Subgroups — #31783039

Order 17536.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₁₉₂	C₂	1	4	yes	maximal, complemented
2	#22577780	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₉₆	—	2	2	no	—
5	C₂×D₁₀₉₆	C₂²	1	4	yes	—
6	#17984085	—	2	2	no	—
7	D₂₁₉₂	—	2	8	no	—
8	C₂×C₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×D₅₄₈	—	2	4	no	—
11	C₂×D₅₄₈	—	4	4	no	—
12	D₁₀₉₆	—	2	8	no	—
13	D₁₀₉₆	—	4	4	no	—
14	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
16	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₃₇	D₄	1	4	yes	complemented
18	C₂²×D₁₃₇	—	4	4	no	—
19	D₅₄₈	—	4	4	no	—
20	D₅₄₈	—	4	4	no	—
21	D₅₄₈	—	8	8	no	—
22	C₂×C₄×C₁₃₇	D₈	1	2	yes	—
23	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
24	C₈×C₁₃₇	D₈	1	2	yes	—
25	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
26	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
27	C₂×D₁₃₇	—	8	4	no	—
28	C₂×D₁₃₇	—	8	8	no	—
29	C₄×C₁₃₇	D₁₆	1	4	yes	complemented
30	C₄×C₁₃₇	#127	1	1	yes	characteristic
31	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
32	C₂×C₁₃₇	#493	1	1	yes	characteristic
33	C₂×C₁₃₇	C₂×D₁₆	1	2	yes	—
34	D₁₃₇	—	16	4	no	—
35	C₁₃₇	#2246	1	1	yes	characteristic, complemented, Sylow
36	#2246	—	137	1	no	maximal, Sylow
37	C₂×D₁₆	—	137	4	no	—
38	#493	—	137	2	no	—
39	C₄×C₁₆	D₁₃₇	1	1	yes	characteristic, complemented
40	C₂×D₈	—	274	2	no	—
41	C₂×D₈	—	137	4	no	—
42	#127	—	274	2	no	—
43	D₁₆	—	274	8	no	—
44	C₂×C₁₆	C₂×D₁₃₇	1	2	yes	—
45	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
46	C₂×D₄	—	274	4	no	—
47	C₂×D₄	—	548	4	no	—
48	D₈	—	274	8	no	—
49	D₈	—	548	4	no	—
50	C₂×C₈	D₅₄₈	1	1	yes	characteristic
51	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
52	C₄²	D₅₄₈	1	1	yes	characteristic
53	C₁₆	D₅₄₈	1	4	yes	complemented
54	C₂³	—	548	4	no	—
55	D₄	—	548	4	no	—
56	D₄	—	548	4	no	—
57	D₄	—	1096	8	no	—
58	C₂×C₄	D₁₀₉₆	1	2	yes	—
59	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
60	C₈	D₁₀₉₆	1	2	yes	—
61	C₈	C₂×D₅₄₈	1	2	yes	—
62	C₂²	—	1096	4	no	—
63	C₂²	—	1096	8	no	—
64	C₂²	C₂×D₁₀₉₆	1	1	yes	characteristic
65	C₄	D₂₁₉₂	1	4	yes	complemented
66	C₄	#17984085	1	1	yes	characteristic
67	C₄	C₂×D₁₀₉₆	1	1	yes	characteristic
68	C₂	—	2192	4	no	—
69	C₂	#22577780	1	1	yes	characteristic
70	C₂	C₂×D₂₁₉₂	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₉₆
commutator, cosocle	C₂×C₈×C₁₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₄×C₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₈
hypercenter	C₄×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#2246
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₁₉₂	C₂	semidirect
2. #22577780	C₂	semidirect
3. C₄×C₁₆×C₁₃₇	C₂	semidirect
17. C₁₆×C₁₃₇	D₄	semidirect
29. C₄×C₁₃₇	D₁₆	semidirect
35. C₁₃₇	#2246	semidirect
39. C₄×C₁₆	D₁₃₇	semidirect
53. C₁₆	D₅₄₈	semidirect
65. C₄	D₂₁₉₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₁₉₂	2
#22577780	2
C₄×C₁₆×C₁₃₇	2
#2246	137