Group #13268527 — subgroups

Subgroups — #13268527

Order 1792.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
2	#11729067	C₂	1	2	yes	maximal
3	#11729066	C₂	1	2	yes	maximal, complemented
4	#11725814	C₂	1	1	yes	maximal, characteristic
5	#11725243	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#500	—	2	2	no	—
7	C₇×#494	C₂²	1	2	yes	—
8	C₂×#6403	—	2	4	no	—
9	C₂×#6402	—	2	4	no	—
10	#89471	C₂²	1	1	yes	characteristic
11	#89467	C₂²	1	1	yes	characteristic
12	#89466	—	2	2	no	—
13	C₇×C₈²	C₂²	1	1	yes	characteristic
14	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
15	#12520	—	7	1	no	maximal, Sylow
16	C₂×C₇×Q₁₆	—	2	2	no	—
17	C₂×C₇×Q₁₆	—	2	2	no	—
18	C₇×C₄⋊Q₈	—	2	2	no	—
19	C₂×Dic₂₈	—	2	2	no	—
20	C₂×D₅₆	—	2	2	no	—
21	#6437	—	2	1	no	—
22	#6432	—	2	1	no	—
23	C₇×#107	—	4	4	no	—
24	C₄×C₇×C₈	D₄	1	2	yes	—
25	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
26	#6403	—	4	8	no	—
27	#6402	—	4	8	no	—
28	#6395	—	4	4	no	—
29	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
30	#2248	—	7	2	no	—
31	#2247	—	7	2	no	—
32	#1713	D₇	1	1	yes	characteristic, complemented
33	#1709	—	7	1	no	—
34	#1311	—	7	1	no	—
35	C₂×C₇×Q₈	—	4	2	no	—
36	C₂×D₂₈	—	4	1	no	—
37	C₂×D₂₈	—	4	2	no	—
38	C₂×Dic₁₄	—	4	1	no	—
39	C₇×Q₁₆	—	4	4	no	—
40	C₇×Q₁₆	—	4	4	no	—
41	C₂×C₇×C₈	D₈	1	4	yes	—
42	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
43	C₇×C₄⋊C₄	—	4	4	no	—
44	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₇	—	4	2	no	—
46	Dic₂₈	—	4	4	no	—
47	D₅₆	—	4	4	no	—
48	C₇⋊C₁₆	D₈	1	8	yes	—
49	C₂×Q₃₂	—	14	4	no	—
50	C₂×SD₃₂	—	14	4	no	—
51	#500	—	2	2	no	—
52	#494	—	7	1	no	—
53	#494	C₂×D₇	1	2	yes	—
54	#493	—	7	1	no	—
55	#486	—	14	2	no	—
56	C₄×C₁₆	—	7	2	no	—
57	C₈²	C₂×D₇	1	1	yes	characteristic
58	C₂²×D₇	—	8	1	no	—
59	C₇×Q₈	—	8	2	no	—
60	C₇×Q₈	—	8	2	no	—
61	C₂×C₄×C₇	—	8	2	no	—
62	C₂×C₄×C₇	#127	1	1	yes	characteristic
63	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
64	C₂×Dic₇	—	8	1	no	—
65	D₂₈	—	8	2	no	—
66	D₂₈	—	8	4	no	—
67	Dic₁₄	—	8	2	no	—
68	C₇×C₈	—	2	4	no	—
69	C₇×C₈	C₂×D₈	1	4	yes	—
70	C₂×Q₁₆	—	2	2	no	—
71	C₂×Q₁₆	—	2	2	no	—
72	C₂×Q₁₆	—	14	2	no	—
73	C₂×D₈	—	14	2	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	14	1	no	—
76	#127	—	14	1	no	—
77	Q₃₂	—	28	8	no	—
78	SD₃₂	—	28	8	no	—
79	C₂×C₁₆	—	7	4	no	—
80	#107	—	4	4	no	—
81	#102	—	28	4	no	—
82	C₄×C₈	C₂²⋊D₇	1	2	yes	—
83	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
84	C₂²×C₇	#493	1	1	yes	characteristic
85	C₂×D₇	—	16	1	no	—
86	C₂×D₇	—	16	2	no	—
87	C₄×C₇	—	16	2	no	—
88	C₄×C₇	#493	1	2	yes	—
89	C₄×C₇	#510	1	4	yes	—
90	Dic₇	—	16	1	no	—
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	28	1	no	—
93	C₂×D₄	—	28	2	no	—
94	C₂×D₄	—	28	1	no	—
95	Q₁₆	—	4	4	no	—
96	Q₁₆	—	4	4	no	—
97	Q₁₆	—	28	4	no	—
98	D₈	—	28	4	no	—
99	C₂×C₈	#1141	1	4	yes	—
100	C₂×C₈	D₄×D₇	1	2	yes	—
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	28	2	no	—
103	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
104	C₁₆	—	7	8	no	—
105	C₂×C₇	#1712	1	1	yes	characteristic
106	C₂×C₇	#2251	1	2	yes	—
107	D₇	—	32	1	no	—
108	C₂³	—	56	1	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	56	2	no	—
112	D₄	—	56	2	no	—
113	D₄	—	56	4	no	—
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	56	1	no	—
116	C₂×C₄	C₂×#1141	1	2	yes	—
117	C₂×C₄	#6503	1	1	yes	characteristic
118	C₈	—	2	4	no	—
119	C₈	D₇×D₈	1	4	yes	—
120	C₇	#12520	1	1	yes	characteristic, complemented, Sylow
121	C₂²	—	112	1	no	—
122	C₂²	—	112	2	no	—
123	C₂²	#89845	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	—	112	1	no	—
126	C₄	#89923	1	2	yes	—
127	C₄	#89951	1	4	yes	—
128	C₂	—	224	1	no	—
129	C₂	#11729030	1	1	yes	characteristic
130	C₂	#11729068	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1713

