Group #22578354 — subgroups

Subgroups — #22578354

Order 8768.

170 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#10587739	C₂	1	2	yes	maximal
2	C₂×C₄⋊#10587739	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#10587739	C₂	1	1	yes	maximal, characteristic
4	(C₂×#10587739)⋊C₄	C₂	1	2	yes	maximal
5	C₄⋊C₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10587739	C₂²	1	2	yes	—
7	C₂²×#10587739	C₂²	1	1	yes	characteristic
8	C₂²×#10587739	C₂²	1	1	yes	characteristic
9	C₄⋊#10587739	—	2	4	no	—
10	C₄⋊#10587739	—	2	2	no	—
11	C₄×#10587739	—	2	2	no	—
12	C₂×C₄×D₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
15	#14555489	C₄	1	2	yes	complemented
16	C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic, complemented
17	C₂×#10587739	—	2	1	no	—
18	C₂×#10587739	—	2	1	no	—
19	C₂×#10587739	—	2	1	no	—
20	C₂×#10587739	—	2	1	no	—
21	C₂×#10587739	—	2	2	no	—
22	C₂×#10587739	—	2	2	no	—
23	C₂×#10587739	—	2	2	no	—
24	C₂×#10587739	D₄	1	2	yes	—
25	C₂×#10587739	Q₈	1	2	yes	—
26	C₂²×D₁₃₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₃₇	C₂×C₄	1	2	yes	—
28	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄×D₁₃₇	—	2	2	no	—
30	C₄×D₁₃₇	—	2	2	no	—
31	C₄×D₁₃₇	D₄	1	2	yes	—
32	C₄×D₁₃₇	Q₈	1	2	yes	—
33	C₂×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₁₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×D₁₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×D₁₃₇	#41	1	1	yes	characteristic
41	C₂×D₁₃₇	#41	1	1	yes	characteristic
42	#10587739	—	2	2	no	—
43	#10587739	—	2	2	no	—
44	#10587739	—	4	2	no	—
45	C₄×C₁₃₇	—	2	2	no	—
46	C₄×C₁₃₇	C₄⋊C₄	1	2	yes	complemented
47	Dic₁₃₇	—	2	2	no	—
48	Dic₁₃₇	C₄⋊C₄	1	2	yes	complemented
49	C₂×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
51	C₂×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
52	D₁₃₇	#121	1	1	yes	characteristic
53	D₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
54	D₁₃₇	#125	1	1	yes	characteristic
55	D₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
56	C₁₃₇	#389	1	1	yes	characteristic, complemented, Sylow
57	#389	—	137	1	no	maximal, Sylow
58	C₂×C₄⋊C₄	—	137	1	no	—
59	C₂×C₄⋊C₄	—	137	2	no	—
60	C₂×C₄⋊C₄	—	137	1	no	—
61	C₂×C₄²	—	137	1	no	—
62	#95	—	137	2	no	—
63	C₂²×C₄	—	137	2	no	—
64	C₂²×C₄	—	137	1	no	—
65	C₂²×C₄	—	137	1	no	—
66	C₂²×C₄	—	137	2	no	—
67	C₂²×C₄	—	137	1	no	—
68	C₄⋊C₄	—	274	4	no	—
69	C₄⋊C₄	—	274	2	no	—
70	C₄⋊C₄	—	137	2	no	—
71	C₄⋊C₄	—	137	1	no	—
72	C₄⋊C₄	#10587739	1	1	yes	characteristic, complemented
73	C₄²	—	274	2	no	—
74	C₂³	—	137	1	no	—
75	C₂×C₄	—	274	2	no	—
76	C₂×C₄	—	274	2	no	—
77	C₂×C₄	—	274	1	no	—
78	C₂×C₄	—	274	1	no	—
79	C₂×C₄	—	274	1	no	—
80	C₂×C₄	—	274	1	no	—
81	C₂×C₄	—	274	2	no	—
82	C₂×C₄	—	274	2	no	—
83	C₂×C₄	—	274	2	no	—
84	C₂×C₄	—	137	2	no	—
85	C₂×C₄	—	137	2	no	—
86	C₂×C₄	—	137	2	no	—
87	C₂×C₄	—	137	1	no	—
88	C₂×C₄	—	137	2	no	—
89	C₂×C₄	—	137	2	no	—
90	C₂×C₄	C₂×#10587739	1	2	yes	—
91	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
92	C₂²	—	137	1	no	—
93	C₂²	—	137	1	no	—
94	C₂²	—	137	1	no	—
95	C₂²	—	137	1	no	—
96	C₂²	—	137	1	no	—
97	C₂²	—	137	1	no	—
98	C₂²	C₂²×#10587739	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	274	2	no	—
101	C₄	—	274	2	no	—
102	C₄	—	274	2	no	—
103	C₄	—	548	2	no	—
104	C₄	—	137	2	no	—
105	C₄	C₄⋊#10587739	1	2	yes	complemented
106	C₂	—	137	1	no	—
107	C₂	—	137	1	no	—
108	C₂	—	137	1	no	—
109	C₂	—	137	1	no	—
110	C₂	C₂×C₄⋊#10587739	1	1	yes	characteristic
111	C₂	D₄×#10587739	1	1	yes	characteristic
112	C₂	Q₈×#10587739	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10587739

