Group #34033972 — subgroups

Subgroups — Q₁₆×#11924040 #34033972

Order 18688.

218 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 89 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	Q₈×#11924040	C₂	1	2	yes	maximal
2	#22626995	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11924040	C₂	1	2	yes	maximal
4	C₈×#11924040	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#10596833	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11924040	—	2	2	no	—
7	C₄⋊#11924040	C₂²	1	2	yes	—
8	C₄×#11924040	—	2	2	no	—
9	C₄×#11924040	C₂²	1	1	yes	characteristic
10	C₄×#11924040	C₂²	1	1	yes	characteristic
11	Q₈×#10596833	C₂²	1	2	yes	—
12	#17989450	C₄	1	1	yes	characteristic
13	Q₈⋊#10596833	C₄	1	2	yes	—
14	C₈×#10596833	C₂²	1	1	yes	characteristic
15	Q₁₆×Dic₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924040	—	2	2	no	—
17	C₂×#11924040	—	2	1	no	—
18	C₂×#11924040	D₄	1	1	yes	characteristic
19	C₂×#11924040	D₄	1	1	yes	characteristic
20	C₄⋊#10596833	—	2	2	no	—
21	C₄⋊#10596833	C₂×C₄	1	2	yes	—
22	C₄×#10596833	—	2	2	no	—
23	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596833	C₂³	1	1	yes	characteristic
25	Q₈×Dic₇₃	C₂×C₄	1	2	yes	—
26	#14556525	C₈	1	1	yes	characteristic
27	#14556515	C₈	1	2	yes	—
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×Q₁₆×C₇₃	C₈	1	1	yes	characteristic
30	#11924040	—	2	2	no	—
31	#11924040	—	2	1	no	—
32	#11924040	—	4	2	no	—
33	#11924040	Q₁₆	1	2	yes	direct product factor
34	C₂×#10596833	—	2	1	no	—
35	C₂×#10596833	—	2	2	no	—
36	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
37	C₂×#10596833	#41	1	1	yes	characteristic
38	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
39	C₄⋊Dic₇₃	—	2	2	no	—
40	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
41	C₄×Dic₇₃	—	2	2	no	—
42	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
44	Q₁₆×C₇₃	C₁₆	1	4	yes	complemented
45	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
46	Q₈×C₇₃	—	2	2	no	—
47	Q₈×C₇₃	C₂×C₁₆	1	4	yes	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	2	no	—
51	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
52	C₂×Dic₇₃	#131	1	1	yes	characteristic
53	#10596833	—	2	2	no	—
54	#10596833	—	2	1	no	—
55	#10596833	—	4	2	no	—
56	#10596833	C₂×Q₁₆	1	1	yes	characteristic
57	#10596833	#135	1	1	yes	characteristic
58	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
59	#10596831	—	2	1	no	—
60	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₇₃	—	2	4	no	—
62	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
63	C₄×C₇₃	#504	1	1	yes	characteristic
64	Dic₇₃	—	2	1	no	—
65	Dic₇₃	—	4	2	no	—
66	Dic₇₃	C₄×Q₁₆	1	1	yes	characteristic
67	Dic₇₃	#443	1	1	yes	characteristic
68	C₁₆×Q₁₆	—	73	1	no	maximal, Sylow
69	C₂×C₇₃	C₈×Q₁₆	1	1	yes	characteristic
70	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
71	C₂×C₇₃	#2170	1	1	yes	characteristic
72	Q₈×C₁₆	—	73	2	no	—
73	C₈×Q₁₆	—	73	1	no	—
74	#1370	—	73	1	no	—
75	Q₈⋊C₁₆	—	73	2	no	—
76	C₈×C₁₆	—	73	1	no	—
77	C₇₃	C₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
78	C₈×Q₈	—	73	2	no	—
79	C₄×Q₁₆	—	73	1	no	—
80	C₄⋊C₁₆	—	73	2	no	—
81	C₄⋊C₁₆	—	146	2	no	—
82	C₄×C₁₆	—	73	1	no	—
83	C₄×C₁₆	—	73	1	no	—
84	C₄×C₁₆	—	146	2	no	—
85	#335	—	73	1	no	—
86	Q₈⋊C₈	—	73	2	no	—
87	C₈²	—	73	1	no	—
88	C₂×Q₁₆	#10596842	1	1	yes	characteristic
89	C₄×Q₈	—	73	2	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	146	2	no	—
93	C₂×C₁₆	—	146	1	no	—
94	#107	—	73	1	no	—
95	C₄⋊C₈	—	73	2	no	—
96	C₄⋊C₈	—	146	2	no	—
97	#103	—	73	2	no	—
98	C₄×C₈	—	73	1	no	—
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	—	73	1	no	—
101	C₄×C₈	—	146	2	no	—
102	C₂×Q₈	C₂×#10596842	1	2	yes	—
103	Q₁₆	#11924040	1	4	yes	direct product factor
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	146	2	no	—
108	C₂×C₈	—	146	1	no	—
109	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
110	C₄⋊C₄	—	73	2	no	—
111	C₄⋊C₄	—	146	2	no	—
112	C₄²	—	73	1	no	—
113	C₄²	—	146	2	no	—
114	C₁₆	—	292	2	no	—
115	C₁₆	—	73	2	no	—
116	C₁₆	—	146	1	no	—
117	C₁₆	—	146	2	no	—
118	Q₈	—	2	2	no	—
119	Q₈	C₂×#11924040	1	4	yes	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	146	2	no	—
124	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
125	C₈	—	292	2	no	—
126	C₈	—	73	1	no	—
127	C₈	—	73	1	no	—
128	C₈	—	146	1	no	—
129	C₈	—	146	2	no	—
130	C₈	—	146	1	no	—
131	C₈	C₂×#11924040	1	2	yes	—
132	C₂²	D₄×#10596842	1	1	yes	characteristic
133	C₄	—	2	4	no	—
134	C₄	—	292	2	no	—
135	C₄	—	73	1	no	—
136	C₄	—	73	1	no	—
137	C₄	—	146	1	no	—
138	C₄	C₂²×#11924040	1	1	yes	characteristic
139	C₄	#17989708	1	1	yes	characteristic
140	C₂	Q₁₆×#10596842	1	1	yes	characteristic
141	C₂	D₄×#11924040	1	1	yes	characteristic
142	C₂	#22627150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11924040

