Subgroups — #27172237

Order 12928.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265977	C₂	1	3	yes	maximal, complemented
2	#20265145	C₂	1	3	yes	maximal, complemented
3	#338×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13108074	C₂²	1	3	yes	—
5	#15692288	—	2	6	no	—
6	#15692283	—	2	6	no	—
7	#15692277	—	2	6	no	—
8	C₂²×#11717579	C₂²	1	1	yes	characteristic
9	#15692167	—	2	6	no	—
10	C₂×M₁₆×C₁₀₁	C₂²	1	3	yes	—
11	#13108074	—	2	6	no	—
12	#13108074	—	2	6	no	—
13	C₂×Dic₂₀₂	D₄	1	3	yes	—
14	C₂×D₄₀₄	D₄	1	3	yes	—
15	C₂×C₂²⋊D₁₀₁	—	2	3	no	—
16	C₂×C₄×D₁₀₁	—	2	3	no	—
17	Dic₄₀₄	—	2	6	no	—
18	#13108063	—	2	12	no	—
19	D₈₀₈	—	2	6	no	—
20	C₂×#11717579	—	2	3	no	—
21	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₀₁	—	2	6	no	—
23	C₂×C₈×C₁₀₁	D₄	1	6	yes	—
24	C₂³×C₁₀₁	#41	1	1	yes	characteristic
25	C₂²×D₁₀₁	—	2	3	no	—
26	C₂×C₄×C₁₀₁	C₂×D₄	1	3	yes	—
27	C₂×C₄×C₁₀₁	C₂×D₄	1	3	yes	—
28	C₂²⋊D₁₀₁	—	4	6	no	—
29	C₂×Dic₁₀₁	—	2	3	no	—
30	D₄₀₄	—	2	6	no	—
31	C₄×D₁₀₁	—	4	3	no	—
32	C₄×D₁₀₁	—	4	3	no	—
33	Dic₂₀₂	—	2	6	no	—
34	C₈×C₁₀₁	—	2	6	no	—
35	#11717579	—	4	1	no	—
36	C₂²×C₁₀₁	—	4	1	no	—
37	C₂²×C₁₀₁	#121	1	3	yes	—
38	C₂×D₁₀₁	—	4	6	no	—
39	C₂×D₁₀₁	—	4	3	no	—
40	C₄×C₁₀₁	#120	1	3	yes	—
41	C₄×C₁₀₁	#127	1	1	yes	characteristic
42	Dic₁₀₁	—	4	3	no	—
43	C₂×C₁₀₁	—	2	3	no	—
44	C₂×C₁₀₁	#392	1	1	yes	characteristic
45	D₁₀₁	—	8	3	no	—
46	#2017	—	101	1	no	maximal, Sylow
47	C₁₀₁	#2017	1	1	yes	characteristic, complemented, Sylow
48	#575	—	101	3	no	—
49	#408	—	101	3	no	—
50	#338	D₁₀₁	1	1	yes	characteristic, complemented
51	C₂×#41	—	101	3	no	—
52	#137	—	202	6	no	—
53	AGL(1,ℤ/8)	—	202	6	no	—
54	#135	—	202	6	no	—
55	C₂×M₁₆	C₂×D₁₀₁	1	3	yes	—
56	C₂²×C₈	—	101	1	no	—
57	C₂²⋊C₈	—	202	6	no	—
58	#41	—	202	6	no	—
59	#41	—	202	6	no	—
60	C₂×Q₈	—	101	3	no	—
61	C₂×D₄	—	101	3	no	—
62	C₂×D₄	—	202	3	no	—
63	C₂²×C₄	—	202	3	no	—
64	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
65	Q₁₆	—	202	6	no	—
66	SD₁₆	—	202	12	no	—
67	D₈	—	202	6	no	—
68	M₁₆	—	2	6	no	—
69	C₂×C₈	—	202	3	no	—
70	C₂×C₈	D₄₀₄	1	6	yes	—
71	C₂³	—	202	3	no	—
72	C₂³	#13108075	1	1	yes	characteristic
73	Q₈	—	202	6	no	—
74	D₄	—	404	6	no	—
75	D₄	—	202	6	no	—
76	C₂×C₄	—	404	3	no	—
77	C₂×C₄	—	404	3	no	—
78	C₂×C₄	—	202	3	no	—
79	C₂×C₄	D₄×D₁₀₁	1	3	yes	—
80	C₂×C₄	C₂×D₄₀₄	1	3	yes	—
81	C₈	—	2	6	no	—
82	C₈	—	404	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	404	6	no	—
85	C₂²	—	404	3	no	—
86	C₂²	#15692230	1	3	yes	—
87	C₄	—	404	3	no	—
88	C₄	#15692225	1	3	yes	—
89	C₄	#15692252	1	1	yes	characteristic
90	C₂	—	2	3	no	—
91	C₂	—	808	3	no	—
92	C₂	#20265087	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15692252
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#338×C₁₀₁	C₂
socle	C₂×C₁₀₁	#392
hypercenter	#338	D₁₀₁
hypocenter	C₁₀₁	#2017
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #20265977	C₂	semidirect
2. #20265145	C₂	semidirect
3. #338×C₁₀₁	C₂	semidirect
47. C₁₀₁	#2017	semidirect
50. #338	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265977	2
#20265145	2
#338×C₁₀₁	2
#2017	101