Group #89548 — subgroups

Subgroups — #89548

Order 448.

188 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 45 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×D₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#1142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1132	C₂	1	1	yes	maximal, characteristic, complemented
4	#6460	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#102	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#6383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₇	—	2	1	no	—
9	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₇	—	2	1	no	—
11	D₄×D₇	—	2	2	no	—
12	D₄×D₇	—	2	2	no	—
13	C₂×D₂₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
17	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
18	#1142	—	2	2	no	—
19	(C₂×D₇)⋊C₄	—	2	1	no	—
20	C₄⋊Dic₇	—	2	1	no	—
21	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
22	#1132	—	2	2	no	—
23	#450	—	7	1	no	maximal, Sylow
24	C₂³×C₇	—	2	1	no	—
25	C₂²×D₇	—	2	4	no	—
26	C₂²×D₇	—	2	1	no	—
27	C₂²×D₇	—	4	2	no	—
28	C₂²×D₇	—	4	2	no	—
29	C₂²×D₇	D₄	1	1	yes	characteristic
30	C₇×D₄	—	2	1	no	—
31	C₇×D₄	D₄	1	2	yes	—
32	C₂×C₄×C₇	—	2	1	no	—
33	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂²⋊D₇	—	2	4	no	—
35	C₂×Dic₇	—	2	1	no	—
36	C₂×Dic₇	D₄	1	1	yes	characteristic
37	D₂₈	—	2	1	no	—
38	D₂₈	D₄	1	2	yes	—
39	C₄×D₇	—	2	1	no	—
40	C₄×D₇	—	2	1	no	—
41	Dic₁₄	—	2	1	no	—
42	Dic₁₄	—	2	1	no	—
43	C₇×C₈	—	2	1	no	—
44	C₇⋊C₈	—	2	1	no	—
45	C₂²×D₄	—	7	1	no	—
46	C₂×SD₁₆	—	7	1	no	—
47	C₂×SD₁₆	—	7	1	no	—
48	C₂²⋊Q₈	—	7	1	no	—
49	#102	—	7	1	no	—
50	#102	D₇	1	1	yes	characteristic, complemented
51	C₂²⋊C₈	—	7	1	no	—
52	C₂²×C₇	—	2	2	no	—
53	C₂²×C₇	—	4	1	no	—
54	C₂²×C₇	—	4	1	no	—
55	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂×D₇	—	2	1	no	—
57	C₂×D₇	—	2	1	no	—
58	C₂×D₇	—	2	2	no	—
59	C₂×D₇	—	4	8	no	—
60	C₂×D₇	—	4	1	no	—
61	C₂×D₇	—	4	1	no	—
62	C₂×D₇	SD₁₆	1	2	yes	complemented
63	C₄×C₇	—	4	1	no	—
64	C₄×C₇	C₂×D₄	1	1	yes	characteristic
65	C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	Dic₇	—	2	1	no	—
67	Dic₇	—	4	1	no	—
68	C₂⁴	—	14	1	no	—
69	C₂×Q₈	—	7	1	no	—
70	C₂×D₄	—	7	1	no	—
71	C₂×D₄	—	14	1	no	—
72	C₂×D₄	—	14	2	no	—
73	C₂×D₄	—	14	2	no	—
74	C₂×D₄	C₂×D₇	1	1	yes	characteristic
75	C₂²×C₄	—	7	1	no	—
76	SD₁₆	—	14	2	no	—
77	SD₁₆	—	14	2	no	—
78	C₂×C₈	—	7	1	no	—
79	C₂×C₈	C₂×D₇	1	1	yes	characteristic, complemented
80	C₄⋊C₄	—	14	1	no	—
81	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic, complemented
82	C₂²⋊C₄	—	14	1	no	—
83	C₂×C₇	—	4	2	no	—
84	C₂×C₇	#120	1	1	yes	characteristic
85	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
87	D₇	—	2	2	no	—
88	D₇	—	4	2	no	—
89	C₂³	—	2	1	no	—
90	C₂³	—	7	1	no	—
91	C₂³	—	14	4	no	—
92	C₂³	—	14	1	no	—
93	C₂³	—	28	2	no	—
94	C₂³	—	28	2	no	—
95	Q₈	—	14	1	no	—
96	Q₈	—	14	1	no	—
97	D₄	—	2	1	no	—
98	D₄	—	7	2	no	—
99	D₄	—	14	4	no	—
100	D₄	—	14	1	no	—
101	D₄	D₂₈	1	2	yes	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	7	1	no	—
104	C₂×C₄	—	14	1	no	—
105	C₂×C₄	—	14	1	no	—
106	C₂×C₄	—	14	1	no	—
107	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	14	1	no	—
110	C₇	#450	1	1	yes	characteristic, complemented, Sylow
111	C₂²	—	2	2	no	—
112	C₂²	—	4	1	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	7	2	no	—
115	C₂²	—	14	1	no	—
116	C₂²	—	14	1	no	—
117	C₂²	—	14	2	no	—
118	C₂²	—	28	8	no	—
119	C₂²	—	28	1	no	—
120	C₂²	—	28	1	no	—
121	C₂²	D₄×D₇	1	1	yes	characteristic
122	C₄	—	4	1	no	—
123	C₄	—	14	1	no	—
124	C₄	—	28	1	no	—
125	C₄	C₂×D₂₈	1	1	yes	characteristic
126	C₄	D₄×D₇	1	1	yes	characteristic
127	C₂	—	4	2	no	—
128	C₂	—	14	2	no	—
129	C₂	—	28	2	no	—
130	C₂	#6445	1	1	yes	characteristic
131	C₂	#6474	1	1	yes	characteristic
132	C₂	D₇×SD₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×D₇

