Group #26038994 — subgroups

Subgroups — #26038994

Order 11456.

222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 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	#19153113	C₂	1	1	yes	maximal, characteristic, complemented
2	#19153104	C₂	1	2	yes	maximal, complemented
3	#19153093	C₂	1	2	yes	maximal
4	#19153089	C₂	1	2	yes	maximal
5	Q₈×Dic₁₇₉	C₂	1	2	yes	maximal, complemented
6	#19153078	C₂	1	2	yes	maximal, complemented
7	#19153071	C₂	1	2	yes	maximal
8	C₄²×D₁₇₉	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₃₅₈	C₂²	1	2	yes	—
11	C₂×C₄×D₁₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₇₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
15	C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
16	C₄⋊Dic₁₇₉	C₂²	1	4	yes	—
17	Dic₁₇₉⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₇₉)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
20	C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₇₉	C₂²	1	4	yes	—
23	C₂×Q₈×C₁₇₉	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
25	C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
26	C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
27	C₂²×D₁₇₉	C₂³	1	1	yes	characteristic
28	Dic₃₅₈	—	2	4	no	—
29	C₂×Dic₁₇₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₇₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₇₉	C₂³	1	4	yes	—
32	C₂×Dic₁₇₉	C₂³	1	1	yes	characteristic
33	C₄×D₁₇₉	—	2	2	no	—
34	C₄×D₁₇₉	—	2	2	no	—
35	C₄×D₁₇₉	Q₈	1	4	yes	complemented
36	Q₈×C₁₇₉	—	2	4	no	—
37	C₂×C₄×C₁₇₉	C₂³	1	4	yes	—
38	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
41	C₂²×C₁₇₉	C₂⁴	1	1	yes	characteristic
42	C₂×D₁₇₉	—	2	1	no	—
43	C₂×D₁₇₉	—	2	1	no	—
44	C₂×D₁₇₉	C₂×Q₈	1	2	yes	—
45	C₄×C₁₇₉	—	2	4	no	—
46	C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
47	C₄×C₁₇₉	#41	1	2	yes	—
48	C₄×C₁₇₉	#41	1	2	yes	—
49	Dic₁₇₉	—	2	1	no	—
50	Dic₁₇₉	—	2	1	no	—
51	Dic₁₇₉	—	2	4	no	—
52	Dic₁₇₉	C₂×Q₈	1	2	yes	—
53	C₂×C₁₇₉	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₁₇₉	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₇₉	C₂×#41	1	1	yes	characteristic
56	D₁₇₉	—	2	2	no	—
57	C₁₇₉	#533	1	1	yes	characteristic, complemented, Sylow
58	#533	—	179	1	no	maximal, Sylow
59	C₄⋊Q₈	—	179	1	no	—
60	C₄⋊Q₈	D₁₇₉	1	1	yes	characteristic, complemented
61	#125	—	179	2	no	—
62	C₂²⋊Q₈	—	179	2	no	—
63	C₂²⋊Q₈	—	179	2	no	—
64	C₄×Q₈	—	179	2	no	—
65	C₄×Q₈	—	179	2	no	—
66	#117	—	179	2	no	—
67	C₂×C₄²	—	179	1	no	—
68	C₂×Q₈	—	179	2	no	—
69	C₂×Q₈	C₂×D₁₇₉	1	2	yes	—
70	C₂²×C₄	—	179	1	no	—
71	C₂²×C₄	—	179	1	no	—
72	C₂²×C₄	—	179	1	no	—
73	C₄⋊C₄	—	179	4	no	—
74	C₄⋊C₄	—	179	2	no	—
75	C₄⋊C₄	—	179	4	no	—
76	C₄⋊C₄	—	179	2	no	—
77	C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
78	C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
79	C₂²⋊C₄	—	179	4	no	—
80	C₄²	—	179	1	no	—
81	C₄²	—	179	1	no	—
82	C₄²	—	179	1	no	—
83	C₄²	—	179	4	no	—
84	C₄²	C₂×D₁₇₉	1	1	yes	characteristic
85	C₂³	—	179	1	no	—
86	Q₈	—	2	4	no	—
87	Q₈	—	358	4	no	—
88	C₂×C₄	—	358	2	no	—
89	C₂×C₄	—	358	2	no	—
90	C₂×C₄	—	179	1	no	—
91	C₂×C₄	—	179	1	no	—
92	C₂×C₄	—	179	4	no	—
93	C₂×C₄	—	179	1	no	—
94	C₂×C₄	—	179	4	no	—
95	C₂×C₄	C₂²×D₁₇₉	1	4	yes	—
96	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
99	C₂²	—	358	1	no	—
100	C₂²	—	358	1	no	—
101	C₂²	—	179	2	no	—
102	C₂²	C₂³×D₁₇₉	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	—	358	1	no	—
105	C₄	—	358	1	no	—
106	C₄	—	358	4	no	—
107	C₄	—	179	2	no	—
108	C₄	Q₈×D₁₇₉	1	2	yes	—
109	C₄	#15668258	1	2	yes	—
110	C₄	#15668260	1	2	yes	—
111	C₂	—	358	2	no	—
112	C₂	C₂×Q₈×D₁₇₉	1	1	yes	characteristic
113	C₂	C₂×#15668258	1	1	yes	characteristic
114	C₂	C₂×#15668260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19153113

