Group #13186025 — subgroups

Subgroups — C₇×#8809 #13186025

Order 1792.

402 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 222 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×#1842	C₂	1	2	yes	maximal
2	C₇×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#334	C₂	1	4	yes	maximal
4	C₂×C₇×C₄⋊C₈	—	2	4	no	—
5	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
7	C₄³×C₇	C₂²	1	1	yes	characteristic
8	C₇×#336	C₄	1	4	yes	—
9	C₇×#334	—	2	8	no	—
10	#8809	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₇×C₈	—	2	4	no	—
12	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₇	D₄	1	2	yes	—
15	C₂×C₄²×C₇	Q₈	1	2	yes	—
16	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
17	C₇×C₄⋊C₈	—	2	8	no	—
18	C₇×C₄⋊C₈	—	4	8	no	—
19	C₄×C₇×C₈	D₄	1	4	yes	—
20	C₄×C₇×C₈	Q₈	1	4	yes	—
21	#1842	C₂×C₇	1	2	yes	—
22	#1744	C₂×C₇	1	1	yes	characteristic
23	C₂×#334	C₂×C₇	1	4	yes	—
24	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
25	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₇×C₈	—	2	8	no	—
29	C₂×C₇×C₈	—	4	4	no	—
30	C₂×C₇×C₈	—	4	8	no	—
31	C₂×C₇×C₈	C₄⋊C₄	1	8	yes	—
32	C₄²×C₇	—	2	4	no	—
33	C₄²×C₇	—	4	2	no	—
34	C₄²×C₇	SD₁₆	1	8	yes	—
35	C₄²×C₇	C₂×D₄	1	2	yes	—
36	C₄²×C₇	C₂×Q₈	1	2	yes	—
37	C₂×C₄⋊C₈	—	2	4	no	—
38	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
39	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
40	C₄³	C₂²×C₇	1	1	yes	characteristic
41	#336	C₄×C₇	1	4	yes	—
42	#334	—	2	8	no	—
43	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇	—	2	4	no	—
45	C₂×C₄×C₇	—	2	4	no	—
46	C₂×C₄×C₇	—	2	2	no	—
47	C₂×C₄×C₇	#106	1	8	yes	—
48	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
51	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₇	#131	1	4	yes	—
53	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
54	C₇×C₈	—	2	8	no	—
55	C₇×C₈	—	8	4	no	—
56	C₂²×C₈	—	2	4	no	—
57	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
58	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
59	C₂×C₄²	C₇×D₄	1	2	yes	—
60	C₂×C₄²	C₇×Q₈	1	2	yes	—
61	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
62	C₄⋊C₈	—	2	8	no	—
63	C₄⋊C₈	—	4	8	no	—
64	C₄×C₈	C₇×D₄	1	4	yes	—
65	C₄×C₈	C₇×Q₈	1	4	yes	—
66	C₂²×C₇	#384	1	1	yes	characteristic
67	C₂²×C₇	#424	1	2	yes	—
68	C₂²×C₇	C₂×#106	1	2	yes	—
69	C₂²×C₇	#430	1	2	yes	—
70	C₄×C₇	—	2	8	no	—
71	C₄×C₇	—	2	2	no	—
72	C₄×C₇	#424	1	4	yes	—
73	C₄×C₇	#492	1	2	yes	—
74	C₄×C₇	#498	1	2	yes	—
75	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
76	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
77	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
78	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
79	C₂×C₈	—	2	8	no	—
80	C₂×C₈	—	4	4	no	—
81	C₂×C₈	—	4	8	no	—
82	C₂×C₈	C₇×C₄⋊C₄	1	8	yes	—
83	C₄²	—	2	4	no	—
84	C₄²	—	4	2	no	—
85	C₄²	C₇×SD₁₆	1	8	yes	—
86	C₄²	C₂×C₇×D₄	1	2	yes	—
87	C₄²	C₂×C₇×Q₈	1	2	yes	—
88	C₂×C₇	#1570	1	4	yes	—
89	C₂×C₇	#1843	1	1	yes	characteristic
90	C₂×C₇	#1844	1	1	yes	characteristic
91	C₂×C₇	#1849	1	1	yes	characteristic
92	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	C₇×#106	1	8	yes	—
97	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
98	C₂×C₄	C₇×#127	1	1	yes	characteristic
99	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
100	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄	C₇×#131	1	4	yes	—
102	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
103	C₈	—	2	8	no	—
104	C₈	—	8	4	no	—
105	C₇	#8809	1	1	yes	characteristic, direct product factor, Sylow
106	C₂²	C₇×#384	1	1	yes	characteristic
107	C₂²	C₇×#424	1	2	yes	—
108	C₂²	C₂×C₇×#106	1	2	yes	—
109	C₂²	C₇×#430	1	2	yes	—
110	C₄	—	2	8	no	—
111	C₄	—	2	2	no	—
112	C₄	C₇×#424	1	4	yes	—
113	C₄	C₇×#492	1	2	yes	—
114	C₄	C₇×#498	1	2	yes	—
115	C₂	C₇×#1570	1	4	yes	—
116	C₂	C₇×#1843	1	1	yes	characteristic
117	C₂	C₇×#1844	1	1	yes	characteristic
118	C₂	C₇×#1849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1842

