Group #13354041 — subgroups

Subgroups — D₇×#1840 #13354041

Order 1792.

894 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 283 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	D₇×#376	C₂	1	1	yes	maximal, characteristic
2	C₇×#1840	C₂	1	1	yes	maximal, characteristic, complemented
3	#11727192	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×#336	C₂	1	6	yes	maximal
5	#11726362	C₂	1	6	yes	maximal, complemented
6	C₂²×C₈×D₇	C₂²	1	4	yes	—
7	C₂×C₄²×D₇	C₂²	1	3	yes	—
8	C₇×#376	C₂²	1	1	yes	characteristic
9	#89704	C₂²	1	3	yes	—
10	D₇×#95	C₄	1	4	yes	—
11	#89420	C₄	1	4	yes	—
12	C₇×#336	C₂²	1	6	yes	—
13	#89345	C₂²	1	12	yes	—
14	#89319	C₂²	1	6	yes	—
15	C₂×#1840	—	7	1	no	maximal, Sylow
16	C₂²×C₄×D₇	C₂×C₄	1	6	yes	—
17	C₂²×C₄×D₇	C₂³	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂³	1	4	yes	—
19	C₂×C₄²×C₇	C₂³	1	3	yes	—
20	C₂×C₄×Dic₇	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₇	C₂³	1	3	yes	—
22	C₂²×C₇⋊C₈	C₂³	1	4	yes	—
23	C₂×C₈×D₇	—	2	24	no	—
24	C₄²×D₇	—	2	12	no	—
25	C₇×#95	C₂×C₄	1	4	yes	—
26	C₇⋊#95	C₂×C₄	1	12	yes	—
27	C₂×#376	—	7	1	no	—
28	#1840	—	7	1	no	—
29	#1840	—	7	6	no	—
30	#1840	D₇	1	1	yes	characteristic, direct product factor
31	C₂×#336	—	7	6	no	—
32	C₂³×D₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
34	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
35	C₂²×Dic₇	C₂²×C₄	1	6	yes	—
36	C₂²×Dic₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×D₇	—	2	12	no	—
38	C₂×C₄×D₇	—	2	24	no	—
39	C₂×C₄×D₇	#41	1	12	yes	—
40	C₂×C₇×C₈	—	2	12	no	—
41	C₄²×C₇	—	2	6	no	—
42	C₄×Dic₇	—	2	6	no	—
43	C₄×Dic₇	—	2	12	no	—
44	C₂×C₇⋊C₈	—	2	12	no	—
45	C₈×D₇	—	4	16	no	—
46	C₂³×C₈	—	7	4	no	—
47	C₂²×C₄²	—	7	3	no	—
48	#376	—	7	3	no	—
49	#376	—	7	4	no	—
50	#376	C₂×D₇	1	1	yes	characteristic
51	C₂×#95	—	7	4	no	—
52	#336	—	7	12	no	—
53	#336	—	7	6	no	—
54	#336	C₂×D₇	1	6	yes	—
55	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
