Group #13268582 — subgroups

Subgroups — #13268582

Order 1792.

272 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 91 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	#11726812	C₂	1	2	yes	maximal
2	#11726798	C₂	1	2	yes	maximal, complemented
3	#11725819	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725812	C₂	1	1	yes	maximal, characteristic
5	C₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	#89471	—	2	1	no	—
7	#89467	C₂²	1	1	yes	characteristic
8	#89466	—	2	1	no	—
9	#89465	—	2	1	no	—
10	#89454	—	2	1	no	—
11	#89454	C₂²	1	1	yes	characteristic
12	#89454	C₂²	1	2	yes	—
13	C₄×C₇×C₁₆	C₂²	1	2	yes	—
14	C₇×C₈²	C₂²	1	1	yes	characteristic
15	#89304	—	2	4	no	—
16	#89299	—	2	4	no	—
17	#12531	—	7	1	no	maximal, Sylow
18	C₂×Dic₂₈	—	4	2	no	—
19	C₂×D₅₆	—	2	2	no	—
20	C₂×#1132	—	4	2	no	—
21	#6437	—	2	1	no	—
22	#6432	—	2	2	no	—
23	#6432	—	2	1	no	—
24	C₂×C₇×C₁₆	D₄	1	4	yes	—
25	C₄×C₇×C₈	D₄	1	1	yes	characteristic
26	C₄×C₇×C₈	D₄	1	1	yes	characteristic
27	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
28	#6395	—	4	2	no	—
29	#6392	—	2	2	no	—
30	#6392	—	2	2	no	—
31	#6392	—	2	2	no	—
32	#6392	—	4	2	no	—
33	#2252	—	7	2	no	—
34	#2240	—	7	2	no	—
35	#1713	—	7	1	no	—
36	#1706	—	7	1	no	—
37	#1312	D₇	1	1	yes	characteristic, complemented
38	C₂×D₂₈	—	4	1	no	—
39	C₂×D₂₈	—	4	1	no	—
40	C₂×D₂₈	—	4	1	no	—
41	C₂×Dic₁₄	—	4	1	no	—
42	C₂×Dic₁₄	—	4	2	no	—
43	C₂×C₇×C₈	D₈	1	2	yes	—
44	C₂×C₇×C₈	D₈	1	2	yes	—
45	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
46	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
47	C₄⋊Dic₇	—	4	2	no	—
48	C₄⋊Dic₇	—	4	1	no	—
49	C₄⋊Dic₇	—	4	2	no	—
50	C₄⋊Dic₇	—	4	1	no	—
51	Dic₂₈	—	8	4	no	—
52	D₅₆	—	4	4	no	—
53	#1132	—	8	4	no	—
54	C₇×C₁₆	—	2	4	no	—
55	#500	—	7	1	no	—
56	#500	—	7	2	no	—
57	#500	—	14	1	no	—
58	#494	—	14	1	no	—
59	#493	—	7	1	no	—
60	#492	—	14	1	no	—
61	#486	—	14	1	no	—
62	#366	—	14	4	no	—
63	#357	—	14	4	no	—
64	C₄×C₁₆	C₂×D₇	1	2	yes	—
65	C₈²	C₂×D₇	1	1	yes	characteristic
66	C₂²×D₇	—	8	1	no	—
67	C₂×C₄×C₇	#127	1	1	yes	characteristic
68	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
70	C₂×Dic₇	—	8	2	no	—
71	C₂×Dic₇	—	8	1	no	—
72	D₂₈	—	8	2	no	—
73	D₂₈	—	8	2	no	—
74	D₂₈	—	8	2	no	—
75	Dic₁₄	—	8	2	no	—
76	Dic₁₄	—	8	4	no	—
77	C₇×C₈	—	2	2	no	—
78	C₇×C₈	D₁₆	1	4	yes	—
79	C₇×C₈	#137	1	4	yes	—
80	C₂×Q₁₆	—	28	2	no	—
81	C₂×SD₁₆	—	28	2	no	—
82	C₂×D₈	—	14	2	no	—
83	C₄⋊Q₈	—	14	1	no	—
84	C₄⋊Q₈	—	14	2	no	—
85	#127	—	14	1	no	—
86	C₂×C₁₆	D₂₈	1	4	yes	—
87	#107	—	14	2	no	—
88	#107	—	14	2	no	—
89	#107	—	14	2	no	—
90	#107	—	28	2	no	—
91	#102	—	28	2	no	—
92	C₄×C₈	D₂₈	1	1	yes	characteristic
93	C₄×C₈	D₂₈	1	1	yes	characteristic
94	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
95	C₂²×C₇	#493	1	1	yes	characteristic
96	C₂×D₇	—	16	1	no	—
97	C₂×D₇	—	16	1	no	—
98	C₂×D₇	—	16	1	no	—
99	C₄×C₇	#497	1	2	yes	—
100	C₄×C₇	C₂×D₁₆	1	2	yes	—
101	C₄×C₇	#510	1	2	yes	—
102	Dic₇	—	16	1	no	—
103	Dic₇	—	16	2	no	—
104	C₂×Q₈	—	28	1	no	—
105	C₂×Q₈	—	28	2	no	—
106	C₂×D₄	—	28	1	no	—
107	C₂×D₄	—	28	1	no	—
108	C₂×D₄	—	28	1	no	—
109	Q₁₆	—	56	4	no	—
110	SD₁₆	—	56	4	no	—
111	D₈	—	28	4	no	—
112	C₂×C₈	D₅₆	1	2	yes	—
113	C₂×C₈	D₅₆	1	2	yes	—
114	C₂×C₈	C₂×D₂₈	1	2	yes	—
115	C₄⋊C₄	—	28	2	no	—
116	C₄⋊C₄	—	28	1	no	—
117	C₄⋊C₄	—	28	1	no	—
118	C₄⋊C₄	—	28	2	no	—
119	C₄²	C₂×D₂₈	1	1	yes	characteristic
120	C₁₆	—	2	4	no	—
121	C₂×C₇	#1722	1	1	yes	characteristic
122	C₂×C₇	#2246	1	1	yes	characteristic
123	C₂×C₇	#2251	1	1	yes	characteristic
124	D₇	—	32	1	no	—
125	C₂³	—	56	1	no	—
126	Q₈	—	56	2	no	—
127	Q₈	—	56	4	no	—
128	D₄	—	56	2	no	—
129	D₄	—	56	2	no	—
130	D₄	—	56	2	no	—
131	C₂×C₄	—	56	1	no	—
132	C₂×C₄	—	56	2	no	—
133	C₂×C₄	#6437	1	1	yes	characteristic
134	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
135	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	D₁₁₂	1	4	yes	—
138	C₈	#6472	1	4	yes	—
139	C₇	#12531	1	1	yes	characteristic, complemented, Sylow
140	C₂²	—	112	1	no	—
141	C₂²	—	112	1	no	—
142	C₂²	—	112	1	no	—
143	C₂²	#89467	1	1	yes	characteristic
144	C₄	—	112	1	no	—
145	C₄	—	112	2	no	—
146	C₄	#89487	1	2	yes	—
147	C₄	C₂×D₁₁₂	1	2	yes	—
148	C₄	#89681	1	2	yes	—
149	C₂	—	224	1	no	—
150	C₂	#11725852	1	1	yes	characteristic
151	C₂	#11726814	1	1	yes	characteristic
152	C₂	#11726838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11726812

