Subgroups — #31778956

Order 17536.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577202	C₂	1	2	yes	maximal, complemented
2	#492×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#17984007	—	2	2	no	—
4	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
5	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
6	#127×C₁₃₇	C₄	1	1	yes	characteristic
7	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
8	#14555494	—	4	2	no	—
9	C₂×Q₈×C₁₃₇	D₄	1	2	yes	—
10	C₂×D₄×C₁₃₇	—	2	1	no	—
11	C₂×D₄×C₁₃₇	—	4	1	no	—
12	SD₁₆×C₁₃₇	—	4	4	no	—
13	C₂×C₈×C₁₃₇	—	2	1	no	—
14	C₄⋊C₄×C₁₃₇	—	4	1	no	—
15	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	#11764040	—	8	2	no	—
17	C₂³×C₁₃₇	—	4	1	no	—
18	Q₈×C₁₃₇	—	4	2	no	—
19	D₄×C₁₃₇	—	4	2	no	—
20	D₄×C₁₃₇	—	8	2	no	—
21	C₂×C₄×C₁₃₇	—	2	1	no	—
22	C₂×C₄×C₁₃₇	—	2	2	no	—
23	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₈×C₁₃₇	—	2	2	no	—
25	C₂²×C₁₃₇	—	8	2	no	—
26	C₂²×C₁₃₇	—	8	1	no	—
27	C₂²×C₁₃₇	#99	1	1	yes	characteristic
28	C₄×C₁₃₇	—	2	2	no	—
29	C₄×C₁₃₇	—	2	1	no	—
30	C₄×C₁₃₇	—	4	2	no	—
31	C₂×C₁₃₇	—	2	1	no	—
32	C₂×C₁₃₇	—	16	1	no	—
33	C₂×C₁₃₇	#353	1	1	yes	characteristic
34	C₁₃₇	#1410	1	1	yes	characteristic, complemented, Sylow
35	#1410	—	137	1	no	maximal, Sylow
36	#492	D₁₃₇	1	1	yes	characteristic
37	#356	—	137	2	no	—
38	C₂×SD₁₆	—	2	2	no	—
39	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
40	#127	Dic₁₃₇	1	1	yes	characteristic
41	#101	—	274	2	no	—
42	C₄×C₈	Dic₁₃₇	1	1	yes	characteristic
43	C₂×Q₈	C₂²⋊D₁₃₇	1	2	yes	—
44	C₂×D₄	—	2	1	no	—
45	C₂×D₄	—	4	1	no	—
46	SD₁₆	—	4	4	no	—
47	M₁₆	—	548	2	no	—
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
51	C₂³	—	4	1	no	—
52	Q₈	—	4	2	no	—
53	D₄	—	4	2	no	—
54	D₄	—	8	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²⋊Dic₁₃₇	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	1096	2	no	—
60	C₂²	—	8	2	no	—
61	C₂²	—	8	1	no	—
62	C₂²	#17984003	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₂	—	2	1	no	—
67	C₂	—	16	1	no	—
68	C₂	#22577196	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22577196
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₃₇
Fitting	#492×C₁₃₇	C₂
socle	C₂×C₁₃₇	#353
prime radical	#127×C₁₃₇	C₄
cosocle	C₄⋊Q₈×C₁₃₇	C₂²
hypercenter	#492	D₁₃₇
hypocenter	C₁₃₇	#1410
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #492

Complements

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

N	H ≅ G/N	type
1. #22577202	C₂	semidirect
34. C₁₃₇	#1410	semidirect

Maximal subgroups

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

subgroup	index
#22577202	2
#492×C₁₃₇	2
#1410	137