C₂

yes

maximal, characteristic, complemented

#19153104

C₂

yes

maximal, complemented

#19153093

C₂

yes

maximal

#19153089

C₂

yes

maximal

Q₈×Dic₁₇₉

C₂

yes

maximal, complemented

#19153078

C₂

yes

maximal, complemented

#19153071

C₂

yes

maximal

C₄²×D₁₇₉

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₁₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₅₈

C₂²

yes

—

C₂×C₄×D₁₇₉

C₂²

yes

characteristic

C₂×C₄×D₁₇₉

C₂²

yes

characteristic

C₂×C₄×D₁₇₉

C₂²

yes

characteristic

C₄⋊Dic₁₇₉

C₂²

yes

—

C₄⋊Dic₁₇₉

C₂²

yes

—

C₄⋊Dic₁₇₉

C₂²

yes

—

Dic₁₇₉⋊C₄

C₂²

yes

—

(C₂×D₁₇₉)⋊C₄

C₂²

yes

—

C₄×Dic₁₇₉

C₂²

yes

characteristic

C₄×Dic₁₇₉

C₂²

yes

characteristic

C₄×Dic₁₇₉

C₂²

yes

characteristic

C₄×Dic₁₇₉

C₂²

yes

—

C₂×Q₈×C₁₇₉

C₂²

yes

—

C₄⋊C₄×C₁₇₉

C₂²

yes

—

C₄⋊C₄×C₁₇₉

C₂²

yes

—

C₄²×C₁₇₉

C₂²

yes

characteristic

C₂²×D₁₇₉

C₂³

yes

characteristic

Dic₃₅₈

—

C₂×Dic₁₇₉

C₂³

yes

characteristic

C₂×Dic₁₇₉

C₂³

yes

characteristic

C₂×Dic₁₇₉

C₂³

yes

—

C₂×Dic₁₇₉

C₂³

yes

characteristic

C₄×D₁₇₉

—

C₄×D₁₇₉

—

C₄×D₁₇₉

Q₈

yes

complemented

Q₈×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂³

yes

—

C₂×C₄×C₁₇₉

C₂³

yes

characteristic

C₂×C₄×C₁₇₉

C₂³

yes

characteristic

C₂×C₄×C₁₇₉

C₂³

yes

characteristic

C₂²×C₁₇₉

C₂⁴

yes

characteristic

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₂×D₁₇₉

C₂×Q₈

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×Q₈

yes

—

C₄×C₁₇₉

#41

yes

—

C₄×C₁₇₉

#41

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

C₂×Q₈

yes

—

C₂×C₁₇₉

C₂²×Q₈

yes

characteristic

C₂×C₁₇₉

C₂×#41

yes

characteristic

C₂×C₁₇₉

C₂×#41

yes

characteristic

D₁₇₉

—

C₁₇₉

#533

yes

characteristic, complemented, Sylow

#533

—

179

maximal, Sylow

C₄⋊Q₈

—

179

—

C₄⋊Q₈

D₁₇₉

yes

characteristic, complemented

#125

—

179

—

C₂²⋊Q₈

—

179

—

C₂²⋊Q₈

—

179

—

C₄×Q₈

—

179

—

C₄×Q₈

—

179

—

#117

—

179

—

C₂×C₄²

—

179

—

C₂×Q₈

—

179

—

C₂×Q₈

C₂×D₁₇₉

yes

—

C₂²×C₄

—

179

—

C₂²×C₄

—

179

—

C₂²×C₄

—

179

—

C₄⋊C₄

—

179

—

C₄⋊C₄

—

179

—

C₄⋊C₄

—

