Group #13268544 — subgroups

Subgroups — #13268544

Order 1792.

246 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 99 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	#11726800	C₂	1	2	yes	maximal
2	#11726798	C₂	1	2	yes	maximal
3	#11725819	C₂	1	1	yes	maximal, characteristic
4	#11725818	C₂	1	1	yes	maximal, characteristic
5	C₇×#1311	C₂	1	1	yes	maximal, characteristic
6	#89471	—	2	1	no	—
7	#89454	—	2	1	no	—
8	#89454	C₂²	1	1	yes	characteristic
9	#89454	C₂²	1	2	yes	—
10	#89454	C₂²	1	1	yes	characteristic
11	#89453	—	2	1	no	—
12	#89453	—	2	1	no	—
13	C₄×C₇×C₁₆	C₂²	1	2	yes	—
14	C₇×C₈²	C₂²	1	1	yes	characteristic
15	#89300	—	2	4	no	—
16	#89299	—	2	4	no	—
17	#12524	—	7	1	no	maximal, Sylow
18	C₂×Dic₂₈	—	4	2	no	—
19	#6432	—	2	1	no	—
20	#6432	—	2	1	no	—
21	#6432	—	2	2	no	—
22	C₂×C₇×C₁₆	D₄	1	4	yes	—
23	C₄×C₇×C₈	D₄	1	1	yes	characteristic
24	C₄×C₇×C₈	D₄	1	1	yes	characteristic
25	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
26	#6392	—	2	2	no	—
27	#6392	—	2	2	no	—
28	#6392	—	2	2	no	—
29	#6392	—	2	2	no	—
30	#6392	—	4	2	no	—
31	#6390	—	4	2	no	—
32	#6390	—	4	2	no	—
33	#2254	—	7	2	no	—
34	#2252	—	7	2	no	—
35	#1713	—	7	1	no	—
36	#1710	—	7	1	no	—
37	#1311	D₇	1	1	yes	characteristic
38	C₂×Dic₁₄	—	4	2	no	—
39	C₂×Dic₁₄	—	4	1	no	—
40	C₂×Dic₁₄	—	4	1	no	—
41	C₂×C₇×C₈	D₈	1	2	yes	—
42	C₂×C₇×C₈	D₈	1	2	yes	—
43	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
44	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₇	—	4	1	no	—
46	C₄⋊Dic₇	—	4	1	no	—
47	C₄⋊Dic₇	—	4	2	no	—
48	C₄⋊Dic₇	—	4	1	no	—
49	C₄⋊Dic₇	—	4	1	no	—
50	C₄⋊Dic₇	—	4	2	no	—
51	Dic₂₈	—	8	4	no	—
52	C₇×C₁₆	Q₁₆	1	8	yes	complemented
53	#500	—	7	1	no	—
54	#500	—	7	1	no	—
55	#500	—	7	2	no	—
56	#500	—	14	1	no	—
57	#494	—	14	1	no	—
58	#487	—	14	1	no	—
59	#487	—	14	1	no	—
60	#367	—	14	4	no	—
61	#366	—	14	4	no	—
62	C₄×C₁₆	C₂×D₇	1	2	yes	—
63	C₈²	C₂×D₇	1	1	yes	characteristic
64	C₂×C₄×C₇	#127	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
67	C₂×Dic₇	—	8	1	no	—
68	C₂×Dic₇	—	8	2	no	—
69	C₂×Dic₇	—	8	1	no	—
70	Dic₁₄	—	8	4	no	—
71	Dic₁₄	—	8	2	no	—
72	Dic₁₄	—	8	2	no	—
73	C₇×C₈	—	2	2	no	—
74	C₇×C₈	—	2	2	no	—
75	C₇×C₈	C₂×Q₁₆	1	4	yes	—
76	C₂×Q₁₆	—	28	2	no	—
77	C₄⋊Q₈	—	14	1	no	—
78	C₄⋊Q₈	—	14	1	no	—
79	C₄⋊Q₈	—	14	2	no	—
80	C₂×C₁₆	D₂₈	1	4	yes	—
81	#107	—	14	2	no	—
82	#107	—	14	2	no	—
83	#107	—	14	2	no	—
84	#107	—	14	2	no	—
85	#107	—	28	2	no	—
86	#103	—	28	2	no	—
87	#103	—	28	2	no	—
88	C₄×C₈	D₂₈	1	1	yes	characteristic
89	C₄×C₈	D₂₈	1	1	yes	characteristic
90	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
91	C₂²×C₇	#493	1	1	yes	characteristic
92	C₄×C₇	#494	1	2	yes	—
93	C₄×C₇	#509	1	2	yes	—
94	C₄×C₇	#510	1	2	yes	—
95	Dic₇	—	16	2	no	—
96	Dic₇	—	16	1	no	—
97	Dic₇	—	16	1	no	—
98	C₂×Q₈	—	28	1	no	—
99	C₂×Q₈	—	28	1	no	—
100	C₂×Q₈	—	28	2	no	—
101	Q₁₆	—	56	4	no	—
102	C₂×C₈	D₅₆	1	2	yes	—
103	C₂×C₈	D₅₆	1	2	yes	—
104	C₂×C₈	C₂×D₂₈	1	2	yes	—
105	C₄⋊C₄	—	28	1	no	—
106	C₄⋊C₄	—	28	2	no	—
107	C₄⋊C₄	—	28	1	no	—
108	C₄⋊C₄	—	28	1	no	—
109	C₄⋊C₄	—	28	1	no	—
110	C₄⋊C₄	—	28	2	no	—
111	C₄²	C₂×D₂₈	1	1	yes	characteristic
112	C₁₆	Dic₂₈	1	8	yes	complemented
113	C₂×C₇	#1713	1	1	yes	characteristic
114	C₂×C₇	#2250	1	1	yes	characteristic
115	C₂×C₇	#2251	1	1	yes	characteristic
116	Q₈	—	56	4	no	—
117	Q₈	—	56	2	no	—
118	Q₈	—	56	2	no	—
119	C₂×C₄	—	56	1	no	—
120	C₂×C₄	—	56	2	no	—
121	C₂×C₄	—	56	1	no	—
122	C₂×C₄	#6437	1	1	yes	characteristic
123	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
124	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	2	no	—
127	C₈	C₂×Dic₂₈	1	4	yes	—
128	C₇	#12524	1	1	yes	characteristic, complemented, Sylow
129	C₂²	#89467	1	1	yes	characteristic
130	C₄	—	112	2	no	—
131	C₄	—	112	1	no	—
132	C₄	—	112	1	no	—
133	C₄	#89471	1	2	yes	—
134	C₄	#89680	1	2	yes	—
135	C₄	#89681	1	2	yes	—
136	C₂	#11725819	1	1	yes	characteristic
137	C₂	#11726835	1	1	yes	characteristic
138	C₂	#11726838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11726800

