Subgroups — #27193321

Order 13184.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆₄₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₀₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#366×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₂₄	—	2	1	no	—
5	D₁₆₄₈	—	2	1	no	—
6	D₁₆₄₈	C₄	1	2	yes	complemented
7	C₁₀₃⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
9	#107×C₁₀₃	—	2	1	no	—
10	C₂×D₄₁₂	—	4	1	no	—
11	D₈₂₄	—	2	2	no	—
12	D₈₂₄	—	4	1	no	—
13	C₂×C₈×C₁₀₃	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₁₀₃	—	4	1	no	—
15	C₁₆×C₁₀₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₀₃	D₄	1	1	yes	characteristic
17	C₂²×D₁₀₃	—	8	1	no	—
18	C₂×C₄×C₁₀₃	—	8	1	no	—
19	C₂×C₄×C₁₀₃	D₈	1	1	yes	characteristic
20	D₄₁₂	—	4	2	no	—
21	D₄₁₂	—	8	1	no	—
22	C₈×C₁₀₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₀₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₀₃	D₁₆	1	1	yes	characteristic
25	C₂×D₁₀₃	—	8	2	no	—
26	C₂×D₁₀₃	—	16	1	no	—
27	C₂×D₁₀₃	—	16	1	no	—
28	C₄×C₁₀₃	—	16	1	no	—
29	C₄×C₁₀₃	#102	1	1	yes	characteristic
30	C₄×C₁₀₃	SD₃₂	1	1	yes	characteristic
31	C₂×C₁₀₃	#357	1	1	yes	characteristic
32	C₂×C₁₀₃	D₃₂	1	1	yes	characteristic
33	C₂×C₁₀₃	SD₆₄	1	1	yes	characteristic
34	D₁₀₃	—	16	2	no	—
35	#1415	—	103	1	no	maximal, Sylow
36	C₁₀₃	#1415	1	1	yes	characteristic, complemented, Sylow
37	C₂×D₁₆	—	103	1	no	—
38	C₂×C₃₂	—	103	1	no	—
39	#366	D₁₀₃	1	1	yes	characteristic, complemented
40	C₂×D₈	—	206	1	no	—
41	D₁₆	—	103	2	no	—
42	D₁₆	—	206	1	no	—
43	C₂×C₁₆	C₂×D₁₀₃	1	1	yes	characteristic
44	#107	—	2	1	no	—
45	C₃₂	—	206	1	no	—
46	C₂×D₄	—	412	1	no	—
47	D₈	—	412	1	no	—
48	D₈	—	206	2	no	—
49	C₂×C₈	C₂²⋊D₁₀₃	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₁₆	C₄×D₁₀₃	1	1	yes	characteristic
52	C₁₆	D₄₁₂	1	1	yes	characteristic
53	C₂³	—	824	1	no	—
54	D₄	—	824	1	no	—
55	D₄	—	412	2	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	#13110090	1	1	yes	characteristic
58	C₈	(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
59	C₈	#13110092	1	1	yes	characteristic
60	C₂²	—	824	2	no	—
61	C₂²	—	1648	1	no	—
62	C₂²	—	1648	1	no	—
63	C₂²	#15692962	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	#15692941	1	1	yes	characteristic
66	C₄	#15692964	1	1	yes	characteristic
67	C₂	—	1648	2	no	—
68	C₂	#20267591	1	1	yes	characteristic
69	C₂	#20267626	1	1	yes	characteristic
70	C₂	#20267628	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692962
commutator	C₁₆×C₁₀₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₃
Fitting	#366×C₁₀₃	C₂
socle	C₂²×C₁₀₃	D₁₆
prime radical	C₂×D₁₆₄₈	C₂
cosocle	C₂×C₁₆×C₁₀₃	C₂²
hypercenter	#366	D₁₀₃
hypocenter	C₁₀₃	#1415
LCS	G ▷ C₁₆×C₁₀₃ ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₀₃⋊C₃₂	C₂	semidirect
3. #366×C₁₀₃	C₂	semidirect
6. D₁₆₄₈	C₄	semidirect
36. C₁₀₃	#1415	semidirect
39. #366	D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆₄₈	2
C₂×C₁₀₃⋊C₃₂	2
#366×C₁₀₃	2
#1415	103