Subgroups — #31778975

Order 17536.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577835	C₂	1	1	yes	maximal, characteristic
2	#22577237	C₂	1	1	yes	maximal, characteristic, complemented
3	#367×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984110	—	2	1	no	—
5	Dic₁₀₉₆	C₄	1	1	yes	characteristic, complemented
6	#17984033	—	2	1	no	—
7	D₂₁₉₂	C₄	1	1	yes	characteristic, complemented
8	#17983994	—	2	1	no	—
9	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	#107×C₁₃₇	—	2	1	no	—
11	#14555510	—	4	1	no	—
12	Dic₅₄₈	—	2	1	no	—
13	#14555499	—	4	1	no	—
14	D₁₀₉₆	—	2	1	no	—
15	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
16	C₄⋊C₄×C₁₃₇	—	4	1	no	—
17	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
19	Dic₂₇₄	—	4	1	no	—
20	D₅₄₈	—	4	1	no	—
21	C₂²⋊D₁₃₇	—	8	1	no	—
22	C₄×D₁₃₇	—	8	1	no	—
23	C₂×C₄×C₁₃₇	—	8	1	no	—
24	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
25	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₇	D₁₆	1	1	yes	characteristic
28	C₂×D₁₃₇	—	8	1	no	—
29	C₄×C₁₃₇	—	16	1	no	—
30	C₄×C₁₃₇	#102	1	1	yes	characteristic
31	C₄×C₁₃₇	SD₃₂	1	1	yes	characteristic
32	Dic₁₃₇	—	8	1	no	—
33	C₂×C₁₃₇	—	2	1	no	—
34	C₂×C₁₃₇	#357	1	1	yes	characteristic
35	D₁₃₇	—	16	1	no	—
36	C₁₃₇	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	137	1	no	maximal, Sylow
38	#508	—	137	1	no	—
39	M₆₄	—	137	1	no	—
40	#367	D₁₃₇	1	1	yes	characteristic, complemented
41	#135	—	274	1	no	—
42	Q₃₂	—	137	1	no	—
43	SD₃₂	—	274	1	no	—
44	D₁₆	—	137	1	no	—
45	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
46	#107	—	2	1	no	—
47	C₃₂	—	274	1	no	—
48	#41	—	548	1	no	—
49	Q₁₆	—	274	1	no	—
50	SD₁₆	—	548	1	no	—
51	D₈	—	274	1	no	—
52	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
53	C₄⋊C₄	—	4	1	no	—
54	C₁₆	C₄×D₁₃₇	1	1	yes	characteristic
55	C₁₆	D₅₄₈	1	1	yes	characteristic
56	Q₈	—	548	1	no	—
57	D₄	—	548	1	no	—
58	D₄	—	1096	1	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	1096	1	no	—
61	C₂×C₄	#14555495	1	1	yes	characteristic
62	C₈	#14555487	1	1	yes	characteristic
63	C₈	#14555497	1	1	yes	characteristic
64	C₂²	—	1096	1	no	—
65	C₂²	#17984029	1	1	yes	characteristic
66	C₄	—	16	1	no	—
67	C₄	—	1096	1	no	—
68	C₄	#17984008	1	1	yes	characteristic
69	C₄	#17984031	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	2192	1	no	—
72	C₂	#22577203	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22577203
commutator	C₁₆×C₁₃₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₃₇
Fitting	#367×C₁₃₇	C₂
socle	C₂×C₁₃₇	#357
prime radical	#22577835	C₂
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	#367	D₁₃₇
hypocenter	C₁₃₇	#1418
LCS	G ▷ C₁₆×C₁₃₇ ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #367

Complements

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

N	H ≅ G/N	type
2. #22577237	C₂	semidirect
3. #367×C₁₃₇	C₂	semidirect
5. Dic₁₀₉₆	C₄	semidirect
7. D₂₁₉₂	C₄	semidirect
36. C₁₃₇	#1418	semidirect
40. #367	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577835	2
#22577237	2
#367×C₁₃₇	2
#1418	137