Subgroups — #31778740

Order 17536.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577185	C₂	1	2	yes	maximal
2	#423×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#17984028	—	2	2	no	—
4	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
5	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₁₃₇	—	2	2	no	—
8	#14555485	—	4	2	no	—
9	C₂²×C₄×C₁₃₇	—	2	1	no	—
10	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
11	M₁₆×C₁₃₇	—	4	1	no	—
12	M₁₆×C₁₃₇	—	4	1	no	—
13	C₂×C₈×C₁₃₇	—	2	1	no	—
14	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
15	C₄²×C₁₃₇	—	2	1	no	—
16	C₄²×C₁₃₇	C₈	1	1	yes	characteristic
17	C₄²×C₁₃₇	C₈	1	1	yes	characteristic
18	C₂³×C₁₃₇	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₃₇	—	2	1	no	—
20	C₂×C₄×C₁₃₇	—	2	1	no	—
21	C₂×C₄×C₁₃₇	—	2	1	no	—
22	C₂×C₄×C₁₃₇	—	2	1	no	—
23	C₂×C₄×C₁₃₇	—	4	2	no	—
24	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
26	C₈×C₁₃₇	—	2	2	no	—
27	C₈×C₁₃₇	—	4	1	no	—
28	C₂²×C₁₃₇	—	2	1	no	—
29	C₂²×C₁₃₇	—	4	1	no	—
30	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₃₇	—	2	2	no	—
32	C₄×C₁₃₇	—	2	1	no	—
33	C₄×C₁₃₇	—	4	1	no	—
34	C₄×C₁₃₇	#99	1	1	yes	characteristic
35	C₄×C₁₃₇	#101	1	1	yes	characteristic
36	C₂×C₁₃₇	—	2	1	no	—
37	C₂×C₁₃₇	—	4	1	no	—
38	C₂×C₁₃₇	#324	1	1	yes	characteristic
39	C₁₃₇	#1327	1	1	yes	characteristic, complemented, Sylow
40	#1327	—	137	1	no	maximal, Sylow
41	#423	D₁₃₇	1	1	yes	characteristic
42	#349	—	137	2	no	—
43	C₂×M₁₆	Dic₁₃₇	1	1	yes	characteristic
44	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
45	C₂×C₄²	Dic₁₃₇	1	1	yes	characteristic
46	M₃₂	—	274	2	no	—
47	C₄⋊C₈	—	2	2	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
50	M₁₆	—	4	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	C₂²⋊D₁₃₇	1	2	yes	—
54	C₄²	—	2	1	no	—
55	C₄²	#11764040	1	1	yes	characteristic
56	C₄²	#11764040	1	1	yes	characteristic
57	C₁₆	—	548	2	no	—
58	C₂³	#14555494	1	1	yes	characteristic
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	C₂²⋊Dic₁₃₇	1	1	yes	characteristic
65	C₂×C₄	C₂×#11764040	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	4	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂²⋊#11764040	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	—	2	1	no	—
73	C₄	—	4	1	no	—
74	C₄	#17984003	1	1	yes	characteristic
75	C₄	#17984007	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	#22577129	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984003
commutator	C₂×C₄×C₁₃₇	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₁₃₇
Fitting	#423×C₁₃₇	C₂
socle	C₂×C₁₃₇	#324
prime radical	C₂³×C₁₃₇	M₁₆
cosocle	C₂×M₁₆×C₁₃₇	C₂²
hypercenter	#423	D₁₃₇
hypocenter	C₁₃₇	#1327
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #423

Complements

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

N	H ≅ G/N	type
39. C₁₃₇	#1327	semidirect

Maximal subgroups

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

subgroup	index
#22577185	2
#423×C₁₃₇	2
#1327	137