Group #14271872 — subgroups

Subgroups — #14271872

Order 1856.

228 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₂₉	C₂	1	1	yes	maximal, characteristic
3	D₄×Dic₂₉	C₂	1	1	yes	maximal, characteristic
4	#11745286	C₂	1	2	yes	maximal, complemented
5	C₂×#90779	C₂	1	2	yes	maximal, complemented
6	#11745252	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745250	C₂	1	1	yes	maximal, characteristic, complemented
8	#11745240	C₂	1	2	yes	maximal, complemented
9	#11745240	C₂	1	2	yes	maximal, complemented
10	#11745232	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₅₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₂₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₉	C₂²	1	2	yes	—
14	C₂²×Dic₂₉	C₂²	1	2	yes	—
15	C₂²×Dic₂₉	C₂²	1	2	yes	—
16	C₂×Dic₅₈	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₈	C₂²	1	2	yes	—
18	C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
20	C₄²×C₂₉	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₂₉	C₂²	1	2	yes	—
22	C₂²⋊Dic₂₉	C₂²	1	2	yes	—
23	C₄⋊Dic₂₉	C₂²	1	2	yes	—
24	C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
27	#90779	C₂²	1	1	yes	characteristic
28	#90779	C₂²	1	2	yes	—
29	#90779	C₂²	1	2	yes	—
30	#90779	C₂²	1	2	yes	—
31	#90779	C₂²	1	1	yes	characteristic
32	#90779	C₂²	1	2	yes	—
33	C₄×Dic₂₉	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₉	C₂²	1	1	yes	characteristic
35	C₂³×C₂₉	C₂³	1	2	yes	—
36	D₄×C₂₉	Q₈	1	4	yes	complemented
37	C₂×C₄×C₂₉	—	4	2	no	—
38	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₂₉	C₂³	1	2	yes	—
41	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₉	—	2	4	no	—
43	C₂×Dic₂₉	—	4	2	no	—
44	C₂×Dic₂₉	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₉	C₂³	1	1	yes	characteristic
46	C₂×Dic₂₉	C₂³	1	1	yes	characteristic
47	C₂×Dic₂₉	C₂³	1	2	yes	—
48	C₂×Dic₂₉	C₂³	1	2	yes	—
49	C₂×Dic₂₉	C₂³	1	1	yes	characteristic
50	Dic₅₈	—	2	2	no	—
51	Dic₅₈	—	4	2	no	—
52	C₂²×C₂₉	—	2	2	no	—
53	C₂²×C₂₉	—	2	2	no	—
54	C₂²×C₂₉	C₂×Q₈	1	4	yes	—
55	C₂²×C₂₉	C₂⁴	1	1	yes	characteristic
56	C₄×C₂₉	—	2	1	no	—
57	C₄×C₂₉	—	2	2	no	—
58	C₄×C₂₉	—	2	1	no	—
59	C₄×C₂₉	C₂×Q₈	1	2	yes	—
60	Dic₂₉	—	2	1	no	—
61	Dic₂₉	—	2	2	no	—
62	Dic₂₉	—	2	1	no	—
