Group #34034031 — subgroups

Subgroups — C₄×#64408⋊C₁₆ #34034031

Order 18688.

558 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 301 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×#64408⋊C₁₆	C₂	1	4	yes	maximal
2	C₄×#14556686	C₂	1	2	yes	maximal
3	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556686	C₂²	1	4	yes	—
5	#64408⋊C₁₆	C₄	1	16	yes	direct product factor
6	C₄×#11924040	C₄	1	8	yes	complemented
7	C₄×#11924036	C₂²	1	1	yes	characteristic
8	C₂×C₈×#64408	C₄	1	2	yes	—
9	C₂×C₈×#64408	C₂²	1	2	yes	—
10	C₄²×#64408	C₄	1	1	yes	characteristic
11	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
12	#14556686	C₂×C₄	1	16	yes	—
13	C₂×#11924040	C₂×C₄	1	8	yes	—
14	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
15	C₂×#11924036	C₂³	1	1	yes	characteristic
16	C₂×C₄×#64408	C₈	1	4	yes	—
17	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
18	C₄×#10596833	C₂×C₄	1	4	yes	—
19	C₈×#64408	C₈	1	8	yes	—
20	C₈×#64408	C₂×C₄	1	8	yes	—
21	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
22	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596831	C₈	1	1	yes	characteristic
24	C₈×Dic₇₃	C₈	1	2	yes	—
25	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
26	#11924040	C₄²	1	16	yes	complemented
27	C₂²×#64408	C₂×C₈	1	4	yes	—
28	#11924036	C₂×C₈	1	4	yes	—
29	#11924036	C₂²×C₄	1	4	yes	—
30	C₂×#10596833	C₄²	1	2	yes	—
31	C₂×#10596833	C₂²×C₄	1	2	yes	—
32	C₄×#64408	C₁₆	1	16	yes	complemented
33	C₄×#64408	C₄²	1	4	yes	—
34	C₄×#64408	C₂×C₈	1	4	yes	—
35	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
36	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×#10596831	C₂×C₈	1	2	yes	—
38	C₈×D₇₃	C₄²	1	4	yes	—
39	C₈×D₇₃	C₂×C₈	1	4	yes	—
40	C₄×Dic₇₃	C₄²	1	2	yes	—
41	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
43	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
45	C₂×#64408	C₄×C₈	1	8	yes	—
46	C₂×#64408	C₂×C₁₆	1	16	yes	—
47	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
48	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
50	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
51	C₄×D₇₃	C₂×C₁₆	1	8	yes	—
52	C₄×D₇₃	C₂×C₄²	1	2	yes	—
53	C₄×D₇₃	C₂²×C₈	1	2	yes	—
54	#10596833	C₄×C₈	1	4	yes	—
55	#10596833	C₂×C₄²	1	4	yes	—
56	C₈×C₇₃	C₄×C₈	1	4	yes	—
57	#10596831	C₄×C₈	1	4	yes	—
58	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
