Subgroups — #27187535

Order 12928.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266221	C₂	1	2	yes	maximal, complemented
2	#20266106	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266092	C₂	1	2	yes	maximal, complemented
4	#20266064	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265610	C₂	1	1	yes	maximal, characteristic
6	#15692343	—	2	2	no	—
7	D₁₀₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#15692326	C₂²	1	2	yes	—
9	#15692320	—	2	2	no	—
10	C₈×#87938	C₂²	1	1	yes	characteristic
11	#15692278	C₂²	1	2	yes	—
12	#15692198	C₄	1	2	yes	complemented
13	D₁₆₁₆	C₄	1	1	yes	characteristic, complemented
14	C₁₆×D₁₀₁	C₂²	1	1	yes	characteristic
15	Q₃₂×C₁₀₁	C₄	1	1	yes	characteristic, complemented
16	#13108084	—	2	2	no	—
17	#13108084	—	4	2	no	—
18	D₁₀₁⋊C₈	—	4	2	no	—
19	D₁₀₁⋊C₈	D₄	1	1	yes	characteristic
20	C₄×#87938	D₄	1	1	yes	characteristic
21	#13108077	—	2	1	no	—
22	#13108075	—	2	2	no	—
23	#13108061	—	2	2	no	—
24	D₈₀₈	C₂×C₄	1	2	yes	—
25	C₈×D₁₀₁	C₂³	1	1	yes	characteristic
26	#13108049	C₂×C₄	1	1	yes	characteristic
27	Q₁₆×C₁₀₁	C₂×C₄	1	2	yes	—
28	C₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
29	C₂×#87938	—	2	1	no	—
30	Q₈×C₁₀₁	—	2	2	no	—
31	D₄₀₄	—	2	2	no	—
32	D₄₀₄	—	4	2	no	—
33	C₄×D₁₀₁	—	4	2	no	—
34	C₄×D₁₀₁	C₂×D₄	1	1	yes	characteristic
35	#11717581	—	4	2	no	—
36	#11717581	D₈	1	2	yes	—
37	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
38	#11717579	#41	1	1	yes	characteristic
39	C₂×D₁₀₁	—	4	2	no	—
40	C₂×D₁₀₁	#135	1	1	yes	characteristic
41	#87938	—	4	1	no	—
42	C₄×C₁₀₁	—	4	2	no	—
43	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
44	Dic₁₀₁	C₂×D₈	1	1	yes	characteristic
45	C₂×C₁₀₁	C₄×D₈	1	1	yes	characteristic
46	D₁₀₁	—	2	1	no	—
47	D₁₀₁	—	8	2	no	—
48	#2179	—	101	1	no	maximal, Sylow
49	C₁₀₁	#2179	1	1	yes	characteristic, complemented, Sylow
50	#508	—	101	1	no	—
51	#443	—	101	2	no	—
52	#368	—	101	1	no	—
53	#359	—	101	2	no	—
54	#346	—	101	1	no	—
55	#135	—	101	2	no	—
56	#131	—	202	2	no	—
57	Q₃₂	#87938	1	1	yes	characteristic, complemented
58	SD₃₂	—	101	2	no	—
59	D₁₆	—	101	1	no	—
60	C₂×C₁₆	—	101	1	no	—
61	C₂×C₁₆	—	101	1	no	—
62	#108	—	101	2	no	—
63	#104	—	202	2	no	—
64	C₄×C₈	—	101	1	no	—
65	#41	—	202	2	no	—
66	Q₁₆	C₂×#87938	1	2	yes	—
67	SD₁₆	—	202	2	no	—
68	D₈	—	101	2	no	—
69	M₁₆	—	404	2	no	—
70	M₁₆	—	202	2	no	—
71	C₂×C₈	—	101	1	no	—
72	C₂×C₈	—	101	1	no	—
73	C₂×C₈	—	404	2	no	—
74	C₄²	—	101	1	no	—
75	C₁₆	—	101	1	no	—
76	C₁₆	—	202	1	no	—
77	C₁₆	C₂×#87938	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	404	2	no	—
80	D₄	—	202	2	no	—
81	C₂×C₄	—	101	1	no	—
82	C₂×C₄	—	404	2	no	—
83	C₂×C₄	—	202	1	no	—
84	C₈	—	101	2	no	—
85	C₈	—	101	1	no	—
86	C₈	—	404	2	no	—
87	C₈	C₂²×#87938	1	1	yes	characteristic
88	C₂²	—	101	1	no	—
89	C₂²	—	404	2	no	—
90	C₄	—	4	2	no	—
91	C₄	—	101	1	no	—
92	C₄	—	404	1	no	—
93	C₄	D₄×#87938	1	1	yes	characteristic
94	C₂	—	808	2	no	—
95	C₂	—	202	1	no	—
96	C₂	D₈×#87938	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#87938
commutator	C₈×C₁₀₁	C₂²×C₄
Frattini	C₈	C₂²×#87938
Fitting	Q₃₂×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₄×D₈
prime radical	#20265610	C₂
cosocle	C₈×D₁₀₁	C₂³
hypercenter	Q₃₂	#87938
hypocenter	C₁₀₁	#2179
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #20266221	C₂	semidirect
2. #20266106	C₂	semidirect
3. #20266092	C₂	semidirect
4. #20266064	C₂	semidirect
12. #15692198	C₄	semidirect
13. D₁₆₁₆	C₄	semidirect
15. Q₃₂×C₁₀₁	C₄	semidirect
49. C₁₀₁	#2179	semidirect
57. Q₃₂	#87938	semidirect

Maximal subgroups

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

subgroup	index
#20266221	2
#20266106	2
#20266092	2
#20266064	2
#20265610	2
#2179	101