Subgroups — #31778702

Order 17536.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577427	C₂	1	2	yes	maximal
2	#423×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555494	C₄	1	4	yes	—
4	C₄⋊#11764040	—	2	2	no	—
5	C₄⋊#11764040	—	2	2	no	—
6	C₂×M₁₆×C₁₃₇	C₄	1	2	yes	—
7	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₁₃₇	—	2	2	no	—
9	#14555494	—	4	4	no	—
10	C₂×#11764040	—	2	4	no	—
11	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
12	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
13	M₁₆×C₁₃₇	—	4	2	no	—
14	C₂×C₈×C₁₃₇	—	2	2	no	—
15	C₄²×C₁₃₇	D₄	1	2	yes	—
16	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
17	C₄²×C₁₃₇	Q₈	1	1	yes	characteristic
18	#11764040	—	4	4	no	—
19	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	—	2	1	no	—
21	C₂×C₄×C₁₃₇	—	2	2	no	—
22	C₂×C₄×C₁₃₇	—	2	2	no	—
23	C₂×C₄×C₁₃₇	—	2	1	no	—
24	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
28	C₈×C₁₃₇	—	4	2	no	—
29	C₂²×C₁₃₇	—	2	2	no	—
30	C₂²×C₁₃₇	—	2	1	no	—
31	C₂²×C₁₃₇	#95	1	1	yes	characteristic
32	C₄×C₁₃₇	—	2	2	no	—
33	C₄×C₁₃₇	—	2	1	no	—
34	C₄×C₁₃₇	#100	1	2	yes	—
35	C₄×C₁₃₇	#100	1	1	yes	characteristic
36	C₄×C₁₃₇	#101	1	2	yes	—
37	C₄×C₁₃₇	#101	1	1	yes	characteristic
38	C₂×C₁₃₇	—	4	1	no	—
39	C₂×C₁₃₇	#343	1	2	yes	—
40	C₂×C₁₃₇	#343	1	1	yes	characteristic
41	C₁₃₇	#1309	1	1	yes	characteristic, complemented, Sylow
42	#1309	—	137	1	no	maximal, Sylow
43	#423	—	137	2	no	—
44	#423	D₁₃₇	1	1	yes	characteristic
45	C₂×M₁₆	—	137	4	no	—
46	C₂×M₁₆	Dic₁₃₇	1	2	yes	—
47	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
48	C₄⋊C₈	—	2	2	no	—
49	C₄⋊C₈	—	274	2	no	—
50	C₄⋊C₈	—	274	2	no	—
51	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
52	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
53	M₁₆	—	4	2	no	—
54	M₁₆	—	548	4	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	274	4	no	—
57	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
58	C₄²	D₅₄₈	1	1	yes	characteristic
59	C₄²	Dic₂₇₄	1	1	yes	characteristic
60	C₂³	C₄×Dic₁₃₇	1	1	yes	characteristic
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	#14555487	1	2	yes	—
66	C₂×C₄	C₂²⋊Dic₁₃₇	1	1	yes	characteristic
67	C₂×C₄	#14555489	1	2	yes	—
68	C₂×C₄	C₄⋊Dic₁₃₇	1	1	yes	characteristic
69	C₈	—	4	2	no	—
70	C₈	—	548	4	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	2	1	no	—
73	C₂²	#17983995	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	#17984004	1	2	yes	—
77	C₄	#17984005	1	1	yes	characteristic
78	C₄	#17984006	1	2	yes	—
79	C₄	#17984007	1	1	yes	characteristic
80	C₂	—	4	1	no	—
81	C₂	#22577171	1	2	yes	—
82	C₂	#22577172	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17983995
commutator, prime radical	C₂³×C₁₃₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₃₇
Fitting	#423×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#95
cosocle	C₂×C₄²×C₁₃₇	C₂²
hypercenter	#423	D₁₃₇
hypocenter	C₁₃₇	#1309
LCS	G ▷ C₂³×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
41. C₁₃₇	#1309	semidirect

Maximal subgroups

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

subgroup	index
#22577427	2
#423×C₁₃₇	2
#1309	137