Group #13268471 — subgroups

Subgroups — #13268471

Order 1792.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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	#11726820	C₂	1	1	yes	maximal, characteristic
2	#11726819	C₂	1	2	yes	maximal
3	#11726800	C₂	1	1	yes	maximal, characteristic
4	#11725817	C₂	1	2	yes	maximal
5	C₇×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆	—	2	2	no	—
7	#89471	C₂²	1	2	yes	—
8	#89454	C₂²	1	2	yes	—
9	#89453	—	2	2	no	—
10	#89452	—	2	2	no	—
11	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₇×C₈²	C₂²	1	1	yes	characteristic
14	#89300	—	2	2	no	—
15	#89298	—	2	4	no	—
16	#12507	—	7	1	no	maximal, Sylow
17	C₂×Dic₂₈	—	2	2	no	—
18	C₂×Dic₂₈	—	2	2	no	—
19	#6432	—	2	2	no	—
20	#6432	—	2	2	no	—
21	C₂×C₇×C₁₆	D₄	1	2	yes	—
22	C₂×C₇×C₁₆	D₄	1	2	yes	—
23	C₄×C₇×C₈	D₄	1	2	yes	—
24	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
25	#6392	—	2	2	no	—
26	#6392	—	2	2	no	—
27	#6391	—	4	4	no	—
28	#6390	—	4	4	no	—
29	Dic₅₆	—	4	4	no	—
30	#2254	—	7	1	no	—
31	#2247	—	7	1	no	—
32	#2241	—	7	2	no	—
33	#1707	—	7	2	no	—
34	#1311	D₇	1	1	yes	characteristic
35	C₂×Dic₁₄	—	4	2	no	—
36	C₂×Dic₁₄	—	4	2	no	—
37	C₂×C₇×C₈	D₈	1	4	yes	—
38	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
39	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₇	—	4	2	no	—
42	C₄⋊Dic₇	—	4	2	no	—
43	C₄⋊Dic₇	—	4	2	no	—
44	C₄⋊Dic₇	—	4	2	no	—
45	Dic₂₈	—	4	4	no	—
46	Dic₂₈	—	4	4	no	—
47	C₇×C₁₆	—	2	2	no	—
48	C₇×C₁₆	SD₁₆	1	4	yes	—
49	C₂×Q₃₂	—	14	2	no	—
50	#500	—	7	2	no	—
51	#498	—	14	2	no	—
52	#494	—	7	2	no	—
53	#487	—	14	2	no	—
54	#367	—	14	2	no	—
55	#358	—	14	4	no	—
56	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
57	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
58	C₈²	C₂×D₇	1	1	yes	characteristic
59	C₂×C₄×C₇	#127	1	1	yes	characteristic
60	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
61	C₂×Dic₇	—	8	2	no	—
62	C₂×Dic₇	—	8	2	no	—
63	Dic₁₄	—	8	2	no	—
64	Dic₁₄	—	8	2	no	—
65	Dic₁₄	—	8	4	no	—
66	C₇×C₈	—	2	4	no	—
67	C₇×C₈	C₂×SD₁₆	1	2	yes	—
68	C₇×C₈	#135	1	2	yes	—
69	C₂×Q₁₆	—	14	2	no	—
70	C₂×Q₁₆	—	14	2	no	—
71	C₄⋊Q₈	—	14	2	no	—
72	C₄⋊Q₈	—	14	2	no	—
73	Q₃₂	—	28	4	no	—
74	C₂×C₁₆	D₂₈	1	2	yes	—
75	C₂×C₁₆	D₂₈	1	2	yes	—
76	#107	—	14	2	no	—
77	#107	—	14	2	no	—
78	#106	—	28	4	no	—
79	#103	—	28	4	no	—
80	C₄×C₈	D₂₈	1	2	yes	—
81	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
82	C₂²×C₇	#493	1	1	yes	characteristic
83	C₄×C₇	#492	1	1	yes	characteristic
84	C₄×C₇	#495	1	1	yes	characteristic
85	C₄×C₇	#510	1	4	yes	—
86	Dic₇	—	16	2	no	—
87	Dic₇	—	16	2	no	—
88	C₂×Q₈	—	28	2	no	—
89	C₂×Q₈	—	28	2	no	—
90	Q₁₆	—	28	4	no	—
91	Q₁₆	—	28	4	no	—
92	C₂×C₈	D₅₆	1	4	yes	—
93	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
94	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
95	C₄⋊C₄	—	28	2	no	—
96	C₄⋊C₄	—	28	2	no	—
97	C₄⋊C₄	—	28	2	no	—
98	C₄⋊C₄	—	28	2	no	—
99	C₄²	C₂×D₂₈	1	1	yes	characteristic
100	C₁₆	—	2	2	no	—
101	C₁₆	#1132	1	4	yes	—
102	C₂×C₇	#1706	1	1	yes	characteristic
103	C₂×C₇	#2251	1	2	yes	—
104	Q₈	—	56	4	no	—
105	Q₈	—	56	2	no	—
106	Q₈	—	56	2	no	—
107	C₂×C₄	—	56	2	no	—
108	C₂×C₄	—	56	2	no	—
109	C₂×C₄	#6437	1	1	yes	characteristic
110	C₂×C₄	C₂×D₅₆	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	C₂×#1132	1	2	yes	—
113	C₈	#6467	1	2	yes	—
114	C₇	#12507	1	1	yes	characteristic, complemented, Sylow
115	C₂²	#89467	1	1	yes	characteristic
116	C₄	—	112	2	no	—
117	C₄	—	112	2	no	—
118	C₄	#89465	1	1	yes	characteristic
119	C₄	#89468	1	1	yes	characteristic
120	C₄	#89681	1	4	yes	—
121	C₂	#11725812	1	1	yes	characteristic
122	C₂	#11726838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11726820

