Group #44484306 — subgroups

Subgroups — C₄×C₂²⋊(C₂₃₉⋊C₈) #44484306

Order 30592.

436 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 213 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×C₂²⋊(C₂₃₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₂₃₉⋊C₈	C₂	1	2	yes	maximal, complemented
3	#29504527	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄²×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₃₉⋊C₈	C₂²	1	4	yes	—
6	C₂²⋊(C₂₃₉⋊C₈)	C₄	1	8	yes	direct product factor
7	C₄×C₂₃₉⋊C₈	—	2	4	no	—
8	C₂³×C₄×C₂₃₉	C₄	1	2	yes	—
9	C₂³×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₂₃₉	—	2	4	no	—
11	C₂×C₄²×C₂₃₉	C₄	1	2	yes	—
12	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₂₃₉⋊C₈	—	2	4	no	—
15	C₂×C₂₃₉⋊C₈	—	2	4	no	—
16	C₂×C₂₃₉⋊C₈	C₂×C₄	1	8	yes	complemented
17	C₂⁴×C₂₃₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃₉	—	2	4	no	—
19	C₂²×C₄×C₂₃₉	—	2	4	no	—
20	C₂²×C₄×C₂₃₉	—	2	2	no	—
21	C₂²×C₄×C₂₃₉	—	2	2	no	—
22	C₂²×C₄×C₂₃₉	C₈	1	8	yes	complemented
23	C₂²×C₄×C₂₃₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
28	C₄²×C₂₃₉	—	2	4	no	—
29	C₄²×C₂₃₉	—	2	2	no	—
30	C₄²×C₂₃₉	D₄	1	4	yes	—
31	C₂₃₉⋊C₈	—	2	8	no	—
32	C₂³×C₂₃₉	—	2	1	no	—
33	C₂³×C₂₃₉	—	2	1	no	—
34	C₂³×C₂₃₉	—	2	2	no	—
35	C₂³×C₂₃₉	C₄²	1	2	yes	—
36	C₂³×C₂₃₉	C₂×C₈	1	4	yes	—
37	C₂³×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃₉	—	2	1	no	—
39	C₂×C₄×C₂₃₉	—	2	2	no	—
40	C₂×C₄×C₂₃₉	—	2	1	no	—
41	C₂×C₄×C₂₃₉	—	2	2	no	—
42	C₂×C₄×C₂₃₉	—	2	16	no	—
43	C₂×C₄×C₂₃₉	—	2	8	no	—
44	C₂×C₄×C₂₃₉	C₄²	1	2	yes	—
45	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	4	yes	—
46	C₂×C₄×C₂₃₉	C₂×C₈	1	4	yes	—
47	C₂×C₄×C₂₃₉	C₂×C₈	1	4	yes	—
48	C₂×C₄×C₂₃₉	M₁₆	1	4	yes	—
49	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₃₉	#41	1	2	yes	—
52	C₂²×C₂₃₉	—	2	2	no	—
53	C₂²×C₂₃₉	—	2	2	no	—
54	C₂²×C₂₃₉	—	2	4	no	—
55	C₂²×C₂₃₉	—	2	4	no	—
56	C₂²×C₂₃₉	C₄×C₈	1	4	yes	complemented
57	C₂²×C₂₃₉	C₂×C₄²	1	1	yes	characteristic
58	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₃₉	#117	1	1	yes	characteristic
60	C₂²×C₂₃₉	C₂²×C₈	1	2	yes	—
61	C₂²×C₂₃₉	C₂×M₁₆	1	2	yes	—
62	C₄×C₂₃₉	—	2	2	no	—
63	C₄×C₂₃₉	—	2	4	no	—
64	C₄×C₂₃₉	C₂²⋊C₈	1	8	yes	complemented
65	C₄×C₂₃₉	C₄×D₄	1	4	yes	—
66	C₂×C₂₃₉	—	2	4	no	—
67	C₂×C₂₃₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₂₃₉	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂×C₂₃₉	C₄×M₁₆	1	1	yes	characteristic
