Group #13274308 — subgroups

Subgroups — #13274308

Order 1792.

178 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂×#89242	C₂	1	1	yes	maximal, characteristic
2	D₇×#368	C₂	1	2	yes	maximal
3	C₇×#1425	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725399	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725387	C₂	1	2	yes	maximal, complemented
6	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
7	D₇×#108	—	2	2	no	—
8	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
9	C₇×#368	C₂²	1	2	yes	—
10	#89301	C₂²	1	2	yes	—
11	C₂×C₇⋊C₃₂	C₂²	1	1	yes	characteristic
12	#89288	—	2	2	no	—
13	#89242	C₄	1	4	yes	—
14	#13636	—	7	1	no	maximal, Sylow
15	D₇×M₁₆	—	4	2	no	—
16	C₂×C₈×D₇	D₄	1	1	yes	characteristic
17	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
18	C₇×#108	—	2	2	no	—
19	#6396	—	4	2	no	—
20	#6393	—	2	2	no	—
21	C₂×C₇⋊C₁₆	D₄	1	1	yes	characteristic
22	D₇×C₁₆	C₂×C₄	1	2	yes	—
23	D₇×C₁₆	Q₈	1	2	yes	—
24	C₇×C₃₂	C₂×C₄	1	2	yes	—
25	C₇⋊C₃₂	C₂×C₄	1	2	yes	—
26	C₂×M₆₄	—	7	1	no	—
27	C₂×#368	—	7	2	no	—
28	#1425	—	7	1	no	—
29	#1425	—	7	2	no	—
30	#1425	D₇	1	1	yes	characteristic, complemented
31	C₂×C₄×D₇	D₈	1	1	yes	characteristic
32	C₇×M₁₆	—	4	2	no	—
33	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
34	#1136	—	4	2	no	—
35	C₂×C₇⋊C₈	D₈	1	1	yes	characteristic
36	#1131	—	8	2	no	—
37	C₈×D₇	—	8	2	no	—
38	C₈×D₇	C₄⋊C₄	1	2	yes	—
39	C₈×D₇	Q₁₆	1	2	yes	—
40	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
41	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
42	C₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
43	C₂²×C₁₆	—	7	1	no	—
44	C₂×#108	—	14	2	no	—
45	M₆₄	—	7	4	no	—
46	C₂×C₃₂	—	7	1	no	—
47	C₂×C₃₂	C₂×D₇	1	1	yes	characteristic
48	#368	—	7	2	no	—
49	#368	—	14	2	no	—
50	#368	C₂×D₇	1	2	yes	—
51	C₂²×D₇	Q₃₂	1	1	yes	characteristic
52	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₇	Q₃₂	1	1	yes	characteristic
54	C₄×D₇	#107	1	2	yes	—
55	C₄×D₇	D₁₆	1	2	yes	—
56	C₇×C₈	—	8	2	no	—
57	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
59	C₇⋊C₈	—	8	2	no	—
60	C₇⋊C₈	#107	1	2	yes	—
61	C₂×M₁₆	—	28	2	no	—
62	C₂²×C₈	—	7	1	no	—
63	C₂×C₁₆	—	7	1	no	—
64	C₂×C₁₆	—	7	2	no	—
65	C₂×C₁₆	—	7	2	no	—
66	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
67	#108	—	2	2	no	—
68	#108	—	14	2	no	—
69	#108	—	28	2	no	—
70	C₃₂	—	7	2	no	—
71	C₃₂	C₄×D₇	1	2	yes	—
72	C₂²×C₇	C₂×Q₃₂	1	1	yes	characteristic
73	C₂×D₇	—	4	1	no	—
74	C₂×D₇	#366	1	2	yes	—
75	C₄×C₇	C₂×#107	1	1	yes	characteristic
76	C₄×C₇	C₂×D₁₆	1	1	yes	characteristic
77	Dic₇	#366	1	2	yes	—
78	C₂²×C₄	—	7	1	no	—
79	M₁₆	—	4	2	no	—
80	M₁₆	—	28	2	no	—
81	M₁₆	—	56	2	no	—
82	C₂×C₈	—	7	1	no	—
83	C₂×C₈	—	7	2	no	—
84	C₂×C₈	—	7	2	no	—
85	C₂×C₈	—	56	2	no	—
86	C₂×C₈	D₄×D₇	1	1	yes	characteristic
87	C₁₆	—	7	2	no	—
88	C₁₆	C₂×C₄×D₇	1	1	yes	characteristic
89	C₁₆	Q₈×D₇	1	1	yes	characteristic
90	C₂×C₇	—	2	1	no	—
91	C₂×C₇	C₂×#366	1	1	yes	characteristic
92	D₇	—	2	2	no	—
93	C₂³	—	7	1	no	—
94	C₂×C₄	—	7	1	no	—
95	C₂×C₄	—	7	2	no	—
96	C₂×C₄	—	7	2	no	—
97	C₂×C₄	D₇×D₈	1	1	yes	characteristic
98	C₈	—	7	2	no	—
99	C₈	—	8	2	no	—
100	C₈	—	56	2	no	—
101	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
102	C₈	D₇×Q₁₆	1	1	yes	characteristic
103	C₇	#13636	1	1	yes	characteristic, complemented, Sylow
104	C₂²	—	7	2	no	—
105	C₂²	—	28	1	no	—
106	C₂²	D₇×Q₃₂	1	1	yes	characteristic
107	C₄	—	7	2	no	—
108	C₄	D₇×#107	1	1	yes	characteristic
109	C₄	D₇×D₁₆	1	1	yes	characteristic
110	C₂	—	2	1	no	—
111	C₂	—	14	2	no	—
112	C₂	D₇×#366	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#89242

