Subgroups — #27169091

Order 12928.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265256	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#13108049	C₂	1	1	yes	maximal, characteristic, complemented
3	#346×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13108057	—	2	1	no	—
5	C₄×D₄₀₄	C₄	1	1	yes	characteristic
6	C₂×#13108049	—	2	1	no	—
7	C₄⋊#11717579	C₄	1	1	yes	characteristic
8	#15692167	—	2	1	no	—
9	C₂×C₁₆×C₁₀₁	—	2	1	no	—
10	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×D₄₀₄	C₈	1	1	yes	characteristic
12	C₂×C₄×D₁₀₁	—	2	1	no	—
13	#13108057	—	4	1	no	—
14	#13108057	—	4	1	no	—
15	C₂×#11717579	—	2	1	no	—
16	C₄⋊Dic₁₀₁	C₈	1	1	yes	characteristic
17	#13108051	—	2	1	no	—
18	#13108049	—	4	1	no	—
19	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
21	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₁₆×C₁₀₁	—	2	2	no	—
23	C₂²×D₁₀₁	—	2	1	no	—
24	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₀₁	M₁₆	1	1	yes	characteristic
27	C₂×Dic₁₀₁	—	2	1	no	—
28	D₄₀₄	—	4	1	no	—
29	C₄×D₁₀₁	—	4	1	no	—
30	C₄×D₁₀₁	—	4	1	no	—
31	C₈×C₁₀₁	—	2	1	no	—
32	C₈×C₁₀₁	D₈	1	1	yes	characteristic
33	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
34	#11717579	—	4	1	no	—
35	C₂²×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
36	C₂×D₁₀₁	—	4	1	no	—
37	C₂×D₁₀₁	—	4	1	no	—
38	C₂×D₁₀₁	—	4	1	no	—
39	C₄×C₁₀₁	—	2	1	no	—
40	C₄×C₁₀₁	—	2	1	no	—
41	C₄×C₁₀₁	#102	1	1	yes	characteristic
42	C₄×C₁₀₁	#104	1	1	yes	characteristic
43	Dic₁₀₁	—	4	1	no	—
44	C₂×C₁₀₁	D₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₀₁	#349	1	1	yes	characteristic
46	C₂×C₁₀₁	#350	1	1	yes	characteristic
47	D₁₀₁	—	8	1	no	—
48	#1330	—	101	1	no	maximal, Sylow
49	C₁₀₁	#1330	1	1	yes	characteristic, complemented, Sylow
50	#436	—	101	1	no	—
51	C₄⋊C₁₆	—	101	1	no	—
52	#346	D₁₀₁	1	1	yes	characteristic, complemented
53	C₂×M₁₆	—	202	1	no	—
54	C₄×D₄	—	101	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	202	1	no	—
57	C₄⋊C₈	—	101	1	no	—
58	C₂²⋊C₈	—	202	1	no	—
59	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
60	C₂×D₄	—	101	1	no	—
61	C₂²×C₄	—	202	1	no	—
62	M₁₆	—	404	1	no	—
63	M₁₆	—	404	1	no	—
64	C₂×C₈	—	202	1	no	—
65	C₂×C₈	D₄₀₄	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
67	C₄⋊C₄	—	101	1	no	—
68	C₂²⋊C₄	—	202	1	no	—
69	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
70	C₁₆	—	2	2	no	—
71	C₁₆	—	404	1	no	—
72	C₂³	—	202	1	no	—
73	D₄	—	404	1	no	—
74	C₂×C₄	—	404	1	no	—
75	C₂×C₄	—	404	1	no	—
76	C₂×C₄	—	202	1	no	—
77	C₂×C₄	#13108051	1	1	yes	characteristic
78	C₂×C₄	C₈×D₁₀₁	1	1	yes	characteristic
79	C₂×C₄	#13108057	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	404	1	no	—
82	C₈	D₈₀₈	1	1	yes	characteristic
83	C₈	#13108063	1	1	yes	characteristic
84	C₂²	—	404	1	no	—
85	C₂²	—	404	1	no	—
86	C₂²	—	404	1	no	—
87	C₂²	#15692167	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	404	1	no	—
91	C₄	#15692177	1	1	yes	characteristic
92	C₄	#15692183	1	1	yes	characteristic
93	C₂	—	808	1	no	—
94	C₂	#20264906	1	1	yes	characteristic
95	C₂	#20264958	1	1	yes	characteristic
96	C₂	#20264962	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13108051
commutator	C₂×C₄×C₁₀₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	#346×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²⋊C₈
prime radical	C₂×D₄₀₄	C₈
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	#346	D₁₀₁
hypocenter	C₁₀₁	#1330
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄⋊#13108049	C₂	semidirect
3. #346×C₁₀₁	C₂	semidirect
49. C₁₀₁	#1330	semidirect
52. #346	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265256	2
C₄⋊#13108049	2
#346×C₁₀₁	2
#1330	101