Subgroups — #31795797

Order 17536.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578326	C₂	1	2	yes	maximal, complemented
2	#22577779	C₂	1	1	yes	maximal, characteristic
3	#17984162	—	2	2	no	—
4	C₂×#14555495	—	2	2	no	—
5	#17984085	C₄	1	1	yes	characteristic
6	C₄×#11764040	C₂²	1	1	yes	characteristic
7	#127×C₁₃₇	C₄	1	1	yes	characteristic
8	#14555513	—	4	2	no	—
9	C₂×D₅₄₈	—	2	1	no	—
10	C₂×D₅₄₈	—	4	1	no	—
11	#14555495	—	4	2	no	—
12	#14555495	—	4	2	no	—
13	C₂×#11764040	D₄	1	2	yes	—
14	C₂×D₄×C₁₃₇	—	2	1	no	—
15	C₂×D₄×C₁₃₇	—	4	1	no	—
16	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×D₁₃₇	—	4	1	no	—
18	D₅₄₈	—	4	2	no	—
19	D₅₄₈	—	8	2	no	—
20	#11764040	D₈	1	2	yes	—
21	#11764040	SD₁₆	1	2	yes	—
22	C₂³×C₁₃₇	—	4	1	no	—
23	D₄×C₁₃₇	—	4	2	no	—
24	D₄×C₁₃₇	—	8	2	no	—
25	C₂×C₄×C₁₃₇	—	2	1	no	—
26	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₃₇	—	8	2	no	—
28	C₂²×C₁₃₇	—	8	1	no	—
29	C₂²×C₁₃₇	#100	1	1	yes	characteristic
30	C₂×D₁₃₇	—	8	2	no	—
31	C₂×D₁₃₇	—	8	1	no	—
32	C₄×C₁₃₇	—	2	2	no	—
33	C₄×C₁₃₇	#102	1	2	yes	—
34	C₂×C₁₃₇	—	2	1	no	—
35	C₂×C₁₃₇	—	16	1	no	—
36	C₂×C₁₃₇	#331	1	1	yes	characteristic
37	D₁₃₇	—	16	1	no	—
38	C₁₃₇	#1357	1	1	yes	characteristic, complemented, Sylow
39	#1357	—	137	1	no	maximal, Sylow
40	#493	—	137	1	no	—
41	#364	—	137	2	no	—
42	C₂×D₈	—	274	2	no	—
43	#127	—	137	1	no	—
44	#127	#10587739	1	1	yes	characteristic
45	M₃₂	—	274	2	no	—
46	C₄×C₈	—	137	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	274	1	no	—
50	C₂×D₄	—	548	1	no	—
51	D₈	—	548	2	no	—
52	D₈	—	548	2	no	—
53	C₂×C₈	—	137	2	no	—
54	C₄²	C₂×#10587739	1	1	yes	characteristic
55	C₁₆	—	548	2	no	—
56	C₂³	—	4	1	no	—
57	C₂³	—	548	1	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	2	no	—
60	D₄	—	548	2	no	—
61	D₄	—	1096	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
64	C₈	—	137	2	no	—
65	C₈	—	137	2	no	—
66	C₂²	—	8	2	no	—
67	C₂²	—	8	1	no	—
68	C₂²	—	1096	2	no	—
69	C₂²	—	1096	1	no	—
70	C₂²	#17984148	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	#17984150	1	2	yes	—
73	C₂	—	2	1	no	—
74	C₂	—	16	1	no	—
75	C₂	—	2192	1	no	—
76	C₂	#22578261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578261
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄²	C₂×#10587739
Fitting	#127×C₁₃₇	C₄
socle	C₂×C₁₃₇	#331
prime radical	#22577779	C₂
cosocle	C₄×#11764040	C₂²
hypercenter	#127	#10587739
hypocenter	C₁₃₇	#1357
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #22578326	C₂	semidirect
38. C₁₃₇	#1357	semidirect

Maximal subgroups

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

subgroup	index
#22578326	2
#22577779	2
#1357	137