Group #13265685 — subgroups

Subgroups — #13265685

Order 1792.

204 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 73 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	#11727055	C₂	1	2	yes	maximal, complemented
2	#11726839	C₂	1	1	yes	maximal, characteristic
3	C₇×#1369	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725375	C₂	1	2	yes	maximal, complemented
5	#11725317	C₂	1	1	yes	maximal, characteristic, complemented
6	#89679	—	2	2	no	—
7	#89676	C₄	1	1	yes	characteristic
8	#89472	C₂²	1	2	yes	—
9	C₇×#364	C₂²	1	2	yes	—
10	C₇×#346	C₂²	1	1	yes	characteristic
11	#89310	—	2	2	no	—
12	#89305	C₄	1	2	yes	—
13	#89301	C₄	1	1	yes	characteristic
14	#89256	C₂²	1	1	yes	characteristic
15	#89250	C₂²	1	1	yes	characteristic
16	#12068	—	7	1	no	maximal, Sylow
17	#6467	C₂×C₄	1	2	yes	—
18	#6464	—	2	2	no	—
19	C₇×M₃₂	—	2	2	no	—
20	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
23	#6393	C₂×C₄	1	2	yes	—
24	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
25	#6379	—	2	2	no	—
26	C₄×C₇⋊C₈	—	2	1	no	—
27	Dic₅₆	C₈	1	1	yes	characteristic, complemented
28	#6374	C₈	1	2	yes	complemented
29	D₁₁₂	C₈	1	1	yes	characteristic, complemented
30	#6372	—	4	2	no	—
31	D₇×C₁₆	—	4	2	no	—
32	#2179	—	7	1	no	—
33	#2171	—	7	2	no	—
34	#1369	D₇	1	1	yes	characteristic, complemented
35	#1336	—	7	2	no	—
36	#1313	—	7	1	no	—
37	#1157	—	2	2	no	—
38	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇	#41	1	1	yes	characteristic
41	#1136	—	2	2	no	—
42	C₂×C₇⋊C₈	—	4	1	no	—
43	Dic₂₈	C₂×C₈	1	2	yes	—
44	D₅₆	C₂×C₈	1	2	yes	—
45	#1132	—	2	2	no	—
46	#1131	—	4	2	no	—
47	C₈×D₇	—	4	2	no	—
48	C₇×C₁₆	—	2	1	no	—
49	C₇×C₁₆	—	4	2	no	—
50	C₇×C₁₆	C₂×C₈	1	1	yes	characteristic
51	C₇×C₁₆	C₂×C₈	1	1	yes	characteristic
52	C₇⋊C₁₆	—	2	2	no	—
53	#508	—	7	1	no	—
54	#504	—	14	2	no	—
55	#443	—	7	2	no	—
56	#368	—	7	1	no	—
57	#364	C₂×D₇	1	2	yes	—
58	#359	—	7	2	no	—
59	#350	—	14	2	no	—
60	#346	—	7	1	no	—
61	#346	C₂×D₇	1	1	yes	characteristic
62	#322	—	7	1	no	—
63	C₂×C₄×C₇	—	2	1	no	—
64	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
65	C₂²⋊D₇	—	4	2	no	—
66	D₂₈	—	2	2	no	—
67	C₄×D₇	—	4	2	no	—
68	Dic₁₄	—	2	2	no	—
69	C₇×C₈	C₂²×C₈	1	1	yes	characteristic
70	C₇×C₈	#131	1	1	yes	characteristic
71	C₇×C₈	AGL(1,ℤ/8)	1	2	yes	—
72	C₇⋊C₈	—	4	2	no	—
73	C₇⋊C₈	—	8	1	no	—
74	#135	—	7	2	no	—
75	#131	—	14	2	no	—
76	Q₃₂	—	7	1	no	—
77	SD₃₂	—	7	2	no	—
78	D₁₆	—	7	1	no	—
79	M₃₂	—	2	2	no	—
80	M₃₂	—	28	2	no	—
81	C₂×C₁₆	—	7	2	no	—
82	C₂×C₁₆	—	28	2	no	—
83	C₂×C₁₆	C₄×D₇	1	1	yes	characteristic
84	C₂×C₁₆	C₄×D₇	1	1	yes	characteristic
85	#108	—	7	2	no	—
86	#104	—	14	2	no	—
87	C₄×C₈	—	14	1	no	—
88	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
89	C₂²×C₇	#444	1	1	yes	characteristic
90	C₂×D₇	—	4	2	no	—
91	C₄×C₇	—	4	1	no	—
92	C₄×C₇	C₈×D₄	1	1	yes	characteristic
93	C₄×C₇	#442	1	1	yes	characteristic
94	Dic₇	—	4	2	no	—
95	#41	—	14	2	no	—
96	Q₁₆	—	7	2	no	—
97	SD₁₆	—	14	2	no	—
98	D₈	—	7	2	no	—
99	M₁₆	—	14	2	no	—
100	M₁₆	—	28	2	no	—
101	C₂×C₈	—	28	2	no	—
102	C₂×C₈	—	28	1	no	—
103	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
104	C₂×C₈	D₄×D₇	1	1	yes	characteristic
105	C₄²	#1161	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	—	4	2	no	—
108	C₁₆	—	14	2	no	—
109	C₁₆	C₈×D₇	1	1	yes	characteristic
110	C₁₆	C₈×D₇	1	1	yes	characteristic
111	C₂×C₇	—	2	1	no	—
112	C₂×C₇	#1580	1	1	yes	characteristic
113	D₇	—	8	2	no	—
114	Q₈	—	14	2	no	—
115	D₄	—	14	2	no	—
116	D₄	—	28	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	28	2	no	—
119	C₂×C₄	#6456	1	1	yes	characteristic
120	C₈	—	28	2	no	—
121	C₈	—	56	1	no	—
122	C₈	C₂×C₈×D₇	1	1	yes	characteristic
123	C₈	#6470	1	1	yes	characteristic
124	C₈	#6477	1	2	yes	—
125	C₇	#12068	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	28	2	no	—
127	C₂²	#89666	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	28	2	no	—
130	C₄	#89606	1	1	yes	characteristic
131	C₄	#89641	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	56	2	no	—
134	C₂	#11726272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11727055

