Group #26019613 — subgroups

Subgroups — #26019613

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	#19151975	C₂	1	1	yes	maximal, characteristic
2	#19151799	C₂	1	2	yes	maximal, complemented
3	C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151341	C₂	1	2	yes	maximal, complemented
5	#335×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083563	—	2	2	no	—
7	C₂×D₇₁₂	—	2	1	no	—
8	C₂×D₇₁₂	—	2	1	no	—
9	#15668009	C₂²	1	2	yes	—
10	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	#15667941	—	2	1	no	—
13	#15667934	—	2	2	no	—
14	C₈×Dic₈₉	—	2	1	no	—
15	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
18	C₂×D₃₅₆	—	2	2	no	—
19	C₂×D₃₅₆	—	2	2	no	—
20	C₂×D₃₅₆	—	2	2	no	—
21	#13083563	—	4	4	no	—
22	D₇₁₂	—	4	2	no	—
23	D₇₁₂	—	4	2	no	—
24	C₂×#10623494	—	2	1	no	—
25	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
26	#13083553	—	2	1	no	—
27	C₄×Dic₈₉	—	2	1	no	—
28	C₂×C₈×C₈₉	—	2	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	C₂²×D₈₉	—	4	2	no	—
33	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
36	C₂×Dic₈₉	—	2	1	no	—
37	C₂×Dic₈₉	—	2	1	no	—
38	D₃₅₆	—	4	2	no	—
39	D₃₅₆	—	4	2	no	—
40	D₃₅₆	—	4	2	no	—
41	D₃₅₆	—	4	2	no	—
42	D₃₅₆	—	8	2	no	—
43	Dic₁₇₈	—	2	1	no	—
44	Dic₁₇₈	D₈	1	2	yes	complemented
45	C₈×C₈₉	—	2	1	no	—
46	C₈×C₈₉	—	4	2	no	—
47	C₈×C₈₉	SD₁₆	1	2	yes	complemented
48	#10623494	—	4	1	no	—
49	C₂²×C₈₉	#121	1	1	yes	characteristic
50	C₂×D₈₉	—	8	2	no	—
51	C₂×D₈₉	—	8	2	no	—
52	C₂×D₈₉	—	8	2	no	—
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	#460	1	1	yes	characteristic
61	C₂×C₈₉	#466	1	1	yes	characteristic
62	C₂×C₈₉	#472	1	1	yes	characteristic
63	D₈₉	—	16	2	no	—
64	#1668	—	89	1	no	maximal, Sylow
65	C₈₉	#1668	1	1	yes	characteristic, complemented, Sylow
66	#493	—	89	1	no	—
67	#460	—	89	2	no	—
68	C₈×Q₈	—	89	1	no	—
69	#335	D₈₉	1	1	yes	characteristic, complemented
70	#331	—	89	2	no	—
71	C₂×SD₁₆	—	178	2	no	—
72	C₂×D₈	—	178	1	no	—
73	C₂×D₈	—	178	1	no	—
74	#127	—	89	2	no	—
75	C₄×Q₈	—	89	1	no	—
76	C₄⋊C₈	—	89	1	no	—
77	C₄⋊C₈	—	178	1	no	—
78	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
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	—	178	2	no	—
84	C₂×D₄	—	178	2	no	—
85	C₂×D₄	—	178	2	no	—
86	SD₁₆	—	356	4	no	—
87	D₈	—	356	2	no	—
88	D₈	—	356	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	178	1	no	—
91	C₂×C₈	D₃₅₆	1	1	yes	characteristic
92	C₂×C₈	D₃₅₆	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	—	356	2	no	—
98	Q₈	—	89	2	no	—
99	Q₈	—	178	1	no	—
100	D₄	—	356	2	no	—
101	D₄	—	356	2	no	—
102	D₄	—	356	2	no	—
103	D₄	—	356	2	no	—
104	D₄	—	712	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×D₃₅₆	1	1	yes	characteristic
109	C₂×C₄	#13083575	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	356	1	no	—
113	C₈	#13083563	1	2	yes	complemented
114	C₂²	—	712	2	no	—
115	C₂²	—	712	2	no	—
116	C₂²	—	712	2	no	—
117	C₂²	#15667987	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×#13083563	1	1	yes	characteristic
123	C₄	#15668035	1	1	yes	characteristic
124	C₄	#15668040	1	1	yes	characteristic
125	C₂	—	1424	2	no	—
126	C₂	#19151799	1	1	yes	characteristic
127	C₂	#19151835	1	1	yes	characteristic
128	C₂	#19151869	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151975

