Subgroups — #27186259

Order 12928.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266086	C₂	1	1	yes	maximal, characteristic, complemented
2	#20266084	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265564	C₂	1	1	yes	maximal, characteristic
4	#15692316	—	2	1	no	—
5	#15692314	—	2	1	no	—
6	C₂×#13108062	—	2	1	no	—
7	C₂×#13108059	—	2	1	no	—
8	#15692243	C₂²	1	1	yes	characteristic
9	#15692166	C₄	1	1	yes	characteristic, complemented
10	#127×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	#13108062	—	4	2	no	—
16	#13108059	—	4	2	no	—
17	C₂×#11717579	—	2	1	no	—
18	#13108051	—	4	1	no	—
19	C₂×D₄×C₁₀₁	—	2	1	no	—
20	C₂×D₄×C₁₀₁	—	4	1	no	—
21	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₁₀₁	—	4	1	no	—
23	C₂²×D₁₀₁	—	2	1	no	—
24	C₂×#87938	—	8	1	no	—
25	D₄×C₁₀₁	—	4	1	no	—
26	D₄×C₁₀₁	—	4	1	no	—
27	D₄×C₁₀₁	—	8	2	no	—
28	C₂×C₄×C₁₀₁	—	2	1	no	—
29	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₁	—	2	1	no	—
31	D₄₀₄	—	4	1	no	—
32	Dic₂₀₂	—	4	1	no	—
33	#11717581	—	8	1	no	—
34	#11717579	—	2	2	no	—
35	C₂²×C₁₀₁	—	8	2	no	—
36	C₂²×C₁₀₁	—	8	1	no	—
37	C₂²×C₁₀₁	#99	1	1	yes	characteristic
38	C₂×D₁₀₁	—	4	1	no	—
39	C₂×D₁₀₁	—	8	1	no	—
40	#87938	—	16	1	no	—
41	C₄×C₁₀₁	—	2	1	no	—
42	C₄×C₁₀₁	—	4	1	no	—
43	Dic₁₀₁	—	4	1	no	—
44	C₂×C₁₀₁	—	2	1	no	—
45	C₂×C₁₀₁	—	16	1	no	—
46	C₂×C₁₀₁	#353	1	1	yes	characteristic
47	D₁₀₁	—	8	1	no	—
48	#1406	—	101	1	no	maximal, Sylow
49	C₁₀₁	#1406	1	1	yes	characteristic, complemented, Sylow
50	#496	—	101	1	no	—
51	#355	—	101	1	no	—
52	#354	—	101	1	no	—
53	C₂×SD₁₆	—	202	1	no	—
54	C₂×D₈	—	202	1	no	—
55	#127	#87938	1	1	yes	characteristic, complemented
56	#124	—	101	1	no	—
57	#101	—	202	1	no	—
58	#99	—	202	1	no	—
59	#97	—	101	1	no	—
60	C₂×Q₈	—	101	1	no	—
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	—	4	1	no	—
63	C₂×D₄	—	101	1	no	—
64	SD₁₆	—	404	2	no	—
65	D₈	—	404	2	no	—
66	M₁₆	—	404	1	no	—
67	C₂×C₈	—	202	1	no	—
68	C₂²⋊C₄	—	404	1	no	—
69	C₂²⋊C₄	—	404	1	no	—
70	C₄²	C₂×#87938	1	1	yes	characteristic
71	C₂³	—	4	1	no	—
72	C₂³	—	202	1	no	—
73	Q₈	—	404	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	4	1	no	—
76	D₄	—	8	2	no	—
77	D₄	—	404	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	808	1	no	—
80	C₂×C₄	—	202	1	no	—
81	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
82	C₈	—	808	1	no	—
83	C₈	—	202	2	no	—
84	C₂²	—	8	2	no	—
85	C₂²	—	8	1	no	—
86	C₂²	—	808	1	no	—
87	C₂²	—	404	1	no	—
88	C₂²	#15692313	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	1616	1	no	—
92	C₄	—	404	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	16	1	no	—
95	C₂	—	808	1	no	—
96	C₂	#20266081	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20266081
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	C₄²	C₂×#87938
Fitting	#127×C₁₀₁	C₄
socle	C₂×C₁₀₁	#353
prime radical	#20265564	C₂
cosocle	#15692243	C₂²
hypercenter	#127	#87938
hypocenter	C₁₀₁	#1406
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20266086	C₂	semidirect
2. #20266084	C₂	semidirect
9. #15692166	C₄	semidirect
10. #127×C₁₀₁	C₄	semidirect
49. C₁₀₁	#1406	semidirect
55. #127	#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
#20265564	2
#1406	101