C₂

yes

maximal, characteristic, complemented

#11729067

C₂

yes

maximal

#11729066

C₂

yes

maximal, complemented

#11725814

C₂

yes

maximal, characteristic

#11725243

C₂

yes

maximal, characteristic, complemented

C₇×#500

—

C₇×#494

C₂²

yes

—

C₂×#6403

—

C₂×#6402

—

#89471

C₂²

yes

characteristic

#89467

C₂²

yes

characteristic

#89466

—

C₇×C₈²

C₂²

yes

characteristic

C₄×C₇⋊C₁₆

C₂²

yes

—

#12520

—

maximal, Sylow

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₇×C₄⋊Q₈

—

C₂×Dic₂₈

—

C₂×D₅₆

—

#6437

—

#6432

—

C₇×#107

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

—

—

—

C₂×C₇⋊C₁₆

D₄

yes

—

#2248

—

#2247

—

#1713

D₇

yes

characteristic, complemented

#1709

—

#1311

—

C₂×C₇×Q₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₇×Q₁₆

—

C₇×Q₁₆

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

Dic₂₈

—

D₅₆

—

C₇⋊C₁₆

D₈

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

#500

—

#494

—

#494

C₂×D₇

yes

—

#493

—

#486

—

C₄×C₁₆

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

C₂×D₈

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

C₂×C₁₆

—

#107

—

#102

—

C₄×C₈

C₂²⋊D₇

yes

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

#493

yes

—

C₄×C₇

#510

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

C₂×C₈

#1141

yes

—

100

C₂×C₈

D₄×D₇

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

C₂×C₂²⋊D₇

yes

characteristic

104

C₁₆

—

105

C₂×C₇

#1712

yes

characteristic

106

C₂×C₇

#2251

yes

—

107

D₇

—

108

C₂³

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×#1141

yes

—

117

C₂×C₄

#6503

yes

characteristic

118

C₈

—

119

C₈

D₇×D₈

yes

—

120

C₇

#12520

yes

characteristic, complemented, Sylow

121

C₂²

—

112

—

122

C₂²

—

112

—

123

C₂²

#89845

yes

characteristic

124

C₄

—

125

C₄

—

112

—

126

C₄

#89923

yes

—

127

C₄

#89951

yes

—

128

C₂

—

224

—

129

C₂

#11729030

yes

characteristic

130

C₂

#11729068

yes

—

	subgroup	G/H
center	C₂²	#89845
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1713	C₂
socle	C₂²×C₇	#493
prime radical	#89467	C₂²
hypercenter	#1713	D₇
hypocenter	C₇	#12520
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#89845

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 ◁ #1713

N	H ≅ G/N	type
1. C₇×#1713	C₂	semidirect
3. #11729066	C₂	semidirect
5. #11725243	C₂	semidirect
32. #1713	D₇	semidirect
120. C₇	#12520	semidirect

H ≅ G/N

type

1. C₇×#1713

C₂

semidirect

3. #11729066

C₂

semidirect

5. #11725243

C₂

semidirect

32. #1713

D₇

semidirect

120. C₇

#12520

semidirect

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

subgroup	index
C₇×#1713	2
#11729067	2
#11729066	2
#11725814	2
#11725243	2
#12520	7

subgroup

index

Subgroups — #13268527

Conjugacy classes of subgroups

Complements

Maximal subgroups