C₂

yes

maximal, complemented

#11726839

C₂

yes

maximal, characteristic

C₇×#1369

C₂

yes

maximal, characteristic, complemented

#11725375

C₂

yes

maximal, complemented

#11725317

C₂

yes

maximal, characteristic, complemented

#89679

—

#89676

C₄

yes

characteristic

#89472

C₂²

yes

—

C₇×#364

C₂²

yes

—

C₇×#346

C₂²

yes

characteristic

#89310

—

#89305

C₄

yes

—

#89301

C₄

yes

characteristic

#89256

C₂²

yes

characteristic

#89250

C₂²

yes

—

C₂×C₄

yes

—

#6464

—

C₇×M₃₂

—

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

characteristic

#6393

C₂×C₄

yes

—

C₂×C₇⋊C₁₆

D₄

yes

—

#6379

—

C₄×C₇⋊C₈

—

Dic₅₆

C₈

yes

characteristic, complemented

#6374

C₈

yes

complemented

D₁₁₂

C₈

yes

characteristic, complemented

#6372

—

D₇×C₁₆

—

#2179

—

#2171

—

#1369

D₇

yes

characteristic, complemented

#1336

—

#1313

—

#1157

—

C₂×C₇×C₈

C₂²×C₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

C₄²×C₇

yes

—

C₂×C₇⋊C₈

—

Dic₂₈

C₂×C₈

yes

—

D₅₆

C₂×C₈

yes

—

#1132

—

#1131

—

C₈×D₇

—

C₇×C₁₆

—

C₇×C₁₆

—

C₇×C₁₆

C₂×C₈

yes

characteristic

C₇×C₁₆

C₂×C₈

yes

characteristic

C₇⋊C₁₆

—

—

—

—

—

C₂×D₇

yes

—

—

—

—

C₂×D₇

yes

characteristic

#322

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₄×D₄

yes

characteristic

C₂²⋊D₇

—

D₂₈

—

C₄×D₇

—

Dic₁₄

—

C₇×C₈

C₂²×C₈

yes

C₇×C₈

yes

C₇×C₈

yes

—

C₇⋊C₈

—

C₇⋊C₈

—

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇

yes

characteristic

C₂×C₁₆

C₄×D₇

yes

characteristic

#108

—

#104

—

C₄×C₈

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#444

yes

characteristic

C₂×D₇

—

C₄×C₇

—

C₄×C₇

C₈×D₄

yes

characteristic

C₄×C₇

#442

yes

characteristic

Dic₇

—

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₂×C₄×D₇

yes

characteristic

104

C₂×C₈

D₄×D₇

yes

characteristic

105

C₄²

#1161

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

C₈×D₇

yes

characteristic

110

C₁₆

C₈×D₇

yes

characteristic

111

C₂×C₇

—

112

C₂×C₇

#1580

yes

characteristic

113

D₇

—

114

Q₈

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#6456

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

C₂×C₈×D₇

yes

characteristic

123

C₈

#6470

yes

characteristic

124

C₈

#6477

yes

—

125

C₇

#12068

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

#89666

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

#89606

yes

characteristic

131

C₄

#89641

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#11726272

yes

characteristic

	subgroup	G/H
center	C₄	#89641
commutator	C₇×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1369	C₂
socle	C₂×C₇	#1580
prime radical	#89676	C₄
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#1369	D₇
hypocenter	C₇	#12068
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

subgroup

G/H

center

C₄

#89641

commutator

C₇×C₈

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂×C₇

C₄

C₄×C₇×C₈

C₂³

D₇

C₇

G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

N	H ≅ G/N	type
1. #11727055	C₂	semidirect
3. C₇×#1369	C₂	semidirect
4. #11725375	C₂	semidirect
5. #11725317	C₂	semidirect
27. Dic₅₆	C₈	semidirect
28. #6374	C₈	semidirect
29. D₁₁₂	C₈	semidirect
34. #1369	D₇	semidirect
125. C₇	#12068	semidirect

H ≅ G/N

type

1. #11727055

C₂

semidirect

3. C₇×#1369

C₂

semidirect

4. #11725375

C₂

semidirect

5. #11725317

C₂

semidirect

27. Dic₅₆

C₈

semidirect

28. #6374

C₈

semidirect

29. D₁₁₂

C₈

semidirect

34. #1369

D₇

semidirect

125. C₇

#12068

semidirect

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

subgroup	index
#11727055	2
#11726839	2
C₇×#1369	2
#11725375	2
#11725317	2
#12068	7

subgroup

index

Subgroups — #13265685

Conjugacy classes of subgroups

Complements

Maximal subgroups