Subgroups — #27193118

Order 13184.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20267909	C₂	1	1	yes	maximal, characteristic
2	#20267612	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈²×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#15693043	C₄	1	1	yes	characteristic
5	#15693027	—	2	1	no	—
6	#15692961	—	2	1	no	—
7	#15692957	C₄	1	1	yes	characteristic
8	#15692949	—	2	1	no	—
9	C₄×C₈×C₁₀₃	—	2	1	no	—
10	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
11	#13110105	—	2	1	no	—
12	Dic₄₁₂	C₈	1	1	yes	characteristic, complemented
13	#13110094	—	2	1	no	—
14	D₈₂₄	C₈	1	1	yes	characteristic, complemented
15	#13110089	—	2	1	no	—
16	#13110088	—	4	1	no	—
17	C₈×D₁₀₃	—	4	1	no	—
18	C₁₀₃⋊C₁₆	—	4	1	no	—
19	C₂×C₈×C₁₀₃	—	2	2	no	—
20	C₂×C₈×C₁₀₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₃	D₄	1	1	yes	characteristic
22	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₃	—	2	1	no	—
24	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²⋊D₁₀₃	—	4	1	no	—
26	D₄₁₂	—	2	1	no	—
27	C₄×D₁₀₃	—	4	1	no	—
28	Dic₂₀₆	—	2	1	no	—
29	C₈×C₁₀₃	—	2	4	no	—
30	C₈×C₁₀₃	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₀₃	M₁₆	1	1	yes	characteristic
32	C₈×C₁₀₃	D₈	1	1	yes	characteristic
33	C₈×C₁₀₃	SD₁₆	1	1	yes	characteristic
34	C₁₀₃⋊C₈	—	4	1	no	—
35	C₂²×C₁₀₃	#104	1	1	yes	characteristic
36	C₂×D₁₀₃	—	4	1	no	—
37	C₄×C₁₀₃	—	2	2	no	—
38	C₄×C₁₀₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₀₃	#102	1	1	yes	characteristic
40	Dic₁₀₃	—	4	1	no	—
41	C₂×C₁₀₃	—	2	1	no	—
42	C₂×C₁₀₃	D₄⋊C₈	1	1	yes	characteristic
43	D₁₀₃	—	8	1	no	—
44	#1335	—	103	1	no	maximal, Sylow
45	C₁₀₃	#1335	1	1	yes	characteristic, complemented, Sylow
46	#443	—	103	1	no	—
47	#364	—	103	1	no	—
48	C₈²	D₁₀₃	1	1	yes	characteristic, complemented
49	#135	—	103	1	no	—
50	#131	—	206	1	no	—
51	M₃₂	—	206	1	no	—
52	#108	—	103	1	no	—
53	#104	—	206	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
56	#41	—	206	1	no	—
57	Q₁₆	—	103	1	no	—
58	SD₁₆	—	206	1	no	—
59	D₈	—	103	1	no	—
60	M₁₆	—	412	1	no	—
61	M₁₆	—	206	1	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	412	1	no	—
64	C₂×C₈	C₄×D₁₀₃	1	1	yes	characteristic
65	C₂×C₈	D₄₁₂	1	1	yes	characteristic
66	C₄²	C₂²⋊D₁₀₃	1	1	yes	characteristic
67	C₁₆	—	412	1	no	—
68	Q₈	—	206	1	no	—
69	D₄	—	412	1	no	—
70	D₄	—	206	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	412	1	no	—
73	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
74	C₈	—	2	4	no	—
75	C₈	—	412	1	no	—
76	C₈	C₈×D₁₀₃	1	1	yes	characteristic
77	C₈	#13110088	1	1	yes	characteristic
78	C₈	D₈₂₄	1	1	yes	characteristic
79	C₈	#13110094	1	1	yes	characteristic
80	C₂²	—	412	1	no	—
81	C₂²	#15692949	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	412	1	no	—
84	C₄	(C₂×D₁₀₃)⋊C₈	1	1	yes	characteristic
85	C₄	#15692943	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	824	1	no	—
88	C₂	#20267520	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₈₂₄
commutator	C₈×C₁₀₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₃
Fitting	C₈²×C₁₀₃	C₂
socle	C₂×C₁₀₃	D₄⋊C₈
prime radical	#15693043	C₄
cosocle	C₄×C₈×C₁₀₃	C₂²
hypercenter	C₈²	D₁₀₃
hypocenter	C₁₀₃	#1335
LCS	G ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #20267612	C₂	semidirect
3. C₈²×C₁₀₃	C₂	semidirect
12. Dic₄₁₂	C₈	semidirect
14. D₈₂₄	C₈	semidirect
45. C₁₀₃	#1335	semidirect
48. C₈²	D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
#20267909	2
#20267612	2
C₈²×C₁₀₃	2
#1335	103