Group #16834886 — subgroups

Subgroups — #16834886

Order 3712.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#14272185	C₂	1	1	yes	maximal, characteristic
2	#14272099	C₂	1	1	yes	maximal, characteristic
3	#14272017	C₂	1	4	yes	maximal
4	#14271718	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#1181	—	2	1	no	—
6	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	#11745308	C₂²	1	2	yes	—
8	Q₈×Dic₂₉	C₄	1	2	yes	complemented
9	#11745259	—	2	1	no	—
10	#11745254	—	2	1	no	—
11	#11745252	—	2	1	no	—
12	#11745249	—	2	1	no	—
13	C₄²×D₂₉	C₂²	1	1	yes	characteristic
14	#11745231	C₄	1	1	yes	characteristic, complemented
15	(C₂×#1181)⋊C₄	—	2	1	no	—
16	(C₂×#1181)⋊C₄	C₂²	1	2	yes	—
17	#11745219	C₂²	1	2	yes	—
18	#11745218	—	2	1	no	—
19	C₄⋊#6599	—	2	1	no	—
20	C₂²×#1181	—	2	2	no	—
21	C₂×Q₈×C₂₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
23	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
24	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₅₈	—	2	1	no	—
26	C₂×#6599	—	2	2	no	—
27	C₄×#1181	—	2	4	no	—
28	C₄⋊C₄×C₂₉	—	2	1	no	—
29	C₄⋊C₄×C₂₉	—	2	1	no	—
30	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
31	#90781	—	2	2	no	—
32	C₄⋊Dic₂₉	—	2	1	no	—
33	C₄⋊Dic₂₉	—	2	2	no	—
34	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
35	#90779	—	2	2	no	—
36	C₄×Dic₂₉	—	2	2	no	—
37	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₂₉	D₄	1	1	yes	characteristic
39	C₄×Dic₂₉	D₄	1	1	yes	characteristic
40	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
41	C₂×#1181	—	2	4	no	—
42	C₂×#1181	—	4	2	no	—
43	C₂×#1181	—	4	2	no	—
44	Q₈×C₂₉	—	4	2	no	—
45	C₂×C₄×C₂₉	—	2	2	no	—
46	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₂₉	—	2	2	no	—
50	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
53	C₄×D₂₉	—	2	2	no	—
54	C₄×D₂₉	—	2	2	no	—
55	C₄×D₂₉	—	4	1	no	—
56	Dic₅₈	—	4	2	no	—
57	#6599	—	4	2	no	—
58	#1506	—	29	1	no	maximal, Sylow
59	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₂₉	—	2	1	no	—
61	C₂×D₂₉	—	2	1	no	—
62	C₂×D₂₉	#137	1	2	yes	—
63	#1181	—	4	4	no	—
64	C₄×C₂₉	—	2	1	no	—
65	C₄×C₂₉	—	2	1	no	—
66	C₄×C₂₉	—	4	2	no	—
67	C₄×C₂₉	#104	1	2	yes	—
68	Dic₂₉	—	2	1	no	—
69	Dic₂₉	—	2	1	no	—
70	Dic₂₉	—	2	1	no	—
71	Dic₂₉	—	4	2	no	—
72	#533	—	29	1	no	—
73	#432	—	29	1	no	—
74	#376	—	29	1	no	—
75	#328	—	29	4	no	—
76	C₂×C₂₉	#410	1	1	yes	characteristic
77	C₂×C₂₉	#419	1	1	yes	characteristic
78	C₂×C₂₉	C₂×#104	1	1	yes	characteristic
79	D₂₉	—	2	2	no	—
80	C₄⋊Q₈	—	29	1	no	—
81	C₄⋊Q₈	#1181	1	1	yes	characteristic, complemented
82	#125	—	58	1	no	—
83	C₂²⋊Q₈	—	29	2	no	—
84	C₂²⋊Q₈	—	58	1	no	—
85	C₄×Q₈	—	29	2	no	—
86	C₄×Q₈	—	58	1	no	—
87	#117	—	58	1	no	—
88	C₂×C₄²	—	29	1	no	—
89	C₂×C₄²	—	58	1	no	—
90	C₄⋊C₈	—	58	1	no	—
91	C₂²⋊C₈	—	29	2	no	—
92	#97	—	58	1	no	—
93	#95	—	29	2	no	—
94	#95	—	58	1	no	—
95	C₂₉	#1506	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	58	1	no	—
97	C₂×Q₈	C₂×#1181	1	2	yes	—
98	C₂²×C₄	—	29	1	no	—
99	C₂²×C₄	—	29	1	no	—
100	C₂²×C₄	—	29	1	no	—
101	C₂²×C₄	—	58	2	no	—
102	C₂×C₈	—	58	2	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	29	2	no	—
106	C₄⋊C₄	—	58	2	no	—
107	C₄⋊C₄	—	58	2	no	—
108	C₄⋊C₄	—	58	1	no	—
109	C₂²⋊C₄	—	58	2	no	—
110	C₄²	—	29	1	no	—
111	C₄²	—	29	1	no	—
112	C₄²	—	29	1	no	—
113	C₄²	—	58	2	no	—
114	C₄²	—	58	4	no	—
115	C₄²	C₂×#1181	1	1	yes	characteristic
116	C₂³	—	29	1	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	116	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	29	1	no	—
121	C₂×C₄	—	29	1	no	—
122	C₂×C₄	—	29	1	no	—
123	C₂×C₄	—	58	2	no	—
124	C₂×C₄	—	58	2	no	—
125	C₂×C₄	—	58	4	no	—
126	C₂×C₄	—	58	2	no	—
127	C₂×C₄	—	116	1	no	—
128	C₂×C₄	—	116	2	no	—
129	C₂×C₄	—	116	2	no	—
130	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
131	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
132	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
133	C₈	—	116	2	no	—
134	C₂²	—	29	2	no	—
135	C₂²	—	58	1	no	—
136	C₂²	—	58	1	no	—
137	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	—	58	1	no	—
142	C₄	—	58	1	no	—
143	C₄	—	58	1	no	—
144	C₄	—	116	4	no	—
145	C₄	—	116	2	no	—
146	C₄	#11745226	1	2	yes	—
147	C₂	—	58	2	no	—
148	C₂	#14272143	1	1	yes	characteristic
149	C₂	#14272156	1	1	yes	characteristic
150	C₂	C₂×#11745226	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14272185