C₂

yes

maximal, characteristic

C₂×#1142

C₂

yes

maximal, characteristic, complemented

C₂×#1132

C₂

yes

maximal, characteristic, complemented

#6460

C₂

yes

maximal, characteristic, complemented

C₇×#102

C₂

yes

maximal, characteristic, complemented

(C₂×D₇)⋊C₈

C₂

yes

maximal, characteristic, complemented

#6383

C₂

yes

maximal, characteristic, complemented

C₂³×D₇

—

C₂×C₇×D₄

C₂²

yes

characteristic

C₂×C₂²⋊D₇

—

D₄×D₇

—

D₄×D₇

—

C₂×D₂₈

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

characteristic, complemented

C₂×C₇×C₈

C₂²

yes

characteristic, complemented

C₇×C₄⋊C₄

C₂²

yes

characteristic, complemented

#1142

—

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

C₂²

yes

characteristic, complemented

#1132

—

#450

—

maximal, Sylow

C₂³×C₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

D₄

yes

characteristic

C₇×D₄

—

C₇×D₄

D₄

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

D₄

yes

characteristic

D₂₈

—

D₂₈

D₄

yes

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂²⋊Q₈

—

#102

—

#102

D₇

yes

characteristic, complemented

C₂²⋊C₈

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×D₄

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

SD₁₆

yes

complemented

C₄×C₇

—

C₄×C₇

C₂×D₄

yes

characteristic

C₄×C₇

C₂×D₄

yes

characteristic

Dic₇

—

Dic₇

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₇

yes

characteristic

C₂²×C₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

C₂×D₇

yes

characteristic, complemented

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

characteristic, complemented

C₂²⋊C₄

—

C₂×C₇

—

C₂×C₇

#120

yes

characteristic

C₂×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₇

AGL(1,ℤ/8)

yes

characteristic

D₇

—

D₇

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

D₂₈

yes

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²×D₇

yes

characteristic

108

C₈

—

109

C₈

—

110

C₇

#450

yes

characteristic, complemented, Sylow

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

D₄×D₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₂×D₂₈

yes

characteristic

126

C₄

D₄×D₇

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#6445

yes

characteristic

131

C₂

#6474

yes

characteristic

132

C₂

D₇×SD₁₆

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#102	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₄×D₇	C₂
hypercenter	#102	D₇
hypocenter	C₇	#450
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₇

commutator, cosocle

C₂×C₄×C₇

C₂³

Frattini

C₂×C₄

C₂²×D₇

Fitting

C₇×#102

C₂

socle

C₂²×C₇

C₂×D₄

prime radical

C₂×D₄×D₇

C₂

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
2. C₂×#1142	C₂	semidirect
3. C₂×#1132	C₂	semidirect
4. #6460	C₂	semidirect
5. C₇×#102	C₂	semidirect
6. (C₂×D₇)⋊C₈	C₂	semidirect
7. #6383	C₂	semidirect
15. C₂×Dic₁₄	C₂²	semidirect
16. C₂×C₇×C₈	C₂²	semidirect
17. C₇×C₄⋊C₄	C₂²	semidirect
21. C₂×C₇⋊C₈	C₂²	semidirect
50. #102	D₇	semidirect
62. C₂×D₇	SD₁₆	semidirect
79. C₂×C₈	C₂×D₇	semidirect
81. C₄⋊C₄	C₂×D₇	semidirect
110. C₇	#450	semidirect

H ≅ G/N

type

2. C₂×#1142

C₂

semidirect

3. C₂×#1132

C₂

semidirect

4. #6460

C₂

semidirect

5. C₇×#102

C₂

semidirect

6. (C₂×D₇)⋊C₈

C₂

semidirect

7. #6383

C₂

semidirect

15. C₂×Dic₁₄

C₂²

semidirect

16. C₂×C₇×C₈

C₂²

semidirect

17. C₇×C₄⋊C₄

C₂²

semidirect

21. C₂×C₇⋊C₈

C₂²

semidirect

50. #102

D₇

semidirect

62. C₂×D₇

SD₁₆

semidirect

79. C₂×C₈

C₂×D₇

semidirect

81. C₄⋊C₄

C₂×D₇

semidirect

110. C₇

#450

semidirect

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

subgroup	index
C₂×D₄×D₇	2
C₂×#1142	2
C₂×#1132	2
#6460	2
C₇×#102	2
(C₂×D₇)⋊C₈	2
#6383	2
#450	7

subgroup

index

C₂×D₄×D₇

(C₂×D₇)⋊C₈

Subgroups — #89548

Conjugacy classes of subgroups

Complements

Maximal subgroups