Subgroups — #31795808

Order 17536.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578324	C₂	1	1	yes	maximal, characteristic
2	#22578267	C₂	1	1	yes	maximal, characteristic
3	#22577807	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	—	2	1	no	—
5	C₄⋊#11764047	—	2	1	no	—
6	#17984087	—	2	1	no	—
7	#17984022	C₄	1	1	yes	characteristic
8	#17984021	—	2	1	no	—
9	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
10	#107×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×#11764047	—	4	1	no	—
12	#14555513	—	4	1	no	—
13	C₂×Dic₂₇₄	—	4	1	no	—
14	C₂×#11764040	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	—	2	1	no	—
16	#14555489	—	4	1	no	—
17	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄⋊C₄×C₁₃₇	—	2	1	no	—
20	#11764047	—	8	1	no	—
21	Dic₂₇₄	—	8	2	no	—
22	C₂×Dic₁₃₇	—	4	1	no	—
23	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
24	C₂×Dic₁₃₇	SD₁₆	1	1	yes	characteristic
25	#11764040	SD₁₆	1	1	yes	characteristic
26	#11764040	Q₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	—	4	1	no	—
28	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₁₃₇	—	2	1	no	—
30	C₂²×C₁₃₇	#102	1	1	yes	characteristic
31	C₄×C₁₃₇	—	8	1	no	—
32	C₄×C₁₃₇	#101	1	1	yes	characteristic
33	C₄×C₁₃₇	#103	1	1	yes	characteristic
34	Dic₁₃₇	—	2	1	no	—
35	Dic₁₃₇	—	8	1	no	—
36	Dic₁₃₇	SD₃₂	1	1	yes	characteristic
37	Dic₁₃₇	Q₃₂	1	1	yes	characteristic
38	C₂×C₁₃₇	#332	1	1	yes	characteristic
39	C₂×C₁₃₇	#358	1	1	yes	characteristic
40	C₂×C₁₃₇	#361	1	1	yes	characteristic
41	C₁₃₇	#1365	1	1	yes	characteristic, complemented, Sylow
42	#1365	—	137	1	no	maximal, Sylow
43	#500	—	137	1	no	—
44	C₄⋊C₁₆	—	137	1	no	—
45	#335	—	137	1	no	—
46	C₄⋊Q₈	—	274	1	no	—
47	C₂×C₁₆	—	274	1	no	—
48	#107	—	274	1	no	—
49	#107	—	137	1	no	—
50	#107	#10587739	1	1	yes	characteristic
51	C₄⋊C₈	—	274	1	no	—
52	C₄×C₈	—	137	1	no	—
53	C₂×Q₈	—	548	1	no	—
54	C₂×C₈	—	548	1	no	—
55	C₂×C₈	—	137	1	no	—
56	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	274	1	no	—
59	C₄⋊C₄	—	548	1	no	—
60	C₄²	—	137	1	no	—
61	C₁₆	—	548	1	no	—
62	Q₈	—	1096	2	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	548	1	no	—
65	C₂×C₄	—	137	1	no	—
66	C₂×C₄	—	137	1	no	—
67	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	1096	1	no	—
70	C₈	—	137	1	no	—
71	C₈	—	137	1	no	—
72	C₂²	#17984150	1	1	yes	characteristic
73	C₄	—	8	1	no	—
74	C₄	—	274	1	no	—
75	C₄	—	1096	1	no	—
76	C₄	—	137	1	no	—
77	C₄	—	137	1	no	—
78	C₄	#17984149	1	1	yes	characteristic
79	C₄	#17984151	1	1	yes	characteristic
80	C₂	#22578262	1	1	yes	characteristic
81	C₂	#22578316	1	1	yes	characteristic
82	C₂	#22578321	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984150
commutator	C₂×C₈×C₁₃₇	C₂×C₄
Frattini	C₂×C₈	C₂×#10587739
Fitting	#107×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	#102
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	#107	#10587739
hypocenter	C₁₃₇	#1365
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
41. C₁₃₇	#1365	semidirect

Maximal subgroups

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

subgroup	index
#22578324	2
#22578267	2
#22577807	2
#1365	137