C₂

yes

maximal

#11726798

C₂

yes

maximal, complemented

#11725819

C₂

yes

maximal, characteristic, complemented

#11725812

C₂

yes

maximal, characteristic

C₇×#1312

C₂

yes

maximal, characteristic, complemented

#89471

—

#89467

C₂²

yes

—

—

—

C₂²

yes

characteristic

#89454

C₂²

yes

—

C₄×C₇×C₁₆

C₂²

yes

—

C₇×C₈²

C₂²

yes

—

—

—

C₂×Dic₂₈

—

C₂×D₅₆

—

—

—

—

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

—

—

—

—

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₂₈

—

D₅₆

—

#1132

—

C₇×C₁₆

—

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₇

yes

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

D₂₈

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

D₁₆

yes

—

C₇×C₈

#137

yes

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₂₈

yes

—

—

—

—

—

—

C₄×C₈

D₂₈

yes

characteristic

C₄×C₈

D₂₈

yes

characteristic

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

#497

yes

—

100

C₄×C₇

C₂×D₁₆

yes

—

101

C₄×C₇

#510

yes

—

102

Dic₇

—

103

Dic₇

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

Q₁₆

—

110

SD₁₆

—

111

D₈

—

112

C₂×C₈

D₅₆

yes

—

113

C₂×C₈

D₅₆

yes

—

114

C₂×C₈

C₂×D₂₈

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂×D₂₈

yes

characteristic

120

C₁₆

—

121

C₂×C₇

#1722

yes

characteristic

122

C₂×C₇

#2246

yes

characteristic

123

C₂×C₇

#2251

yes

characteristic

124

D₇

—

125

C₂³

—

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#6437

yes

characteristic

134

C₂×C₄

C₂×D₅₆

yes

characteristic

135

C₂×C₄

C₂×D₅₆

yes

characteristic

136

C₈

—

137

C₈

D₁₁₂

yes

—

138

C₈

#6472

yes

—

139

C₇

#12531

yes

characteristic, complemented, Sylow

140

C₂²

—

112

—

141

C₂²

—

112

—

142

C₂²

—

112

—

143

C₂²

#89467

yes

characteristic

144

C₄

—

112

—

145

C₄

—

112

—

146

C₄

#89487

yes

—

147

C₄

C₂×D₁₁₂

yes

—

148

C₄

#89681

yes

—

149

C₂

—

224

—

150

C₂

#11725852

yes

characteristic

151

C₂

#11726814

yes

characteristic

152

C₂

#11726838

yes

characteristic

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1312	C₂
socle	C₂²×C₇	#493
prime radical	#89467	C₂²
hypercenter	#1312	D₇
hypocenter	C₇	#12531
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#89467

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

C₂²

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
2. #11726798	C₂	semidirect
3. #11725819	C₂	semidirect
5. C₇×#1312	C₂	semidirect
37. #1312	D₇	semidirect
139. C₇	#12531	semidirect

H ≅ G/N

type

2. #11726798

C₂

semidirect

3. #11725819

C₂

semidirect

5. C₇×#1312

C₂

semidirect

37. #1312

D₇

semidirect

139. C₇

#12531

semidirect

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

subgroup	index
#11726812	2
#11726798	2
#11725819	2
#11725812	2
C₇×#1312	2
#12531	7

subgroup

index

Subgroups — #13268582

Conjugacy classes of subgroups

Complements

Maximal subgroups