Group #13187828 — subgroups

Subgroups — C₇×#10612 #13187828

Order 1792.

328 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 98 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×#1574	C₂	1	3	yes	maximal, complemented
2	C₇×#1545	C₂	1	3	yes	maximal
3	C₇×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#127	C₂²	1	1	yes	characteristic
5	C₇×#431	C₂²	1	3	yes	—
6	C₇×#423	C₂²	1	3	yes	—
7	C₇×#335	—	2	3	no	—
8	C₇×#334	—	2	3	no	—
9	C₇×#331	—	2	3	no	—
10	C₇×#331	—	2	3	no	—
11	#10612	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₇×D₄	—	2	3	no	—
13	C₂×C₇×M₁₆	—	2	3	no	—
14	C₇×#127	—	2	3	no	—
15	C₇×#127	Q₈	1	2	yes	—
16	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
17	C₇×C₄⋊C₈	—	2	3	no	—
18	C₇×C₄⋊C₈	—	2	3	no	—
19	C₇×C₄⋊C₈	D₄	1	3	yes	—
20	C₇×C₄⋊C₈	D₄	1	3	yes	—
21	C₇×C₂²⋊C₈	—	2	3	no	—
22	C₄×C₇×C₈	—	2	3	no	—
23	#1574	C₂×C₇	1	3	yes	complemented
24	#1545	C₂×C₇	1	3	yes	—
25	#1309	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂⁴×C₇	—	4	1	no	—
27	C₂×C₇×D₄	—	2	6	no	—
28	C₂×C₇×D₄	—	4	3	no	—
29	C₂×C₇×D₄	—	8	3	no	—
30	C₂×C₇×D₄	—	8	3	no	—
31	C₂²×C₄×C₇	#41	1	3	yes	—
32	C₇×M₁₆	—	8	3	no	—
33	C₂×C₇×C₈	—	2	3	no	—
34	C₂×C₇×C₈	—	2	3	no	—
35	C₂×C₇×C₈	—	4	3	no	—
36	C₄²×C₇	C₂×D₄	1	3	yes	—
37	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×#127	C₂²×C₇	1	1	yes	characteristic
39	#431	C₂²×C₇	1	3	yes	—
40	#423	C₂²×C₇	1	3	yes	—
41	#335	—	2	3	no	—
42	#334	—	2	3	no	—
43	#331	—	2	3	no	—
44	#331	—	2	3	no	—
45	C₂³×C₇	—	4	2	no	—
46	C₂³×C₇	—	8	6	no	—
47	C₂³×C₇	#126	1	1	yes	characteristic
48	C₇×D₄	—	8	6	no	—
49	C₇×D₄	—	8	3	no	—
50	C₇×D₄	—	8	3	no	—
51	C₂×C₄×C₇	—	2	3	no	—
52	C₂×C₄×C₇	—	2	3	no	—
53	C₂×C₄×C₇	#121	1	3	yes	—
54	C₂×C₄×C₇	C₂²⋊Q₈	1	3	yes	—
55	C₇×C₈	—	4	3	no	—
56	C₇×C₈	—	4	3	no	—
57	C₇×C₈	—	8	3	no	—
58	C₂²×D₄	—	2	3	no	—
59	C₂×M₁₆	—	2	3	no	—
60	#127	—	2	3	no	—
61	#127	C₇×Q₈	1	2	yes	—
62	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
63	C₄⋊C₈	—	2	3	no	—
64	C₄⋊C₈	—	2	3	no	—
65	C₄⋊C₈	C₇×D₄	1	3	yes	—
66	C₄⋊C₈	C₇×D₄	1	3	yes	—
67	C₂²⋊C₈	—	2	3	no	—
68	C₄×C₈	—	2	3	no	—
69	C₂²×C₇	—	2	3	no	—
70	C₂²×C₇	—	8	6	no	—
71	C₂²×C₇	—	16	4	no	—
72	C₂²×C₇	#396	1	1	yes	characteristic
73	C₄×C₇	—	2	3	no	—
74	C₄×C₇	#471	1	3	yes	—
75	C₄×C₇	#472	1	3	yes	—
76	C₂⁴	—	4	1	no	—
77	C₂×D₄	—	2	6	no	—
78	C₂×D₄	—	4	3	no	—
79	C₂×D₄	—	8	3	no	—
80	C₂×D₄	—	8	3	no	—
81	C₂²×C₄	C₇×#41	1	3	yes	—
82	M₁₆	—	8	3	no	—
83	C₂×C₈	—	2	3	no	—
84	C₂×C₈	—	2	3	no	—
85	C₂×C₈	—	4	3	no	—
86	C₄²	C₂×C₇×D₄	1	3	yes	—
87	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
88	C₂×C₇	—	4	1	no	—
89	C₂×C₇	—	16	2	no	—
90	C₂×C₇	#2063	1	3	yes	—
91	C₂³	—	4	2	no	—
92	C₂³	—	8	6	no	—
93	C₂³	C₇×#126	1	1	yes	characteristic
94	D₄	—	8	6	no	—
95	D₄	—	8	3	no	—
96	D₄	—	8	3	no	—
97	C₂×C₄	—	2	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	C₇×#121	1	3	yes	—
100	C₂×C₄	C₇×C₂²⋊Q₈	1	3	yes	—
101	C₈	—	4	3	no	—
102	C₈	—	4	3	no	—
103	C₈	—	8	3	no	—
104	C₇	#10612	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²	—	2	3	no	—
106	C₂²	—	8	6	no	—
107	C₂²	—	16	4	no	—
108	C₂²	C₇×#396	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	C₇×#471	1	3	yes	—
111	C₄	C₇×#472	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	16	2	no	—
114	C₂	C₇×#2063	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1574

