Subgroups — #27193112

Order 13184.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₈₂₄	C₂	1	1	yes	maximal, characteristic
2	#20267542	C₂	1	1	yes	maximal, characteristic, complemented
3	#346×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₂₄	C₄	1	1	yes	characteristic
5	C₄×D₄₁₂	—	2	1	no	—
6	#15692955	C₄	1	1	yes	characteristic
7	C₄⋊(C₁₀₃⋊C₈)	—	2	1	no	—
8	#15692943	—	2	1	no	—
9	C₂×C₁₆×C₁₀₃	—	2	1	no	—
10	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
11	C₂×D₄₁₂	—	2	1	no	—
12	C₂×C₄×D₁₀₃	—	4	1	no	—
13	D₈₂₄	—	2	1	no	—
14	D₈₂₄	C₈	1	2	yes	complemented
15	C₂×C₁₀₃⋊C₈	—	4	1	no	—
16	C₄⋊Dic₁₀₃	—	2	1	no	—
17	(C₂×D₁₀₃)⋊C₄	—	4	1	no	—
18	C₂×C₈×C₁₀₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₃	D₄	1	1	yes	characteristic
20	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
21	C₁₆×C₁₀₃	—	2	2	no	—
22	C₂²×D₁₀₃	—	4	1	no	—
23	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₃	D₈	1	1	yes	characteristic
25	C₂×C₄×C₁₀₃	SD₁₆	1	1	yes	characteristic
26	C₂×Dic₁₀₃	—	4	1	no	—
27	D₄₁₂	—	2	2	no	—
28	D₄₁₂	—	4	1	no	—
29	C₄×D₁₀₃	—	8	2	no	—
30	C₈×C₁₀₃	—	2	1	no	—
31	C₈×C₁₀₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₀₃	M₁₆	1	1	yes	characteristic
33	C₁₀₃⋊C₈	—	8	1	no	—
34	C₂²×C₁₀₃	#102	1	1	yes	characteristic
35	C₂×D₁₀₃	—	4	2	no	—
36	C₂×D₁₀₃	—	8	1	no	—
37	C₂×D₁₀₃	—	8	1	no	—
38	C₄×C₁₀₃	—	2	1	no	—
39	C₄×C₁₀₃	—	2	1	no	—
40	C₄×C₁₀₃	C₂²⋊C₈	1	1	yes	characteristic
41	C₄×C₁₀₃	#104	1	1	yes	characteristic
42	Dic₁₀₃	—	8	1	no	—
43	C₂×C₁₀₃	D₄⋊C₈	1	1	yes	characteristic
44	C₂×C₁₀₃	#360	1	1	yes	characteristic
45	C₂×C₁₀₃	#361	1	1	yes	characteristic
46	D₁₀₃	—	8	2	no	—
47	#1333	—	103	1	no	maximal, Sylow
48	C₁₀₃	#1333	1	1	yes	characteristic, complemented, Sylow
49	C₄×D₈	—	103	1	no	—
50	#346	D₁₀₃	1	1	yes	characteristic, complemented
51	#334	—	103	1	no	—
52	C₂×D₈	—	103	1	no	—
53	C₄×D₄	—	206	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	#107	—	103	1	no	—
56	C₄⋊C₈	—	206	1	no	—
57	#102	—	206	1	no	—
58	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
59	C₂×D₄	—	206	1	no	—
60	C₂²×C₄	—	412	1	no	—
61	D₈	—	103	2	no	—
62	D₈	—	206	1	no	—
63	C₂×C₈	—	412	1	no	—
64	C₂×C₈	C₄×D₁₀₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₀₃	1	1	yes	characteristic
66	C₄⋊C₄	—	206	1	no	—
67	C₂²⋊C₄	—	412	1	no	—
68	C₄²	D₄₁₂	1	1	yes	characteristic
69	C₁₆	—	2	2	no	—
70	C₂³	—	412	1	no	—
71	D₄	—	412	1	no	—
72	D₄	—	206	2	no	—
73	C₂×C₄	—	824	2	no	—
74	C₂×C₄	—	412	1	no	—
75	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	D₈₂₄	1	1	yes	characteristic
77	C₂×C₄	#13110094	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	824	1	no	—
80	C₈	C₈×D₁₀₃	1	1	yes	characteristic
81	C₈	#13110088	1	1	yes	characteristic
82	C₂²	—	824	1	no	—
83	C₂²	—	824	1	no	—
84	C₂²	—	412	2	no	—
85	C₂²	#15692943	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	824	1	no	—
89	C₄	(C₂×D₁₀₃)⋊C₈	1	1	yes	characteristic
90	C₄	#15692949	1	1	yes	characteristic
91	C₂	—	824	2	no	—
92	C₂	#20267520	1	1	yes	characteristic
93	C₂	#20267602	1	1	yes	characteristic
94	C₂	#20267605	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692943
commutator	C₈×C₁₀₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₃
Fitting	#346×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#102
prime radical	C₂×D₈₂₄	C₄
cosocle	C₄×C₈×C₁₀₃	C₂²
hypercenter	#346	D₁₀₃
hypocenter	C₁₀₃	#1333
LCS	G ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #20267542	C₂	semidirect
3. #346×C₁₀₃	C₂	semidirect
14. D₈₂₄	C₈	semidirect
48. C₁₀₃	#1333	semidirect
50. #346	D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₈₂₄	2
#20267542	2
#346×C₁₀₃	2
#1333	103