C₂

yes

maximal, characteristic

#19151799

C₂

yes

maximal, complemented

C₈×Dic₁₇₈

C₂

yes

maximal, characteristic, complemented

#19151341

C₂

yes

maximal, complemented

#335×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083563

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

characteristic

#15667941

—

#15667934

—

C₈×Dic₈₉

—

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083563

—

D₇₁₂

—

D₇₁₂

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₄

yes

characteristic

#13083553

—

C₄×Dic₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

D₈

yes

complemented

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

SD₁₆

yes

complemented

#10623494

—

C₂²×C₈₉

#121

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

C₂×SD₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#1668

—

maximal, Sylow

C₈₉

#1668

yes

characteristic, complemented, Sylow

#493

—

#460

—

C₈×Q₈

—

#335

D₈₉

yes

characteristic, complemented

#331

—

C₂×SD₁₆

—

178

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

#127

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

—

#102

—

178

—

C₄×C₈

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

SD₁₆

—

356

—

D₈

—

356

—

D₈

—

356

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

characteristic

C₂×C₈

D₃₅₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

356

—

Q₈

—

Q₈

—

178

—

100

D₄

—

356

—

101

D₄

—

356

—

102

D₄

—

356

—

103

D₄

—

356

—

104

D₄

—

712

—

105

C₂×C₄

—

178

—

106

C₂×C₄

—

178

—

107

C₂×C₄

D₄×D₈₉

yes

characteristic

108

C₂×C₄

C₂×D₃₅₆

yes

characteristic

109

C₂×C₄

#13083575

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

356

—

113

C₈

#13083563

yes

complemented

114

C₂²

—

712

—

115

C₂²

—

712

—

116

C₂²

—

712

—

117

C₂²

#15667987

yes

characteristic

118

C₄

—

119

C₄

—

356

—

120

C₄

—

178

—

121

C₄

D₈×D₈₉

yes

characteristic

122

C₄

C₂×#13083563

yes

characteristic

123

C₄

#15668035

yes

characteristic

124

C₄

#15668040

yes

characteristic

125

C₂

—

1424

—

126

C₂

#19151799

yes

characteristic

127

C₂

#19151835

yes

characteristic

128

C₂

#19151869

yes

characteristic

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#335×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	#19151975	C₂
hypercenter	#335	D₈₉
hypocenter	C₈₉	#1668
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#15667987

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#335×C₈₉

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
2. #19151799	C₂	semidirect
3. C₈×Dic₁₇₈	C₂	semidirect
4. #19151341	C₂	semidirect
5. #335×C₈₉	C₂	semidirect
44. Dic₁₇₈	D₈	semidirect
47. C₈×C₈₉	SD₁₆	semidirect
65. C₈₉	#1668	semidirect
69. #335	D₈₉	semidirect
113. C₈	#13083563	semidirect

H ≅ G/N

type

2. #19151799

C₂

semidirect

3. C₈×Dic₁₇₈

C₂

semidirect

4. #19151341

C₂

semidirect

5. #335×C₈₉

C₂

semidirect

44. Dic₁₇₈

D₈

semidirect

47. C₈×C₈₉

SD₁₆

semidirect

65. C₈₉

#1668

semidirect

69. #335

D₈₉

semidirect

113. C₈

#13083563

semidirect

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

subgroup	index
#19151975	2
#19151799	2
C₈×Dic₁₇₈	2
#19151341	2
#335×C₈₉	2
#1668	89

subgroup

index

C₈×Dic₁₇₈

#335×C₈₉

Subgroups — #26019613

Conjugacy classes of subgroups

Complements

Maximal subgroups