Subgroups — #31778968

Order 17536.

82 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	#22577229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×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	complemented
7	#17984022	—	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	C₄⋊Dic₁₃₇	—	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	C₂×Dic₁₃₇	—	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	D₁₆	1	1	yes	characteristic
25	C₂×D₁₃₇	—	8	2	no	—
26	C₂×D₁₃₇	—	16	1	no	—
27	C₂×D₁₃₇	—	16	1	no	—
28	C₄×C₁₃₇	#102	1	1	yes	characteristic
29	C₄×C₁₃₇	SD₃₂	1	1	yes	characteristic
30	Dic₁₃₇	—	16	1	no	—
31	C₂×C₁₃₇	#357	1	1	yes	characteristic
32	C₂×C₁₃₇	D₃₂	1	1	yes	characteristic
33	C₂×C₁₃₇	SD₆₄	1	1	yes	characteristic
34	D₁₃₇	—	16	2	no	—
35	C₁₃₇	#1415	1	1	yes	characteristic, complemented, Sylow
36	#1415	—	137	1	no	maximal, Sylow
37	C₂×D₁₆	—	137	1	no	—
38	C₂×C₃₂	D₁₃₇	1	1	yes	characteristic, complemented
39	#366	—	137	1	no	—
40	C₂×D₈	—	274	1	no	—
41	D₁₆	—	274	1	no	—
42	D₁₆	—	137	2	no	—
43	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
44	#107	—	274	1	no	—
45	C₃₂	—	2	1	no	—
46	C₂×D₄	—	548	1	no	—
47	D₈	—	274	2	no	—
48	D₈	—	548	1	no	—
49	C₂×C₈	D₅₄₈	1	1	yes	characteristic
50	C₄⋊C₄	—	548	1	no	—
51	C₁₆	C₄×D₁₃₇	1	1	yes	characteristic
52	C₁₆	C₂²⋊D₁₃₇	1	1	yes	characteristic
53	C₂³	—	1096	1	no	—
54	D₄	—	548	2	no	—
55	D₄	—	1096	1	no	—
56	C₂×C₄	—	1096	1	no	—
57	C₂×C₄	D₁₀₉₆	1	1	yes	characteristic
58	C₈	#14555487	1	1	yes	characteristic
59	C₈	#14555499	1	1	yes	characteristic
60	C₂²	—	1096	2	no	—
61	C₂²	—	2192	1	no	—
62	C₂²	—	2192	1	no	—
63	C₂²	D₂₁₉₂	1	1	yes	characteristic
64	C₄	—	2192	1	no	—
65	C₄	#17984010	1	1	yes	characteristic
66	C₄	#17984033	1	1	yes	characteristic
67	C₂	—	2192	2	no	—
68	C₂	#22577205	1	1	yes	characteristic
69	C₂	D₄₃₈₄	1	1	yes	characteristic
70	C₂	#22577242	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₁₉₂
commutator	C₁₆×C₁₃₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₃₇
Fitting	C₂×C₃₂×C₁₃₇	C₂
socle	C₂²×C₁₃₇	D₁₆
prime radical	C₂×D₂₁₉₂	C₂
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	C₂×C₃₂	D₁₃₇
hypocenter	C₁₃₇	#1415
LCS	G ▷ C₁₆×C₁₃₇ ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #22577229	C₂	semidirect
3. C₂×C₃₂×C₁₃₇	C₂	semidirect
6. D₂₁₉₂	C₄	semidirect
35. C₁₃₇	#1415	semidirect
38. 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×D₂₁₉₂	2
#22577229	2
C₂×C₃₂×C₁₃₇	2
#1415	137