56	C₂²×D₇	#117	1	6	yes	—
57	C₂²×D₇	#131	1	8	yes	—
58	C₂×C₄×C₇	—	2	6	no	—
59	C₂×C₄×C₇	—	2	12	no	—
60	C₂×C₄×C₇	C₂×#41	1	6	yes	—
61	C₂×Dic₇	—	2	6	no	—
62	C₂×Dic₇	—	2	12	no	—
63	C₂×Dic₇	#117	1	6	yes	—
64	C₄×D₇	—	4	24	no	—
65	C₄×D₇	#126	1	16	yes	—
66	C₇×C₈	—	4	4	no	—
67	C₇⋊C₈	—	4	4	no	—
68	C₂³×C₄	—	7	1	no	—
69	C₂³×C₄	—	7	6	no	—
70	C₂²×C₈	—	7	4	no	—
71	C₂²×C₈	—	14	24	no	—
72	C₂²×C₈	C₂²×D₇	1	4	yes	—
73	C₂×C₄²	—	7	6	no	—
74	C₂×C₄²	—	7	3	no	—
75	C₂×C₄²	—	14	12	no	—
76	C₂×C₄²	C₂²×D₇	1	3	yes	—
77	#95	—	7	12	no	—
78	#95	C₄×D₇	1	4	yes	—
79	C₂²×C₇	C₂×#117	1	3	yes	—
80	C₂²×C₇	C₂×#131	1	4	yes	—
81	C₂×D₇	#383	1	4	yes	—
82	C₂×D₇	#433	1	24	yes	—
83	C₄×C₇	—	4	6	no	—
84	C₄×C₇	C₂×#126	1	4	yes	—
85	Dic₇	—	4	6	no	—
86	Dic₇	#383	1	4	yes	—
87	C₂⁴	—	7	1	no	—
88	C₂²×C₄	—	7	1	no	—
89	C₂²×C₄	—	7	12	no	—
90	C₂²×C₄	—	7	6	no	—
91	C₂²×C₄	—	14	12	no	—
92	C₂²×C₄	—	14	24	no	—
93	C₂²×C₄	C₂×C₄×D₇	1	6	yes	—
94	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
95	C₂×C₈	—	2	12	no	—
96	C₂×C₈	—	14	12	no	—
97	C₂×C₈	—	28	16	no	—
98	C₄²	—	2	6	no	—
99	C₄²	—	14	12	no	—
100	C₄²	—	14	6	no	—
101	C₂×C₇	C₂×#383	1	1	yes	characteristic
102	C₂×C₇	C₂×#433	1	6	yes	—
103	D₇	#1840	1	8	yes	direct product factor
104	C₂³	—	7	8	no	—
105	C₂³	—	7	6	no	—
106	C₂³	C₂²×C₄×D₇	1	1	yes	characteristic
107	C₂×C₄	—	2	12	no	—
108	C₂×C₄	—	2	6	no	—
109	C₂×C₄	—	7	6	no	—
110	C₂×C₄	—	7	16	no	—
111	C₂×C₄	—	14	12	no	—
112	C₂×C₄	—	14	6	no	—
113	C₂×C₄	—	28	24	no	—
114	C₂×C₄	D₇×#41	1	6	yes	—
115	C₈	—	4	4	no	—
116	C₈	—	28	4	no	—
117	C₇	C₂×#1840	1	1	yes	characteristic, complemented, Sylow
118	C₂²	—	7	4	no	—
119	C₂²	—	7	24	no	—
120	C₂²	D₇×#117	1	3	yes	—
121	C₂²	D₇×#131	1	4	yes	—
122	C₄	—	4	6	no	—
123	C₄	—	7	4	no	—
124	C₄	—	28	6	no	—
125	C₄	D₇×#126	1	4	yes	—
126	C₂	—	7	8	no	—
127	C₂	D₇×#383	1	1	yes	characteristic
128	C₂	D₇×#433	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₇×#376

