Group #13268524 — subgroups

Subgroups — #13268524

Order 1792.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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	#11726815	C₂	1	2	yes	maximal, complemented
2	#11726814	C₂	1	2	yes	maximal, complemented
3	#11725816	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725814	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6374	—	2	4	no	—
7	C₂×D₁₁₂	—	2	4	no	—
8	#89471	C₂²	1	1	yes	characteristic
9	#89467	—	2	2	no	—
10	#89467	C₂²	1	1	yes	characteristic
11	#89467	C₂²	1	2	yes	—
12	#89466	—	2	2	no	—
13	C₄×C₇×C₁₆	C₂²	1	2	yes	—
14	C₇×C₈²	C₂²	1	1	yes	characteristic
15	#12519	—	7	1	no	maximal, Sylow
16	C₂×Dic₂₈	—	2	2	no	—
17	C₂×D₅₆	—	2	2	no	—
18	C₂×D₅₆	—	2	2	no	—
19	C₂×D₅₆	—	2	2	no	—
20	C₂×D₅₆	—	4	4	no	—
21	#6437	—	2	1	no	—
22	#6437	—	2	2	no	—
23	#6432	—	2	1	no	—
24	C₂×C₇×C₁₆	D₄	1	4	yes	—
25	C₄×C₇×C₈	D₄	1	2	yes	—
26	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
27	#6395	—	4	4	no	—
28	#6374	—	4	8	no	—
29	D₁₁₂	—	4	8	no	—
30	#2248	—	7	2	no	—
31	#2246	—	7	2	no	—
32	#1712	—	7	1	no	—
33	#1709	—	7	1	no	—
34	#1311	D₇	1	1	yes	characteristic, complemented
35	C₂×D₂₈	—	4	1	no	—
36	C₂×D₂₈	—	4	2	no	—
37	C₂×D₂₈	—	4	4	no	—
38	C₂×D₂₈	—	4	2	no	—
39	C₂×Dic₁₄	—	4	1	no	—
40	C₂×C₇×C₈	D₈	1	4	yes	—
41	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
42	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₇	—	4	2	no	—
44	Dic₂₈	—	4	4	no	—
45	D₅₆	—	4	4	no	—
46	D₅₆	—	4	4	no	—
47	D₅₆	—	4	4	no	—
48	D₅₆	—	8	8	no	—
49	C₇×C₁₆	D₈	1	8	yes	complemented
50	C₂×SD₃₂	—	14	4	no	—
51	C₂×D₁₆	—	14	4	no	—
52	#494	—	7	1	no	—
53	#493	—	7	1	no	—
54	#493	—	7	2	no	—
55	#493	—	14	2	no	—
56	#486	—	14	2	no	—
57	C₄×C₁₆	C₂×D₇	1	2	yes	—
58	C₈²	C₂×D₇	1	1	yes	characteristic
59	C₂²×D₇	—	8	2	no	—
60	C₂²×D₇	—	8	1	no	—
61	C₂×C₄×C₇	#127	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
63	C₂×Dic₇	—	8	1	no	—
64	D₂₈	—	8	2	no	—
65	D₂₈	—	8	2	no	—
66	D₂₈	—	8	2	no	—
67	D₂₈	—	8	8	no	—
68	D₂₈	—	8	4	no	—
69	Dic₁₄	—	8	2	no	—
70	C₇×C₈	—	2	4	no	—
71	C₇×C₈	C₂×D₈	1	4	yes	—
72	C₂×Q₁₆	—	14	2	no	—
73	C₂×D₈	—	14	2	no	—
74	C₂×D₈	—	14	2	no	—
75	C₂×D₈	—	14	2	no	—
76	C₂×D₈	—	28	4	no	—
77	C₄⋊Q₈	—	14	1	no	—
78	#127	—	14	1	no	—
79	#127	—	14	2	no	—
80	SD₃₂	—	28	8	no	—
81	D₁₆	—	28	8	no	—
82	C₂×C₁₆	D₂₈	1	4	yes	—
83	#102	—	28	4	no	—
84	C₄×C₈	D₂₈	1	2	yes	—
85	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
86	C₂²×C₇	#493	1	1	yes	characteristic
87	C₂×D₇	—	16	1	no	—
88	C₂×D₇	—	16	2	no	—
89	C₂×D₇	—	16	4	no	—
90	C₂×D₇	—	16	2	no	—
91	C₄×C₇	#493	1	2	yes	—
92	C₄×C₇	#509	1	4	yes	—
93	Dic₇	—	16	1	no	—
94	C₂×Q₈	—	28	1	no	—
95	C₂×D₄	—	28	1	no	—
96	C₂×D₄	—	28	2	no	—
97	C₂×D₄	—	28	2	no	—
98	C₂×D₄	—	28	4	no	—
99	Q₁₆	—	28	4	no	—
100	D₈	—	28	4	no	—
101	D₈	—	28	4	no	—
102	D₈	—	28	4	no	—
103	D₈	—	56	8	no	—
104	C₂×C₈	D₅₆	1	4	yes	—
105	C₂×C₈	C₂×D₂₈	1	2	yes	—
106	C₄⋊C₄	—	28	2	no	—
107	C₄²	C₂×D₂₈	1	1	yes	characteristic
108	C₁₆	D₅₆	1	8	yes	complemented
109	C₂×C₇	#1712	1	1	yes	characteristic
110	C₂×C₇	#2250	1	2	yes	—
111	D₇	—	32	2	no	—
112	D₇	—	32	1	no	—
113	C₂³	—	56	1	no	—
114	C₂³	—	56	2	no	—
115	Q₈	—	56	2	no	—
116	D₄	—	56	8	no	—
117	D₄	—	56	4	no	—
118	D₄	—	56	2	no	—
119	D₄	—	56	2	no	—
120	D₄	—	56	2	no	—
121	C₂×C₄	—	56	1	no	—
122	C₂×C₄	#6437	1	1	yes	characteristic
123	C₂×C₄	C₂×D₅₆	1	2	yes	—
124	C₈	—	2	4	no	—
125	C₈	C₂×D₅₆	1	4	yes	—
126	C₇	#12519	1	1	yes	characteristic, complemented, Sylow
127	C₂²	—	112	4	no	—
128	C₂²	—	112	2	no	—
129	C₂²	—	112	1	no	—
130	C₂²	—	112	2	no	—
131	C₂²	#89467	1	1	yes	characteristic
132	C₄	—	112	1	no	—
133	C₄	#89467	1	2	yes	—
134	C₄	#89680	1	4	yes	—
135	C₂	—	224	2	no	—
136	C₂	—	224	1	no	—
137	C₂	#11725816	1	1	yes	characteristic
138	C₂	#11726835	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11726815

