Group #11927166 — subgroups

Subgroups — #11927166

Order 1216.

216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#10597393	C₂	1	1	yes	maximal, characteristic, complemented
3	#10597392	C₂	1	1	yes	maximal, characteristic, complemented
4	#10597342	C₂	1	2	yes	maximal, complemented
5	#10597341	C₂	1	1	yes	maximal, characteristic
6	#10597340	C₂	1	2	yes	maximal, complemented
7	#10597334	C₂	1	1	yes	maximal, characteristic, complemented
8	#10597322	C₂	1	2	yes	maximal, complemented
9	#10597321	C₂	1	2	yes	maximal, complemented
10	C₄×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#10597318	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×D₇₆	C₂²	1	1	yes	characteristic
14	C₂×D₇₆	C₂²	1	2	yes	—
15	C₂×C₄×D₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₉	C₂²	1	2	yes	—
17	C₂×Dic₃₈	C₂²	1	2	yes	—
18	C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
20	C₄²×C₁₉	C₂²	1	1	yes	characteristic
21	C₄²×C₁₉	C₂²	1	2	yes	—
22	(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
27	(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
28	(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
29	C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
30	Dic₁₉⋊C₄	C₂²	1	2	yes	—
31	Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
32	Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
33	Dic₁₉⋊C₄	C₂²	1	2	yes	—
34	C₄×Dic₁₉	C₂²	1	1	yes	characteristic
35	C₂²×D₁₉	C₂³	1	1	yes	characteristic
36	C₂²×D₁₉	C₂³	1	2	yes	—
37	Q₈×C₁₉	—	2	2	no	—
38	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₉	C₂³	1	2	yes	—
40	C₂×C₄×C₁₉	C₂³	1	2	yes	—
41	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₉	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₉	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₉	C₂³	1	2	yes	—
47	D₇₆	—	2	2	no	—
48	D₇₆	—	4	2	no	—
49	C₄×D₁₉	—	2	2	no	—
50	C₄×D₁₉	—	4	2	no	—
51	Dic₃₈	—	4	2	no	—
52	C₂²×C₁₉	C₂⁴	1	1	yes	characteristic
53	C₂×D₁₉	—	2	1	no	—
54	C₂×D₁₉	—	2	2	no	—
55	C₂×D₁₉	—	2	2	no	—
56	C₂×D₁₉	—	2	1	no	—
57	C₂×D₁₉	—	2	1	no	—
58	C₂×D₁₉	—	2	2	no	—
59	C₄×C₁₉	—	2	2	no	—
60	C₄×C₁₉	—	2	2	no	—
61	C₄×C₁₉	—	2	1	no	—
62	C₄×C₁₉	—	2	1	no	—
63	C₄×C₁₉	#41	1	2	yes	—
64	Dic₁₉	—	2	1	no	—
65	Dic₁₉	—	2	1	no	—
66	Dic₁₉	—	2	2	no	—
67	Dic₁₉	—	2	1	no	—
68	#542	—	19	1	no	maximal, Sylow
69	C₂×C₁₉	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
72	D₁₉	—	4	1	no	—
73	D₁₉	—	4	2	no	—
74	C₄⋊Q₈	—	19	1	no	—
75	#126	—	19	2	no	—
76	#124	—	19	1	no	—
77	#124	—	19	2	no	—
78	#123	—	19	2	no	—
79	C₂²⋊Q₈	—	19	1	no	—
80	C₂²⋊Q₈	—	19	2	no	—
81	#121	—	19	1	no	—
82	C₄×Q₈	D₁₉	1	1	yes	characteristic, complemented
83	C₄×D₄	—	19	1	no	—
84	#117	—	19	1	no	—
85	C₁₉	#542	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₈	—	19	2	no	—
87	C₂×Q₈	C₂×D₁₉	1	1	yes	characteristic
88	C₂×D₄	—	19	1	no	—
89	C₂×D₄	—	19	2	no	—
90	C₂²×C₄	—	19	1	no	—
91	C₂²×C₄	—	19	2	no	—
92	C₄⋊C₄	—	19	1	no	—
93	C₄⋊C₄	—	19	1	no	—
94	C₄⋊C₄	—	19	2	no	—
95	C₄⋊C₄	—	19	2	no	—
96	C₄⋊C₄	—	19	1	no	—
97	C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₁₉	1	2	yes	—
99	C₂²⋊C₄	—	19	1	no	—
100	C₂²⋊C₄	—	19	1	no	—
101	C₂²⋊C₄	—	19	2	no	—
102	C₂²⋊C₄	—	19	2	no	—
103	C₂²⋊C₄	—	19	2	no	—
104	C₂²⋊C₄	—	19	2	no	—
105	C₂²⋊C₄	—	19	2	no	—
106	C₄²	—	19	1	no	—
107	C₄²	C₂×D₁₉	1	1	yes	characteristic
108	C₄²	C₂×D₁₉	1	2	yes	—
109	C₂³	—	19	1	no	—
110	C₂³	—	19	2	no	—
111	Q₈	—	2	2	no	—
112	Q₈	—	76	2	no	—
113	D₄	—	38	2	no	—
114	D₄	—	76	2	no	—
115	C₂×C₄	—	19	1	no	—
116	C₂×C₄	—	19	1	no	—
117	C₂×C₄	—	19	1	no	—
118	C₂×C₄	—	19	2	no	—
119	C₂×C₄	—	38	2	no	—
120	C₂×C₄	—	76	2	no	—
121	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₁₉	1	2	yes	—
123	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₁₉	1	2	yes	—
125	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
126	C₂²	—	38	1	no	—
127	C₂²	—	38	2	no	—
128	C₂²	—	38	1	no	—
129	C₂²	—	38	2	no	—
130	C₂²	—	38	2	no	—
131	C₂²	—	38	1	no	—
132	C₂²	C₂³×D₁₉	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	38	1	no	—
138	C₄	—	38	1	no	—
139	C₄	—	38	2	no	—
140	C₄	—	38	1	no	—
141	C₄	#64541	1	2	yes	—
142	C₂	—	76	1	no	—
143	C₂	—	76	2	no	—
144	C₂	C₂×#64541	1	1	yes	characteristic
145	C₂	#10597433	1	1	yes	characteristic
146	C₂	#10597435	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×C₁₉

