Group #11725844 — subgroups

Subgroups — #11725844

Order 896.

198 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 69 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	#89466	C₂	1	1	yes	maximal, characteristic
2	#89465	C₂	1	1	yes	maximal, characteristic
3	#89465	C₂	1	1	yes	maximal, characteristic
4	#89454	C₂	1	1	yes	maximal, characteristic, complemented
5	#89453	C₂	1	1	yes	maximal, characteristic, complemented
6	#89452	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#1132	—	2	2	no	—
9	C₂×#1132	—	2	2	no	—
10	#6437	C₂²	1	1	yes	characteristic
11	#6432	C₂²	1	1	yes	characteristic
12	#6432	C₂²	1	1	yes	characteristic
13	#6432	C₂²	1	1	yes	characteristic
14	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	#6395	—	2	2	no	—
18	#6392	—	2	2	no	—
19	#6391	—	2	2	no	—
20	#6390	—	2	2	no	—
21	#1714	—	7	1	no	maximal, Sylow
22	C₂×D₂₈	—	2	1	no	—
23	C₂×D₂₈	—	2	1	no	—
24	C₂×D₂₈	—	2	1	no	—
25	C₂×Dic₁₄	—	2	1	no	—
26	C₂×Dic₁₄	—	2	1	no	—
27	C₂×Dic₁₄	—	2	1	no	—
28	C₂×C₇×C₈	D₄	1	2	yes	—
29	C₂×C₇×C₈	D₄	1	2	yes	—
30	C₂×C₇×C₈	D₄	1	2	yes	—
31	C₄²×C₇	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₇	—	2	1	no	—
33	C₄⋊Dic₇	—	2	1	no	—
34	C₄⋊Dic₇	—	2	1	no	—
35	C₄⋊Dic₇	—	2	1	no	—
36	C₄⋊Dic₇	—	2	1	no	—
37	C₄⋊Dic₇	—	2	1	no	—
38	#1132	—	4	4	no	—
39	#1132	—	4	4	no	—
40	#500	—	7	1	no	—
41	#498	—	7	1	no	—
42	#492	—	7	1	no	—
43	#492	—	7	1	no	—
44	#487	—	7	1	no	—
45	#486	—	7	1	no	—
46	#322	D₇	1	1	yes	characteristic, complemented
47	C₂²×D₇	—	4	1	no	—
48	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₇	—	4	1	no	—
52	C₂×Dic₇	—	4	1	no	—
53	C₂×Dic₇	—	4	1	no	—
54	D₂₈	—	4	2	no	—
55	D₂₈	—	4	2	no	—
56	D₂₈	—	4	2	no	—
57	Dic₁₄	—	4	2	no	—
58	Dic₁₄	—	4	2	no	—
59	Dic₁₄	—	4	2	no	—
60	C₇×C₈	—	2	2	no	—
61	C₇×C₈	—	2	2	no	—
62	C₇×C₈	SD₁₆	1	4	yes	complemented
63	C₂×SD₁₆	—	14	2	no	—
64	C₂×SD₁₆	—	14	2	no	—
65	C₄⋊Q₈	—	7	1	no	—
66	C₄⋊Q₈	—	7	1	no	—
67	C₄⋊Q₈	—	7	1	no	—
68	#127	—	7	1	no	—
69	#107	—	14	2	no	—
70	#106	—	14	2	no	—
71	#103	—	14	2	no	—
72	#102	—	14	2	no	—
73	C₄×C₈	C₂×D₇	1	1	yes	characteristic
74	C₄×C₈	C₂×D₇	1	1	yes	characteristic
75	C₄×C₈	C₂×D₇	1	1	yes	characteristic
76	C₂²×C₇	#127	1	1	yes	characteristic
77	C₂×D₇	—	8	1	no	—
78	C₂×D₇	—	8	1	no	—
79	C₂×D₇	—	8	1	no	—
80	C₄×C₇	C₂×SD₁₆	1	2	yes	—
81	C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
82	C₄×C₇	#137	1	2	yes	—
83	Dic₇	—	8	1	no	—
84	Dic₇	—	8	1	no	—
85	Dic₇	—	8	1	no	—
86	C₂×Q₈	—	14	1	no	—
87	C₂×Q₈	—	14	1	no	—
88	C₂×Q₈	—	14	1	no	—
89	C₂×D₄	—	14	1	no	—
90	C₂×D₄	—	14	1	no	—
91	C₂×D₄	—	14	1	no	—
92	SD₁₆	—	28	4	no	—
93	SD₁₆	—	28	4	no	—
94	C₂×C₈	D₂₈	1	2	yes	—
95	C₂×C₈	D₂₈	1	2	yes	—
96	C₂×C₈	D₂₈	1	2	yes	—
97	C₄⋊C₄	—	14	1	no	—
98	C₄⋊C₄	—	14	1	no	—
99	C₄⋊C₄	—	14	1	no	—
100	C₄⋊C₄	—	14	1	no	—
101	C₄⋊C₄	—	14	1	no	—
102	C₄⋊C₄	—	14	1	no	—
103	C₄²	C₂²×D₇	1	1	yes	characteristic
104	C₂×C₇	#492	1	1	yes	characteristic
105	C₂×C₇	#496	1	1	yes	characteristic
106	C₂×C₇	#497	1	1	yes	characteristic
107	D₇	—	16	1	no	—
108	C₂³	—	28	1	no	—
109	Q₈	—	28	2	no	—
110	Q₈	—	28	2	no	—
111	Q₈	—	28	2	no	—
112	D₄	—	28	2	no	—
113	D₄	—	28	2	no	—
114	D₄	—	28	2	no	—
115	C₂×C₄	—	28	1	no	—
116	C₂×C₄	—	28	1	no	—
117	C₂×C₄	—	28	1	no	—
118	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
119	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
120	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	2	no	—
123	C₈	#1132	1	4	yes	complemented
124	C₇	#1714	1	1	yes	characteristic, complemented, Sylow
125	C₂²	—	56	1	no	—
126	C₂²	—	56	1	no	—
127	C₂²	—	56	1	no	—
128	C₂²	#6437	1	1	yes	characteristic
129	C₄	—	56	1	no	—
130	C₄	—	56	1	no	—
131	C₄	—	56	1	no	—
132	C₄	C₂×#1132	1	2	yes	—
133	C₄	#6471	1	2	yes	—
134	C₄	#6472	1	2	yes	—
135	C₂	—	112	1	no	—
136	C₂	#89465	1	1	yes	characteristic
137	C₂	#89484	1	1	yes	characteristic
138	C₂	#89487	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#89466