C₂

yes

maximal, characteristic

D₇×#368

C₂

yes

maximal

C₇×#1425

C₂

yes

maximal, characteristic, complemented

#11725399

C₂

yes

maximal, characteristic, complemented

#11725387

C₂

yes

maximal, complemented

C₂×D₇×C₁₆

C₂²

yes

characteristic

D₇×#108

—

C₂×C₇×C₃₂

C₂²

yes

characteristic

C₇×#368

C₂²

yes

—

#89301

C₂²

yes

—

C₂×C₇⋊C₃₂

C₂²

yes

characteristic

#89288

—

#89242

C₄

yes

—

#13636

—

maximal, Sylow

D₇×M₁₆

—

C₂×C₈×D₇

D₄

yes

characteristic

C₂×C₇×C₁₆

C₂³

yes

—

—

—

C₂×C₇⋊C₁₆

D₄

yes

characteristic

D₇×C₁₆

C₂×C₄

yes

—

D₇×C₁₆

Q₈

yes

—

C₇×C₃₂

C₂×C₄

yes

—

C₇⋊C₃₂

C₂×C₄

yes

—

C₂×M₆₄

—

—

—

—

D₇

yes

characteristic, complemented

C₂×C₄×D₇

D₈

yes

characteristic

C₇×M₁₆

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

#1136

—

C₂×C₇⋊C₈

D₈

yes

characteristic

#1131

—

C₈×D₇

—

C₈×D₇

C₄⋊C₄

yes

—

C₈×D₇

Q₁₆

yes

—

C₇×C₁₆

C₂²×C₄

yes

characteristic

C₇×C₁₆

C₂×Q₈

yes

characteristic

C₇⋊C₁₆

C₄⋊C₄

yes

—

C₂²×C₁₆

—

C₂×#108

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇

yes

—

—

C₂×D₇

yes

—

C₂²×D₇

Q₃₂

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×Dic₇

Q₃₂

yes

characteristic

C₄×D₇

#107

yes

—

C₄×D₇

D₁₆

yes

—

C₇×C₈

—

C₇×C₈

C₂×C₄⋊C₄

yes

characteristic

C₇×C₈

C₂×Q₁₆

yes

characteristic

C₇⋊C₈

—

C₇⋊C₈

#107

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₇

yes

—

—

—

C₃₂

—

C₃₂

C₄×D₇

yes

—

C₂²×C₇

C₂×Q₃₂

yes

characteristic

C₂×D₇

—

C₂×D₇

#366

yes

—

C₄×C₇

C₂×#107

yes

characteristic

C₄×C₇

C₂×D₁₆

yes

characteristic

Dic₇

#366

yes

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₇

yes

characteristic

C₁₆

—

C₁₆

C₂×C₄×D₇

yes

characteristic

C₁₆

Q₈×D₇

yes

characteristic

C₂×C₇

—

C₂×C₇

C₂×#366

yes

characteristic

D₇

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₇×D₈

yes

characteristic

C₈

—

C₈

—

100

C₈

—

101

C₈

D₇×C₄⋊C₄

yes

characteristic

102

C₈

D₇×Q₁₆

yes

characteristic

103

C₇

#13636

yes

characteristic, complemented, Sylow

104

C₂²

—

105

C₂²

—

106

C₂²

D₇×Q₃₂

yes

characteristic

107

C₄

—

108

C₄

D₇×#107

yes

characteristic

109

C₄

D₇×D₁₆

yes

characteristic

110

C₂

—

111

C₂

—

112

C₂

D₇×#366

yes

characteristic

	subgroup	G/H
center	C₄	D₇×D₁₆
commutator	C₇×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇
Fitting	C₇×#1425	C₂
socle	C₂×C₇	C₂×#366
prime radical	C₂²×D₇	Q₃₂
cosocle	C₂×C₇×C₁₆	C₂³
hypercenter	#1425	D₇
hypocenter	C₇	#13636
LCS	G ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

subgroup

G/H

center

C₄

D₇×D₁₆

commutator

C₇×C₁₆

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇

C₂

C₂×C₇

C₂²×D₇

Q₃₂

cosocle

C₂×C₇×C₁₆

C₂³

D₇

C₇

G ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

N	H ≅ G/N	type
3. C₇×#1425	C₂	semidirect
4. #11725399	C₂	semidirect
5. #11725387	C₂	semidirect
30. #1425	D₇	semidirect
103. C₇	#13636	semidirect

H ≅ G/N

type

3. C₇×#1425

C₂

semidirect

4. #11725399

C₂

semidirect

5. #11725387

C₂

semidirect

30. #1425

D₇

semidirect

103. C₇

#13636

semidirect

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

subgroup	index
C₂×#89242	2
D₇×#368	2
C₇×#1425	2
#11725399	2
#11725387	2
#13636	7

subgroup

index

Subgroups — #13274308

Conjugacy classes of subgroups

Complements

Maximal subgroups