59	C₂×D₇₃	C₂×C₄×C₈	1	2	yes	—
60	C₂×D₇₃	C₂²×C₁₆	1	4	yes	—
61	#64408	C₄×C₁₆	1	16	yes	complemented
62	C₄×C₇₃	C₄×C₁₆	1	4	yes	complemented
63	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
64	Dic₇₃	C₄×C₁₆	1	4	yes	complemented
65	Dic₇₃	C₄³	1	1	yes	characteristic
66	Dic₇₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄²×C₁₆	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
69	C₂×C₇₃	C₂×C₄×C₁₆	1	2	yes	—
70	D₇₃	C₂×C₄×C₁₆	1	4	yes	—
71	C₂×C₄×C₁₆	—	73	2	no	—
72	C₂×C₄×C₁₆	—	73	4	no	—
73	C₄²×C₈	—	73	1	no	—
74	C₇₃	C₄²×C₁₆	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₁₆	—	73	4	no	—
76	C₂×C₄×C₈	—	73	2	no	—
77	C₂×C₄×C₈	—	73	1	no	—
78	C₂×C₄×C₈	—	73	2	no	—
79	C₂×C₄×C₈	—	73	1	no	—
80	C₄³	—	73	1	no	—
81	C₄×C₁₆	—	73	8	no	—
82	C₄×C₁₆	—	73	16	no	—
83	C₂²×C₈	—	73	1	no	—
84	C₂²×C₈	—	73	1	no	—
85	C₂²×C₈	—	73	2	no	—
86	C₂×C₄²	—	73	2	no	—
87	C₂×C₄²	—	73	4	no	—
88	C₂×C₄²	—	73	1	no	—
89	C₂×C₁₆	—	73	8	no	—
90	C₂×C₁₆	—	73	16	no	—
91	C₄×C₈	—	73	8	no	—
92	C₄×C₈	—	73	4	no	—
93	C₄×C₈	—	73	8	no	—
94	C₄×C₈	—	73	2	no	—
95	C₄×C₈	—	73	1	no	—
96	C₄×C₈	#10596842	1	1	yes	characteristic
97	C₂²×C₄	—	73	4	no	—
98	C₂²×C₄	—	73	2	no	—
99	C₂²×C₄	—	73	1	no	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	—	73	2	no	—
102	C₂×C₈	—	73	4	no	—
103	C₂×C₈	—	73	4	no	—
104	C₂×C₈	—	73	2	no	—
105	C₂×C₈	—	73	4	no	—
106	C₂×C₈	—	73	4	no	—
107	C₂×C₈	C₂×#10596842	1	2	yes	—
108	C₄²	—	73	16	no	—
109	C₄²	—	73	4	no	—
110	C₄²	—	73	4	no	—
111	C₄²	—	73	2	no	—
112	C₄²	—	73	1	no	—
113	C₄²	C₂×#10596842	1	1	yes	characteristic
114	C₁₆	—	73	16	no	—
115	C₂³	—	73	1	no	—
116	C₂×C₄	—	73	8	no	—
117	C₂×C₄	—	73	16	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	73	2	no	—
120	C₂×C₄	—	73	8	no	—
121	C₂×C₄	—	73	2	no	—
122	C₂×C₄	—	73	2	no	—
123	C₂×C₄	C₄×#10596842	1	2	yes	—
124	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
125	C₈	—	73	4	no	—
126	C₈	—	73	4	no	—
127	C₈	—	73	4	no	—
128	C₈	C₄×#10596842	1	4	yes	—
129	C₂²	—	73	4	no	—
130	C₂²	—	73	2	no	—
131	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
132	C₄	—	73	16	no	—
133	C₄	—	73	4	no	—
134	C₄	—	73	1	no	—
135	C₄	—	73	1	no	—
136	C₄	#64408⋊C₁₆	1	4	yes	direct product factor
137	C₄	C₂×C₄×#10596842	1	2	yes	—
138	C₂	—	73	4	no	—
139	C₂	C₄²×#10596842	1	1	yes	characteristic
140	C₂	C₂×#64408⋊C₁₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64408⋊C₁₆