C₂

yes

maximal, characteristic

#89465

C₂

yes

maximal, characteristic

#89465

C₂

yes

maximal, characteristic

#89454

C₂

yes

maximal, characteristic, complemented

#89453

C₂

yes

maximal, characteristic, complemented

#89452

C₂

yes

maximal, characteristic, complemented

C₇×#322

C₂

yes

maximal, characteristic, complemented

C₂×#1132

—

C₂×#1132

—

#6437

C₂²

yes

characteristic

#6432

C₂²

yes

characteristic

#6432

C₂²

yes

characteristic

#6432

C₂²

yes

characteristic

C₄×C₇×C₈

C₂²

yes

characteristic

C₄×C₇×C₈

C₂²

yes

characteristic

C₄×C₇×C₈

C₂²

yes

—

—

—

—

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₄

yes

—

C₂×C₇×C₈

D₄

yes

—

C₂×C₇×C₈

D₄

yes

—

C₄²×C₇

C₂³

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

—

—

—

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂²×D₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

D₂₈

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

SD₁₆

yes

complemented

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

C₄×C₈

C₂×D₇

yes

characteristic

C₄×C₈

C₂×D₇

yes

characteristic

C₄×C₈

C₂×D₇

yes

characteristic

C₂²×C₇

#127

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

C₂×SD₁₆

yes

—

C₄×C₇

AGL(1,ℤ/8)

yes

—

C₄×C₇

#137

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

D₂₈

yes

—

C₂×C₈

D₂₈

yes

—

C₂×C₈

D₂₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

C₂²×D₇

yes

characteristic

104

C₂×C₇

#492

yes

characteristic

105

C₂×C₇

#496

yes

characteristic

106

C₂×C₇

#497

yes

characteristic

107

D₇

—

108

C₂³

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×D₂₈

yes

characteristic

119

C₂×C₄

C₂×D₂₈

yes

characteristic

120

C₂×C₄

C₂×D₂₈

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

#1132

yes

complemented

124

C₇

#1714

yes

characteristic, complemented, Sylow

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#6437

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂×#1132

yes

—

133

C₄

#6471

yes

—

134

C₄

#6472

yes

—

135

C₂

—

112

—

136

C₂

#89465

yes

characteristic

137

C₂

#89484

yes

characteristic

138

C₂

#89487

yes

characteristic

	subgroup	G/H
center	C₂²	#6437
commutator, cosocle	C₄²×C₇	C₂³
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#322	C₂
socle	C₂²×C₇	#127
prime radical	#6437	C₂²
hypercenter	#322	D₇
hypocenter	C₇	#1714
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#6437

commutator, cosocle

C₄²×C₇

C₂³

Frattini

C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂²

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #89454	C₂	semidirect
5. #89453	C₂	semidirect
6. #89452	C₂	semidirect
7. C₇×#322	C₂	semidirect
46. #322	D₇	semidirect
62. C₇×C₈	SD₁₆	semidirect
123. C₈	#1132	semidirect
124. C₇	#1714	semidirect

H ≅ G/N

type

4. #89454

C₂

semidirect

5. #89453

C₂

semidirect

6. #89452

C₂

semidirect

7. C₇×#322

C₂

semidirect

46. #322

D₇

semidirect

62. C₇×C₈

SD₁₆

semidirect

123. C₈

#1132

semidirect

124. C₇

#1714

semidirect

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

subgroup	index
#89466	2
#89465	2
#89454	2
#89453	2
#89452	2
C₇×#322	2
#1714	7

subgroup

index

Subgroups — #11725844

Conjugacy classes of subgroups

Complements

Maximal subgroups