Subgroups — #31778932

Order 17536.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577819	C₂	1	1	yes	maximal, characteristic
2	C₂×#17983994	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984094	C₄	1	1	yes	characteristic
5	#17984028	C₄	1	1	yes	characteristic
6	#17984027	—	2	1	no	—
7	C₁₆×D₁₃₇	—	2	1	no	—
8	#17983994	—	2	1	no	—
9	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₃₇	—	2	1	no	—
11	#14555510	C₈	1	1	yes	characteristic
12	#14555494	C₈	1	1	yes	characteristic
13	#14555493	—	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	Dic₂₇₄	C₁₆	1	1	yes	characteristic
20	D₅₄₈	C₁₆	1	1	yes	characteristic
21	C₂²⋊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	M₃₂	1	1	yes	characteristic
28	C₂×D₁₃₇	—	2	1	no	—
29	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₃₇	—	2	1	no	—
32	C₂×C₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₃₇	—	4	1	no	—
35	C₁₃₇	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	137	1	no	maximal, Sylow
37	#504	—	137	1	no	—
38	M₆₄	D₁₃₇	1	1	yes	characteristic, complemented
39	C₂×C₃₂	—	137	1	no	—
40	#131	—	137	1	no	—
41	M₃₂	—	274	1	no	—
42	M₃₂	—	137	1	no	—
43	C₂×C₁₆	—	274	1	no	—
44	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	274	1	no	—
47	#41	—	137	1	no	—
48	M₁₆	—	274	1	no	—
49	M₁₆	—	137	1	no	—
50	C₂×C₈	—	274	1	no	—
51	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
52	C₁₆	—	274	1	no	—
53	C₁₆	C₂²⋊D₁₃₇	1	1	yes	characteristic
54	C₁₆	D₅₄₈	1	1	yes	characteristic
55	Q₈	—	137	1	no	—
56	D₄	—	274	1	no	—
57	D₄	—	137	1	no	—
58	C₂×C₄	—	274	1	no	—
59	C₂×C₄	C₈×D₁₃₇	1	1	yes	characteristic
60	C₈	—	274	1	no	—
61	C₈	#14555487	1	1	yes	characteristic
62	C₈	#14555493	1	1	yes	characteristic
63	C₂²	—	274	1	no	—
64	C₂²	#17984027	1	1	yes	characteristic
65	C₄	—	274	1	no	—
66	C₄	#17984000	1	1	yes	characteristic
67	C₄	C₁₆×D₁₃₇	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	548	1	no	—
70	C₂	#22577182	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₃₇
Fitting	M₆₄×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂²⋊C₁₆
prime radical	#14555510	C₈
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	M₆₄	D₁₃₇
hypocenter	C₁₃₇	#1401
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#17983994	C₂	semidirect
3. M₆₄×C₁₃₇	C₂	semidirect
35. C₁₃₇	#1401	semidirect
38. M₆₄	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577819	2
C₂×#17983994	2
M₆₄×C₁₃₇	2
#1401	137