C₂

yes

maximal, characteristic

#14272099

C₂

yes

maximal, characteristic

#14272017

C₂

yes

maximal

#14271718

C₂

yes

maximal, characteristic

C₂×C₄×#1181

—

C₂₉×C₄⋊Q₈

C₄

yes

characteristic, complemented

#11745308

C₂²

yes

—

Q₈×Dic₂₉

C₄

yes

—

—

—

—

C₄²×D₂₉

C₂²

yes

characteristic

#11745231

C₄

yes

characteristic, complemented

(C₂×#1181)⋊C₄

—

(C₂×#1181)⋊C₄

C₂²

yes

—

#11745219

C₂²

yes

—

#11745218

—

C₄⋊#6599

—

C₂²×#1181

—

C₂×Q₈×C₂₉

C₂×C₄

yes

—

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₂×#6599

—

C₄×#1181

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

#90781

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90779

—

C₄×Dic₂₉

—

C₄×Dic₂₉

C₂×C₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₂²×D₂₉

C₂×D₄

yes

—

—

—

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

—

Dic₅₈

—

#6599

—

#1506

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

#137

yes

—

#1181

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#104

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

—

—

—

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

D₂₉

—

C₄⋊Q₈

—

C₄⋊Q₈

#1181

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#95

—

#95

—

C₂₉

#1506

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×#1181

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂×#1181

yes

characteristic

116

C₂³

—

117

Q₈

—

118

Q₈

—

116

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

116

—

128

C₂×C₄

—

116

—

129

C₂×C₄

—

116

—

130

C₂×C₄

C₂²⋊#1181

yes

characteristic

131

C₂×C₄

C₂²⋊#1181

yes

characteristic

132

C₂×C₄

C₂²×#1181

yes

characteristic

133

C₈

—

116

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×C₂²⋊#1181

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

116

—

145

C₄

—

116

—

146

C₄

#11745226

yes

—

147

C₂

—

148

C₂

#14272143

yes

characteristic

149

C₂

#14272156

yes

characteristic

150

C₂

C₂×#11745226

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#1181
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂₉×C₄⋊Q₈	C₄
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₉	C₂×D₄
cosocle	C₂×C₄×D₂₉	C₂³
hypercenter	C₄⋊Q₈	#1181
hypocenter	C₂₉	#1506
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#1181

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#1181

Fitting

C₂₉×C₄⋊Q₈

C₄

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₉

C₂×D₄

cosocle

C₂×C₄×D₂₉

C₂³

C₄⋊Q₈

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
6. C₂₉×C₄⋊Q₈	C₄	semidirect
8. Q₈×Dic₂₉	C₄	semidirect
14. #11745231	C₄	semidirect
81. C₄⋊Q₈	#1181	semidirect
95. C₂₉	#1506	semidirect

H ≅ G/N

type

6. C₂₉×C₄⋊Q₈

C₄

semidirect

8. Q₈×Dic₂₉

C₄

semidirect

14. #11745231

C₄

semidirect

81. C₄⋊Q₈

#1181

semidirect

95. C₂₉

#1506

semidirect

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

subgroup	index
#14272185	2
#14272099	2
#14272017	2
#14271718	2
#1506	29

subgroup

index

Subgroups — #16834886

Conjugacy classes of subgroups

Complements

Maximal subgroups