Group #26019609 — subgroups

Subgroups — #26019609

Order 11392.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19151795	C₂	1	2	yes	maximal, complemented
2	C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151351	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151342	C₂	1	2	yes	maximal, complemented
5	#493×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083562	—	2	2	no	—
7	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
8	C₄⋊#10623494	C₂²	1	1	yes	characteristic
9	C₄⋊#10623494	C₂²	1	2	yes	—
10	#15667941	—	2	1	no	—
11	#15667933	—	2	2	no	—
12	C₈×Dic₈₉	—	2	1	no	—
13	C₂×D₈×C₈₉	—	2	1	no	—
14	C₂×D₈×C₈₉	—	2	1	no	—
15	#127×C₈₉	C₂²	1	2	yes	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
18	#13083562	—	4	4	no	—
19	C₂×#10623494	—	2	2	no	—
20	C₂×#10623494	—	2	1	no	—
21	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
22	#13083553	—	2	1	no	—
23	C₄×Dic₈₉	—	2	1	no	—
24	C₂×D₄×C₈₉	—	2	2	no	—
25	C₂×D₄×C₈₉	—	2	2	no	—
26	C₂×D₄×C₈₉	—	2	2	no	—
27	D₈×C₈₉	—	4	2	no	—
28	D₈×C₈₉	—	4	2	no	—
29	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₂³×C₈₉	—	4	2	no	—
33	D₄×C₈₉	—	4	2	no	—
34	D₄×C₈₉	—	4	2	no	—
35	D₄×C₈₉	—	4	2	no	—
36	D₄×C₈₉	—	4	2	no	—
37	D₄×C₈₉	—	8	2	no	—
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
41	C₂×Dic₈₉	—	2	1	no	—
42	C₂×Dic₈₉	—	2	1	no	—
43	Dic₁₇₈	—	2	1	no	—
44	Dic₁₇₈	D₈	1	2	yes	complemented
45	C₈×C₈₉	—	2	1	no	—
46	C₈×C₈₉	SD₁₆	1	2	yes	complemented
47	#10623494	—	4	2	no	—
48	#10623494	—	4	1	no	—
49	C₂²×C₈₉	—	8	2	no	—
50	C₂²×C₈₉	—	8	2	no	—
51	C₂²×C₈₉	—	8	2	no	—
52	C₂²×C₈₉	#121	1	1	yes	characteristic
53	C₄×C₈₉	—	2	1	no	—
54	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
55	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₄×C₈₉	#135	1	1	yes	characteristic
57	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₈₉	—	2	2	no	—
59	Dic₈₉	—	4	1	no	—
60	C₂×C₈₉	—	16	2	no	—
61	C₂×C₈₉	#460	1	1	yes	characteristic
62	C₂×C₈₉	#466	1	1	yes	characteristic
63	C₂×C₈₉	#472	1	1	yes	characteristic
64	#1668	—	89	1	no	maximal, Sylow
65	C₈₉	#1668	1	1	yes	characteristic, complemented, Sylow
66	#493	D₈₉	1	1	yes	characteristic
67	#460	—	89	2	no	—
68	C₈×Q₈	—	89	1	no	—
69	#335	—	89	1	no	—
70	#331	—	89	2	no	—
71	C₂×SD₁₆	—	178	2	no	—
72	C₂×D₈	—	2	1	no	—
73	C₂×D₈	—	2	1	no	—
74	#127	C₂×D₈₉	1	2	yes	—
75	C₄×Q₈	—	89	1	no	—
76	C₄⋊C₈	—	89	1	no	—
77	C₄⋊C₈	—	89	2	no	—
78	C₄⋊C₈	—	178	1	no	—
79	#102	—	178	2	no	—
80	C₄×C₈	—	178	1	no	—
81	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
82	C₂×Q₈	—	89	1	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	SD₁₆	—	356	4	no	—
87	D₈	—	4	2	no	—
88	D₈	—	4	2	no	—
89	C₂×C₈	—	178	2	no	—
90	C₂×C₈	—	178	1	no	—
91	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
93	C₄⋊C₄	—	89	1	no	—
94	C₄⋊C₄	—	178	1	no	—
95	C₄²	—	178	1	no	—
96	C₄²	C₂²×D₈₉	1	1	yes	characteristic
97	C₂³	—	4	2	no	—
98	Q₈	—	89	2	no	—
99	Q₈	—	178	1	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	8	2	no	—
105	C₂×C₄	—	178	1	no	—
106	C₂×C₄	—	178	1	no	—
107	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
109	C₂×C₄	#13083573	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	356	2	no	—
112	C₈	—	356	1	no	—
113	C₈	#13083562	1	2	yes	complemented
114	C₂²	—	8	2	no	—
115	C₂²	—	8	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	#15667983	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	356	1	no	—
120	C₄	—	178	2	no	—
121	C₄	D₈×D₈₉	1	1	yes	characteristic
122	C₄	C₂×#13083562	1	1	yes	characteristic
123	C₄	#15668031	1	1	yes	characteristic
124	C₄	#15668038	1	1	yes	characteristic
125	C₂	—	16	2	no	—
126	C₂	#19151795	1	1	yes	characteristic
127	C₂	#19151831	1	1	yes	characteristic
128	C₂	#19151865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151795

