Subgroups — #22577608

Order 8768.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14555498	C₂	1	2	yes	maximal
2	#17984088	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#11764040	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984011	C₂	1	2	yes	maximal, complemented
5	C₄×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₇₄	C₂²	1	2	yes	—
7	#14555498	—	2	4	no	—
8	C₂×#11764040	C₂²	1	2	yes	—
9	C₄⋊Dic₁₃₇	—	2	1	no	—
10	C₄⋊Dic₁₃₇	—	2	1	no	—
11	C₂×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₃₇	—	2	1	no	—
13	C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
15	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	Dic₂₇₄	—	2	2	no	—
17	Dic₂₇₄	—	2	2	no	—
18	C₂×Dic₁₃₇	—	2	2	no	—
19	#11764040	—	2	2	no	—
20	C₂³×C₁₃₇	—	2	1	no	—
21	D₄×C₁₃₇	—	2	1	no	—
22	D₄×C₁₃₇	D₄	1	2	yes	—
23	C₂×C₄×C₁₃₇	—	2	1	no	—
24	C₂×C₄×C₁₃₇	—	4	2	no	—
25	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₂²×C₁₃₇	—	2	2	no	—
29	C₂²×C₁₃₇	—	4	1	no	—
30	C₂²×C₁₃₇	—	4	1	no	—
31	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₃₇	—	2	1	no	—
33	C₄×C₁₃₇	—	4	1	no	—
34	C₄×C₁₃₇	SD₁₆	1	2	yes	complemented
35	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₃₇	#41	1	1	yes	characteristic
37	Dic₁₃₇	—	4	2	no	—
38	C₂×C₁₃₇	—	4	2	no	—
39	C₂×C₁₃₇	#121	1	1	yes	characteristic
40	C₂×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₁₃₇	#137	1	1	yes	characteristic
42	C₁₃₇	#461	1	1	yes	characteristic, complemented, Sylow
43	#461	—	137	1	no	maximal, Sylow
44	C₂×SD₁₆	—	137	2	no	—
45	C₄⋊Q₈	—	137	1	no	—
46	C₄×D₄	D₁₃₇	1	1	yes	characteristic
47	C₄⋊C₈	—	137	1	no	—
48	#103	—	137	2	no	—
49	C₂×Q₈	—	137	2	no	—
50	C₂×D₄	C₂×D₁₃₇	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	SD₁₆	—	274	4	no	—
53	C₂×C₈	—	137	2	no	—
54	C₄⋊C₄	—	274	1	no	—
55	C₄⋊C₄	—	274	1	no	—
56	C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
59	C₂³	—	2	1	no	—
60	Q₈	—	274	2	no	—
61	Q₈	—	274	2	no	—
62	D₄	—	2	1	no	—
63	D₄	D₅₄₈	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	274	2	no	—
67	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
70	C₈	—	274	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	548	2	no	—
78	C₄	#14555498	1	2	yes	complemented
79	C₄	C₂×D₅₄₈	1	1	yes	characteristic
80	C₄	#14555510	1	1	yes	characteristic
81	C₂	—	4	2	no	—
82	C₂	#17984062	1	1	yes	characteristic
83	C₂	C₂×#14555498	1	1	yes	characteristic
84	C₂	#17984118	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₄×D₄×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₄×D₄	D₁₃₇
hypocenter	C₁₃₇	#461
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #17984088	C₂	semidirect
3. C₄⋊#11764040	C₂	semidirect
4. #17984011	C₂	semidirect
34. C₄×C₁₃₇	SD₁₆	semidirect
42. C₁₃₇	#461	semidirect
78. C₄	#14555498	semidirect