C₂

yes

maximal

#11726798

C₂

yes

maximal

#11725819

C₂

yes

maximal, characteristic

#11725818

C₂

yes

maximal, characteristic

C₇×#1311

C₂

yes

maximal, characteristic

#89471

—

#89454

—

#89454

C₂²

yes

characteristic

#89454

C₂²

yes

—

#89454

C₂²

yes

characteristic

#89453

—

#89453

—

C₄×C₇×C₁₆

C₂²

yes

—

C₇×C₈²

C₂²

yes

—

—

—

C₂×Dic₂₈

—

#6432

—

#6432

—

#6432

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

—

—

—

—

—

—

—

—

—

—

—

D₇

yes

characteristic

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₂₈

—

C₇×C₁₆

Q₁₆

yes

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₇

yes

—

C₈²

C₂×D₇

yes

characteristic

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

C₂×Q₁₆

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₂₈

yes

—

—

—

—

—

—

—

—

C₄×C₈

D₂₈

yes

characteristic

C₄×C₈

D₂₈

yes

characteristic

C₄×C₈

C₂²×D₇

yes

C₂²×C₇

yes

C₄×C₇

yes

—

C₄×C₇

#509

yes

—

C₄×C₇

#510

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

Q₁₆

—

102

C₂×C₈

D₅₆

yes

—

103

C₂×C₈

D₅₆

yes

—

104

C₂×C₈

C₂×D₂₈

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

C₂×D₂₈

yes

characteristic

112

C₁₆

Dic₂₈

yes

complemented

113

C₂×C₇

#1713

yes

characteristic

114

C₂×C₇

#2250

yes

characteristic

115

C₂×C₇

#2251

yes

characteristic

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#6437

yes

characteristic

123

C₂×C₄

C₂×D₅₆

yes

characteristic

124

C₂×C₄

C₂×D₅₆

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

C₂×Dic₂₈

yes

—

128

C₇

#12524

yes

characteristic, complemented, Sylow

129

C₂²

#89467

yes

characteristic

130

C₄

—

112

—

131

C₄

—

112

—

132

C₄

—

112

—

133

C₄

#89471

yes

—

134

C₄

#89680

yes

—

135

C₄

#89681

yes

—

136

C₂

#11725819

yes

characteristic

137

C₂

#11726835

yes

characteristic

138

C₂

#11726838

yes

characteristic

	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	#12524
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
52. C₇×C₁₆	Q₁₆	semidirect
112. C₁₆	Dic₂₈	semidirect
128. C₇	#12524	semidirect

H ≅ G/N

type

52. C₇×C₁₆

Q₁₆

semidirect

112. C₁₆

Dic₂₈

semidirect

128. C₇

#12524

semidirect

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

subgroup	index
#11726800	2
#11726798	2
#11725819	2
#11725818	2
C₇×#1311	2
#12524	7

subgroup

index

Subgroups — #13268544

Conjugacy classes of subgroups

Complements

Maximal subgroups