C₂

yes

maximal

C₄×#14556686

C₂

yes

maximal

C₄×C₈×#64408

C₂

yes

maximal, characteristic

C₂×#14556686

C₂²

yes

—

#64408⋊C₁₆

C₄

yes

direct product factor

C₄×#11924040

C₄

yes

complemented

C₄×#11924036

C₂²

yes

characteristic

C₂×C₈×#64408

C₄

yes

—

C₂×C₈×#64408

C₂²

yes

—

C₄²×#64408

C₄

yes

characteristic

C₄×C₈×D₇₃

C₄

yes

characteristic

#14556686

C₂×C₄

yes

—

C₂×#11924040

C₂×C₄

yes

—

C₂×#11924036

C₂×C₄

yes

characteristic

C₂×#11924036

C₂³

yes

characteristic

C₂×C₄×#64408

C₈

yes

—

C₂×C₄×#64408

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₈×#64408

C₈

yes

—

C₈×#64408

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₈

yes

characteristic

#11924040

C₄²

yes

complemented

C₂²×#64408

C₂×C₈

yes

—

#11924036

C₂×C₈

yes

—

#11924036

C₂²×C₄

yes

—

C₂×#10596833

C₄²

yes

—

C₂×#10596833

C₂²×C₄

yes

—

C₄×#64408

C₁₆

yes

complemented

C₄×#64408

C₄²

yes

—

C₄×#64408

C₂×C₈

yes

—

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

C₄²

yes

—

C₈×D₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₄²

yes

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂²×C₈

yes

characteristic

C₂×#64408

C₄×C₈

yes

—

C₂×#64408

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

—

C₂×Dic₇₃

C₂×C₄²

yes

characteristic

C₄×D₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

C₂×C₄²

yes

—

C₄×D₇₃

C₂²×C₈

yes

—

#10596833

C₄×C₈

yes

—

#10596833

C₂×C₄²

yes

—

C₈×C₇₃

C₄×C₈

yes

—

#10596831

C₄×C₈

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

C₂×C₄×C₈

yes

—

C₂×D₇₃

C₂²×C₁₆

yes

—

#64408

C₄×C₁₆

yes

complemented

C₄×C₇₃

C₄×C₁₆

yes

complemented

C₄×C₇₃

C₂×C₄×C₈

yes

—

Dic₇₃

C₄×C₁₆

yes

complemented

Dic₇₃

C₄³

yes

characteristic

Dic₇₃

C₂×C₄×C₈

yes

characteristic

C₄²×C₁₆

—

maximal, Sylow

C₂×C₇₃

C₄²×C₈

yes

characteristic

C₂×C₇₃

C₂×C₄×C₁₆

yes

—

D₇₃

C₂×C₄×C₁₆

yes

—

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

—

C₄²×C₈

—

C₇₃

C₄²×C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#10596842

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂×#10596842

yes

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂×#10596842

yes

characteristic

114

C₁₆

—

115

C₂³

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₄×#10596842

yes

—

124

C₂×C₄

C₂²×#10596842

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

C₄×#10596842

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×C₄×#10596842

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#64408⋊C₁₆

yes

direct product factor

137

C₄

C₂×C₄×#10596842

yes

—

138

C₂

—

139

C₂

C₄²×#10596842

yes

characteristic

140

C₂

C₂×#64408⋊C₁₆

yes

—

	subgroup	G/H
center, hypercenter	C₄×C₈	#10596842
commutator, hypocenter	C₇₃	C₄²×C₁₆
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#11924036	C₂³
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₄×C₈

subgroup

G/H

center, hypercenter

C₄×C₈

#10596842

commutator, hypocenter

C₇₃

C₄²×C₁₆

Frattini

C₂×C₄

C₂²×#10596842

Fitting

C₄×C₈×C₇₃

C₈

socle

C₂²×C₇₃

C₂×C₄×C₈

prime radical

C₂²×D₇₃

C₂²×C₈

C₂³

G ▷ C₇₃

C₁ ◁ C₄×C₈

N	H ≅ G/N	type
5. #64408⋊C₁₆	C₄	direct (N × H)
6. C₄×#11924040	C₄	semidirect
26. #11924040	C₄²	semidirect
32. C₄×#64408	C₁₆	semidirect
61. #64408	C₄×C₁₆	semidirect
62. C₄×C₇₃	C₄×C₁₆	semidirect
64. Dic₇₃	C₄×C₁₆	semidirect
74. C₇₃	C₄²×C₁₆	semidirect
136. C₄	#64408⋊C₁₆	direct (N × H)

H ≅ G/N

type

5. #64408⋊C₁₆

C₄

direct (N × H)

6. C₄×#11924040

C₄

semidirect

26. #11924040

C₄²

semidirect

32. C₄×#64408

C₁₆

semidirect

61. #64408

C₄×C₁₆

semidirect

62. C₄×C₇₃

C₄×C₁₆

semidirect

64. Dic₇₃

C₄×C₁₆

semidirect

74. C₇₃

C₄²×C₁₆

semidirect

136. C₄

#64408⋊C₁₆

direct (N × H)

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

subgroup	index
C₂×#64408⋊C₁₆	2
C₄×#14556686	2
C₄×C₈×#64408	2
C₄²×C₁₆	73

subgroup

index

C₂×#64408⋊C₁₆

C₄×#14556686

C₄×C₈×#64408

C₄²×C₁₆

Subgroups — C₄×#64408⋊C₁₆ #34034031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#64408⋊C16 #34034031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#64408⋊C₁₆ #34034031