C₂

yes

maximal

C₂×C₄⋊#10587739

C₂

yes

maximal, characteristic

C₂×C₄×#10587739

C₂

yes

maximal, characteristic

(C₂×#10587739)⋊C₄

C₂

yes

maximal

C₄⋊C₄×D₁₃₇

C₂

yes

maximal, characteristic

C₂²×#10587739

C₂²

yes

—

C₂²×#10587739

C₂²

yes

characteristic

C₂²×#10587739

C₂²

yes

—

—

—

C₂×C₄×D₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₄

yes

characteristic, complemented

#14555489

C₄

yes

complemented

C₄⋊C₄×C₁₃₇

C₄

yes

characteristic, complemented

—

—

—

—

—

—

—

D₄

yes

—

C₂×#10587739

Q₈

yes

—

C₂²×D₁₃₇

C₂³

yes

characteristic

C₂×Dic₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

D₄

yes

—

C₄×D₁₃₇

Q₈

yes

—

C₂×C₄×C₁₃₇

C₂×C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×D₁₃₇

C₂×D₄

yes

characteristic

C₂×D₁₃₇

C₂×D₄

yes

characteristic

C₂×D₁₃₇

C₂×Q₈

yes

characteristic

C₂×D₁₃₇

C₂×Q₈

yes

C₂×D₁₃₇

yes

C₂×D₁₃₇

yes

—

—

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄⋊C₄

yes

complemented

Dic₁₃₇

—

Dic₁₃₇

C₄⋊C₄

yes

complemented

C₂×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₁₃₇

C₄×D₄

yes

characteristic

C₂×C₁₃₇

C₄×Q₈

yes

characteristic

D₁₃₇

#121

yes

characteristic

D₁₃₇

C₂²⋊Q₈

yes

characteristic

D₁₃₇

#125

yes

characteristic

D₁₃₇

C₄⋊Q₈

yes

characteristic

C₁₃₇

#389

yes

characteristic, complemented, Sylow

#389

—

137

maximal, Sylow

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄²

—

137

—

#95

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

#10587739

yes

characteristic, complemented

C₄²

—

274

—

C₂³

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂×#10587739

yes

—

C₂×C₄

C₂×#10587739

yes

characteristic

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

C₂²×#10587739

yes

characteristic

C₄

—

100

C₄

—

274

—

101

C₄

—

274

—

102

C₄

—

274

—

103

C₄

—

548

—

104

C₄

—

137

—

105

C₄

C₄⋊#10587739

yes

complemented

106

C₂

—

137

—

107

C₂

—

137

—

108

C₂

—

137

—

109

C₂

—

137

—

110

C₂

C₂×C₄⋊#10587739

yes

characteristic

111

C₂

D₄×#10587739

yes

characteristic

112

C₂

Q₈×#10587739

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂²×C₄
Fitting	C₄⋊C₄×C₁₃₇	C₄
hypercenter	C₄⋊C₄	#10587739
hypocenter	C₁₃₇	#389
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂²×C₄

Fitting

C₄⋊C₄×C₁₃₇

C₄

C₄⋊C₄

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₄⋊C₄

N	H ≅ G/N	type
14. C₄⋊Dic₁₃₇	C₄	semidirect
15. #14555489	C₄	semidirect
16. C₄⋊C₄×C₁₃₇	C₄	semidirect
46. C₄×C₁₃₇	C₄⋊C₄	semidirect
48. Dic₁₃₇	C₄⋊C₄	semidirect
56. C₁₃₇	#389	semidirect
72. C₄⋊C₄	#10587739	semidirect
105. C₄	C₄⋊#10587739	semidirect

H ≅ G/N

type

14. C₄⋊Dic₁₃₇

C₄

semidirect

15. #14555489

C₄

semidirect

16. C₄⋊C₄×C₁₃₇

C₄

semidirect

46. C₄×C₁₃₇

C₄⋊C₄

semidirect

48. Dic₁₃₇

C₄⋊C₄

semidirect

56. C₁₃₇

#389

semidirect

72. C₄⋊C₄

#10587739

semidirect

105. C₄

C₄⋊#10587739

semidirect

Subgroups — #22578354

Conjugacy classes of subgroups

Complements