C₂

yes

maximal, characteristic, complemented

#10597393

C₂

yes

maximal, characteristic, complemented

#10597392

C₂

yes

maximal, characteristic, complemented

#10597342

C₂

yes

maximal, complemented

#10597341

C₂

yes

maximal, characteristic

#10597340

C₂

yes

maximal, complemented

#10597334

C₂

yes

maximal, characteristic, complemented

#10597322

C₂

yes

maximal, complemented

#10597321

C₂

yes

maximal, complemented

C₄×D₇₆

C₂

yes

maximal, characteristic, complemented

#10597318

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₁₉

C₂²

yes

characteristic

C₂×D₇₆

C₂²

yes

characteristic

C₂×D₇₆

C₂²

yes

—

C₂×C₄×D₁₉

C₂²

yes

characteristic

C₂×C₄×D₁₉

C₂²

yes

—

C₂×Dic₃₈

C₂²

yes

—

C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₉

C₂²

yes

—

C₄²×C₁₉

C₂²

yes

characteristic

C₄²×C₁₉

C₂²

yes

—

(C₂×D₁₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₄⋊Dic₁₉

C₂²

yes

characteristic

Dic₁₉⋊C₄

C₂²

yes

—

Dic₁₉⋊C₄

C₂²

yes

characteristic

Dic₁₉⋊C₄

C₂²

yes

characteristic

Dic₁₉⋊C₄

C₂²

yes

—

C₄×Dic₁₉

C₂²

yes

characteristic

C₂²×D₁₉

C₂³

yes

characteristic

C₂²×D₁₉

C₂³

yes

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₄×C₁₉

C₂³

yes

—

C₂×C₄×C₁₉

C₂³

yes

—

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂×Dic₁₉

C₂³

yes

characteristic

C₂×Dic₁₉

C₂³

yes

characteristic

C₂×Dic₁₉

C₂³

yes

characteristic

C₂×Dic₁₉

C₂³

yes

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

C₄×D₁₉

—

Dic₃₈

—

C₂²×C₁₉

C₂⁴

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#41

yes

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

#542

—

maximal, Sylow

C₂×C₁₉

C₂×#41

yes

characteristic

C₂×C₁₉

H(5,𝔽₂)

yes

characteristic

C₂×C₁₉

2₋¹⁺⁴

yes

characteristic

D₁₉

—

D₁₉

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

D₁₉

yes

characteristic, complemented

C₄×D₄

—

#117

—

C₁₉

#542

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×D₁₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₉

yes

characteristic

C₄⋊C₄

C₂×D₁₉

yes

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

C₂×D₁₉

yes

characteristic

108

C₄²

C₂×D₁₉

yes

—

109

C₂³

—

110

C₂³

—

111

Q₈

—

112

Q₈

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×D₁₉

yes

characteristic

122

C₂×C₄

C₂²×D₁₉

yes

—

123

C₂×C₄

C₂²×D₁₉

yes

characteristic

124

C₂×C₄

C₂²×D₁₉

yes

—

125

C₂×C₄

C₂²×D₁₉

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×D₁₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#64541

yes

—

142

C₂

—

143

C₂

—

144

C₂

C₂×#64541

yes

characteristic

145

C₂

#10597433

yes

characteristic

146

C₂

#10597435

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁴
Fitting	C₄×Q₈×C₁₉	C₂
prime radical	#10597341	C₂
hypercenter	C₄×Q₈	D₁₉
hypocenter	C₁₉	#542
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉

commutator, socle, cosocle

C₂²×C₁₉

C₂⁴

Fitting

C₄×Q₈×C₁₉

C₂

prime radical

#10597341

C₂

hypercenter

C₄×Q₈

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. C₄×Q₈×C₁₉	C₂	semidirect
2. #10597393	C₂	semidirect
3. #10597392	C₂	semidirect
4. #10597342	C₂	semidirect
6. #10597340	C₂	semidirect
7. #10597334	C₂	semidirect
8. #10597322	C₂	semidirect
9. #10597321	C₂	semidirect
10. C₄×D₇₆	C₂	semidirect
11. #10597318	C₂	semidirect
82. C₄×Q₈	D₁₉	semidirect
85. C₁₉	#542	semidirect

H ≅ G/N

type

1. C₄×Q₈×C₁₉

C₂

semidirect

2. #10597393

C₂

semidirect

3. #10597392

C₂

semidirect

4. #10597342

C₂

semidirect

6. #10597340

C₂

semidirect

7. #10597334

C₂

semidirect

8. #10597322

C₂

semidirect

9. #10597321

C₂

semidirect

10. C₄×D₇₆

C₂

semidirect

11. #10597318

C₂

semidirect

82. C₄×Q₈

D₁₉

semidirect

85. C₁₉

#542

semidirect

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

subgroup	index
C₄×Q₈×C₁₉	2
#10597393	2
#10597392	2
#10597342	2
#10597341	2
#10597340	2
#10597334	2
#10597322	2
#10597321	2
C₄×D₇₆	2
#10597318	2
#542	19

subgroup

index

C₄×Q₈×C₁₉

C₄×D₇₆

Subgroups — #11927166

Conjugacy classes of subgroups

Complements

Maximal subgroups