Subgroups — #31795747

Order 17536.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578292	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578264	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	—	2	1	no	—
5	C₄⋊#11764047	—	2	1	no	—
6	D₄×Dic₁₃₇	—	2	1	no	—
7	#17984022	C₄	1	1	yes	characteristic
8	#17984009	—	2	1	no	—
9	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×D₈×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×#11764047	—	4	1	no	—
12	#14555513	—	2	2	no	—
13	C₂²×Dic₁₃₇	—	4	1	no	—
14	C₂×#11764040	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	—	2	1	no	—
16	C₂²⋊Dic₁₃₇	—	4	1	no	—
17	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
18	C₂×D₄×C₁₃₇	—	2	1	no	—
19	D₈×C₁₃₇	—	2	1	no	—
20	D₈×C₁₃₇	C₈	1	2	yes	complemented
21	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	#11764047	—	8	1	no	—
23	C₂×Dic₁₃₇	—	4	1	no	—
24	C₂×Dic₁₃₇	—	8	2	no	—
25	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
26	C₂×Dic₁₃₇	SD₁₆	1	1	yes	characteristic
27	#11764040	—	2	1	no	—
28	C₂³×C₁₃₇	—	4	1	no	—
29	D₄×C₁₃₇	—	2	2	no	—
30	D₄×C₁₃₇	—	4	1	no	—
31	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
33	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
34	C₂²×C₁₃₇	—	4	2	no	—
35	C₂²×C₁₃₇	—	8	1	no	—
36	C₂²×C₁₃₇	—	8	1	no	—
37	C₂²×C₁₃₇	#102	1	1	yes	characteristic
38	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₃₇	#104	1	1	yes	characteristic
40	Dic₁₃₇	—	2	1	no	—
41	Dic₁₃₇	—	2	1	no	—
42	Dic₁₃₇	—	8	1	no	—
43	C₂×C₁₃₇	—	8	2	no	—
44	C₂×C₁₃₇	D₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₃₇	#360	1	1	yes	characteristic
46	C₂×C₁₃₇	#361	1	1	yes	characteristic
47	C₁₃₇	#1333	1	1	yes	characteristic, complemented, Sylow
48	#1333	—	137	1	no	maximal, Sylow
49	C₄×D₈	—	137	1	no	—
50	#346	—	137	1	no	—
51	#334	—	137	1	no	—
52	C₂×D₈	#10587739	1	1	yes	characteristic
53	C₄×D₄	—	274	1	no	—
54	C₂×C₁₆	—	274	1	no	—
55	#107	—	137	1	no	—
56	C₄⋊C₈	—	274	1	no	—
57	#102	—	274	1	no	—
58	C₄×C₈	—	137	1	no	—
59	C₂×D₄	—	2	1	no	—
60	C₂²×C₄	—	548	1	no	—
61	D₈	—	2	1	no	—
62	D₈	#11764047	1	2	yes	complemented
63	C₂×C₈	—	548	1	no	—
64	C₂×C₈	—	137	1	no	—
65	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
66	C₄⋊C₄	—	274	1	no	—
67	C₂²⋊C₄	—	548	1	no	—
68	C₄²	—	137	1	no	—
69	C₁₆	—	274	2	no	—
70	C₂³	—	4	1	no	—
71	D₄	—	2	2	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	548	1	no	—
74	C₂×C₄	—	1096	2	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	1	no	—
77	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
78	C₈	—	274	1	no	—
79	C₈	—	1096	1	no	—
80	C₈	C₂×#11764047	1	1	yes	characteristic
81	C₈	#14555519	1	1	yes	characteristic
82	C₂²	—	4	2	no	—
83	C₂²	—	8	1	no	—
84	C₂²	—	8	1	no	—
85	C₂²	#17984150	1	1	yes	characteristic
86	C₄	—	274	1	no	—
87	C₄	—	274	1	no	—
88	C₄	—	1096	1	no	—
89	C₄	C₂²⋊#11764047	1	1	yes	characteristic
90	C₄	#17984152	1	1	yes	characteristic
91	C₂	—	8	2	no	—
92	C₂	D₄⋊#11764047	1	1	yes	characteristic
93	C₂	#22578319	1	1	yes	characteristic
94	C₂	#22578321	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984150
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₂×C₈	C₂×#10587739
Fitting, prime radical	C₂×D₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	#102
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	C₂×D₈	#10587739
hypocenter	C₁₃₇	#1333
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #22578292	C₂	semidirect
2. #22578264	C₂	semidirect
20. D₈×C₁₃₇	C₈	semidirect
47. C₁₃₇	#1333	semidirect
62. D₈	#11764047	semidirect

Maximal subgroups

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

subgroup	index
#22578292	2
#22578264	2
D₈×Dic₁₃₇	2
#1333	137