Subgroups — #31778747

Order 17536.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14555485	C₂	1	1	yes	maximal, characteristic, complemented
2	#22577179	C₂	1	1	yes	maximal, characteristic, complemented
3	#436×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555485	—	2	1	no	—
5	C₂×#14555485	—	2	1	no	—
6	C₂×M₁₆×C₁₃₇	—	2	1	no	—
7	C₄×D₄×C₁₃₇	C₄	1	1	yes	characteristic
8	C₄⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
9	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
10	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	#14555485	—	2	2	no	—
12	#14555485	—	4	1	no	—
13	C₂×D₄×C₁₃₇	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₃₇	—	2	1	no	—
15	M₁₆×C₁₃₇	—	4	1	no	—
16	M₁₆×C₁₃₇	—	4	1	no	—
17	C₂×C₈×C₁₃₇	—	2	1	no	—
18	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
21	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
22	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₁₃₇	—	2	1	no	—
24	D₄×C₁₃₇	—	4	1	no	—
25	C₂×C₄×C₁₃₇	—	2	1	no	—
26	C₂×C₄×C₁₃₇	—	4	1	no	—
27	C₂×C₄×C₁₃₇	—	4	1	no	—
28	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
31	C₈×C₁₃₇	—	2	1	no	—
32	C₈×C₁₃₇	—	4	1	no	—
33	C₈×C₁₃₇	D₈	1	1	yes	characteristic
34	C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₁₃₇	—	4	1	no	—
36	C₂²×C₁₃₇	—	4	1	no	—
37	C₂²×C₁₃₇	—	4	1	no	—
38	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₃₇	—	2	1	no	—
40	C₄×C₁₃₇	—	2	1	no	—
41	C₄×C₁₃₇	—	4	1	no	—
42	C₄×C₁₃₇	#102	1	1	yes	characteristic
43	C₄×C₁₃₇	#104	1	1	yes	characteristic
44	C₂×C₁₃₇	—	8	1	no	—
45	C₂×C₁₃₇	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₁₃₇	#349	1	1	yes	characteristic
47	C₂×C₁₃₇	#350	1	1	yes	characteristic
48	C₁₃₇	#1330	1	1	yes	characteristic, complemented, Sylow
49	#1330	—	137	1	no	maximal, Sylow
50	#436	D₁₃₇	1	1	yes	characteristic
51	C₄⋊C₁₆	—	137	1	no	—
52	#346	—	137	1	no	—
53	C₂×M₁₆	—	2	1	no	—
54	C₄×D₄	Dic₁₃₇	1	1	yes	characteristic
55	C₂×C₁₆	—	274	1	no	—
56	C₂×C₁₆	—	274	1	no	—
57	C₄⋊C₈	Dic₁₃₇	1	1	yes	characteristic
58	C₂²⋊C₈	—	2	1	no	—
59	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
60	C₂×D₄	#11764040	1	1	yes	characteristic
61	C₂²×C₄	—	2	1	no	—
62	M₁₆	—	4	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
67	C₄⋊C₄	#11764040	1	1	yes	characteristic
68	C₂²⋊C₄	—	2	1	no	—
69	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
70	C₁₆	—	274	2	no	—
71	C₁₆	—	548	1	no	—
72	C₂³	—	2	1	no	—
73	D₄	—	4	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	C₂²⋊Dic₁₃₇	1	1	yes	characteristic
78	C₂×C₄	C₂×#11764040	1	1	yes	characteristic
79	C₂×C₄	#14555494	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	4	1	no	—
82	C₈	#14555495	1	1	yes	characteristic
83	C₈	#14555498	1	1	yes	characteristic
84	C₂²	—	4	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	C₂²⋊#11764040	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	#17984009	1	1	yes	characteristic
92	C₄	#17984015	1	1	yes	characteristic
93	C₂	—	8	1	no	—
94	C₂	D₄⋊#11764040	1	1	yes	characteristic
95	C₂	#22577185	1	1	yes	characteristic
96	C₂	#22577187	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₃₇
Fitting	#436×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²⋊C₈
prime radical	C₂×D₄×C₁₃₇	C₈
cosocle	C₄×C₈×C₁₃₇	C₂²
hypercenter	#436	D₁₃₇
hypocenter	C₁₃₇	#1330
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₄⋊#14555485	C₂	semidirect
2. #22577179	C₂	semidirect
48. C₁₃₇	#1330	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14555485	2
#22577179	2
#436×C₁₃₇	2
#1330	137