Subgroups — #22578275

Order 8768.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984167	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10587739	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10587739	—	2	1	no	—
5	C₄⋊#10587739	C₄	1	2	yes	complemented
6	C₂²⋊#10587739	—	2	1	no	—
7	C₄×#10587739	—	2	1	no	—
8	C₄×#10587739	—	2	1	no	—
9	C₄×#10587739	C₄	1	2	yes	complemented
10	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
11	#14555494	C₄	1	1	yes	characteristic, complemented
12	#14555493	—	2	1	no	—
13	C₈×D₁₃₇	—	2	1	no	—
14	M₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#10587739	—	2	4	no	—
16	C₂×#10587739	—	2	2	no	—
17	C₂×#10587739	—	2	2	no	—
18	C₂²×D₁₃₇	D₄	1	1	yes	characteristic
19	C₂×Dic₁₃₇	Q₈	1	1	yes	characteristic
20	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₁₃₇	D₄	1	1	yes	characteristic
23	C₄×D₁₃₇	D₄	1	1	yes	characteristic
24	#11764040	—	2	1	no	—
25	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₈×C₁₃₇	—	2	1	no	—
27	C₂²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×D₁₃₇	—	2	2	no	—
29	C₂×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	#10587739	—	2	4	no	—
32	#10587739	—	4	2	no	—
33	C₄×C₁₃₇	C₄²	1	1	yes	characteristic
34	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
36	Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₁₃₇	—	2	1	no	—
38	C₂×C₁₃₇	#95	1	1	yes	characteristic
39	D₁₃₇	—	2	1	no	—
40	D₁₃₇	#104	1	2	yes	—
41	C₁₃₇	#339	1	1	yes	characteristic, complemented, Sylow
42	#339	—	137	1	no	maximal, Sylow
43	C₂×M₁₆	—	137	1	no	—
44	#117	—	137	1	no	—
45	C₂×C₄²	—	137	1	no	—
46	C₂²×C₄	—	274	1	no	—
47	C₂²×C₄	—	137	1	no	—
48	M₁₆	—	274	1	no	—
49	M₁₆	—	137	1	no	—
50	M₁₆	#10587739	1	1	yes	characteristic, complemented
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	2	no	—
57	C₂³	—	137	1	no	—
58	C₂×C₄	—	274	4	no	—
59	C₂×C₄	—	274	2	no	—
60	C₂×C₄	—	274	2	no	—
61	C₂×C₄	—	137	1	no	—
62	C₂×C₄	—	137	1	no	—
63	C₂×C₄	—	137	1	no	—
64	C₂×C₄	—	137	1	no	—
65	C₂×C₄	—	137	1	no	—
66	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	274	1	no	—
69	C₂²	—	274	2	no	—
70	C₂²	—	137	1	no	—
71	C₂²	—	137	1	no	—
72	C₂²	C₄⋊#10587739	1	1	yes	characteristic
73	C₄	—	274	4	no	—
74	C₄	—	548	2	no	—
75	C₄	—	137	1	no	—
76	C₄	—	137	1	no	—
77	C₄	C₄×#10587739	1	1	yes	characteristic
78	C₄	C₂²⋊#10587739	1	1	yes	characteristic
79	C₂	—	2	1	no	—
80	C₂	—	274	1	no	—
81	C₂	—	137	2	no	—
82	C₂	(C₂×#10587739)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#10587739
commutator	C₄×C₁₃₇	C₄²
Frattini	C₂×C₄	C₂×#10587739
Fitting	M₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	#95
hypercenter	M₁₆	#10587739
hypocenter	C₁₃₇	#339
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄⋊#10587739	C₄	semidirect
9. C₄×#10587739	C₄	semidirect
11. #14555494	C₄	semidirect
14. M₁₆×C₁₃₇	C₄	semidirect
41. C₁₃₇	#339	semidirect
50. M₁₆	#10587739	semidirect