C₂

yes

maximal, characteristic

#11726819

C₂

yes

maximal

#11726800

C₂

yes

maximal, characteristic

#11725817

C₂

yes

maximal

C₇×#1311

C₂

yes

maximal, characteristic

C₂×Dic₅₆

—

#89471

C₂²

yes

—

#89454

C₂²

yes

—

#89453

—

#89452

—

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₇×C₈²

C₂²

yes

—

—

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

#6432

—

#6432

—

C₂×C₇×C₁₆

D₄

yes

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

—

—

—

—

Dic₅₆

—

—

—

—

—

D₇

yes

characteristic

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₂₈

—

Dic₂₈

—

C₇×C₁₆

—

C₇×C₁₆

SD₁₆

yes

—

C₂×Q₃₂

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₇

yes

characteristic

C₄×C₁₆

C₂×D₇

yes

characteristic

C₈²

C₂×D₇

yes

characteristic

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

C₂×SD₁₆

yes

—

C₇×C₈

#135

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₂₈

yes

—

C₂×C₁₆

D₂₈

yes

—

—

—

—

—

C₄×C₈

D₂₈

yes

—

C₄×C₈

C₂²×D₇

yes

C₂²×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₅₆

yes

—

C₂×C₈

C₂×D₂₈

yes

characteristic

C₂×C₈

C₂×D₂₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₂₈

yes

characteristic

100

C₁₆

—

101

C₁₆

#1132

yes

—

102

C₂×C₇

#1706

yes

characteristic

103

C₂×C₇

#2251

yes

—

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#6437

yes

characteristic

110

C₂×C₄

C₂×D₅₆

yes

—

111

C₈

—

112

C₈

C₂×#1132

yes

—

113

C₈

#6467

yes

—

114

C₇

#12507

yes

characteristic, complemented, Sylow

115

C₂²

#89467

yes

characteristic

116

C₄

—

112

—

117

C₄

—

112

—

118

C₄

#89465

yes

characteristic

119

C₄

#89468

yes

characteristic

120

C₄

#89681

yes

—

121

C₂

#11725812

yes

characteristic

122

C₂

#11726838

yes

—

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1311	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#1311	D₇
hypocenter	C₇	#12507
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#89467

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
114. C₇	#12507	semidirect

H ≅ G/N

type

114. C₇

#12507

semidirect

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

subgroup	index
#11726820	2
#11726819	2
#11726800	2
#11725817	2
C₇×#1311	2
#12507	7

subgroup

index

Subgroups — #13268471

Conjugacy classes of subgroups

Complements

Maximal subgroups