Subgroups — #22578274

Order 8768.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#17984167	C₂	1	1	yes	maximal, characteristic
3	C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#14555520	C₄	1	2	yes	complemented
5	#14555519	—	2	1	no	—
6	#14555518	—	2	1	no	—
7	C₄⋊#10587739	C₄	1	2	yes	complemented
8	C₂²⋊#10587739	—	2	1	no	—
9	C₄×#10587739	—	2	1	no	—
10	C₂×C₄×D₁₃₇	—	2	1	no	—
11	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₃₇	—	2	1	no	—
13	C₄×Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
14	C₄²×C₁₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#10587739	—	2	2	no	—
16	#11764047	—	2	2	no	—
17	C₂²×D₁₃₇	D₄	1	1	yes	characteristic
18	C₂×Dic₁₃₇	—	2	1	no	—
19	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
20	C₄×D₁₃₇	—	2	4	no	—
21	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄×D₁₃₇	D₄	1	1	yes	characteristic
24	C₄×D₁₃₇	Q₈	1	1	yes	characteristic
25	C₂×C₄×C₁₃₇	—	2	1	no	—
26	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×D₁₃₇	—	2	2	no	—
29	C₂×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
31	#10587739	—	4	2	no	—
32	C₄×C₁₃₇	—	2	2	no	—
33	C₄×C₁₃₇	C₄²	1	1	yes	characteristic
34	C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
35	Dic₁₃₇	—	2	2	no	—
36	Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
37	Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₁₃₇	—	2	1	no	—
39	C₂×C₁₃₇	#95	1	1	yes	characteristic
40	D₁₃₇	—	2	1	no	—
41	D₁₃₇	#104	1	2	yes	—
42	C₁₃₇	#339	1	1	yes	characteristic, complemented, Sylow
43	#339	—	137	1	no	maximal, Sylow
44	C₂×M₁₆	—	137	1	no	—
45	#117	—	137	1	no	—
46	C₂×C₄²	—	137	1	no	—
47	C₂²×C₄	—	274	1	no	—
48	C₂²×C₄	—	137	1	no	—
49	M₁₆	—	274	1	no	—
50	M₁₆	—	137	2	no	—
51	C₂×C₈	—	274	1	no	—
52	C₄⋊C₄	—	137	2	no	—
53	C₂²⋊C₄	—	274	1	no	—
54	C₄²	—	274	1	no	—
55	C₄²	—	274	1	no	—
56	C₄²	—	137	1	no	—
57	C₄²	#10587739	1	1	yes	characteristic, complemented
58	C₂³	—	137	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	274	4	no	—
61	C₂×C₄	—	274	1	no	—
62	C₂×C₄	—	274	2	no	—
63	C₂×C₄	—	137	1	no	—
64	C₂×C₄	—	137	1	no	—
65	C₂×C₄	—	137	1	no	—
66	C₂×C₄	—	137	1	no	—
67	C₂×C₄	—	137	1	no	—
68	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
69	C₈	—	274	2	no	—
70	C₂²	—	274	2	no	—
71	C₂²	—	137	1	no	—
72	C₂²	—	137	1	no	—
73	C₂²	C₂²⋊#10587739	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	274	2	no	—
76	C₄	—	548	2	no	—
77	C₄	—	137	1	no	—
78	C₄	—	137	1	no	—
79	C₄	C₄×#10587739	1	1	yes	characteristic
80	C₄	C₄⋊#10587739	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	274	1	no	—
83	C₂	—	137	2	no	—
84	C₂	(C₂×#10587739)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#10587739
commutator	C₄×C₁₃₇	C₄²
Frattini	C₂×C₄	C₂×#10587739
Fitting	C₄²×C₁₃₇	C₄
socle	C₂×C₁₃₇	#95
hypercenter	C₄²	#10587739
hypocenter	C₁₃₇	#339
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. #14555520	C₄	semidirect
7. C₄⋊#10587739	C₄	semidirect
13. C₄×Dic₁₃₇	C₄	semidirect
14. C₄²×C₁₃₇	C₄	semidirect
42. C₁₃₇	#339	semidirect
57. C₄²	#10587739	semidirect