Subgroups — #31783104

Order 17536.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577835	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₁₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577242	C₂	1	2	yes	maximal, complemented
4	D₄₃₈₄	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984110	—	2	1	no	—
7	C₂×D₁₀₉₆	—	2	1	no	—
8	Dic₁₀₉₆	C₂²	1	1	yes	characteristic, complemented
9	#17984033	—	2	1	no	—
10	D₂₁₉₂	—	2	1	no	—
11	D₂₁₉₂	C₂²	1	2	yes	—
12	D₂₁₉₂	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₃₇	C₂²	1	2	yes	complemented
15	#14555510	—	4	1	no	—
16	C₂×D₅₄₈	—	4	1	no	—
17	Dic₅₄₈	—	2	1	no	—
18	#14555499	—	4	1	no	—
19	D₁₀₉₆	—	2	2	no	—
20	D₁₀₉₆	—	2	1	no	—
21	D₁₀₉₆	—	4	1	no	—
22	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
24	C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₂²×D₁₃₇	—	8	1	no	—
26	Dic₂₇₄	—	4	1	no	—
27	D₅₄₈	—	4	2	no	—
28	D₅₄₈	—	4	1	no	—
29	D₅₄₈	—	8	1	no	—
30	C₂²⋊D₁₃₇	—	8	1	no	—
31	C₄×D₁₃₇	—	8	1	no	—
32	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
33	C₈×C₁₃₇	D₈	1	1	yes	characteristic
34	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₃₇	D₁₆	1	1	yes	characteristic
36	C₂×D₁₃₇	—	8	1	no	—
37	C₂×D₁₃₇	—	8	2	no	—
38	C₂×D₁₃₇	—	16	2	no	—
39	C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
40	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
41	Dic₁₃₇	—	8	1	no	—
42	C₂×C₁₃₇	—	2	1	no	—
43	C₂×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
44	D₁₃₇	—	16	1	no	—
45	D₁₃₇	—	16	2	no	—
46	C₁₃₇	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	137	1	no	maximal, Sylow
48	#508	—	137	1	no	—
49	C₂×D₁₆	—	137	1	no	—
50	SD₆₄	—	137	2	no	—
51	D₃₂	—	137	2	no	—
52	M₆₄	D₁₃₇	1	1	yes	characteristic, complemented
53	#135	—	274	1	no	—
54	C₂×D₈	—	274	1	no	—
55	Q₃₂	—	137	1	no	—
56	SD₃₂	—	274	1	no	—
57	D₁₆	—	274	1	no	—
58	D₁₆	—	137	2	no	—
59	D₁₆	—	137	1	no	—
60	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
61	C₃₂	C₂×D₁₃₇	1	2	yes	complemented
62	#41	—	548	1	no	—
63	C₂×D₄	—	548	1	no	—
64	Q₁₆	—	274	1	no	—
65	SD₁₆	—	548	1	no	—
66	D₈	—	274	2	no	—
67	D₈	—	274	1	no	—
68	D₈	—	548	1	no	—
69	C₂×C₈	D₅₄₈	1	1	yes	characteristic
70	C₁₆	D₅₄₈	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₃₇	1	1	yes	characteristic
72	C₂³	—	1096	1	no	—
73	Q₈	—	548	1	no	—
74	D₄	—	548	2	no	—
75	D₄	—	548	1	no	—
76	D₄	—	1096	1	no	—
77	D₄	—	1096	1	no	—
78	C₂×C₄	—	1096	1	no	—
79	C₂×C₄	D₁₀₉₆	1	1	yes	characteristic
80	C₈	D₁₀₉₆	1	1	yes	characteristic
81	C₈	C₂×D₅₄₈	1	1	yes	characteristic
82	C₂²	—	1096	1	no	—
83	C₂²	—	1096	2	no	—
84	C₂²	—	2192	2	no	—
85	C₂²	D₂₁₉₂	1	1	yes	characteristic
86	C₄	—	1096	1	no	—
87	C₄	D₂₁₉₂	1	1	yes	characteristic
88	C₄	C₂×D₁₀₉₆	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2192	2	no	—
91	C₂	—	2192	1	no	—
92	C₂	C₂×D₂₁₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₁₉₂
commutator, cosocle	C₁₆×C₁₃₇	C₂³
Frattini	C₁₆	C₂²×D₁₃₇
Fitting	M₆₄×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂×D₁₆
hypercenter	M₆₄	D₁₃₇
hypocenter	C₁₃₇	#2263
LCS	G ▷ C₁₆×C₁₃₇ ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #22577835	C₂	semidirect
2. C₂×D₂₁₉₂	C₂	semidirect
3. #22577242	C₂	semidirect
4. D₄₃₈₄	C₂	semidirect
5. M₆₄×C₁₃₇	C₂	semidirect
8. Dic₁₀₉₆	C₂²	semidirect
12. D₂₁₉₂	C₂²	semidirect
14. C₃₂×C₁₃₇	C₂²	semidirect
46. C₁₃₇	#2263	semidirect
52. M₆₄	D₁₃₇	semidirect
61. C₃₂	C₂×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577835	2
C₂×D₂₁₉₂	2
#22577242	2
D₄₃₈₄	2
M₆₄×C₁₃₇	2
#2263	137