Subgroups — #31783098

Order 17536.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577835	C₂	1	2	yes	maximal, complemented
2	Dic₂₁₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577242	C₂	1	2	yes	maximal, complemented
4	D₄₃₈₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984110	—	2	2	no	—
7	Dic₁₀₉₆	C₂²	1	2	yes	—
8	#17984033	—	2	2	no	—
9	D₂₁₉₂	C₂²	1	2	yes	—
10	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
13	#14555510	—	4	2	no	—
14	Dic₅₄₈	—	2	2	no	—
15	#14555499	—	4	2	no	—
16	D₁₀₉₆	—	2	2	no	—
17	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
18	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
19	C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
20	Dic₂₇₄	—	4	2	no	—
21	D₅₄₈	—	4	2	no	—
22	C₂²⋊D₁₃₇	—	8	2	no	—
23	C₄×D₁₃₇	—	8	2	no	—
24	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
25	C₈×C₁₃₇	D₈	1	1	yes	characteristic
26	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₁₃₇	D₁₆	1	1	yes	characteristic
28	C₂×D₁₃₇	—	8	2	no	—
29	C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
30	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
31	Dic₁₃₇	—	8	2	no	—
32	C₂×C₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
34	D₁₃₇	—	16	2	no	—
35	C₁₃₇	#2262	1	1	yes	characteristic, complemented, Sylow
36	#2262	—	137	1	no	maximal, Sylow
37	#508	—	137	2	no	—
38	Q₆₄	—	137	1	no	—
39	SD₆₄	—	137	2	no	—
40	D₃₂	—	137	1	no	—
41	C₂×C₃₂	D₁₃₇	1	1	yes	characteristic, complemented
42	#135	—	274	2	no	—
43	Q₃₂	—	137	2	no	—
44	SD₃₂	—	274	2	no	—
45	D₁₆	—	137	2	no	—
46	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
47	C₃₂	C₂×D₁₃₇	1	1	yes	characteristic
48	C₃₂	C₂×D₁₃₇	1	1	yes	characteristic
49	#41	—	548	2	no	—
50	Q₁₆	—	274	2	no	—
51	SD₁₆	—	548	2	no	—
52	D₈	—	274	2	no	—
53	C₂×C₈	D₅₄₈	1	1	yes	characteristic
54	C₁₆	D₅₄₈	1	1	yes	characteristic
55	C₁₆	C₂²×D₁₃₇	1	1	yes	characteristic
56	Q₈	—	548	2	no	—
57	D₄	—	548	2	no	—
58	D₄	—	1096	2	no	—
59	C₂×C₄	—	1096	2	no	—
60	C₂×C₄	D₁₀₉₆	1	1	yes	characteristic
61	C₈	D₁₀₉₆	1	1	yes	characteristic
62	C₈	C₂×D₅₄₈	1	1	yes	characteristic
63	C₂²	—	1096	2	no	—
64	C₂²	D₂₁₉₂	1	1	yes	characteristic
65	C₄	—	1096	2	no	—
66	C₄	D₂₁₉₂	1	1	yes	characteristic
67	C₄	C₂×D₁₀₉₆	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	2192	2	no	—
70	C₂	C₂×D₂₁₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₁₉₂
commutator, cosocle	C₁₆×C₁₃₇	C₂³
Frattini	C₁₆	C₂²×D₁₃₇
Fitting	C₂×C₃₂×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂×D₁₆
hypercenter	C₂×C₃₂	D₁₃₇
hypocenter	C₁₃₇	#2262
LCS	G ▷ C₁₆×C₁₃₇ ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #22577835	C₂	semidirect
2. Dic₂₁₉₂	C₂	semidirect
3. #22577242	C₂	semidirect
4. D₄₃₈₄	C₂	semidirect
5. C₂×C₃₂×C₁₃₇	C₂	semidirect
35. C₁₃₇	#2262	semidirect
41. C₂×C₃₂	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577835	2
Dic₂₁₉₂	2
#22577242	2
D₄₃₈₄	2
C₂×C₃₂×C₁₃₇	2
#2262	137