Group #26030592 — subgroups

Subgroups — C₂×#19151967 #26030592

Order 11392.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15668012	C₂	1	1	yes	maximal, characteristic
2	C₂×#15668004	C₂	1	1	yes	maximal, characteristic
3	#19151967	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15667937	C₂	1	4	yes	maximal
5	C₂×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	#15668012	—	2	2	no	—
8	#15668012	C₂²	1	4	yes	—
9	#15668004	—	2	2	no	—
10	#15668004	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₈₉	—	2	2	no	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×#13083553	—	2	2	no	—
14	#15667937	C₂²	1	16	yes	—
15	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	#97×C₈₉	C₂²	1	4	yes	—
18	C₂×Dic₁₇₈	—	2	8	no	—
19	C₂×Dic₁₇₈	—	2	2	no	—
20	C₂×Dic₁₇₈	C₂³	1	4	yes	—
21	C₂²×Dic₈₉	—	2	2	no	—
22	C₂²×Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	—	2	2	no	—
24	C₄⋊Dic₈₉	—	2	8	no	—
25	C₄⋊Dic₈₉	C₂³	1	4	yes	—
26	#13083553	—	2	8	no	—
27	C₂²×C₄×C₈₉	D₄	1	2	yes	—
28	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₈₉	—	2	4	no	—
30	C₂×C₈×C₈₉	C₂³	1	4	yes	—
31	C₄²×C₈₉	D₄	1	2	yes	—
32	C₄²×C₈₉	C₂³	1	2	yes	—
33	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	—	2	4	no	—
35	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	#41	1	4	yes	—
38	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
39	C₂×Dic₈₉	—	2	4	no	—
40	C₂×Dic₈₉	—	2	4	no	—
41	C₂×Dic₈₉	—	4	4	no	—
42	C₂×Dic₈₉	—	4	4	no	—
43	Dic₁₇₈	—	2	8	no	—
44	Dic₁₇₈	—	4	4	no	—
45	C₈×C₈₉	—	2	4	no	—
46	C₂²×C₈₉	#124	1	2	yes	—
47	C₂²×C₈₉	#137	1	4	yes	—
48	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₈₉	—	2	4	no	—
50	C₄×C₈₉	#124	1	2	yes	—
51	C₄×C₈₉	C₂×#41	1	2	yes	—
52	Dic₈₉	—	4	4	no	—
53	Dic₈₉	—	4	4	no	—
54	C₂×C₈₉	#491	1	4	yes	complemented
55	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
56	C₂×C₈₉	C₂×#137	1	2	yes	—
57	C₂×#491	—	89	1	no	maximal, Sylow
58	C₈₉	C₂×#491	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄⋊Q₈	—	89	1	no	—
60	C₂×#125	—	89	1	no	—
61	#491	—	89	8	no	—
62	C₂×#103	—	89	4	no	—
63	C₂×#97	D₈₉	1	1	yes	characteristic
64	C₂²×Q₈	—	89	2	no	—
65	C₂²×C₈	C₂×D₈₉	1	2	yes	—
66	C₄⋊Q₈	—	89	4	no	—
67	C₄⋊Q₈	—	178	2	no	—
68	#125	—	89	4	no	—
69	#125	—	178	2	no	—
70	C₂×C₄⋊C₄	—	89	2	no	—
71	C₂×C₄⋊C₄	—	178	2	no	—
72	C₂×C₄⋊C₄	—	178	2	no	—
73	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
74	#103	—	89	16	no	—
75	#97	C₂×D₈₉	1	4	yes	—
76	C₂×Q₈	—	89	4	no	—
77	C₂×Q₈	—	178	8	no	—
78	C₂×Q₈	—	178	2	no	—
79	C₂²×C₄	—	178	2	no	—
80	C₂²×C₄	—	178	2	no	—
81	C₂²×C₄	D₃₅₆	1	2	yes	—
82	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂²×D₈₉	1	4	yes	—
85	C₄⋊C₄	—	89	4	no	—
86	C₄⋊C₄	—	178	2	no	—
87	C₄⋊C₄	—	178	8	no	—
88	C₄⋊C₄	—	178	8	no	—
89	C₄²	D₃₅₆	1	2	yes	—
90	C₄²	C₂²×D₈₉	1	2	yes	—
91	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
92	Q₈	—	356	4	no	—
93	Q₈	—	178	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	356	4	no	—
96	C₂×C₄	—	356	4	no	—
97	C₂×C₄	—	178	4	no	—
98	C₂×C₄	—	178	4	no	—
99	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
100	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
101	C₂×C₄	#13083574	1	4	yes	—
102	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	#15668000	1	2	yes	—
105	C₂²	#15668045	1	4	yes	—
106	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	356	4	no	—
109	C₄	—	356	4	no	—
110	C₄	#15668000	1	2	yes	—
111	C₄	C₂×#13083574	1	2	yes	—
112	C₂	#19151967	1	4	yes	direct product factor
113	C₂	C₂×#15668000	1	1	yes	characteristic
114	C₂	C₂×#15668045	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

C₂×#15668004

C₂

yes

maximal, characteristic

#19151967

C₂

yes

maximal, direct product factor

C₂×#15667937

C₂

yes

maximal

C₂×#97×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668012

—

#15668012

C₂²

yes

—

#15668004

—

#15668004

C₂²

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

—

#15667937

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#97×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

#124

yes

—

C₂²×C₈₉

#137

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

C₄×C₈₉

C₂×#41

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×#491

—

maximal, Sylow

C₈₉

C₂×#491

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#125

—

#491

—

C₂×#103

—

C₂×#97

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₈₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

#125

—

#125

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

#97

C₂×D₈₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

D₃₅₆

yes

—

C₄²

C₂²×D₈₉

yes

—

C₂³

C₂×D₃₅₆

yes

characteristic

Q₈

—

356

—

Q₈

—

178

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×D₃₅₆

yes

—

100

C₂×C₄

C₂×D₃₅₆

yes

characteristic

101

C₂×C₄

#13083574

yes

—

102

C₂×C₄

C₂³×D₈₉

yes

characteristic

103

C₈

—

104

C₂²

#15668000

yes

—

105

C₂²

#15668045

yes

—

106

C₂²

C₂²×D₃₅₆

yes

characteristic

107

C₄

—

108

C₄

—

356

—

109

C₄

—

356

—

110

C₄

#15668000

yes

—

111

C₄

C₂×#13083574

yes

—

112

C₂

#19151967

yes

direct product factor

113

C₂

C₂×#15668000

yes

characteristic

114

C₂

C₂×#15668045

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#97×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×#97	D₈₉
hypocenter	C₈₉	C₂×#491
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×#97×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

hypercenter

C₂×#97

D₈₉

hypocenter

C₈₉

C₂×#491

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
3. #19151967	C₂	direct (N × H)
54. C₂×C₈₉	#491	semidirect
58. C₈₉	C₂×#491	semidirect
112. C₂	#19151967	direct (N × H)

H ≅ G/N

type

3. #19151967

C₂

direct (N × H)

54. C₂×C₈₉

#491

semidirect

58. C₈₉

C₂×#491

semidirect

112. C₂

#19151967

direct (N × H)

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

subgroup	index
C₂×#15668012	2
C₂×#15668004	2
#19151967	2
C₂×#15667937	2
C₂×#97×C₈₉	2
C₂×#491	89

subgroup

index

C₂×#97×C₈₉

Subgroups — C₂×#19151967 #26030592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19151967 #26030592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19151967 #26030592