C₂

yes

maximal, complemented

C₈×Dic₁₇₈

C₂

yes

maximal, characteristic, complemented

#19151351

C₂

yes

maximal, characteristic, complemented

#19151342

C₂

yes

maximal, complemented

#493×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083562

—

C₄×Dic₁₇₈

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

#15667941

—

#15667933

—

C₈×Dic₈₉

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

#127×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

D₄

yes

—

—

—

C₄⋊Dic₈₉

D₄

yes

characteristic

#13083553

—

C₄×Dic₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

D₈

yes

complemented

C₈×C₈₉

—

C₈×C₈₉

SD₁₆

yes

complemented

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

C₂×SD₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#493

D₈₉

yes

characteristic

#460

—

C₈×Q₈

—

#335

—

#331

—

C₂×SD₁₆

—

178

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₈₉

yes

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

#102

—

178

—

C₄×C₈

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

356

—

D₈

—

D₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

178

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

178

—

106

C₂×C₄

—

178

—

107

C₂×C₄

D₄×D₈₉

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

109

C₂×C₄

#13083573

yes

characteristic

110

C₈

—

111

C₈

—

356

—

112

C₈

—

356

—

113

C₈

#13083562

yes

complemented

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#15667983

yes

characteristic

118

C₄

—

119

C₄

—

356

—

120

C₄

—

178

—

121

C₄

D₈×D₈₉

yes

characteristic

122

C₄

C₂×#13083562

yes

characteristic

123

C₄

#15668031

yes

characteristic

124

C₄

#15668038

yes

characteristic

125

C₂

—

126

C₂

#19151795

yes

characteristic

127

C₂

#19151831

yes

characteristic

128

C₂

#19151865

yes

characteristic

	subgroup	G/H
center	C₂²	#15667983
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting, prime radical	#493×C₈₉	C₂
socle	C₂²×C₈₉	#121
hypercenter	#493	D₈₉
hypocenter	C₈₉	#1668
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#15667983

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting, prime radical

#493×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #19151795	C₂	semidirect
2. C₈×Dic₁₇₈	C₂	semidirect
3. #19151351	C₂	semidirect
4. #19151342	C₂	semidirect
44. Dic₁₇₈	D₈	semidirect
46. C₈×C₈₉	SD₁₆	semidirect
65. C₈₉	#1668	semidirect
113. C₈	#13083562	semidirect

H ≅ G/N

type

1. #19151795

C₂

semidirect

2. C₈×Dic₁₇₈

C₂

semidirect

3. #19151351

C₂

semidirect

4. #19151342

C₂

semidirect

44. Dic₁₇₈

D₈

semidirect

46. C₈×C₈₉

SD₁₆

semidirect

65. C₈₉

#1668

semidirect

113. C₈

#13083562

semidirect

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

subgroup	index
#19151795	2
C₈×Dic₁₇₈	2
#19151351	2
#19151342	2
#493×C₈₉	2
#1668	89

subgroup

index

C₈×Dic₁₇₈

#493×C₈₉

Subgroups — #26019609

Conjugacy classes of subgroups

Complements

Maximal subgroups