Subgroups — #31778928

Order 17536.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984027	C₂	1	1	yes	maximal, characteristic
2	#22577237	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
5	#17984027	—	2	1	no	—
6	#17984027	—	2	1	no	—
7	C₂×#14555485	C₄	1	1	yes	characteristic
8	#17983994	—	2	1	no	—
9	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₃₇	—	2	1	no	—
11	C₂×C₄×D₁₃₇	C₈	1	1	yes	characteristic
12	C₂×#11764040	C₈	1	1	yes	characteristic
13	C₈×D₁₃₇	—	2	1	no	—
14	C₈×D₁₃₇	—	2	1	no	—
15	#14555485	—	2	1	no	—
16	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
18	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
19	C₂²×D₁₃₇	C₁₆	1	1	yes	characteristic
20	C₂×Dic₁₃₇	C₁₆	1	1	yes	characteristic
21	C₄×D₁₃₇	—	2	1	no	—
22	C₄×D₁₃₇	—	2	1	no	—
23	#11764040	—	2	1	no	—
24	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
28	C₂×D₁₃₇	—	2	1	no	—
29	C₂×D₁₃₇	—	4	1	no	—
30	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
32	Dic₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	—	2	1	no	—
34	C₂×C₁₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
35	D₁₃₇	—	4	1	no	—
36	C₁₃₇	#1400	1	1	yes	characteristic, complemented, Sylow
37	#1400	—	137	1	no	maximal, Sylow
38	C₂×M₃₂	—	137	1	no	—
39	M₆₄	—	137	1	no	—
40	M₆₄	D₁₃₇	1	1	yes	characteristic, complemented
41	C₂²×C₈	—	137	1	no	—
42	M₃₂	—	274	1	no	—
43	M₃₂	—	274	1	no	—
44	C₂×C₁₆	—	137	1	no	—
45	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
46	C₃₂	—	2	1	no	—
47	C₃₂	—	274	1	no	—
48	C₂²×C₄	—	137	1	no	—
49	C₂×C₈	—	274	1	no	—
50	C₂×C₈	—	274	1	no	—
51	C₂×C₈	—	137	1	no	—
52	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
53	C₁₆	—	274	1	no	—
54	C₁₆	C₂²⋊D₁₃₇	1	1	yes	characteristic
55	C₁₆	D₅₄₈	1	1	yes	characteristic
56	C₂³	—	137	1	no	—
57	C₂×C₄	—	274	1	no	—
58	C₂×C₄	—	274	1	no	—
59	C₂×C₄	—	137	1	no	—
60	C₂×C₄	C₈×D₁₃₇	1	1	yes	characteristic
61	C₈	—	274	1	no	—
62	C₈	#14555487	1	1	yes	characteristic
63	C₈	#14555493	1	1	yes	characteristic
64	C₂²	—	274	1	no	—
65	C₂²	—	548	1	no	—
66	C₂²	C₁₆×D₁₃₇	1	1	yes	characteristic
67	C₄	—	274	1	no	—
68	C₄	#17984000	1	1	yes	characteristic
69	C₄	#17984027	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	548	1	no	—
72	C₂	#22577182	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14555487
commutator	C₂²×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₃₇
Fitting	M₆₄×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂²⋊C₁₆
prime radical	C₂²×D₁₃₇	C₁₆
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	M₆₄	D₁₃₇
hypocenter	C₁₃₇	#1400
LCS	G ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #22577237	C₂	semidirect
3. M₆₄×C₁₃₇	C₂	semidirect
36. C₁₃₇	#1400	semidirect
40. M₆₄	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984027	2
#22577237	2
M₆₄×C₁₃₇	2
#1400	137