Subgroups — #31795752

Order 17536.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578325	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#11764047	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577519	C₂	1	1	yes	maximal, characteristic
4	#17984162	—	2	1	no	—
5	C₄×#11764047	—	2	1	no	—
6	#17984110	C₄	1	1	yes	characteristic
7	#17984096	—	2	1	no	—
8	#17984014	—	2	1	no	—
9	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
10	#108×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×#11764047	—	2	2	no	—
12	#14555513	—	4	1	no	—
13	#14555510	—	2	1	no	—
14	Dic₅₄₈	C₈	1	1	yes	characteristic, complemented
15	#14555499	—	2	1	no	—
16	D₁₀₉₆	C₈	1	1	yes	characteristic, complemented
17	#14555493	—	4	1	no	—
18	C₂×#11764040	D₄	1	1	yes	characteristic
19	C₈×D₁₃₇	—	4	1	no	—
20	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
21	M₁₆×C₁₃₇	—	2	1	no	—
22	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	#11764047	—	2	4	no	—
24	Dic₂₇₄	—	2	1	no	—
25	D₅₄₈	—	2	1	no	—
26	C₂²⋊D₁₃₇	—	4	1	no	—
27	C₂×Dic₁₃₇	—	2	1	no	—
28	C₄×D₁₃₇	—	4	1	no	—
29	#11764040	D₈	1	1	yes	characteristic
30	#11764040	SD₁₆	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₈×C₁₃₇	—	4	1	no	—
33	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
34	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
35	C₂²×C₁₃₇	#104	1	1	yes	characteristic
36	C₂×D₁₃₇	—	4	1	no	—
37	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
38	C₄×C₁₃₇	#102	1	1	yes	characteristic
39	Dic₁₃₇	—	2	2	no	—
40	Dic₁₃₇	—	4	1	no	—
41	C₂×C₁₃₇	—	2	1	no	—
42	C₂×C₁₃₇	D₄⋊C₈	1	1	yes	characteristic
43	D₁₃₇	—	8	1	no	—
44	C₁₃₇	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	137	1	no	maximal, Sylow
46	#443	—	137	1	no	—
47	#364	—	137	1	no	—
48	C₈²	—	137	1	no	—
49	#135	—	137	1	no	—
50	#131	—	274	1	no	—
51	M₃₂	—	274	1	no	—
52	#108	#10587739	1	1	yes	characteristic
53	#104	—	274	1	no	—
54	C₄×C₈	—	274	1	no	—
55	C₄×C₈	—	137	1	no	—
56	#41	—	274	1	no	—
57	Q₁₆	—	137	1	no	—
58	SD₁₆	—	274	1	no	—
59	D₈	—	137	1	no	—
60	M₁₆	—	2	1	no	—
61	M₁₆	—	548	1	no	—
62	C₂×C₈	—	274	2	no	—
63	C₂×C₈	—	548	1	no	—
64	C₂×C₈	—	137	1	no	—
65	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
66	C₄²	—	137	1	no	—
67	C₁₆	—	548	1	no	—
68	Q₈	—	274	1	no	—
69	D₄	—	274	1	no	—
70	D₄	—	548	1	no	—
71	C₂×C₄	—	274	1	no	—
72	C₂×C₄	—	548	1	no	—
73	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
74	C₈	—	4	1	no	—
75	C₈	—	274	4	no	—
76	C₈	—	137	1	no	—
77	C₈	—	137	1	no	—
78	C₈	#14555518	1	1	yes	characteristic
79	C₈	#14555520	1	1	yes	characteristic
80	C₂²	—	548	1	no	—
81	C₂²	#17984153	1	1	yes	characteristic
82	C₄	—	274	2	no	—
83	C₄	—	548	1	no	—
84	C₄	#17984145	1	1	yes	characteristic
85	C₄	#17984150	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1096	1	no	—
88	C₂	#22578251	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984150
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₂×C₈	C₂×#10587739
Fitting	#108×C₁₃₇	C₄
socle	C₂×C₁₃₇	D₄⋊C₈
prime radical	#17984110	C₄
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	#108	#10587739
hypocenter	C₁₃₇	#1335
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
1. #22578325	C₂	semidirect
2. C₈×#11764047	C₂	semidirect
14. Dic₅₄₈	C₈	semidirect
16. D₁₀₉₆	C₈	semidirect
44. C₁₃₇	#1335	semidirect

Maximal subgroups

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

subgroup	index
#22578325	2
C₈×#11764047	2
#22577519	2
#1335	137