179

—

C₄⋊C₄

—

179

—

C₄⋊C₄

C₂×D₁₇₉

yes

—

C₄⋊C₄

C₂×D₁₇₉

yes

—

C₂²⋊C₄

—

179

—

C₄²

—

179

—

C₄²

—

179

—

C₄²

—

179

—

C₄²

—

179

—

C₄²

C₂×D₁₇₉

yes

characteristic

C₂³

—

179

—

Q₈

—

Q₈

—

358

—

C₂×C₄

—

358

—

C₂×C₄

—

358

—

C₂×C₄

—

179

—

C₂×C₄

—

179

—

C₂×C₄

—

179

—

C₂×C₄

—

179

—

C₂×C₄

—

179

—

C₂×C₄

C₂²×D₁₇₉

yes

—

C₂×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₄

C₂²×D₁₇₉

yes

characteristic

C₂²

—

358

—

100

C₂²

—

358

—

101

C₂²

—

179

—

102

C₂²

C₂³×D₁₇₉

yes

characteristic

103

C₄

—

104

C₄

—

358

—

105

C₄

—

358

—

106

C₄

—

358

—

107

C₄

—

179

—

108

C₄

Q₈×D₁₇₉

yes

—

109

C₄

#15668258

yes

—

110

C₄

#15668260

yes

—

111

C₂

—

358

—

112

C₂

C₂×Q₈×D₁₇₉

yes

characteristic

113

C₂

C₂×#15668258

yes

characteristic

114

C₂

C₂×#15668260

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇₉
commutator, socle, cosocle	C₂²×C₁₇₉	C₂⁴
Fitting	C₄⋊Q₈×C₁₇₉	C₂
prime radical	C₂²×D₁₇₉	C₂³
hypercenter	C₄⋊Q₈	D₁₇₉
hypocenter	C₁₇₉	#533
LCS	G ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₇₉

commutator, socle, cosocle

C₂²×C₁₇₉

C₂⁴

Fitting

C₄⋊Q₈×C₁₇₉

C₂

prime radical

C₂²×D₁₇₉

C₂³

hypercenter

C₄⋊Q₈

D₁₇₉

C₁₇₉

G ▷ C₂²×C₁₇₉ ▷ C₁₇₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #19153113	C₂	semidirect
2. #19153104	C₂	semidirect
5. Q₈×Dic₁₇₉	C₂	semidirect
6. #19153078	C₂	semidirect
9. C₄⋊Q₈×C₁₇₉	C₂	semidirect
35. C₄×D₁₇₉	Q₈	semidirect
57. C₁₇₉	#533	semidirect
60. C₄⋊Q₈	D₁₇₉	semidirect

H ≅ G/N

type

1. #19153113

C₂

semidirect

2. #19153104

C₂

semidirect

5. Q₈×Dic₁₇₉

C₂

semidirect

6. #19153078

C₂

semidirect

9. C₄⋊Q₈×C₁₇₉

C₂

semidirect

35. C₄×D₁₇₉

Q₈

semidirect

57. C₁₇₉

#533

semidirect

60. C₄⋊Q₈

D₁₇₉

semidirect

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

subgroup	index
#19153113	2
#19153104	2
#19153093	2
#19153089	2
Q₈×Dic₁₇₉	2
#19153078	2
#19153071	2
C₄²×D₁₇₉	2
C₄⋊Q₈×C₁₇₉	2
#533	179

subgroup

index

Q₈×Dic₁₇₉

C₄²×D₁₇₉

C₄⋊Q₈×C₁₇₉

#533

179

Subgroups — #26038994

Conjugacy classes of subgroups

Complements

Maximal subgroups