Group #13265682 — subgroups

Subgroups — #13265682

Order 1792.

246 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 79 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	#11726832	C₂	1	1	yes	maximal, characteristic
2	#11725831	C₂	1	2	yes	maximal, complemented
3	C₇×#1313	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725322	C₂	1	2	yes	maximal, complemented
5	#11725319	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₂	C₄	1	1	yes	characteristic
7	C₄×D₅₆	C₂²	1	2	yes	—
8	C₈×D₂₈	—	2	2	no	—
9	C₇×#346	C₂²	1	1	yes	characteristic
10	C₇×#346	C₂²	1	1	yes	characteristic
11	C₇×#322	C₂²	1	1	yes	characteristic
12	#89304	C₄	1	2	yes	—
13	#89300	C₄	1	1	yes	characteristic
14	#89254	—	2	2	no	—
15	#89252	C₂²	1	2	yes	—
16	#12067	—	7	1	no	maximal, Sylow
17	C₂×D₅₆	C₂×C₄	1	2	yes	—
18	C₂×C₈×D₇	—	4	2	no	—
19	C₄×D₂₈	—	2	2	no	—
20	C₂×C₇×C₁₆	—	2	1	no	—
21	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₄×C₇×C₈	D₄	1	2	yes	—
24	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
25	#6395	—	2	2	no	—
26	(C₂×D₇)⋊C₈	—	4	2	no	—
27	#6392	C₂×C₄	1	2	yes	—
28	C₄⋊(C₇⋊C₈)	—	2	2	no	—
29	D₁₁₂	C₈	1	4	yes	complemented
30	#2177	—	7	1	no	—
31	#1580	—	7	2	no	—
32	#1368	—	7	1	no	—
33	#1333	—	7	2	no	—
34	#1313	D₇	1	1	yes	characteristic, complemented
35	C₂×D₂₈	—	2	2	no	—
36	C₂×C₄×D₇	—	4	2	no	—
37	C₂×C₇×C₈	—	2	2	no	—
38	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₇×C₈	#41	1	1	yes	characteristic
40	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
41	(C₂×D₇)⋊C₄	—	4	2	no	—
42	C₄⋊Dic₇	—	2	2	no	—
43	C₂×C₇⋊C₈	—	4	2	no	—
44	D₅₆	—	2	2	no	—
45	D₅₆	C₂×C₈	1	4	yes	complemented
46	C₈×D₇	—	8	4	no	—
47	C₇×C₁₆	—	2	1	no	—
48	C₇×C₁₆	—	4	1	no	—
49	C₇×C₁₆	C₂×C₈	1	2	yes	complemented
50	C₂×D₁₆	—	7	1	no	—
51	C₄×D₈	—	7	2	no	—
52	C₈×D₄	—	14	2	no	—
53	#367	—	7	1	no	—
54	#357	—	7	2	no	—
55	#346	C₂×D₇	1	1	yes	characteristic
56	#346	C₂×D₇	1	1	yes	characteristic
57	#334	—	7	2	no	—
58	D₄⋊C₈	—	14	2	no	—
59	#322	C₂×D₇	1	1	yes	characteristic
60	C₂²×D₇	—	4	2	no	—
61	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₂×Dic₇	—	4	2	no	—
64	D₂₈	—	2	4	no	—
65	D₂₈	—	4	2	no	—
66	C₄×D₇	—	8	2	no	—
67	C₄×D₇	—	8	2	no	—
68	C₇×C₈	—	2	1	no	—
69	C₇×C₈	—	4	2	no	—
70	C₇×C₈	C₂²×C₈	1	1	yes	characteristic
71	C₇×C₈	#131	1	1	yes	characteristic
72	C₇⋊C₈	—	8	2	no	—
73	C₂×D₈	—	7	2	no	—
74	C₂²×C₈	—	28	2	no	—
75	C₄×D₄	—	14	2	no	—
76	D₁₆	—	7	4	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	C₄×D₇	1	1	yes	characteristic
79	C₂×C₁₆	C₄×D₇	1	1	yes	characteristic
80	#107	—	7	2	no	—
81	C₄⋊C₈	—	14	2	no	—
82	#102	—	14	2	no	—
83	C₂²⋊C₈	—	28	2	no	—
84	C₄×C₈	D₂₈	1	2	yes	—
85	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
86	C₂²×C₇	#442	1	1	yes	characteristic
87	C₂×D₇	—	4	4	no	—
88	C₂×D₇	—	8	2	no	—
89	C₂×D₇	—	8	2	no	—
90	C₄×C₇	—	2	2	no	—
91	C₄×C₇	C₈×D₄	1	1	yes	characteristic
92	C₄×C₇	#444	1	1	yes	characteristic
93	Dic₇	—	8	2	no	—
94	C₂×D₄	—	14	2	no	—
95	C₂²×C₄	—	28	2	no	—
96	D₈	—	7	4	no	—
97	D₈	—	14	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	28	2	no	—
100	C₂×C₈	—	56	4	no	—
101	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
102	C₂×C₈	#1157	1	1	yes	characteristic
103	C₄⋊C₄	—	14	2	no	—
104	C₂²⋊C₄	—	28	2	no	—
105	C₄²	C₂×D₂₈	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	—	4	1	no	—
108	C₁₆	C₈×D₇	1	2	yes	complemented
109	C₂×C₇	#1580	1	1	yes	characteristic
110	C₂×C₇	AGL(1,ℤ/16)	1	2	yes	—
111	D₇	—	8	4	no	—
112	C₂³	—	28	2	no	—
113	D₄	—	14	4	no	—
114	D₄	—	28	2	no	—
115	C₂×C₄	—	28	2	no	—
116	C₂×C₄	—	56	2	no	—
117	C₂×C₄	—	56	2	no	—
118	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
119	C₂×C₄	#6471	1	2	yes	—
120	C₈	—	2	1	no	—
121	C₈	—	4	2	no	—
122	C₈	—	56	2	no	—
123	C₈	C₂×C₈×D₇	1	1	yes	characteristic
124	C₈	#6464	1	1	yes	characteristic
125	C₇	#12067	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	28	4	no	—
127	C₂²	—	56	2	no	—
128	C₂²	—	56	2	no	—
129	C₂²	#89483	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	56	2	no	—
132	C₄	C₈×D₂₈	1	1	yes	characteristic
133	C₄	#89489	1	1	yes	characteristic
134	C₂	—	56	4	no	—
135	C₂	#11725831	1	1	yes	characteristic
136	C₂	#11726845	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11726832