70	C₂×C₂₃₉	C₂×C₂²⋊C₈	1	2	yes	—
71	C₂×C₂₃₉	#431	1	2	yes	—
72	C₂₃₉	C₄×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
73	C₄×C₂²⋊C₈	—	239	1	no	maximal, Sylow
74	C₂²×C₄²	D₂₃₉	1	1	yes	characteristic
75	C₂×C₂²⋊C₈	—	239	2	no	—
76	C₂×C₄×C₈	—	239	2	no	—
77	#336	—	239	2	no	—
78	C₂³×C₄	Dic₂₃₉	1	2	yes	—
79	C₂³×C₄	C₂×D₂₃₉	1	1	yes	characteristic
80	C₂²×C₈	—	239	4	no	—
81	C₂×C₄²	—	2	4	no	—
82	C₂×C₄²	Dic₂₃₉	1	2	yes	—
83	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
85	C₂²⋊C₈	—	239	8	no	—
86	C₄×C₈	—	478	4	no	—
87	C₂⁴	C₂×Dic₂₃₉	1	1	yes	characteristic
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	C₂₃₉⋊C₈	1	8	yes	complemented
93	C₂²×C₄	C₄×D₂₃₉	1	4	yes	—
94	C₂²×C₄	C₂×Dic₂₃₉	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₂₃₉	1	2	yes	—
96	C₂²×C₄	C₂×Dic₂₃₉	1	2	yes	—
97	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
98	C₂×C₈	—	478	4	no	—
99	C₂×C₈	—	478	4	no	—
100	C₂×C₈	—	239	8	no	—
101	C₄²	—	2	4	no	—
102	C₄²	—	2	2	no	—
103	C₄²	C₂²⋊D₂₃₉	1	4	yes	—
104	C₂³	—	2	1	no	—
105	C₂³	—	2	1	no	—
106	C₂³	—	2	2	no	—
107	C₂³	C₄×Dic₂₃₉	1	2	yes	—
108	C₂³	C₂×C₂₃₉⋊C₈	1	4	yes	—
109	C₂³	C₂²×Dic₂₃₉	1	1	yes	characteristic
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	16	no	—
115	C₂×C₄	—	2	8	no	—
116	C₂×C₄	C₄×Dic₂₃₉	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₂₃₉	1	4	yes	—
118	C₂×C₄	C₂×C₂₃₉⋊C₈	1	4	yes	—
119	C₂×C₄	C₂×C₂₃₉⋊C₈	1	4	yes	—
120	C₂×C₄	#16838532	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₂₃₉	1	2	yes	—
122	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	2	yes	—
123	C₂×C₄	#16838548	1	2	yes	—
124	C₈	—	478	8	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	4	no	—
129	C₂²	C₄×C₂₃₉⋊C₈	1	4	yes	complemented
130	C₂²	C₂×C₄×Dic₂₃₉	1	1	yes	characteristic
131	C₂²	C₂×C₂²⋊Dic₂₃₉	1	1	yes	characteristic
132	C₂²	#21439919	1	1	yes	characteristic
133	C₂²	C₂²×C₂₃₉⋊C₈	1	2	yes	—
134	C₂²	C₂×#16838532	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	—	2	4	no	—
137	C₄	C₂²⋊(C₂₃₉⋊C₈)	1	8	yes	direct product factor
138	C₄	C₄×C₂²⋊D₂₃₉	1	4	yes	—
139	C₂	—	2	4	no	—
140	C₂	C₄×C₂²⋊Dic₂₃₉	1	1	yes	characteristic
141	C₂	C₂×C₄×C₂₃₉⋊C₈	1	1	yes	characteristic
142	C₂	C₄×#16838532	1	1	yes	characteristic
143	C₂	C₂×C₂²⋊(C₂₃₉⋊C₈)	1	2	yes	—
144	C₂	#29504825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₂₃₉⋊C₈)