C₂

yes

maximal, complemented

C₇×#1545

C₂

yes

maximal

C₇×#1309

C₂

yes

maximal, characteristic, complemented

C₂×C₇×#127

C₂²

yes

characteristic

C₇×#431

C₂²

yes

—

C₇×#423

C₂²

yes

—

—

—

—

—

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₇×D₄

—

C₂×C₇×M₁₆

—

C₇×#127

—

C₇×#127

Q₈

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

D₄

yes

—

C₇×C₄⋊C₈

D₄

yes

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

—

#1574

C₂×C₇

yes

complemented

#1545

C₂×C₇

yes

—

#1309

C₂×C₇

yes

characteristic, complemented

C₂⁴×C₇

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂²×C₄×C₇

#41

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×Q₈

yes

characteristic

C₂×#127

C₂²×C₇

yes

characteristic

#431

C₂²×C₇

yes

—

#423

C₂²×C₇

yes

—

—

—

—

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

#126

yes

characteristic

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#121

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₂²×D₄

—

C₂×M₁₆

—

#127

—

#127

C₇×Q₈

yes

—

C₂×C₄²

C₂³×C₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₇×D₄

yes

—

C₄⋊C₈

C₇×D₄

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#396

yes

characteristic

C₄×C₇

—

C₄×C₇

#471

yes

—

C₄×C₇

#472

yes

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₇×#41

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂×C₇×D₄

yes

—

C₄²

C₂×C₇×Q₈

yes

characteristic

C₂×C₇

—

C₂×C₇

—

C₂×C₇

#2063

yes

—

C₂³

—

C₂³

—

C₂³

C₇×#126

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₇×#121

yes

—

100

C₂×C₄

C₇×C₂²⋊Q₈

yes

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₇

#10612

yes

characteristic, direct product factor, Sylow

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₇×#396

yes

characteristic

109

C₄

—

110

C₄

C₇×#471

yes

—

111

C₄

C₇×#472

yes

—

112

C₂

—

113

C₂

—

114

C₂

C₇×#2063

yes

—

	subgroup	G/H
center, socle	C₂²×C₇	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂×C₇×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₇

#396

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇

prime radical

C₂×C₇×#127

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
1. C₇×#1574	C₂	semidirect
3. C₇×#1309	C₂	semidirect
11. #10612	C₇	direct (N × H)
23. #1574	C₂×C₇	semidirect
25. #1309	C₂×C₇	semidirect
104. C₇	#10612	direct (N × H)

H ≅ G/N

type

1. C₇×#1574

C₂

semidirect

3. C₇×#1309

C₂

semidirect

11. #10612

C₇

direct (N × H)

23. #1574

C₂×C₇

semidirect

25. #1309

C₂×C₇

semidirect

104. C₇

#10612

direct (N × H)

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

subgroup	index
C₇×#1574	2
C₇×#1545	2
C₇×#1309	2
#10612	7

subgroup

index

Subgroups — C₇×#10612 #13187828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#10612 #13187828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10612 #13187828