C₂

yes

maximal, characteristic

#11725831

C₂

yes

maximal, complemented

C₇×#1313

C₂

yes

maximal, characteristic, complemented

#11725322

C₂

yes

maximal, complemented

#11725319

C₂

yes

maximal, characteristic, complemented

C₂×D₁₁₂

C₄

yes

characteristic

C₄×D₅₆

C₂²

yes

—

C₈×D₂₈

—

C₇×#346

C₂²

yes

characteristic

C₇×#346

C₂²

yes

characteristic

C₇×#322

C₂²

yes

characteristic

#89304

C₄

yes

—

#89300

C₄

yes

characteristic

#89254

—

#89252

C₂²

yes

—

#12067

—

maximal, Sylow

C₂×D₅₆

C₂×C₄

yes

—

C₂×C₈×D₇

—

C₄×D₂₈

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

characteristic

#6395

—

(C₂×D₇)⋊C₈

—

#6392

C₂×C₄

yes

—

C₄⋊(C₇⋊C₈)

—

D₁₁₂

C₈

yes

—

—

—

—

D₇

yes

characteristic, complemented

C₂×D₂₈

—

C₂×C₄×D₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂²×C₄

yes

characteristic

C₂×C₇×C₈

#41

yes

characteristic

C₄²×C₇

C₂×D₄

yes

characteristic

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

D₅₆

—

D₅₆

C₂×C₈

yes

complemented

C₈×D₇

—

C₇×C₁₆

—

C₇×C₁₆

—

C₇×C₁₆

C₂×C₈

yes

complemented

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

#367

—

#357

—

#346

C₂×D₇

yes

characteristic

#346

C₂×D₇

yes

characteristic

#334

—

D₄⋊C₈

—

#322

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

C₄×D₄

yes

characteristic

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

C₄×D₇

—

C₄×D₇

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

C₂²×C₈

yes

characteristic

C₇×C₈

#131

yes

characteristic

C₇⋊C₈

—

C₂×D₈

—

C₂²×C₈

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇

yes

characteristic

C₂×C₁₆

C₄×D₇

yes

characteristic

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

D₂₈

yes

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#442

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

C₈×D₄

yes

characteristic

C₄×C₇

#444

yes

characteristic

Dic₇

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×C₄×D₇

yes

characteristic

102

C₂×C₈

#1157

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

C₂×D₂₈

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

C₈×D₇

yes

complemented

109

C₂×C₇

#1580

yes

characteristic

110

C₂×C₇

AGL(1,ℤ/16)

yes

—

111

D₇

—

112

C₂³

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₄×D₂₈

yes

characteristic

119

C₂×C₄

#6471

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

C₂×C₈×D₇

yes

characteristic

124

C₈

#6464

yes

characteristic

125

C₇

#12067

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#89483

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

C₈×D₂₈

yes

characteristic

133

C₄

#89489

yes

characteristic

134

C₂

—

135

C₂

#11725831

yes

characteristic

136

C₂

#11726845

yes

—

	subgroup	G/H
center	C₂²	#89483
commutator	C₇×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1313	C₂
socle	C₂²×C₇	#442
prime radical	C₂×D₁₁₂	C₄
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#1313	D₇
hypocenter	C₇	#12067
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#89483

commutator

C₇×C₈

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

C₂×D₁₁₂

C₄

cosocle

C₄×C₇×C₈

C₂³

D₇

C₇

G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
2. #11725831	C₂	semidirect
3. C₇×#1313	C₂	semidirect
4. #11725322	C₂	semidirect
5. #11725319	C₂	semidirect
29. D₁₁₂	C₈	semidirect
34. #1313	D₇	semidirect
45. D₅₆	C₂×C₈	semidirect
49. C₇×C₁₆	C₂×C₈	semidirect
108. C₁₆	C₈×D₇	semidirect
125. C₇	#12067	semidirect

H ≅ G/N

type

2. #11725831

C₂

semidirect

3. C₇×#1313

C₂

semidirect

4. #11725322

C₂

semidirect

5. #11725319

C₂

semidirect

29. D₁₁₂

C₈

semidirect

34. #1313

D₇

semidirect

45. D₅₆

C₂×C₈

semidirect

49. C₇×C₁₆

C₂×C₈

semidirect

108. C₁₆

C₈×D₇

semidirect

125. C₇

#12067

semidirect

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

subgroup	index
#11726832	2
#11725831	2
C₇×#1313	2
#11725322	2
#11725319	2
#12067	7

subgroup

index

Subgroups — #13265682

Conjugacy classes of subgroups

Complements

Maximal subgroups