938	1	3	yes	—
70	C₂³	—	101	6	no	—
71	C₂³	—	101	4	no	—
72	C₂³	—	101	4	no	—
73	C₂³	C₂²×#87938	1	1	yes	characteristic
74	Q₈	—	2	8	no	—
75	Q₈	—	202	24	no	—
76	C₂×C₄	—	2	6	no	—
77	C₂×C₄	—	101	6	no	—
78	C₂×C₄	—	404	4	no	—
79	C₂×C₄	—	404	12	no	—
80	C₂×C₄	—	202	24	no	—
81	C₂×C₄	—	202	24	no	—
82	C₂×C₄	—	202	6	no	—
83	C₂×C₄	C₂²⋊#87938	1	6	yes	—
84	C₂²	—	101	4	no	—
85	C₂²	—	101	6	no	—
86	C₂²	—	101	12	no	—
87	C₂²	—	101	6	no	—
88	C₂²	C₂×C₂²⋊#87938	1	3	yes	—
89	C₂²	Q₈×#87938	1	4	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	404	8	no	—
92	C₄	—	202	6	no	—
93	C₂	—	101	4	no	—
94	C₂	—	101	4	no	—
95	C₂	#20266118	1	1	yes	characteristic
96	C₂	(C₂×#87938)⋊Q₈	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87938
commutator, socle	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₂²×Q₈×C₁₀₁	C₄
prime radical, cosocle	C₂³×D₁₀₁	C₂³
hypercenter	C₂²×Q₈	#87938
hypocenter	C₁₀₁	#1446
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
5. C₂²×Dic₂₀₂	C₄	semidirect
8. C₂²×Q₈×C₁₀₁	C₄	semidirect
11. C₂²×#87938	Q₈	semidirect
49. C₁₀₁	#1446	semidirect
54. C₂²×Q₈	#87938	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×#87938)⋊C₄	2
C₂²×Q₈×D₁₀₁	2
#1446	101