Group #11927258 — subgroups

Subgroups — #11927258

Order 1216.

236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₁₉×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	#10597385	C₂	1	2	yes	maximal, complemented
3	#10597384	C₂	1	1	yes	maximal, characteristic, complemented
4	#10597382	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#10597372	C₂	1	1	yes	maximal, characteristic, complemented
8	#10597338	C₂	1	1	yes	maximal, characteristic, complemented
9	#10597334	C₂	1	1	yes	maximal, characteristic, complemented
10	#10597331	C₂	1	2	yes	maximal, complemented
11	#10597323	C₂	1	2	yes	maximal, complemented
12	C₂×D₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₁₉	C₂²	1	2	yes	—
14	C₂²×C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊D₁₉	C₂²	1	2	yes	—
17	C₂²×Dic₁₉	C₂²	1	2	yes	—
18	C₂×C₄×D₁₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₈	C₂²	1	2	yes	—
20	C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₉	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₁₉	C₂²	1	2	yes	—
25	C₂²⋊Dic₁₉	C₂²	1	2	yes	—
26	(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
30	Dic₁₉⋊C₄	C₂²	1	2	yes	—
31	Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₉	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₉	C₂²	1	2	yes	—
34	C₄×Dic₁₉	C₂²	1	2	yes	—
35	C₂³×C₁₉	C₂³	1	1	yes	characteristic
36	C₂³×C₁₉	C₂³	1	2	yes	—
37	C₂²×D₁₉	C₂³	1	1	yes	characteristic
38	D₄×C₁₉	—	2	2	no	—
39	D₄×C₁₉	—	4	2	no	—
40	C₂×C₄×C₁₉	—	2	2	no	—
41	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₉	C₂³	1	2	yes	—
43	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²⋊D₁₉	—	2	2	no	—
45	C₂²⋊D₁₉	—	4	2	no	—
46	C₂×Dic₁₉	—	2	4	no	—
47	C₂×Dic₁₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₉	C₂³	1	2	yes	—
49	C₂×Dic₁₉	C₂³	1	2	yes	—
50	C₂×Dic₁₉	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₉	C₂³	1	1	yes	characteristic
52	C₄×D₁₉	—	2	2	no	—
53	Dic₃₈	—	4	2	no	—
54	C₂²×C₁₉	—	2	2	no	—
55	C₂²×C₁₉	—	2	1	no	—
56	C₂²×C₁₉	—	2	1	no	—
57	C₂²×C₁₉	—	2	2	no	—
58	C₂²×C₁₉	—	2	2	no	—
59	C₂²×C₁₉	—	2	1	no	—
60	C₂²×C₁₉	C₂⁴	1	1	yes	characteristic
61	C₂×D₁₉	—	2	1	no	—
62	C₂×D₁₉	—	2	1	no	—
63	C₂×D₁₉	—	2	1	no	—
64	C₄×C₁₉	—	2	1	no	—
65	C₄×C₁₉	—	2	2	no	—
66	C₄×C₁₉	#41	1	2	yes	—
67	Dic₁₉	—	2	1	no	—