C₂

yes

maximal

#22626995

C₂

yes

maximal, characteristic

Q₈⋊#11924040

C₂

yes

maximal

C₈×#11924040

C₂

yes

maximal, characteristic

Q₁₆×#10596833

C₂

yes

maximal, characteristic

C₄⋊#11924040

—

C₄⋊#11924040

C₂²

yes

—

C₄×#11924040

—

C₄×#11924040

C₂²

yes

characteristic

C₄×#11924040

C₂²

yes

characteristic

Q₈×#10596833

C₂²

yes

—

#17989450

C₄

yes

characteristic

Q₈⋊#10596833

C₄

yes

—

C₈×#10596833

C₂²

yes

characteristic

Q₁₆×Dic₇₃

C₄

yes

—

—

D₄

yes

characteristic

C₂×#11924040

D₄

yes

characteristic

C₄⋊#10596833

—

C₄⋊#10596833

C₂×C₄

yes

—

C₄×#10596833

—

C₄×#10596833

C₂×C₄

yes

characteristic

C₄×#10596833

C₂³

yes

characteristic

Q₈×Dic₇₃

C₂×C₄

yes

—

#14556525

C₈

yes

characteristic

#14556515

C₈

yes

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₂×Q₁₆×C₇₃

C₈

yes

—

—

—

Q₁₆

yes

direct product factor

C₂×#10596833

—

C₂×#10596833

—

C₂×#10596833

C₂×D₄

yes

yes

C₂×C₈

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₈

yes

—

Q₁₆×C₇₃

C₁₆

yes

complemented

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄×D₄

yes

C₂×Dic₇₃

yes

—

—

—

C₂×Q₁₆

yes

yes

C₈×C₇₃

C₂×C₁₆

yes

—

#10596831

—

C₂²×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₄×Q₁₆

yes

characteristic

Dic₇₃

#443

yes

characteristic

C₁₆×Q₁₆

—

maximal, Sylow

C₂×C₇₃

C₈×Q₁₆

yes

characteristic

C₂×C₇₃

D₄×C₁₆

yes

characteristic

C₂×C₇₃

#2170

yes

characteristic

Q₈×C₁₆

—

C₈×Q₁₆

—

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₇₃

C₁₆×Q₁₆

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

146

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×Q₁₆

#10596842

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

#103

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

146

—

102

C₂×Q₈

C₂×#10596842

yes

—

103

Q₁₆

#11924040

yes

direct product factor

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₂×C₈

—

146

—

109

C₂×C₈

C₂×#10596842

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

146

—

112

C₄²

—

113

C₄²

—

146

—

114

C₁₆

—

292

—

115

C₁₆

—

116

C₁₆

—

146

—

117

C₁₆

—

146

—

118

Q₈

—

119

Q₈

C₂×#11924040

yes

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

146

—

124

C₂×C₄

C₂²×#10596842

yes

characteristic

125

C₈

—

292

—

126

C₈

—

127

C₈

—

128

C₈

—

146

—

129

C₈

—

146

—

130

C₈

—

146

—

131

C₈

C₂×#11924040

yes

—

132

C₂²

D₄×#10596842

yes

characteristic

133

C₄

—

134

C₄

—

292

—

135

C₄

—

136

C₄

—

137

C₄

—

146

—

138

C₄

C₂²×#11924040

yes

characteristic

139

C₄

#17989708

yes

characteristic

140

C₂

Q₁₆×#10596842

yes

characteristic

141

C₂

D₄×#11924040

yes

characteristic

142

C₂

#22627150

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#10596842
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₂×Q₁₆×C₇₃	C₈
socle, prime radical	C₂²×C₇₃	C₈×D₄
cosocle	C₄×#10596833	C₂³
hypercenter	C₂×Q₁₆	#10596842
hypocenter	C₇₃	C₁₆×Q₁₆
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#10596842

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#10596842

Fitting

C₂×Q₁₆×C₇₃

C₈

socle, prime radical

C₂²×C₇₃

C₈×D₄

C₂³

C₂×Q₁₆

C₇₃

C₁₆×Q₁₆

LCS

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
33. #11924040	Q₁₆	direct (N × H)
44. Q₁₆×C₇₃	C₁₆	semidirect
77. C₇₃	C₁₆×Q₁₆	semidirect
103. Q₁₆	#11924040	direct (N × H)

H ≅ G/N

type

33. #11924040

Q₁₆

direct (N × H)

44. Q₁₆×C₇₃

C₁₆

semidirect

77. C₇₃

C₁₆×Q₁₆

semidirect

103. Q₁₆

#11924040

direct (N × H)

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

subgroup	index
Q₈×#11924040	2
#22626995	2
Q₈⋊#11924040	2
C₈×#11924040	2
Q₁₆×#10596833	2
C₁₆×Q₁₆	73

subgroup

index

Q₁₆×#10596833

C₁₆×Q₁₆

Subgroups — Q₁₆×#11924040 #34033972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q16×#11924040 #34033972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#11924040 #34033972