C₂

yes

maximal

C₇×#1744

C₂

yes

maximal, characteristic

C₂×C₇×#334

C₂

yes

maximal

C₂×C₇×C₄⋊C₈

—

C₂×C₇×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

—

C₄³×C₇

C₂²

yes

characteristic

C₇×#336

C₄

yes

—

C₇×#334

—

#8809

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₇×C₈

—

C₂²×C₇×C₈

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

Q₈

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

Q₈

yes

—

#1842

C₂×C₇

yes

—

#1744

C₂×C₇

yes

characteristic

C₂×#334

C₂×C₇

yes

—

C₂²×C₄×C₇

C₄⋊C₄

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×Q₈

yes

characteristic

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

SD₁₆

yes

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×Q₈

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₇

yes

—

C₂×C₄×C₈

C₂²×C₇

yes

—

C₄³

C₂²×C₇

yes

characteristic

#336

C₄×C₇

yes

—

#334

—

C₂³×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#106

yes

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₄⋊Q₈

yes

—

C₂×C₄×C₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇

#131

yes

—

C₂×C₄×C₇

C₂×SD₁₆

yes

—

C₇×C₈

—

C₇×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₇

yes

—

C₂×C₄²

C₂×C₄×C₇

yes

—

C₂×C₄²

C₇×D₄

yes

—

C₂×C₄²

C₇×Q₈

yes

—

C₂×C₄²

C₂³×C₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₇×D₄

yes

—

C₄×C₈

C₇×Q₈

yes

—

C₂²×C₇

#384

yes

characteristic

C₂²×C₇

#424

yes

—

C₂²×C₇

C₂×#106

yes

—

C₂²×C₇

#430

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#424

yes

—

C₄×C₇

#492

yes

—

C₄×C₇

#498

yes

—

C₂²×C₄

C₇×C₄⋊C₄

yes

—

C₂²×C₄

C₂²×C₄×C₇

yes

characteristic

C₂²×C₄

C₂×C₇×D₄

yes

characteristic

C₂²×C₄

C₂×C₇×Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₇×C₄⋊C₄

yes

—

C₄²

—

C₄²

—

C₄²

C₇×SD₁₆

yes

—

C₄²

C₂×C₇×D₄

yes

—

C₄²

C₂×C₇×Q₈

yes

—

C₂×C₇

#1570

yes

—

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

C₂³

C₂×C₇×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₇×#106

yes

—

C₂×C₄

C₂×C₇×C₄⋊C₄

yes

—

C₂×C₄

C₇×#127

yes

characteristic

C₂×C₄

C₇×C₄⋊Q₈

yes

—

100

C₂×C₄

C₇×C₄⋊Q₈

yes

characteristic

101

C₂×C₄

C₇×#131

yes

—

102

C₂×C₄

C₂×C₇×SD₁₆

yes

—

103

C₈

—

104

C₈

—

105

C₇

#8809

yes

characteristic, direct product factor, Sylow

106

C₂²

C₇×#384

yes

characteristic

107

C₂²

C₇×#424

yes

—

108

C₂²

C₂×C₇×#106

yes

—

109

C₂²

C₇×#430

yes

—

110

C₄

—

111

C₄

—

112

C₄

C₇×#424

yes

—

113

C₄

C₇×#492

yes

—

114

C₄

C₇×#498

yes

—

115

C₂

C₇×#1570

yes

—

116

C₂

C₇×#1843

yes

characteristic

117

C₂

C₇×#1844

yes

characteristic

118

C₂

C₇×#1849

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle, prime radical	C₂³×C₇	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

subgroup

G/H

center

C₂²×C₄×C₇

C₂×D₄

commutator

C₂²×C₄

C₂²×C₄×C₇

Frattini, cosocle

C₂×C₄²

C₂³×C₇

socle, prime radical

C₂³×C₇

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

N	H ≅ G/N	type
10. #8809	C₇	direct (N × H)
105. C₇	#8809	direct (N × H)

H ≅ G/N

type

10. #8809

C₇

direct (N × H)

105. C₇

#8809

direct (N × H)

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

subgroup	index
C₇×#1842	2
C₇×#1744	2
C₂×C₇×#334	2
#8809	7

subgroup

index

C₇×#1842

C₇×#1744

C₂×C₇×#334

#8809

Subgroups — C₇×#8809 #13186025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#8809 #13186025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8809 #13186025