Group #13265798 — subgroups

Subgroups — #13265798

Order 1792.

206 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄×Dic₅₆	C₂	1	1	yes	maximal, characteristic
2	#11726318	C₂	1	2	yes	maximal
3	C₇×#1372	C₂	1	1	yes	maximal, characteristic
4	#11725304	C₂	1	1	yes	maximal, characteristic
5	#11725278	C₂	1	2	yes	maximal
6	C₂×Dic₅₆	C₄	1	1	yes	characteristic
7	#89603	—	2	2	no	—
8	C₄×Dic₂₈	C₂²	1	2	yes	—
9	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₇×#335	C₂²	1	2	yes	—
11	#89299	C₄	1	1	yes	characteristic
12	#89298	C₄	1	2	yes	—
13	#89279	—	2	2	no	—
14	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	#12085	—	7	1	no	maximal, Sylow
17	C₂×Dic₂₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₄	—	2	2	no	—
19	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₇×C₄⋊C₈	—	2	2	no	—
22	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
23	#6392	C₂×C₄	1	2	yes	—
24	#6390	—	2	2	no	—
25	#6389	—	4	2	no	—
26	Dic₇⋊C₈	—	4	2	no	—
27	C₂×C₇⋊C₁₆	—	2	1	no	—
28	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
29	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
30	Dic₅₆	—	2	2	no	—
31	C₄×Q₃₂	—	7	1	no	—
32	#1591	—	7	2	no	—
33	#1372	D₇	1	1	yes	characteristic
34	#1332	—	7	2	no	—
35	#1311	—	7	1	no	—
36	C₂×Dic₁₄	—	2	2	no	—
37	C₂×C₇×C₈	—	4	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	C₄⋊Dic₇	—	2	2	no	—
42	Dic₇⋊C₄	—	4	2	no	—
43	C₄×Dic₇	—	4	2	no	—
44	C₂×C₇⋊C₈	—	2	1	no	—
45	C₂×C₇⋊C₈	D₈	1	2	yes	—
46	Dic₂₈	—	2	2	no	—
47	Dic₂₈	—	2	2	no	—
48	C₇×C₁₆	—	2	1	no	—
49	C₇×C₁₆	M₁₆	1	2	yes	—
50	C₇⋊C₁₆	—	2	2	no	—
51	C₂×Q₃₂	—	7	1	no	—
52	#446	—	14	2	no	—
53	C₄×Q₁₆	—	7	2	no	—
54	#366	—	7	1	no	—
55	#358	—	7	2	no	—
56	C₄×C₁₆	—	7	1	no	—
57	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
58	#335	C₂×D₇	1	2	yes	—
59	Q₈⋊C₈	—	14	2	no	—
60	C₈²	—	7	1	no	—
61	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇	#135	1	1	yes	characteristic
64	C₂×Dic₇	—	4	2	no	—
65	C₂×Dic₇	—	4	2	no	—
66	Dic₁₄	—	4	2	no	—
67	Dic₁₄	—	4	2	no	—
68	C₇×C₈	—	2	1	no	—
69	C₇×C₈	—	8	2	no	—
70	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
71	C₇×C₈	#131	1	1	yes	characteristic
72	C₇⋊C₈	—	2	2	no	—
73	C₇⋊C₈	—	4	1	no	—
74	C₂×Q₁₆	—	7	2	no	—
75	C₄×Q₈	—	14	2	no	—
76	Q₃₂	—	14	2	no	—
77	C₂×C₁₆	—	14	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	—	2	2	no	—
82	C₄⋊C₈	—	28	2	no	—
83	#103	—	14	2	no	—
84	#97	—	28	2	no	—
85	C₄×C₈	—	7	1	no	—
86	C₄×C₈	—	7	1	no	—
87	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
88	C₂²×C₇	C₄×D₈	1	1	yes	characteristic
89	C₄×C₇	—	2	1	no	—
90	C₄×C₇	#436	1	1	yes	characteristic
91	C₄×C₇	#443	1	1	yes	characteristic
92	C₄×C₇	#510	1	2	yes	—
93	Dic₇	—	8	2	no	—
94	Dic₇	—	8	2	no	—
95	C₂×Q₈	—	14	2	no	—
96	Q₁₆	—	14	2	no	—
97	Q₁₆	—	14	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	7	2	no	—
100	C₂×C₈	—	14	1	no	—
101	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
102	C₂×C₈	#1161	1	1	yes	characteristic
103	C₄⋊C₄	—	14	2	no	—
104	C₄⋊C₄	—	28	2	no	—
105	C₄²	—	28	2	no	—
106	C₄²	D₄×D₇	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	—	14	2	no	—
109	C₁₆	#1131	1	2	yes	—
110	C₂×C₇	#1589	1	1	yes	characteristic
111	C₂×C₇	#2176	1	1	yes	characteristic
112	C₂×C₇	#2179	1	1	yes	characteristic
113	Q₈	—	28	2	no	—
114	Q₈	—	28	2	no	—
115	C₂×C₄	—	28	2	no	—
116	C₂×C₄	—	28	2	no	—
117	C₂×C₄	#6456	1	1	yes	characteristic
118	C₂×C₄	D₇×D₈	1	1	yes	characteristic
119	C₂×C₄	#6482	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	8	2	no	—
122	C₈	—	14	2	no	—
123	C₈	—	28	1	no	—
124	C₈	C₂×#1131	1	1	yes	characteristic
125	C₈	#6470	1	1	yes	characteristic
126	C₇	#12085	1	1	yes	characteristic, complemented, Sylow
127	C₂²	#89644	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	56	2	no	—
130	C₄	—	56	2	no	—
131	C₄	#89610	1	1	yes	characteristic
132	C₄	#89667	1	1	yes	characteristic
133	C₄	#89687	1	2	yes	—
134	C₂	#11726317	1	1	yes	characteristic
135	C₂	#11727092	1	1	yes	characteristic
136	C₂	#11727112	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₅₆