68	Dic₁₉	—	2	2	no	—
69	Dic₁₉	—	2	1	no	—
70	Dic₁₉	—	2	2	no	—
71	Dic₁₉	#41	1	2	yes	—
72	#555	—	19	1	no	maximal, Sylow
73	C₂×C₁₉	—	4	2	no	—
74	C₂×C₁₉	—	4	1	no	—
75	C₂×C₁₉	C₂×#41	1	1	yes	characteristic
76	C₂×C₁₉	C₂×#41	1	1	yes	characteristic
77	C₂×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
78	D₁₉	—	4	1	no	—
79	C₄⋊Q₈	—	19	1	no	—
80	#124	—	19	2	no	—
81	#124	—	19	2	no	—
82	#121	—	19	1	no	—
83	#121	—	19	2	no	—
84	#121	D₁₉	1	1	yes	characteristic, complemented
85	C₄×D₄	—	19	1	no	—
86	C₄×D₄	—	19	1	no	—
87	#117	—	19	1	no	—
88	#117	—	19	1	no	—
89	#117	—	19	2	no	—
90	C₁₉	#555	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	19	2	no	—
92	C₂×D₄	—	19	1	no	—
93	C₂×D₄	—	19	2	no	—
94	C₂×D₄	C₂×D₁₉	1	1	yes	characteristic
95	C₂×D₄	C₂×D₁₉	1	2	yes	—
96	C₂²×C₄	—	19	2	no	—
97	C₂²×C₄	—	19	1	no	—
98	C₂²×C₄	C₂×D₁₉	1	1	yes	characteristic
99	C₄⋊C₄	—	19	1	no	—
100	C₄⋊C₄	—	19	2	no	—
101	C₄⋊C₄	—	19	1	no	—
102	C₄⋊C₄	—	19	1	no	—
103	C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
104	C₂²⋊C₄	—	19	2	no	—
105	C₂²⋊C₄	—	19	2	no	—
106	C₂²⋊C₄	—	19	1	no	—
107	C₂²⋊C₄	—	19	2	no	—
108	C₂²⋊C₄	—	19	2	no	—
109	C₂²⋊C₄	—	19	1	no	—
110	C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
111	C₄²	—	19	2	no	—
112	C₄²	—	19	2	no	—
113	C₄²	—	19	1	no	—
114	C₂³	—	19	1	no	—
115	C₂³	C₂²×D₁₉	1	1	yes	characteristic
116	C₂³	C₂²×D₁₉	1	2	yes	—
117	Q₈	—	76	2	no	—
118	D₄	—	2	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	38	2	no	—
121	D₄	—	76	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	19	1	no	—
124	C₂×C₄	—	19	1	no	—
125	C₂×C₄	—	19	2	no	—
126	C₂×C₄	—	19	1	no	—
127	C₂×C₄	—	19	2	no	—
128	C₂×C₄	—	38	4	no	—
129	C₂×C₄	—	38	2	no	—
130	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₁₉	1	2	yes	—
132	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	38	1	no	—
140	C₂²	—	38	1	no	—
141	C₂²	—	38	1	no	—
142	C₂²	C₂³×D₁₉	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	19	2	no	—
146	C₄	—	38	1	no	—
147	C₄	—	38	2	no	—
148	C₄	—	38	1	no	—
149	C₄	—	38	2	no	—
150	C₄	#64543	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	76	1	no	—
154	C₂	C₂×#64543	1	1	yes	characteristic
155	C₂	#10597430	1	1	yes	characteristic
156	C₂	#41×D₁₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#121

