Subgroups — #31795809

Order 17536.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578326	C₂	1	1	yes	maximal, characteristic
2	#22578326	C₂	1	1	yes	maximal, characteristic
3	#22577802	C₂	1	1	yes	maximal, characteristic
4	#17984162	—	2	1	no	—
5	#17984162	—	2	1	no	—
6	#17984080	C₄	1	1	yes	characteristic
7	#17984021	—	2	1	no	—
8	#17984019	—	2	1	no	—
9	C₄×#11764040	C₂²	1	1	yes	characteristic
10	#125×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555513	—	4	1	no	—
12	#14555513	—	4	1	no	—
13	C₂×#11764040	D₄	1	1	yes	characteristic
14	C₂×#11764040	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	—	2	1	no	—
16	#14555489	—	4	1	no	—
17	C₄⋊C₄×C₁₃₇	—	2	1	no	—
18	C₄⋊C₄×C₁₃₇	—	4	1	no	—
19	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₃₇	—	4	1	no	—
21	#11764040	SD₁₆	1	1	yes	characteristic
22	#11764040	SD₁₆	1	1	yes	characteristic
23	#11764040	Q₁₆	1	1	yes	characteristic
24	#11764040	Q₁₆	1	1	yes	characteristic
25	C₂×C₄×C₁₃₇	—	2	1	no	—
26	C₂×C₄×C₁₃₇	—	4	1	no	—
27	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₁₃₇	#100	1	1	yes	characteristic
29	C₄×C₁₃₇	—	4	1	no	—
30	C₄×C₁₃₇	—	8	1	no	—
31	C₄×C₁₃₇	#103	1	1	yes	characteristic
32	C₄×C₁₃₇	#103	1	1	yes	characteristic
33	Dic₁₃₇	—	8	1	no	—
34	C₂×C₁₃₇	—	2	1	no	—
35	C₂×C₁₃₇	#333	1	1	yes	characteristic
36	C₁₃₇	#1366	1	1	yes	characteristic, complemented, Sylow
37	#1366	—	137	1	no	maximal, Sylow
38	#499	—	137	1	no	—
39	#364	—	137	1	no	—
40	#364	—	137	1	no	—
41	#125	—	137	1	no	—
42	#125	#10587739	1	1	yes	characteristic
43	M₃₂	—	274	1	no	—
44	M₃₂	—	274	1	no	—
45	#107	—	274	1	no	—
46	#106	—	274	1	no	—
47	C₄×C₈	—	137	1	no	—
48	C₂×C₈	—	137	1	no	—
49	C₂×C₈	—	137	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	4	1	no	—
52	C₄⋊C₄	—	274	1	no	—
53	C₄⋊C₄	—	548	1	no	—
54	C₄²	C₂×#10587739	1	1	yes	characteristic
55	C₁₆	—	548	1	no	—
56	C₁₆	—	548	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	548	1	no	—
60	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
61	C₈	—	137	1	no	—
62	C₈	—	137	1	no	—
63	C₈	—	137	1	no	—
64	C₈	—	137	1	no	—
65	C₂²	#17984148	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	8	1	no	—
68	C₄	—	1096	1	no	—
69	C₄	#17984151	1	1	yes	characteristic
70	C₄	#17984151	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#22578263	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578263
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄²	C₂×#10587739
Fitting	#125×C₁₃₇	C₄
socle	C₂×C₁₃₇	#333
prime radical	C₂²×C₁₃₇	#100
cosocle	C₄×#11764040	C₂²
hypercenter	#125	#10587739
hypocenter	C₁₃₇	#1366
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
36. C₁₃₇	#1366	semidirect

Maximal subgroups

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

subgroup	index
#22578326	2
#22577802	2
#1366	137