C₂

yes

maximal

C₂×C₄×C₂₃₉⋊C₈

C₂

yes

maximal, complemented

#29504527

C₂

yes

maximal, complemented

C₂²×C₄²×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×C₂₃₉⋊C₈

C₂²

yes

—

C₂²⋊(C₂₃₉⋊C₈)

C₄

yes

direct product factor

C₄×C₂₃₉⋊C₈

—

C₂³×C₄×C₂₃₉

C₄

yes

—

C₂³×C₄×C₂₃₉

C₂²

yes

characteristic

C₂×C₄²×C₂₃₉

—

C₂×C₄²×C₂₃₉

C₄

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

C₂×C₄

yes

complemented

C₂⁴×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

C₈

yes

complemented

C₂²×C₄×C₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₄²×C₂₃₉

—

C₄²×C₂₃₉

—

C₄²×C₂₃₉

D₄

yes

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

—

C₂³×C₂₃₉

—

C₂³×C₂₃₉

—

C₂³×C₂₃₉

C₄²

yes

—

C₂³×C₂₃₉

C₂×C₈

yes

—

C₂³×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₄²

yes

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂×C₈

yes

—

C₂×C₄×C₂₃₉

C₂×C₈

yes

—

C₂×C₄×C₂₃₉

M₁₆

yes

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₂×C₄×C₂₃₉

#41

yes

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₄×C₈

yes

complemented

C₂²×C₂₃₉

C₂×C₄²

yes

characteristic

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃₉

#117

yes

characteristic

C₂²×C₂₃₉

C₂²×C₈

yes

—

C₂²×C₂₃₉

C₂×M₁₆

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂²⋊C₈

yes

complemented

C₄×C₂₃₉

C₄×D₄

yes

—

C₂×C₂₃₉

—

C₂×C₂₃₉

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₂₃₉

C₂×C₄×C₈

yes

characteristic

C₂×C₂₃₉

C₄×M₁₆

yes

characteristic

C₂×C₂₃₉

C₂×C₂²⋊C₈

yes

—

C₂×C₂₃₉

#431

yes

—

C₂₃₉

C₄×C₂²⋊C₈

yes

characteristic, complemented, Sylow

C₄×C₂²⋊C₈

—

239

maximal, Sylow

C₂²×C₄²

D₂₃₉

yes

characteristic

C₂×C₂²⋊C₈

—

239

—

C₂×C₄×C₈

—

239

—

#336

—

239

—

C₂³×C₄

Dic₂₃₉

yes

—

C₂³×C₄

C₂×D₂₃₉

yes

characteristic

C₂²×C₈

—

239

—

C₂×C₄²

—

C₂×C₄²

Dic₂₃₉

yes

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

C₂²⋊C₈

—

239

—

C₄×C₈

—

478

—

C₂⁴

C₂×Dic₂₃₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂₃₉⋊C₈

yes

complemented

C₂²×C₄

C₄×D₂₃₉

yes

—

C₂²×C₄

C₂×Dic₂₃₉

yes

characteristic

C₂²×C₄

C₂×Dic₂₃₉

yes

—

C₂²×C₄

C₂×Dic₂₃₉

yes

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

478

—

C₂×C₈

—

478

—

100

C₂×C₈

—

239

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂²⋊D₂₃₉

yes

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₄×Dic₂₃₉

yes

—

108

C₂³

C₂×C₂₃₉⋊C₈

yes

—

109

C₂³

C₂²×Dic₂₃₉

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄×Dic₂₃₉

yes

—

117

C₂×C₄

C₂²⋊Dic₂₃₉

yes

—

118

C₂×C₄

C₂×C₂₃₉⋊C₈

yes

—

119

C₂×C₄

C₂×C₂₃₉⋊C₈

yes

—

120

C₂×C₄

#16838532

yes

—

121

C₂×C₄

C₂×C₄×D₂₃₉

yes

—

122

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

—

123

C₂×C₄

#16838548

yes

—

124

C₈

—

478

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₄×C₂₃₉⋊C₈

yes

complemented

130

C₂²

C₂×C₄×Dic₂₃₉

yes

characteristic

131

C₂²

C₂×C₂²⋊Dic₂₃₉

yes

characteristic

132

C₂²

#21439919

yes

characteristic

133

C₂²

C₂²×C₂₃₉⋊C₈

yes

—

134

C₂²

C₂×#16838532

yes

—

135

C₄

—

136

C₄

—

137

C₄

C₂²⋊(C₂₃₉⋊C₈)

