Subgroups — #88138

Order 416.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#6103	C₂	1	1	yes	maximal, characteristic, complemented
7	#6101	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×D₁₃	C₂²	1	1	yes	characteristic
10	C₂²×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²⋊D₁₃	—	2	2	no	—
14	C₂²⋊D₁₃	—	2	2	no	—
15	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
17	D₅₂	—	2	2	no	—
18	C₄×D₁₃	—	2	2	no	—
19	C₂²×C₁₃	—	2	1	no	—
20	C₂²×C₁₃	—	2	1	no	—
21	C₂²×C₁₃	—	2	1	no	—
22	C₂²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×D₁₃	—	2	1	no	—
24	C₂×D₁₃	—	2	1	no	—
25	C₂×D₁₃	—	2	1	no	—
26	C₂×D₁₃	—	2	1	no	—
27	C₂×D₁₃	—	2	1	no	—
28	C₂×D₁₃	D₄	1	2	yes	complemented
29	C₄×C₁₃	—	2	1	no	—
30	C₄×C₁₃	—	2	1	no	—
31	Dic₁₃	—	2	1	no	—
32	Dic₁₃	D₄	1	2	yes	complemented
33	#121	—	13	1	no	maximal, Sylow
34	C₂×C₁₃	—	4	1	no	—
35	C₂×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₁₃	#41	1	1	yes	characteristic
38	D₁₃	—	2	2	no	—
39	D₁₃	—	4	1	no	—
40	C₂×D₄	—	13	1	no	—
41	C₂×D₄	—	13	1	no	—
42	C₂×D₄	—	13	1	no	—
43	C₂²×C₄	—	13	1	no	—
44	C₄⋊C₄	—	13	1	no	—
45	C₂²⋊C₄	—	13	1	no	—
46	C₂²⋊C₄	D₁₃	1	1	yes	characteristic, complemented
47	C₁₃	#121	1	1	yes	characteristic, complemented, Sylow
48	C₂³	—	13	1	no	—
49	C₂³	—	13	1	no	—
50	C₂³	C₂×D₁₃	1	1	yes	characteristic
51	D₄	—	26	2	no	—
52	D₄	—	26	2	no	—
53	D₄	—	26	2	no	—
54	C₂×C₄	—	13	1	no	—
55	C₂×C₄	—	13	1	no	—
56	C₂×C₄	—	26	2	no	—
57	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
58	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	13	2	no	—
63	C₂²	—	26	1	no	—
64	C₂²	—	26	1	no	—
65	C₂²	—	26	1	no	—
66	C₂²	—	26	1	no	—
67	C₂²	—	26	1	no	—
68	C₂²	C₂²×D₁₃	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	—	13	2	no	—
72	C₄	—	26	1	no	—
73	C₂	—	4	1	no	—
74	C₂	—	26	2	no	—
75	C₂	—	52	1	no	—
76	C₂	#6127	1	1	yes	characteristic
77	C₂	D₄×D₁₃	1	1	yes	characteristic
78	C₂	D₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂³
Fitting	C₁₃×C₂²⋊C₄	C₂
hypercenter	C₂²⋊C₄	D₁₃
hypocenter	C₁₃	#121
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊D₁₃	C₂	semidirect
2. C₂×C₂²⋊D₁₃	C₂	semidirect
3. C₂×D₅₂	C₂	semidirect
4. C₂×C₄×D₁₃	C₂	semidirect
5. C₁₃×C₂²⋊C₄	C₂	semidirect
6. #6103	C₂	semidirect
7. #6101	C₂	semidirect
28. C₂×D₁₃	D₄	semidirect
32. Dic₁₃	D₄	semidirect
46. C₂²⋊C₄	D₁₃	semidirect
47. C₁₃	#121	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₁₃	2
C₂×D₅₂	2
C₂×C₄×D₁₃	2
C₁₃×C₂²⋊C₄	2
#6103	2
#6101	2
#121	13