Subgroups — #31795738

Order 17536.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578299	C₂	1	2	yes	maximal
2	#22577429	C₂	1	1	yes	maximal, characteristic
3	#17984162	—	2	2	no	—
4	C₂×#14555494	C₂²	1	1	yes	characteristic
5	#17984047	C₄	1	1	yes	characteristic
6	#17984017	—	2	2	no	—
7	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
8	#14555513	—	4	2	no	—
9	#14555494	—	4	1	no	—
10	C₂×#11764040	D₄	1	2	yes	—
11	C₄⋊Dic₁₃₇	—	2	1	no	—
12	C₂²⋊Dic₁₃₇	C₈	1	2	yes	—
13	C₄×Dic₁₃₇	—	2	1	no	—
14	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₇	—	2	1	no	—
16	C₂×C₈×C₁₃₇	—	4	1	no	—
17	C₂×Dic₁₃₇	—	2	2	no	—
18	#11764040	—	2	2	no	—
19	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	—	2	1	no	—
21	C₂×C₄×C₁₃₇	—	2	1	no	—
22	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
24	C₈×C₁₃₇	—	4	1	no	—
25	C₂²×C₁₃₇	—	2	1	no	—
26	C₂²×C₁₃₇	—	4	1	no	—
27	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₁₃₇	—	2	1	no	—
29	C₄×C₁₃₇	#99	1	1	yes	characteristic
30	C₄×C₁₃₇	#101	1	1	yes	characteristic
31	Dic₁₃₇	—	4	2	no	—
32	C₂×C₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	—	4	1	no	—
34	C₂×C₁₃₇	#324	1	1	yes	characteristic
35	C₁₃₇	#1328	1	1	yes	characteristic, complemented, Sylow
36	#1328	—	137	1	no	maximal, Sylow
37	#424	—	137	1	no	—
38	#349	—	137	2	no	—
39	C₂×M₁₆	—	137	1	no	—
40	C₂²×C₈	#10587739	1	1	yes	characteristic
41	#117	—	137	1	no	—
42	M₃₂	—	274	2	no	—
43	C₄⋊C₈	—	274	2	no	—
44	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
45	M₁₆	—	548	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	—	137	2	no	—
49	C₄⋊C₄	—	274	1	no	—
50	C₂²⋊C₄	—	137	2	no	—
51	C₄²	—	274	1	no	—
52	C₁₆	—	548	2	no	—
53	C₂³	#14555518	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	274	2	no	—
57	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
58	C₂×C₄	#14555520	1	1	yes	characteristic
59	C₈	—	4	1	no	—
60	C₈	—	274	2	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#17984145	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	548	2	no	—
66	C₄	#17984146	1	1	yes	characteristic
67	C₄	#17984149	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#22578249	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984146
commutator, prime radical	C₂³×C₁₃₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#324
cosocle	C₂×#14555494	C₂²
hypercenter	C₂²×C₈	#10587739
hypocenter	C₁₃₇	#1328
LCS	G ▷ C₂³×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
35. C₁₃₇	#1328	semidirect

Maximal subgroups

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

subgroup	index
#22578299	2
#22577429	2
#1328	137