Group #26021053 — subgroups

Subgroups — #26021053

Order 11392.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15667935	C₂	1	4	yes	maximal
3	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
4	#384×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
6	C₂²×#10623494	C₂²	1	2	yes	—
7	#15668012	—	2	2	no	—
8	#15668012	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₈₉	—	2	2	no	—
10	#15667935	—	2	8	no	—
11	C₄×#10623494	—	2	2	no	—
12	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₇₈	—	2	4	no	—
16	C₂×Dic₁₇₈	—	2	4	no	—
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₁₇₈	C₂×C₄	1	4	yes	—
19	C₂²×Dic₈₉	—	2	2	no	—
20	C₂×#10623494	—	2	2	no	—
21	C₂×#10623494	—	2	2	no	—
22	C₂×#10623494	D₄	1	4	yes	—
23	C₄⋊Dic₈₉	—	2	8	no	—
24	C₂²×C₄×C₈₉	—	2	2	no	—
25	C₂²×C₄×C₈₉	D₄	1	2	yes	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₈₉	—	2	4	no	—
28	C₄⋊C₄×C₈₉	—	4	4	no	—
29	C₄²×C₈₉	C₂×C₄	1	2	yes	—
30	C₄²×C₈₉	D₄	1	2	yes	—
31	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	4	2	no	—
33	C₂×C₄×C₈₉	—	4	4	no	—
34	C₂×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₈₉	SD₁₆	1	4	yes	—
36	C₂×C₄×C₈₉	Q₁₆	1	4	yes	—
37	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	#41	1	2	yes	—
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂×Dic₈₉	—	4	2	no	—
43	C₂×Dic₈₉	—	4	2	no	—
44	Dic₁₇₈	—	2	8	no	—
45	Dic₁₇₈	—	4	4	no	—
46	#10623494	—	2	4	no	—
47	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₈₉	#124	1	1	yes	characteristic
49	C₂²×C₈₉	#127	1	1	yes	characteristic
50	C₂²×C₈₉	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
52	C₄×C₈₉	—	8	2	no	—
53	C₄×C₈₉	#103	1	8	yes	complemented
54	C₄×C₈₉	C₄×D₄	1	2	yes	—
55	C₄×C₈₉	#121	1	2	yes	—
56	Dic₈₉	—	4	4	no	—
57	C₂×C₈₉	#390	1	1	yes	characteristic
58	C₂×C₈₉	C₂×#103	1	2	yes	—
59	C₂×C₈₉	#487	1	2	yes	—
60	C₂×C₈₉	#492	1	1	yes	characteristic
61	C₂×C₈₉	#494	1	1	yes	characteristic
62	#1956	—	89	1	no	maximal, Sylow
63	C₈₉	#1956	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	89	1	no	—
65	C₂×#103	—	89	4	no	—
66	C₂×C₄×C₈	—	89	1	no	—
67	#384	D₈₉	1	1	yes	characteristic
68	C₂²×Q₈	—	89	2	no	—
69	C₂²×C₈	—	89	2	no	—
70	C₄⋊Q₈	—	89	4	no	—
71	C₄⋊Q₈	—	178	2	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	178	2	no	—
74	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
75	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
76	#103	—	178	8	no	—
77	C₄×C₈	—	178	2	no	—
78	C₂×Q₈	—	89	4	no	—
79	C₂×Q₈	—	178	4	no	—
80	C₂×Q₈	—	178	4	no	—
81	C₂×Q₈	—	178	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	178	2	no	—
84	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
86	C₂×C₈	—	89	4	no	—
87	C₂×C₈	—	178	2	no	—
88	C₂×C₈	—	178	2	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	178	8	no	—
92	C₄²	C₄×D₈₉	1	2	yes	—
93	C₄²	D₃₅₆	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
95	Q₈	—	356	4	no	—
96	Q₈	—	178	8	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	356	2	no	—
100	C₂×C₄	—	356	2	no	—
101	C₂×C₄	—	178	4	no	—
102	C₂×C₄	#13083551	1	4	yes	—
103	C₂×C₄	#13083562	1	4	yes	—
104	C₂×C₄	#13083564	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
106	C₂×C₄	D₄×D₈₉	1	2	yes	—
107	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
108	C₂×C₄	#13083575	1	2	yes	—
109	C₈	—	178	4	no	—
110	C₂²	C₂×#13083551	1	1	yes	characteristic
111	C₂²	#15668001	1	1	yes	characteristic
112	C₂²	#15668008	1	1	yes	characteristic
113	C₂²	C₂×#13083562	1	2	yes	—
114	C₂²	C₂×#13083564	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	356	4	no	—
117	C₄	#15667935	1	8	yes	complemented
118	C₄	#15667974	1	2	yes	—
119	C₄	#15667987	1	2	yes	—
120	C₂	#19151499	1	1	yes	characteristic
121	C₂	C₂×#15667935	1	2	yes	—
122	C₂	#19151950	1	2	yes	—
123	C₂	#19151970	1	1	yes	characteristic
124	C₂	#19151977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