C₂

yes

maximal, characteristic

C₇×#1840

C₂

yes

maximal, characteristic, complemented

#11727192

C₂

yes

maximal, characteristic, complemented

D₇×#336

C₂

yes

maximal

#11726362

C₂

yes

maximal, complemented

C₂²×C₈×D₇

C₂²

yes

—

C₂×C₄²×D₇

C₂²

yes

—

C₇×#376

C₂²

yes

characteristic

#89704

C₂²

yes

—

D₇×#95

C₄

yes

—

#89420

C₄

yes

—

C₇×#336

C₂²

yes

—

#89345

C₂²

yes

—

#89319

C₂²

yes

—

C₂×#1840

—

maximal, Sylow

C₂²×C₄×D₇

C₂×C₄

yes

—

C₂²×C₄×D₇

C₂³

yes

characteristic

C₂²×C₇×C₈

C₂³

yes

—

C₂×C₄²×C₇

C₂³

yes

—

C₂×C₄×Dic₇

C₂×C₄

yes

—

C₂×C₄×Dic₇

C₂³

yes

—

C₂²×C₇⋊C₈

C₂³

yes

—

C₂×C₈×D₇

—

C₄²×D₇

—

C₇×#95

C₂×C₄

yes

—

C₇⋊#95

C₂×C₄

yes

—

—

—

—

D₇

yes

characteristic, direct product factor

C₂×#336

—

C₂³×D₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

C₂²×C₄

yes

—

C₂²×C₄×C₇

C₂⁴

yes

characteristic

C₂²×Dic₇

C₂²×C₄

yes

—

C₂²×Dic₇

C₂²×C₄

yes

characteristic

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

#41

yes

—

C₂×C₇×C₈

—

C₄²×C₇

—

C₄×Dic₇

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

C₈×D₇

—

C₂³×C₈

—

C₂²×C₄²

—

#376

—

#376

—

#376

C₂×D₇

yes

characteristic

C₂×#95

—

#336

—

#336

—

#336

C₂×D₇

yes

—

C₂³×C₇

C₂³×C₄

yes

characteristic

C₂²×D₇

#117

yes

—

C₂²×D₇

#131

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

#117

yes

—

C₄×D₇

—

C₄×D₇

#126

yes

—

C₇×C₈

—

C₇⋊C₈

—

C₂³×C₄

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₇

yes

—

#95

—

#95

C₄×D₇

yes

—

C₂²×C₇

C₂×#117

yes

—

C₂²×C₇

C₂×#131

yes

—

C₂×D₇

#383

yes

—

C₂×D₇

#433

yes

—

C₄×C₇

—

C₄×C₇

C₂×#126

yes

—

Dic₇

—

Dic₇

#383

yes

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₇

yes

—

C₂²×C₄

C₂³×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₂×C₇

C₂×#383

yes

characteristic

102

C₂×C₇

C₂×#433

yes

—

103

D₇

#1840

yes

direct product factor

104

C₂³

—

105

C₂³

—

106

C₂³

C₂²×C₄×D₇

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

D₇×#41

yes

—

115

C₈

—

116

C₈

—

117

C₇

C₂×#1840

yes

characteristic, complemented, Sylow

118

C₂²

—

119

C₂²

—

120

C₂²

D₇×#117

yes

—

121

C₂²

D₇×#131

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

D₇×#126

yes

—

126

C₂

—

127

C₂

D₇×#383

yes

characteristic

128

C₂

D₇×#433

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇
commutator, socle	C₂³×C₇	C₂³×C₄
Fitting	C₇×#1840	C₂
prime radical	C₂³×D₇	C₂²×C₄
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	#1840	D₇
hypocenter	C₇	C₂×#1840
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1840

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₇

commutator, socle

C₂³×C₇

C₂³×C₄

Fitting

C₇×#1840

C₂

prime radical

C₂³×D₇

C₂²×C₄

cosocle

C₂²×C₄×C₇

C₂⁴

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂²×C₄ ◁ #1840

N	H ≅ G/N	type
2. C₇×#1840	C₂	semidirect
3. #11727192	C₂	semidirect
5. #11726362	C₂	semidirect
30. #1840	D₇	direct (N × H)
103. D₇	#1840	direct (N × H)
117. C₇	C₂×#1840	semidirect

H ≅ G/N

type

2. C₇×#1840

C₂

semidirect

3. #11727192

C₂

semidirect

5. #11726362

C₂

semidirect

30. #1840

D₇

direct (N × H)

103. D₇

#1840

direct (N × H)

117. C₇

C₂×#1840

semidirect

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

subgroup	index
D₇×#376	2
C₇×#1840	2
#11727192	2
D₇×#336	2
#11726362	2
C₂×#1840	7

subgroup

index

Subgroups — D₇×#1840 #13354041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×#1840 #13354041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#1840 #13354041