C₂

yes

maximal, characteristic

#11726318

C₂

yes

maximal

C₇×#1372

C₂

yes

maximal, characteristic

#11725304

C₂

yes

maximal, characteristic

#11725278

C₂

yes

maximal

C₂×Dic₅₆

C₄

yes

characteristic

#89603

—

C₄×Dic₂₈

C₂²

yes

—

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₇×#335

C₂²

yes

—

#89299

C₄

yes

characteristic

#89298

C₄

yes

—

#89279

—

C₄×C₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₇⋊C₈

C₂²

yes

characteristic

#12085

—

maximal, Sylow

C₂×Dic₂₈

C₂×C₄

yes

—

C₄×Dic₁₄

—

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₇×C₄⋊C₈

—

C₄×C₇×C₈

C₂³

yes

characteristic

#6392

C₂×C₄

yes

—

#6390

—

#6389

—

Dic₇⋊C₈

—

C₂×C₇⋊C₁₆

—

C₄×C₇⋊C₈

D₄

yes

characteristic

C₄×C₇⋊C₈

D₄

yes

characteristic

Dic₅₆

—

C₄×Q₃₂

—

#1591

—

#1372

D₇

yes

characteristic

#1332

—

#1311

—

C₂×Dic₁₄

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂²×C₄

yes

characteristic

C₂×C₇×C₈

#41

yes

characteristic

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

D₈

yes

—

Dic₂₈

—

Dic₂₈

—

C₇×C₁₆

—

C₇×C₁₆

M₁₆

yes

—

C₇⋊C₁₆

—

C₂×Q₃₂

—

#446

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇

yes

characteristic

#335

C₂×D₇

yes

—

Q₈⋊C₈

—

C₈²

—

C₂×C₄×C₇

C₄×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

#135

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

C₂×M₁₆

yes

characteristic

C₇×C₈

#131

yes

characteristic

C₇⋊C₈

—

C₇⋊C₈

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇

yes

characteristic

C₂×C₁₆

C₄×D₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

C₄×D₈

yes

characteristic

C₄×C₇

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×C₄×D₇

yes

characteristic

102

C₂×C₈

#1161

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

—

106

C₄²

D₄×D₇

yes

characteristic

107

C₁₆

—

108

C₁₆

—

109

C₁₆

#1131

yes

—

110

C₂×C₇

#1589

yes

characteristic

111

C₂×C₇

#2176

yes

characteristic

112

C₂×C₇

#2179

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#6456

yes

characteristic

118

C₂×C₄

D₇×D₈

yes

characteristic

119

C₂×C₄

#6482

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×#1131

yes

characteristic

125

C₈

#6470

yes

characteristic

126

C₇

#12085

yes

characteristic, complemented, Sylow

127

C₂²

#89644

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#89610

yes

characteristic

132

C₄

#89667

yes

characteristic

133

C₄

#89687

yes

—

134

C₂

#11726317

yes

characteristic

135

C₂

#11727092

yes

characteristic

136

C₂

#11727112

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₇×D₈
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1372	C₂
socle, prime radical	C₂²×C₇	C₄×D₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#1372	D₇
hypocenter	C₇	#12085
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂×C₄

D₇×D₈

commutator

C₂×C₇×C₈

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇

Fitting

C₇×#1372

C₂

socle, prime radical

C₂²×C₇

C₄×D₈

cosocle

C₄×C₇×C₈

C₂³

D₇

C₇

G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
126. C₇	#12085	semidirect

H ≅ G/N

type

126. C₇

#12085

semidirect

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

subgroup	index
C₄×Dic₅₆	2
#11726318	2
C₇×#1372	2
#11725304	2
#11725278	2
#12085	7

subgroup

index

C₄×Dic₅₆

Subgroups — #13265798

Conjugacy classes of subgroups

Complements

Maximal subgroups