Group #13244249 — subgroups

Subgroups — #13244249

Order 1792.

330 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 147 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×#1726	C₂	1	1	yes	maximal, characteristic
2	#11727480	C₂	1	2	yes	maximal
3	#11725466	C₂	1	2	yes	maximal
4	#11725420	C₂	1	2	yes	maximal
5	C₂×C₇×#97	C₄	1	2	yes	—
6	C₂×C₇×#97	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#376	C₂²	1	1	yes	characteristic
10	#89697	C₂²	1	4	yes	—
11	#89249	C₄	1	8	yes	complemented
12	#8140	—	7	1	no	maximal, Sylow
13	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
18	C₇×#97	—	2	4	no	—
19	C₇×#97	—	2	2	no	—
20	C₄×C₇×C₈	—	2	4	no	—
21	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
22	C₇×#95	—	2	2	no	—
23	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₇⋊C₈)	—	2	4	no	—
25	C₄×C₇⋊C₈	C₂×C₄	1	8	yes	—
26	#1726	D₇	1	1	yes	characteristic
27	#1452	—	7	2	no	—
28	#1275	—	7	2	no	—
29	#1274	—	7	2	no	—
30	C₂²×C₄×C₇	—	2	2	no	—
31	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇	#41	1	2	yes	—
33	C₂×C₇×C₈	—	2	2	no	—
34	C₂×C₇×C₈	—	2	2	no	—
35	C₂×C₇×C₈	—	2	2	no	—
36	C₂×C₇×C₈	—	2	2	no	—
37	C₂×C₇×C₈	C₄²	1	2	yes	—
38	C₂×C₇×C₈	C₄²	1	2	yes	—
39	C₂×C₇×C₈	M₁₆	1	4	yes	—
40	C₄²×C₇	—	2	2	no	—
41	C₄²×C₇	M₁₆	1	4	yes	—
42	C₄²×C₇	C₂²×C₄	1	2	yes	—
43	C₄²×C₇	#41	1	2	yes	—
44	C₂×C₇⋊C₈	—	2	8	no	—
45	#431	—	7	4	no	—
46	C₂×#97	Dic₇	1	2	yes	—
47	C₂×#97	C₂×D₇	1	1	yes	characteristic
48	C₂×C₄×C₈	Dic₇	1	2	yes	—
49	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
50	#376	C₂×D₇	1	1	yes	characteristic
51	#322	—	7	8	no	—
52	C₂³×C₇	#117	1	1	yes	characteristic
53	C₂×C₄×C₇	—	2	4	no	—
54	C₂×C₄×C₇	—	4	2	no	—
55	C₂×C₄×C₇	—	4	4	no	—
56	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₇	#117	1	4	yes	—
58	C₂×C₄×C₇	#117	1	1	yes	characteristic
59	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
60	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
61	C₇×C₈	—	2	2	no	—
62	C₇×C₈	—	2	2	no	—
63	C₇×C₈	—	2	4	no	—
64	C₇⋊C₈	—	4	8	no	—
65	C₂²×C₈	C₂×Dic₇	1	2	yes	—
66	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
67	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
68	C₂×C₄²	C₂×Dic₇	1	2	yes	—
69	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
70	C₄⋊C₈	—	14	4	no	—
71	C₂²⋊C₈	—	14	4	no	—
72	#97	—	2	4	no	—
73	#97	—	2	2	no	—
74	C₄×C₈	—	2	4	no	—
75	C₄×C₈	—	7	8	no	—
76	C₄×C₈	C₄×D₇	1	4	yes	—
77	#95	—	2	2	no	—
78	C₂²×C₇	—	2	2	no	—
79	C₂²×C₇	#376	1	1	yes	characteristic
80	C₂²×C₇	#432	1	2	yes	—
81	C₄×C₇	—	2	4	no	—
82	C₄×C₇	—	4	4	no	—
83	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
84	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
85	C₄×C₇	#432	1	2	yes	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	#1157	1	2	yes	—
88	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	14	8	no	—
94	C₂×C₈	#1136	1	4	yes	—
95	C₂×C₈	C₄×Dic₇	1	2	yes	—
96	C₂×C₈	C₄×Dic₇	1	2	yes	—
97	C₄²	—	2	2	no	—
98	C₄²	#1136	1	4	yes	—
99	C₄²	C₂×C₄×D₇	1	2	yes	—
100	C₄²	#1157	1	2	yes	—
101	C₂×C₇	—	2	2	no	—
102	C₂×C₇	#1745	1	1	yes	characteristic
103	C₂×C₇	#1747	1	2	yes	—
104	C₂³	#6489	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	#6435	1	4	yes	—
109	C₂×C₄	C₂×#1136	1	2	yes	—
110	C₂×C₄	C₂×#1136	1	2	yes	—
111	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
112	C₂×C₄	#6489	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	—	28	8	no	—
117	C₇	#8140	1	1	yes	characteristic, complemented, Sylow
118	C₂²	—	2	2	no	—
119	C₂²	#89698	1	2	yes	—
120	C₂²	#89704	1	1	yes	characteristic
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	C₄×#1136	1	1	yes	characteristic
124	C₄	C₄×#1136	1	1	yes	characteristic
125	C₄	#89698	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	#11727143	1	1	yes	characteristic
128	C₂	#11727649	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1726

