Subgroups — #31795750

Order 17536.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578292	C₂	1	1	yes	maximal, characteristic
2	#22578264	C₂	1	1	yes	maximal, characteristic
3	#22577497	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	—	2	1	no	—
5	C₄⋊#11764047	—	2	1	no	—
6	C₂×Dic₅₄₈	C₄	1	1	yes	characteristic
7	#17984053	—	2	1	no	—
8	#17984011	—	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	—	2	2	no	—
13	C₂×Dic₂₇₄	—	2	1	no	—
14	Dic₅₄₈	—	2	1	no	—
15	Dic₅₄₈	C₈	1	2	yes	complemented
16	C₂×#11764040	D₄	1	1	yes	characteristic
17	#14555489	—	4	1	no	—
18	C₄×Dic₁₃₇	—	4	1	no	—
19	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₁₃₇	—	2	1	no	—
22	#11764047	—	8	1	no	—
23	Dic₂₇₄	—	2	2	no	—
24	Dic₂₇₄	—	4	1	no	—
25	C₂×Dic₁₃₇	—	4	1	no	—
26	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
27	C₂×Dic₁₃₇	SD₁₆	1	1	yes	characteristic
28	#11764040	—	2	1	no	—
29	C₂×C₄×C₁₃₇	—	4	1	no	—
30	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₃₇	#102	1	1	yes	characteristic
34	C₄×C₁₃₇	—	8	1	no	—
35	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₁₃₇	#104	1	1	yes	characteristic
37	Dic₁₃₇	—	2	1	no	—
38	Dic₁₃₇	—	2	1	no	—
39	Dic₁₃₇	—	4	2	no	—
40	C₂×C₁₃₇	D₄⋊C₈	1	1	yes	characteristic
41	C₂×C₁₃₇	#360	1	1	yes	characteristic
42	C₂×C₁₃₇	#362	1	1	yes	characteristic
43	C₁₃₇	#1334	1	1	yes	characteristic, complemented, Sylow
44	#1334	—	137	1	no	maximal, Sylow
45	C₄×Q₁₆	—	137	1	no	—
46	#346	—	137	1	no	—
47	#334	—	137	1	no	—
48	C₂×Q₁₆	—	137	1	no	—
49	C₄×Q₈	—	274	1	no	—
50	C₂×C₁₆	—	274	1	no	—
51	#107	#10587739	1	1	yes	characteristic
52	C₄⋊C₈	—	274	1	no	—
53	#103	—	274	1	no	—
54	C₄×C₈	—	137	1	no	—
55	C₂×Q₈	—	274	1	no	—
56	Q₁₆	—	274	1	no	—
57	Q₁₆	—	137	2	no	—
58	C₂×C₈	—	548	1	no	—
59	C₂×C₈	—	137	1	no	—
60	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
61	C₄⋊C₄	—	2	1	no	—
62	C₄⋊C₄	—	548	1	no	—
63	C₄²	—	548	1	no	—
64	C₄²	—	137	1	no	—
65	C₁₆	—	274	2	no	—
66	Q₈	—	274	2	no	—
67	Q₈	—	548	1	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	548	1	no	—
70	C₂×C₄	—	137	1	no	—
71	C₂×C₄	—	137	1	no	—
72	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
73	C₈	—	274	1	no	—
74	C₈	—	1096	1	no	—
75	C₈	#14555518	1	1	yes	characteristic
76	C₈	#14555520	1	1	yes	characteristic
77	C₂²	#17984150	1	1	yes	characteristic
78	C₄	—	8	1	no	—
79	C₄	—	274	1	no	—
80	C₄	—	274	1	no	—
81	C₄	—	548	2	no	—
82	C₄	#17984145	1	1	yes	characteristic
83	C₄	#17984153	1	1	yes	characteristic
84	C₂	#22578251	1	1	yes	characteristic
85	C₂	#22578319	1	1	yes	characteristic
86	C₂	#22578322	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984150
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₂×C₈	C₂×#10587739
Fitting	#107×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	#102
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	#107	#10587739
hypocenter	C₁₃₇	#1334
LCS	G ▷ C₈×C₁₃₇ ▷ 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
15. Dic₅₄₈	C₈	semidirect
43. C₁₃₇	#1334	semidirect

Maximal subgroups

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

subgroup	index
#22578292	2
#22578264	2
#22577497	2
#1334	137