63	Dic₂₉	—	2	2	no	—
64	Dic₂₉	—	2	1	no	—
65	Dic₂₉	#41	1	2	yes	—
66	#554	—	29	1	no	maximal, Sylow
67	C₂×C₂₉	—	2	4	no	—
68	C₂×C₂₉	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₂₉	C₂×#41	1	1	yes	characteristic
70	C₂×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
71	C₄⋊Q₈	—	29	1	no	—
72	#125	—	29	1	no	—
73	#125	—	29	1	no	—
74	C₂²⋊Q₈	—	29	2	no	—
75	C₂²⋊Q₈	—	29	2	no	—
76	C₂²⋊Q₈	—	29	2	no	—
77	C₄×Q₈	—	29	1	no	—
78	C₄×D₄	—	29	1	no	—
79	C₄×D₄	—	29	1	no	—
80	C₄×D₄	D₂₉	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	29	2	no	—
82	C₂₉	#554	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	29	1	no	—
84	C₂×Q₈	—	29	2	no	—
85	C₂×D₄	C₂×D₂₉	1	1	yes	characteristic
86	C₂²×C₄	—	29	2	no	—
87	C₂²×C₄	—	29	2	no	—
88	C₂²×C₄	C₂×D₂₉	1	2	yes	—
89	C₄⋊C₄	—	29	1	no	—
90	C₄⋊C₄	—	29	2	no	—
91	C₄⋊C₄	—	29	2	no	—
92	C₄⋊C₄	—	29	2	no	—
93	C₄⋊C₄	—	29	1	no	—
94	C₄⋊C₄	—	29	1	no	—
95	C₄⋊C₄	—	29	1	no	—
96	C₄⋊C₄	—	29	1	no	—
97	C₄⋊C₄	—	29	2	no	—
98	C₄⋊C₄	—	29	2	no	—
99	C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
100	C₂²⋊C₄	—	29	2	no	—
101	C₂²⋊C₄	—	29	2	no	—
102	C₂²⋊C₄	C₂×D₂₉	1	2	yes	—
103	C₄²	—	29	1	no	—
104	C₄²	—	29	1	no	—
105	C₄²	C₂×D₂₉	1	1	yes	characteristic
106	C₂³	C₂²×D₂₉	1	2	yes	—
107	Q₈	—	58	2	no	—
108	Q₈	—	116	2	no	—
109	D₄	Dic₅₈	1	4	yes	complemented
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	29	1	no	—
112	C₂×C₄	—	29	1	no	—
113	C₂×C₄	—	29	1	no	—
114	C₂×C₄	—	29	1	no	—
115	C₂×C₄	—	29	2	no	—
116	C₂×C₄	—	29	2	no	—
117	C₂×C₄	—	58	4	no	—
118	C₂×C₄	—	116	2	no	—
119	C₂×C₄	C₂²×D₂₉	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₂₉	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₂₉	1	2	yes	—
122	C₂×C₄	C₂²×D₂₉	1	1	yes	characteristic
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	C₂×Dic₅₈	1	4	yes	—
126	C₂²	C₂³×D₂₉	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	29	2	no	—
131	C₄	—	58	1	no	—
132	C₄	—	58	2	no	—
133	C₄	—	58	1	no	—
134	C₄	—	58	2	no	—
135	C₄	—	58	1	no	—
136	C₄	C₂×Dic₅₈	1	2	yes	—
137	C₂	—	2	4	no	—
138	C₂	C₂²×Dic₅₈	1	1	yes	characteristic
139	C₂	#11745359	1	1	yes	characteristic
140	C₂	#41×D₂₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄×C₂₉

