Subgroups — #31778980

Order 17536.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₁₉₂	C₂	1	1	yes	maximal, characteristic
2	#22577233	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₉₆	—	2	1	no	—
5	D₂₁₉₂	—	2	1	no	—
6	D₂₁₉₂	C₄	1	2	yes	—
7	#17984024	—	2	1	no	—
8	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₃₂×C₁₃₇	—	2	1	no	—
10	C₂×D₅₄₈	—	4	1	no	—
11	D₁₀₉₆	—	2	2	no	—
12	D₁₀₉₆	—	4	1	no	—
13	#14555494	—	4	1	no	—
14	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
15	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
17	C₂²×D₁₃₇	—	8	1	no	—
18	D₅₄₈	—	4	2	no	—
19	D₅₄₈	—	8	1	no	—
20	#11764040	—	8	1	no	—
21	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
22	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₃₇	SD₃₂	1	1	yes	characteristic
25	C₂×D₁₃₇	—	8	2	no	—
26	C₂×D₁₃₇	—	16	2	no	—
27	C₄×C₁₃₇	#102	1	1	yes	characteristic
28	C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
29	C₂×C₁₃₇	—	2	1	no	—
30	C₂×C₁₃₇	#357	1	1	yes	characteristic
31	D₁₃₇	—	16	2	no	—
32	C₁₃₇	#1419	1	1	yes	characteristic, complemented, Sylow
33	#1419	—	137	1	no	maximal, Sylow
34	C₂×D₁₆	—	137	1	no	—
35	M₆₄	D₁₃₇	1	1	yes	characteristic, complemented
36	#368	—	137	1	no	—
37	C₂×D₈	—	274	1	no	—
38	D₁₆	—	274	1	no	—
39	D₁₆	—	137	2	no	—
40	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
41	#108	—	274	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	548	1	no	—
44	D₈	—	274	2	no	—
45	D₈	—	548	1	no	—
46	M₁₆	—	548	1	no	—
47	C₂×C₈	D₅₄₈	1	1	yes	characteristic
48	C₁₆	C₄×D₁₃₇	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₃₇	1	1	yes	characteristic
50	C₂³	—	1096	1	no	—
51	D₄	—	548	2	no	—
52	D₄	—	1096	1	no	—
53	C₂×C₄	D₁₀₉₆	1	1	yes	characteristic
54	C₈	—	1096	1	no	—
55	C₈	#14555487	1	1	yes	characteristic
56	C₈	#14555499	1	1	yes	characteristic
57	C₂²	—	1096	2	no	—
58	C₂²	—	2192	2	no	—
59	C₂²	#17984033	1	1	yes	characteristic
60	C₄	#17984010	1	1	yes	characteristic
61	C₄	D₂₁₉₂	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	2192	2	no	—
64	C₂	#22577205	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22577205
commutator	C₁₆×C₁₃₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₃₇
Fitting	M₆₄×C₁₃₇	C₂
socle	C₂×C₁₃₇	#357
prime radical	C₂×D₂₁₉₂	C₂
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	M₆₄	D₁₃₇
hypocenter	C₁₃₇	#1419
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. #22577233	C₂	semidirect
3. M₆₄×C₁₃₇	C₂	semidirect
32. C₁₃₇	#1419	semidirect
35. 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×D₂₁₉₂	2
#22577233	2
M₆₄×C₁₃₇	2
#1419	137