C₂×#15667935

C₂

yes

maximal

C₂×C₄×#10623494

C₂

yes

maximal, characteristic

#384×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

#15668012

—

#15668012

C₄

yes

complemented

C₂×C₄⋊Dic₈₉

—

#15667935

—

C₄×#10623494

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

D₄

yes

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×SD₁₆

yes

—

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₄×C₈₉

—

C₄×C₈₉

#103

yes

complemented

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

#121

yes

—

Dic₈₉

—

C₂×C₈₉

#390

yes

characteristic

C₂×C₈₉

C₂×#103

yes

—

C₂×C₈₉

#487

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

—

#384

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

178

—

C₄×C₈

—

178

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄²

C₄×D₈₉

yes

—

C₄²

D₃₅₆

yes

—

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

Q₈

—

356

—

Q₈

—

178

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

100

C₂×C₄

—

356

—

101

C₂×C₄

—

178

—

102

C₂×C₄

#13083551

yes

—

103

C₂×C₄

#13083562

yes

—

104

C₂×C₄

#13083564

yes

—

105

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

106

C₂×C₄

D₄×D₈₉

yes

—

107

C₂×C₄

C₂×D₃₅₆

yes

characteristic

108

C₂×C₄

#13083575

yes

—

109

C₈

—

178

—

110

C₂²

C₂×#13083551

yes

characteristic

111

C₂²

#15668001

yes

characteristic

112

C₂²

#15668008

yes

characteristic

113

C₂²

C₂×#13083562

yes

—

114

C₂²

C₂×#13083564

yes

—

115

C₄

—

116

C₄

—

356

—

117

C₄

#15667935

yes

complemented

118

C₄

#15667974

yes

—

119

C₄

#15667987

yes

—

120

C₂

#19151499

yes

characteristic

121

C₂

C₂×#15667935

yes

—

122

C₂

#19151950

yes

—

123

C₂

#19151970

yes

characteristic

124

C₂

#19151977

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#384×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#384	D₈₉
hypocenter	C₈₉	#1956
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#384×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #15668012	C₄	semidirect
53. C₄×C₈₉	#103	semidirect
63. C₈₉	#1956	semidirect
117. C₄	#15667935	semidirect

H ≅ G/N

type

8. #15668012

C₄

semidirect

53. C₄×C₈₉

#103

semidirect

63. C₈₉

#1956

semidirect

117. C₄

#15667935

semidirect

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

subgroup	index
C₂×#15668012	2
C₂×#15667935	2
C₂×C₄×#10623494	2
#384×C₈₉	2
#1956	89

subgroup

index

#384×C₈₉

Subgroups — #26021053

Conjugacy classes of subgroups

Complements

Maximal subgroups