C₂

yes

maximal, characteristic

#11727480

C₂

yes

maximal

#11725466

C₂

yes

maximal

#11725420

C₂

yes

maximal

C₂×C₇×#97

C₄

yes

—

C₂×C₇×#97

C₂²

yes

characteristic

C₂×C₄×C₇×C₈

C₄

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#376

C₂²

yes

characteristic

#89697

C₂²

yes

—

#89249

C₄

yes

complemented

#8140

—

maximal, Sylow

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×#97

—

C₇×#97

—

C₄×C₇×C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₇×#95

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

#1726

D₇

yes

—

—

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄²

yes

—

C₂×C₇×C₈

C₄²

yes

—

C₂×C₇×C₈

M₁₆

yes

—

C₄²×C₇

—

C₄²×C₇

M₁₆

yes

—

C₄²×C₇

C₂²×C₄

yes

—

C₄²×C₇

#41

yes

—

C₂×C₇⋊C₈

—

#431

—

C₂×#97

Dic₇

yes

—

C₂×#97

C₂×D₇

yes

characteristic

C₂×C₄×C₈

Dic₇

yes

—

C₂×C₄×C₈

C₂×D₇

yes

characteristic

#376

C₂×D₇

yes

characteristic

#322

—

C₂³×C₇

#117

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#117

yes

characteristic

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×C₈

C₂×Dic₇

yes

—

C₂²×C₈

C₂×Dic₇

yes

characteristic

C₂²×C₈

C₂×Dic₇

yes

characteristic

C₂×C₄²

C₂×Dic₇

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇

yes

—

#95

—

C₂²×C₇

—

C₂²×C₇

#376

yes

characteristic

C₂²×C₇

#432

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₄×M₁₆

yes

characteristic

C₄×C₇

C₄×M₁₆

yes

characteristic

C₄×C₇

#432

yes

—

C₂²×C₄

—

C₂²×C₄

#1157

yes

—

C₂²×C₄

C₂²×Dic₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#1136

yes

—

C₂×C₈

C₄×Dic₇

yes

—

C₂×C₈

C₄×Dic₇

yes

—

C₄²

—

C₄²

#1136

yes

—

C₄²

C₂×C₄×D₇

yes

—

100

C₄²

#1157

yes

—

101

C₂×C₇

—

102

C₂×C₇

#1745

yes

characteristic

103

C₂×C₇

#1747

yes

—

104

C₂³

#6489

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#6435

yes

—

109

C₂×C₄

C₂×#1136

yes

—

110

C₂×C₄

C₂×#1136

yes

—

111

C₂×C₄

C₂×C₄×Dic₇

yes

characteristic

112

C₂×C₄

#6489

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₇

#8140

yes

characteristic, complemented, Sylow

118

C₂²

—

119

C₂²

#89698

yes

—

120

C₂²

#89704

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

C₄×#1136

yes

characteristic

124

C₄

C₄×#1136

yes

characteristic

125

C₄

#89698

yes

—

126

C₂

—

127

C₂

#11727143

yes

characteristic

128

C₂

#11727649

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1726	C₂
socle	C₂²×C₇	#376
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1726	D₇
hypocenter	C₇	#8140
LCS	G ▷ C₂×C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #1726

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₇

commutator

C₂×C₄×C₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₇

C₁ ◁ C₂×C₄ ◁ #1726

N	H ≅ G/N	type
11. #89249	C₄	semidirect
117. C₇	#8140	semidirect

H ≅ G/N

type

11. #89249

C₄

semidirect

117. C₇

#8140

semidirect

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

subgroup	index
C₇×#1726	2
#11727480	2
#11725466	2
#11725420	2
#8140	7

subgroup

index

Subgroups — #13244249

Conjugacy classes of subgroups

Complements

Maximal subgroups