Subgroups — #31795748

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	#22577485	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	—	2	1	no	—
5	C₄⋊#11764047	—	2	1	no	—
6	C₂×D₁₀₉₆	C₄	1	1	yes	characteristic
7	#17984050	—	2	1	no	—
8	#17984008	—	2	1	no	—
9	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
10	#107×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×#11764047	—	4	1	no	—
12	#14555513	—	2	2	no	—
13	C₂×D₅₄₈	—	2	1	no	—
14	C₂×C₄×D₁₃₇	—	4	1	no	—
15	D₁₀₉₆	—	2	1	no	—
16	D₁₀₉₆	C₈	1	2	yes	complemented
17	C₂×#11764040	D₄	1	1	yes	characteristic
18	#14555487	—	4	1	no	—
19	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₁₃₇	—	2	1	no	—
22	#11764047	—	8	1	no	—
23	C₂²×D₁₃₇	—	4	1	no	—
24	D₅₄₈	—	2	2	no	—
25	D₅₄₈	—	4	1	no	—
26	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
27	C₂×Dic₁₃₇	SD₁₆	1	1	yes	characteristic
28	C₄×D₁₃₇	—	8	2	no	—
29	#11764040	—	2	1	no	—
30	C₂×C₄×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	#102	1	1	yes	characteristic
35	C₂×D₁₃₇	—	4	2	no	—
36	C₂×D₁₃₇	—	8	1	no	—
37	C₂×D₁₃₇	—	8	1	no	—
38	C₄×C₁₃₇	—	8	1	no	—
39	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₁₃₇	#104	1	1	yes	characteristic
41	Dic₁₃₇	—	2	1	no	—
42	Dic₁₃₇	—	2	1	no	—
43	C₂×C₁₃₇	D₄⋊C₈	1	1	yes	characteristic
44	C₂×C₁₃₇	#360	1	1	yes	characteristic
45	C₂×C₁₃₇	#361	1	1	yes	characteristic
46	D₁₃₇	—	8	2	no	—
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	—	137	1	no	—
53	C₄×D₄	—	274	1	no	—
54	C₂×C₁₆	—	274	1	no	—
55	#107	#10587739	1	1	yes	characteristic
56	C₄⋊C₈	—	274	1	no	—
57	#102	—	274	1	no	—
58	C₄×C₈	—	137	1	no	—
59	C₂×D₄	—	274	1	no	—
60	C₂²×C₄	—	548	1	no	—
61	D₈	—	274	1	no	—
62	D₈	—	137	2	no	—
63	C₂×C₈	—	548	1	no	—
64	C₂×C₈	—	137	1	no	—
65	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
66	C₄⋊C₄	—	2	1	no	—
67	C₂²⋊C₄	—	548	1	no	—
68	C₄²	—	137	1	no	—
69	C₁₆	—	274	2	no	—
70	C₂³	—	548	1	no	—
71	D₄	—	274	2	no	—
72	D₄	—	548	1	no	—
73	C₂×C₄	—	4	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	#14555518	1	1	yes	characteristic
81	C₈	#14555520	1	1	yes	characteristic
82	C₂²	—	548	2	no	—
83	C₂²	—	1096	1	no	—
84	C₂²	—	1096	1	no	—
85	C₂²	#17984150	1	1	yes	characteristic
86	C₄	—	8	1	no	—
87	C₄	—	274	1	no	—
88	C₄	—	274	1	no	—
89	C₄	#17984145	1	1	yes	characteristic
90	C₄	#17984153	1	1	yes	characteristic
91	C₂	—	1096	2	no	—
92	C₂	#22578251	1	1	yes	characteristic
93	C₂	#22578320	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	#107×C₁₃₇	C₄
socle	C₂²×C₁₃₇	#102
prime radical	C₂×D₁₀₉₆	C₄
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	#107	#10587739
hypocenter	C₁₃₇	#1333
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
1. #22578292	C₂	semidirect
2. #22578264	C₂	semidirect
16. D₁₀₉₆	C₈	semidirect
47. C₁₃₇	#1333	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
#22577485	2
#1333	137