C₂

yes

maximal, complemented

#11726814

C₂

yes

maximal, complemented

#11725816

C₂

yes

maximal, characteristic, complemented

#11725814

C₂

yes

maximal, characteristic, complemented

C₇×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#6374

—

C₂×D₁₁₂

—

#89471

C₂²

yes

characteristic

#89467

—

#89467

C₂²

yes

characteristic

#89467

C₂²

yes

—

#89466

—

C₄×C₇×C₁₆

C₂²

yes

—

C₇×C₈²

C₂²

yes

characteristic

#12519

—

maximal, Sylow

C₂×Dic₂₈

—

C₂×D₅₆

—

C₂×D₅₆

—

C₂×D₅₆

—

C₂×D₅₆

—

#6437

—

#6437

—

#6432

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

characteristic

#6395

—

#6374

—

D₁₁₂

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

Dic₂₈

—

D₅₆

—

D₅₆

—

D₅₆

—

D₅₆

—

C₇×C₁₆

D₈

yes

complemented

C₂×SD₃₂

—

C₂×D₁₆

—

—

—

—

—

—

C₄×C₁₆

C₂×D₇

yes

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

D₂₈

—

D₂₈

—

D₂₈

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

C₂×D₈

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

D₂₈

yes

—

#102

—

C₄×C₈

D₂₈

yes

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

#493

yes

—

C₄×C₇

#509

yes

—

Dic₇

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

100

D₈

—

101

D₈

—

102

D₈

—

103

D₈

—

104

C₂×C₈

D₅₆

yes

—

105

C₂×C₈

C₂×D₂₈

yes

—

106

C₄⋊C₄

—

107

C₄²

C₂×D₂₈

yes

characteristic

108

C₁₆

D₅₆

yes

complemented

109

C₂×C₇

#1712

yes

characteristic

110

C₂×C₇

#2250

yes

—

111

D₇

—

112

D₇

—

113

C₂³

—

114

C₂³

—

115

Q₈

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

#6437

yes

characteristic

123

C₂×C₄

C₂×D₅₆

yes

—

124

C₈

—

125

C₈

C₂×D₅₆

yes

—

126

C₇

#12519

yes

characteristic, complemented, Sylow

127

C₂²

—

112

—

128

C₂²

—

112

—

129

C₂²

—

112

—

130

C₂²

—

112

—

131

C₂²

#89467

yes

characteristic

132

C₄

—

112

—

133

C₄

#89467

yes

—

134

C₄

#89680

yes

—

135

C₂

—

224

—

136

C₂

—

224

—

137

C₂

#11725816

yes

characteristic

138

C₂

#11726835

yes

—

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1311	C₂
socle	C₂²×C₇	#493
hypercenter	#1311	D₇
hypocenter	C₇	#12519
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#89467

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #11726815	C₂	semidirect
2. #11726814	C₂	semidirect
3. #11725816	C₂	semidirect
4. #11725814	C₂	semidirect
5. C₇×#1311	C₂	semidirect
34. #1311	D₇	semidirect
49. C₇×C₁₆	D₈	semidirect
108. C₁₆	D₅₆	semidirect
126. C₇	#12519	semidirect

H ≅ G/N

type

1. #11726815

C₂

semidirect

2. #11726814

C₂

semidirect

3. #11725816

C₂

semidirect

4. #11725814

C₂

semidirect

5. C₇×#1311

C₂

semidirect

34. #1311

D₇

semidirect

49. C₇×C₁₆

D₈

semidirect

108. C₁₆

D₅₆

semidirect

126. C₇

#12519

semidirect

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

subgroup	index
#11726815	2
#11726814	2
#11725816	2
#11725814	2
C₇×#1311	2
#12519	7

subgroup

index

Subgroups — #13268524

Conjugacy classes of subgroups

Complements

Maximal subgroups