yes

direct product factor

138

C₄

C₄×C₂²⋊D₂₃₉

yes

—

139

C₂

—

140

C₂

C₄×C₂²⋊Dic₂₃₉

yes

characteristic

141

C₂

C₂×C₄×C₂₃₉⋊C₈

yes

characteristic

142

C₂

C₄×#16838532

yes

characteristic

143

C₂

C₂×C₂²⋊(C₂₃₉⋊C₈)

yes

—

144

C₂

#29504825

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₂₃₉
commutator	C₂×C₂₃₉	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₂²×C₄²×C₂₃₉	C₂
socle	C₂³×C₂₃₉	C₂²×C₄
prime radical	C₂⁴×C₂₃₉	C₂×C₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂²×C₄²	D₂₃₉
hypocenter	C₂₃₉	C₄×C₂²⋊C₈
LCS	G ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

subgroup

G/H

center

C₂×C₄²

C₂×D₂₃₉

commutator

C₂×C₂₃₉

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₂²×C₄²×C₂₃₉

C₂

socle

C₂³×C₂₃₉

C₂²×C₄

prime radical

C₂⁴×C₂₃₉

C₂×C₄

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₂²×C₄²

D₂₃₉

hypocenter

C₂₃₉

C₄×C₂²⋊C₈

LCS

G ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

N	H ≅ G/N	type
2. C₂×C₄×C₂₃₉⋊C₈	C₂	semidirect
3. #29504527	C₂	semidirect
6. C₂²⋊(C₂₃₉⋊C₈)	C₄	direct (N × H)
16. C₂×C₂₃₉⋊C₈	C₂×C₄	semidirect
22. C₂²×C₄×C₂₃₉	C₈	semidirect
56. C₂²×C₂₃₉	C₄×C₈	semidirect
64. C₄×C₂₃₉	C₂²⋊C₈	semidirect
72. C₂₃₉	C₄×C₂²⋊C₈	semidirect
92. C₂²×C₄	C₂₃₉⋊C₈	semidirect
129. C₂²	C₄×C₂₃₉⋊C₈	semidirect
137. C₄	C₂²⋊(C₂₃₉⋊C₈)	direct (N × H)

H ≅ G/N

type

2. C₂×C₄×C₂₃₉⋊C₈

C₂

semidirect

3. #29504527

C₂

semidirect

6. C₂²⋊(C₂₃₉⋊C₈)

C₄

direct (N × H)

16. C₂×C₂₃₉⋊C₈

C₂×C₄

semidirect

22. C₂²×C₄×C₂₃₉

C₈

semidirect

56. C₂²×C₂₃₉

C₄×C₈

semidirect

64. C₄×C₂₃₉

C₂²⋊C₈

semidirect

72. C₂₃₉

C₄×C₂²⋊C₈

semidirect

92. C₂²×C₄

C₂₃₉⋊C₈

semidirect

129. C₂²

C₄×C₂₃₉⋊C₈

semidirect

137. C₄

C₂²⋊(C₂₃₉⋊C₈)

direct (N × H)

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

subgroup	index
C₂×C₂²⋊(C₂₃₉⋊C₈)	2
C₂×C₄×C₂₃₉⋊C₈	2
#29504527	2
C₂²×C₄²×C₂₃₉	2
C₄×C₂²⋊C₈	239

subgroup

index

C₂×C₂²⋊(C₂₃₉⋊C₈)

C₂×C₄×C₂₃₉⋊C₈

#29504527

C₂²×C₄²×C₂₃₉

C₄×C₂²⋊C₈

239

Subgroups — C₄×C₂²⋊(C₂₃₉⋊C₈) #44484306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×C22⋊(C239⋊C8) #44484306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊(C₂₃₉⋊C₈) #44484306