C₂

yes

maximal, characteristic, complemented

#10597385

C₂

yes

maximal, complemented

#10597384

C₂

yes

maximal, characteristic, complemented

#10597382

C₂

yes

maximal, complemented

D₄×Dic₁₉

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₉

C₂

yes

maximal, characteristic, complemented

#10597372

C₂

yes

maximal, characteristic, complemented

#10597338

C₂

yes

maximal, characteristic, complemented

#10597334

C₂

yes

maximal, characteristic, complemented

#10597331

C₂

yes

maximal, complemented

#10597323

C₂

yes

maximal, complemented

C₂×D₄×C₁₉

C₂²

yes

characteristic

C₂×D₄×C₁₉

C₂²

yes

—

C₂²×C₄×C₁₉

C₂²

yes

characteristic

C₂×C₂²⋊D₁₉

C₂²

yes

characteristic

C₂×C₂²⋊D₁₉

C₂²

yes

—

C₂²×Dic₁₉

C₂²

yes

—

C₂×C₄×D₁₉

C₂²

yes

characteristic

C₂×Dic₃₈

C₂²

yes

—

C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₉

C₂²

yes

—

C₂²⋊Dic₁₉

C₂²

yes

—

C₂²⋊Dic₁₉

C₂²

yes

characteristic

C₂²⋊Dic₁₉

C₂²

yes

—

C₂²⋊Dic₁₉

C₂²

yes

—

(C₂×D₁₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉)⋊C₄

C₂²

yes

characteristic

C₄⋊Dic₁₉

C₂²

yes

characteristic

C₄⋊Dic₁₉

C₂²

yes

characteristic

Dic₁₉⋊C₄

C₂²

yes

—

Dic₁₉⋊C₄

C₂²

yes

characteristic

C₄×Dic₁₉

C₂²

yes

characteristic

C₄×Dic₁₉

C₂²

yes

—

C₄×Dic₁₉

C₂²

yes

—

C₂³×C₁₉

C₂³

yes

characteristic

C₂³×C₁₉

C₂³

yes

—

C₂²×D₁₉

C₂³

yes

characteristic

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₄×C₁₉

C₂³

yes

—

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂²⋊D₁₉

—

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

C₂³

yes

characteristic

C₂×Dic₁₉

C₂³

yes

—

C₂×Dic₁₉

C₂³

yes

—

C₂×Dic₁₉

C₂³

yes

characteristic

C₂×Dic₁₉

C₂³

yes

characteristic

C₄×D₁₉

—

Dic₃₈

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂⁴

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#41

yes

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

#41

yes

—

#555

—

maximal, Sylow

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

C₂×#41

yes

characteristic

C₂×C₁₉

C₂×#41

yes

characteristic

C₂×C₁₉

H(5,𝔽₂)

yes

characteristic

D₁₉

—

C₄⋊Q₈

—

—

—

—

—

D₁₉

yes

characteristic, complemented

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

#117

—

C₁₉

#555

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₁₉

yes

characteristic

C₂×D₄

C₂×D₁₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₁₉

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂×D₁₉

yes

characteristic

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

C₂×D₁₉

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂³

—

115

C₂³

C₂²×D₁₉

yes

characteristic

116

C₂³

C₂²×D₁₉

yes

—

117

Q₈

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²×D₁₉

yes

characteristic

131

C₂×C₄

C₂²×D₁₉

yes

—

132

C₂×C₄

C₂²×D₁₉

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×D₁₉

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#64543

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

C₂×#64543

yes

characteristic

155

C₂

#10597430

yes

characteristic

156

C₂

#41×D₁₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁴
Fitting	C₁₉×#121	C₂
hypercenter	#121	D₁₉
hypocenter	C₁₉	#555
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉

commutator, socle, cosocle

C₂²×C₁₉

C₂⁴

Fitting

C₁₉×#121

C₂

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. C₁₉×#121	C₂	semidirect
2. #10597385	C₂	semidirect
3. #10597384	C₂	semidirect
4. #10597382	C₂	semidirect
5. D₄×Dic₁₉	C₂	semidirect
6. C₄×C₂²⋊D₁₉	C₂	semidirect
7. #10597372	C₂	semidirect
8. #10597338	C₂	semidirect
9. #10597334	C₂	semidirect
10. #10597331	C₂	semidirect
11. #10597323	C₂	semidirect
84. #121	D₁₉	semidirect
90. C₁₉	#555	semidirect

H ≅ G/N

type

1. C₁₉×#121

C₂

semidirect

2. #10597385

C₂

semidirect

3. #10597384

C₂

semidirect

4. #10597382

C₂

semidirect

5. D₄×Dic₁₉

C₂

semidirect

6. C₄×C₂²⋊D₁₉

C₂

semidirect

7. #10597372

C₂

semidirect

8. #10597338

C₂

semidirect

9. #10597334

C₂

semidirect

10. #10597331

C₂

semidirect

11. #10597323

C₂

semidirect

84. #121

D₁₉

semidirect

90. C₁₉

#555

semidirect

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

subgroup	index
C₁₉×#121	2
#10597385	2
#10597384	2
#10597382	2
D₄×Dic₁₉	2
C₄×C₂²⋊D₁₉	2
#10597372	2
#10597338	2
#10597334	2
#10597331	2
#10597323	2
#555	19

subgroup

index

C₁₉×#121

#10597385

#10597384

#10597382

D₄×Dic₁₉

C₄×C₂²⋊D₁₉

Subgroups — #11927258

Conjugacy classes of subgroups

Complements

Maximal subgroups