Subgroups — #31783109

Order 17536.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577835	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₀₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₁₉₂	C₂	1	2	yes	maximal, complemented
4	#22577242	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984110	—	2	1	no	—
7	C₂×Dic₅₄₈	—	2	1	no	—
8	Dic₁₀₉₆	—	2	1	no	—
9	Dic₁₀₉₆	C₂²	1	2	yes	—
10	Dic₁₀₉₆	C₂²	1	1	yes	characteristic
11	#17984033	—	2	1	no	—
12	D₂₁₉₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₃₇	C₂²	1	2	yes	—
15	#14555510	—	4	1	no	—
16	C₂×Dic₂₇₄	—	4	1	no	—
17	Dic₅₄₈	—	2	1	no	—
18	Dic₅₄₈	—	2	2	no	—
19	Dic₅₄₈	—	4	1	no	—
20	#14555499	—	4	1	no	—
21	D₁₀₉₆	—	2	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	Dic₂₇₄	—	4	1	no	—
26	Dic₂₇₄	—	4	2	no	—
27	Dic₂₇₄	—	8	1	no	—
28	D₅₄₈	—	4	1	no	—
29	C₂²⋊D₁₃₇	—	8	1	no	—
30	C₂×Dic₁₃₇	—	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	C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
38	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
39	Dic₁₃₇	—	8	1	no	—
40	Dic₁₃₇	—	8	2	no	—
41	C₂×C₁₃₇	—	2	1	no	—
42	C₂×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
43	D₁₃₇	—	16	1	no	—
44	C₁₃₇	#2264	1	1	yes	characteristic, complemented, Sylow
45	#2264	—	137	1	no	maximal, Sylow
46	#508	—	137	1	no	—
47	C₂×Q₃₂	—	137	1	no	—
48	Q₆₄	—	137	2	no	—
49	SD₆₄	—	137	2	no	—
50	M₆₄	D₁₃₇	1	1	yes	characteristic, complemented
51	#135	—	274	1	no	—
52	C₂×Q₁₆	—	274	1	no	—
53	Q₃₂	—	274	1	no	—
54	Q₃₂	—	137	2	no	—
55	Q₃₂	—	137	1	no	—
56	SD₃₂	—	274	1	no	—
57	D₁₆	—	137	1	no	—
58	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
59	C₃₂	C₂×D₁₃₇	1	2	yes	—
60	#41	—	548	1	no	—
61	C₂×Q₈	—	548	1	no	—
62	Q₁₆	—	274	1	no	—
63	Q₁₆	—	274	2	no	—
64	Q₁₆	—	548	1	no	—
65	SD₁₆	—	548	1	no	—
66	D₈	—	274	1	no	—
67	C₂×C₈	D₅₄₈	1	1	yes	characteristic
68	C₁₆	D₅₄₈	1	1	yes	characteristic
69	C₁₆	C₂²×D₁₃₇	1	1	yes	characteristic
70	Q₈	—	548	1	no	—
71	Q₈	—	548	2	no	—
72	Q₈	—	1096	1	no	—
73	D₄	—	548	1	no	—
74	D₄	—	1096	1	no	—
75	C₂×C₄	—	1096	1	no	—
76	C₂×C₄	—	1096	1	no	—
77	C₂×C₄	D₁₀₉₆	1	1	yes	characteristic
78	C₈	D₁₀₉₆	1	1	yes	characteristic
79	C₈	C₂×D₅₄₈	1	1	yes	characteristic
80	C₂²	—	1096	1	no	—
81	C₂²	D₂₁₉₂	1	1	yes	characteristic
82	C₄	—	1096	1	no	—
83	C₄	—	1096	2	no	—
84	C₄	D₂₁₉₂	1	1	yes	characteristic
85	C₄	C₂×D₁₀₉₆	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2192	1	no	—
88	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
prime radical	#22577835	C₂
hypercenter	M₆₄	D₁₃₇
hypocenter	C₁₃₇	#2264
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
2. C₂×Dic₁₀₉₆	C₂	semidirect
3. Dic₂₁₉₂	C₂	semidirect
4. #22577242	C₂	semidirect
5. M₆₄×C₁₃₇	C₂	semidirect
44. C₁₃₇	#2264	semidirect
50. M₆₄	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×Dic₁₀₉₆	2
Dic₂₁₉₂	2
#22577242	2
M₆₄×C₁₃₇	2
#2264	137