C₂

yes

maximal, characteristic

D₄×Dic₂₉

C₂

yes

maximal, characteristic

D₄×Dic₂₉

C₂

yes

maximal, characteristic

#11745286

C₂

yes

maximal, complemented

C₂×#90779

C₂

yes

maximal, complemented

#11745252

C₂

yes

maximal, characteristic, complemented

#11745250

C₂

yes

maximal, characteristic, complemented

#11745240

C₂

yes

maximal, complemented

#11745240

C₂

yes

maximal, complemented

#11745232

C₂

yes

maximal, characteristic, complemented

C₄×Dic₅₈

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₂₉

C₂²

yes

characteristic

C₂²×C₄×C₂₉

C₂²

yes

—

C₂²×Dic₂₉

C₂²

yes

—

C₂²×Dic₂₉

C₂²

yes

—

C₂×Dic₅₈

C₂²

yes

characteristic

C₂×Dic₅₈

C₂²

yes

—

C₄⋊C₄×C₂₉

C₂²

yes

characteristic

C₂²⋊C₄×C₂₉

C₂²

yes

—

C₄²×C₂₉

C₂²

yes

characteristic

C₂²⋊Dic₂₉

C₂²

yes

—

C₂²⋊Dic₂₉

C₂²

yes

—

C₄⋊Dic₂₉

C₂²

yes

—

C₄⋊Dic₂₉

C₂²

yes

characteristic

C₄⋊Dic₂₉

C₂²

yes

characteristic

C₄⋊Dic₂₉

C₂²

yes

characteristic

#90779

C₂²

yes

characteristic

#90779

C₂²

yes

—

#90779

C₂²

yes

—

#90779

C₂²

yes

—

#90779

C₂²

yes

characteristic

#90779

C₂²

yes

—

C₄×Dic₂₉

C₂²

yes

characteristic

C₄×Dic₂₉

C₂²

yes

characteristic

C₂³×C₂₉

C₂³

yes

—

D₄×C₂₉

Q₈

yes

complemented

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂³

yes

characteristic

C₂×C₄×C₂₉

C₂³

yes

characteristic

C₂×C₄×C₂₉

C₂³

yes

—

C₂×C₄×C₂₉

C₂³

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂³

yes

characteristic

C₂×Dic₂₉

C₂³

yes

characteristic

C₂×Dic₂₉

C₂³

yes

characteristic

C₂×Dic₂₉

C₂³

yes

—

C₂×Dic₂₉

C₂³

yes

—

C₂×Dic₂₉

C₂³

yes

characteristic

Dic₅₈

—

Dic₅₈

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×Q₈

yes

—

C₂²×C₂₉

C₂⁴

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂×Q₈

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

#41

yes

—

#554

—

maximal, Sylow

C₂×C₂₉

—

C₂×C₂₉

C₂²×Q₈

yes

characteristic

C₂×C₂₉

C₂×#41

yes

characteristic

C₂×C₂₉

H(5,𝔽₂)

yes

characteristic

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

D₂₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂₉

#554

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₂₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₉

yes

characteristic

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

C₂×D₂₉

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×D₂₉

yes

characteristic

106

C₂³

C₂²×D₂₉

yes

—

107

Q₈

—

108

Q₈

—

116

—

109

D₄

Dic₅₈

yes

complemented

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

116

—

119

C₂×C₄

C₂²×D₂₉

yes

characteristic

120

C₂×C₄

C₂²×D₂₉

yes

characteristic

121

C₂×C₄

C₂²×D₂₉

yes

—

122

C₂×C₄

C₂²×D₂₉

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×Dic₅₈

yes

—

126

C₂²

C₂³×D₂₉

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×Dic₅₈

yes

—

137

C₂

—

138

C₂

C₂²×Dic₅₈

yes

characteristic

139

C₂

#11745359

yes

characteristic

140

C₂

#41×D₂₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₉
commutator, socle, cosocle	C₂²×C₂₉	C₂⁴
Fitting	C₄×D₄×C₂₉	C₂
prime radical	C₂×D₄×C₂₉	C₂²
hypercenter	C₄×D₄	D₂₉
hypocenter	C₂₉	#554
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₉

commutator, socle, cosocle

C₂²×C₂₉

C₂⁴

Fitting

C₄×D₄×C₂₉

C₂

prime radical

C₂×D₄×C₂₉

C₂²

hypercenter

C₄×D₄

D₂₉

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂₉

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
4. #11745286	C₂	semidirect
5. C₂×#90779	C₂	semidirect
6. #11745252	C₂	semidirect
7. #11745250	C₂	semidirect
8. #11745240	C₂	semidirect
9. #11745240	C₂	semidirect
10. #11745232	C₂	semidirect
11. C₄×Dic₅₈	C₂	semidirect
36. D₄×C₂₉	Q₈	semidirect
82. C₂₉	#554	semidirect
109. D₄	Dic₅₈	semidirect

H ≅ G/N

type

4. #11745286

C₂

semidirect

5. C₂×#90779

C₂

semidirect

6. #11745252

C₂

semidirect

7. #11745250

C₂

semidirect

8. #11745240

C₂

semidirect

9. #11745240

C₂

semidirect

10. #11745232

C₂

semidirect

11. C₄×Dic₅₈

C₂

semidirect

36. D₄×C₂₉

Q₈

semidirect

82. C₂₉

#554

semidirect

109. D₄

Dic₅₈

semidirect

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

subgroup	index
C₄×D₄×C₂₉	2
D₄×Dic₂₉	2
#11745286	2
C₂×#90779	2
#11745252	2
#11745250	2
#11745240	2
#11745232	2
C₄×Dic₅₈	2
#554	29

subgroup

index

C₄×D₄×C₂₉

D₄×Dic₂₉

C₄×Dic₅₈

Subgroups — #14271872

Conjugacy classes of subgroups

Complements

Maximal subgroups