Subgroups — #31783045

Order 17536.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984033	C₂	1	4	yes	maximal, complemented
2	#22577783	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577780	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₅₄₈	—	2	1	no	—
6	C₂×Dic₅₄₈	C₂²	1	2	yes	—
7	C₂×D₁₀₉₆	—	2	1	no	—
8	C₂×D₁₀₉₆	C₂²	1	2	yes	—
9	#17984088	—	2	1	no	—
10	#17984085	—	2	1	no	—
11	#17984033	—	2	8	no	—
12	C₂×C₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₇₄	—	2	2	no	—
15	C₂×D₅₄₈	—	2	2	no	—
16	C₂×D₅₄₈	—	4	2	no	—
17	Dic₅₄₈	—	2	4	no	—
18	Dic₅₄₈	—	4	2	no	—
19	D₁₀₉₆	—	2	4	no	—
20	D₁₀₉₆	—	4	2	no	—
21	C₄⋊Dic₁₃₇	—	4	2	no	—
22	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
25	C₁₆×C₁₃₇	D₄	1	4	yes	complemented
26	C₂²×D₁₃₇	—	4	2	no	—
27	Dic₂₇₄	—	4	2	no	—
28	Dic₂₇₄	—	4	2	no	—
29	D₅₄₈	—	4	2	no	—
30	D₅₄₈	—	4	2	no	—
31	D₅₄₈	—	8	4	no	—
32	C₂×Dic₁₃₇	—	4	2	no	—
33	C₂×C₄×C₁₃₇	D₈	1	2	yes	—
34	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₃₇	D₈	1	2	yes	—
36	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
37	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
38	C₂×D₁₃₇	—	8	2	no	—
39	C₂×D₁₃₇	—	8	4	no	—
40	C₄×C₁₃₇	SD₃₂	1	4	yes	complemented
41	C₄×C₁₃₇	#127	1	1	yes	characteristic
42	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
43	Dic₁₃₇	—	8	2	no	—
44	C₂×C₁₃₇	#493	1	1	yes	characteristic
45	C₂×C₁₃₇	C₂×SD₃₂	1	2	yes	—
46	D₁₃₇	—	16	2	no	—
47	C₁₃₇	#2248	1	1	yes	characteristic, complemented, Sylow
48	#2248	—	137	1	no	maximal, Sylow
49	C₂×SD₃₂	—	137	4	no	—
50	#494	—	137	1	no	—
51	#493	—	137	1	no	—
52	C₄×C₁₆	D₁₃₇	1	1	yes	characteristic, complemented
53	C₂×Q₁₆	—	274	1	no	—
54	C₂×Q₁₆	—	137	2	no	—
55	C₂×D₈	—	274	1	no	—
56	C₂×D₈	—	137	2	no	—
57	C₄⋊Q₈	—	274	1	no	—
58	#127	—	274	1	no	—
59	SD₃₂	—	274	8	no	—
60	C₂×C₁₆	C₂×D₁₃₇	1	2	yes	—
61	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
62	C₂×Q₈	—	274	2	no	—
63	C₂×D₄	—	274	2	no	—
64	C₂×D₄	—	548	2	no	—
65	Q₁₆	—	274	4	no	—
66	Q₁₆	—	548	2	no	—
67	D₈	—	274	4	no	—
68	D₈	—	548	2	no	—
69	C₂×C₈	D₅₄₈	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
71	C₄⋊C₄	—	548	2	no	—
72	C₄²	D₅₄₈	1	1	yes	characteristic
73	C₁₆	D₅₄₈	1	4	yes	complemented
74	C₂³	—	548	2	no	—
75	Q₈	—	548	2	no	—
76	Q₈	—	548	2	no	—
77	D₄	—	548	2	no	—
78	D₄	—	548	2	no	—
79	D₄	—	1096	4	no	—
80	C₂×C₄	—	548	2	no	—
81	C₂×C₄	D₁₀₉₆	1	2	yes	—
82	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
83	C₈	D₁₀₉₆	1	2	yes	—
84	C₈	C₂×D₅₄₈	1	2	yes	—
85	C₂²	—	1096	2	no	—
86	C₂²	—	1096	4	no	—
87	C₂²	C₂×D₁₀₉₆	1	1	yes	characteristic
88	C₄	—	1096	2	no	—
89	C₄	#17984033	1	4	yes	complemented
90	C₄	#17984085	1	1	yes	characteristic
91	C₄	C₂×D₁₀₉₆	1	1	yes	characteristic
92	C₂	—	2192	2	no	—
93	C₂	#22577780	1	1	yes	characteristic
94	C₂	C₂×#17984033	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₉₆
commutator, cosocle	C₂×C₈×C₁₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₄×C₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₈
prime radical	#22577780	C₂
hypercenter	C₄×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#2248
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#17984033	C₂	semidirect
2. #22577783	C₂	semidirect
4. C₄×C₁₆×C₁₃₇	C₂	semidirect
25. C₁₆×C₁₃₇	D₄	semidirect
40. C₄×C₁₃₇	SD₃₂	semidirect
47. C₁₃₇	#2248	semidirect
52. C₄×C₁₆	D₁₃₇	semidirect
73. C₁₆	D₅₄₈	semidirect
89. C₄	#17984033	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984033	2
#22577783	2
#22577780	2
C₄×C₁₆×C₁₃₇	2
#2248	137