Subgroups — #31778926

Order 17536.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#17983994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
5	C₂×#14555485	C₄	1	1	yes	characteristic
6	C₁₆×D₁₃₇	—	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	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	—	2	1	no	—
30	C₂×D₁₃₇	C₃₂	1	2	yes	complemented
31	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
33	Dic₁₃₇	—	2	1	no	—
34	C₂×C₁₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
35	C₂×C₁₃₇	C₂×C₃₂	1	1	yes	characteristic
36	C₂×C₁₃₇	M₆₄	1	1	yes	characteristic
37	D₁₃₇	—	2	2	no	—
38	C₁₃₇	C₂²⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
39	C₂²⋊C₃₂	—	137	1	no	maximal, Sylow
40	C₂²×C₁₆	—	137	1	no	—
41	C₂×C₃₂	—	137	1	no	—
42	C₂×C₃₂	D₁₃₇	1	1	yes	characteristic, complemented
43	C₂²×C₈	—	137	1	no	—
44	C₂×C₁₆	—	274	1	no	—
45	C₂×C₁₆	—	274	1	no	—
46	C₂×C₁₆	—	137	1	no	—
47	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
48	C₃₂	—	2	1	no	—
49	C₃₂	—	274	1	no	—
50	C₂²×C₄	—	137	1	no	—
51	C₂×C₈	—	274	1	no	—
52	C₂×C₈	—	274	1	no	—
53	C₂×C₈	—	137	1	no	—
54	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
55	C₁₆	—	274	1	no	—
56	C₁₆	C₂²⋊D₁₃₇	1	1	yes	characteristic
57	C₁₆	D₅₄₈	1	1	yes	characteristic
58	C₂³	—	137	1	no	—
59	C₂×C₄	—	274	1	no	—
60	C₂×C₄	—	274	1	no	—
61	C₂×C₄	—	137	1	no	—
62	C₂×C₄	C₈×D₁₃₇	1	1	yes	characteristic
63	C₈	—	274	1	no	—
64	C₈	#14555487	1	1	yes	characteristic
65	C₈	#14555493	1	1	yes	characteristic
66	C₂²	—	274	1	no	—
67	C₂²	—	274	1	no	—
68	C₂²	—	137	2	no	—
69	C₂²	C₁₆×D₁₃₇	1	1	yes	characteristic
70	C₄	—	274	1	no	—
71	C₄	#17984000	1	1	yes	characteristic
72	C₄	#17984027	1	1	yes	characteristic
73	C₂	—	274	2	no	—
74	C₂	#22577182	1	1	yes	characteristic
75	C₂	C₃₂×D₁₃₇	1	1	yes	characteristic
76	C₂	#22577236	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₁₃₇
commutator	C₂×C₁₃₇	C₂×C₃₂
Fitting	C₂×C₃₂×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₁₆
prime radical	C₂²×D₁₃₇	C₁₆
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	C₂×C₃₂	D₁₃₇
hypocenter	C₁₃₇	C₂²⋊C₃₂
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#17983994	C₂	semidirect
3. C₂×C₃₂×C₁₃₇	C₂	semidirect
30. C₂×D₁₃₇	C₃₂	semidirect
38. C₁₃₇	C₂²⋊C₃₂	semidirect
42. C₂×C₃₂	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×D₁₃₇	2
C₂×#17983994	2
C₂×C₃₂×C₁₃₇	2
C₂²⋊C₃₂	137