Group #34011905 — subgroups

Subgroups — #34011905

Order 18688.

206 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 88 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	#22626346	C₂	1	2	yes	maximal
2	#22624960	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11924031	C₂	1	2	yes	maximal
4	C₈×#11924031	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11924031	—	2	2	no	—
7	C₄⋊#11924031	C₂²	1	2	yes	—
8	#17989475	—	2	2	no	—
9	C₄×#11924031	C₂²	1	1	yes	characteristic
10	C₄×#11924031	C₂²	1	1	yes	characteristic
11	Q₈×#10596831	C₂²	1	2	yes	—
12	#17988340	C₄	1	1	yes	characteristic
13	Q₈⋊#10596831	C₄	1	2	yes	—
14	C₈×#10596831	C₂²	1	1	yes	characteristic
15	C₄×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924031	—	2	2	no	—
17	C₂×#11924031	—	2	1	no	—
18	C₂×#11924031	D₄	1	1	yes	characteristic
19	C₂×#11924031	D₄	1	1	yes	characteristic
20	C₄⋊#10596831	—	2	2	no	—
21	C₄⋊#10596831	C₂×C₄	1	2	yes	—
22	C₄×#10596831	—	2	2	no	—
23	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596831	C₂³	1	1	yes	characteristic
25	C₂×Q₁₆×C₇₃	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
27	#107×C₇₃	C₈	1	1	yes	characteristic
28	#103×C₇₃	C₈	1	2	yes	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	#11924031	—	2	2	no	—
31	#11924031	—	2	1	no	—
32	#11924031	—	4	2	no	—
33	#11924031	Q₁₆	1	2	yes	complemented
34	C₂×#10596831	—	2	1	no	—
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
37	C₂×#10596831	#41	1	1	yes	characteristic
38	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
39	Q₁₆×C₇₃	—	2	2	no	—
40	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₄⋊C₄×C₇₃	—	2	2	no	—
43	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
44	C₄²×C₇₃	—	2	2	no	—
45	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	Q₈×C₇₃	—	2	2	no	—
47	Q₈×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
53	C₈×C₇₃	—	2	1	no	—
54	C₈×C₇₃	M₃₂	1	2	yes	—
55	#10596831	—	2	2	no	—
56	#10596831	—	2	1	no	—
57	#10596831	—	4	2	no	—
58	#10596831	C₂×Q₁₆	1	1	yes	characteristic
59	#10596831	#135	1	1	yes	characteristic
60	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₇₃	—	2	1	no	—
62	C₄×C₇₃	—	4	2	no	—
63	C₄×C₇₃	—	4	2	no	—
64	C₄×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
65	C₄×C₇₃	#443	1	1	yes	characteristic
66	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
67	C₄×C₇₃	#504	1	1	yes	characteristic
68	#12054	—	73	1	no	maximal, Sylow
69	C₂×C₇₃	C₈×Q₁₆	1	1	yes	characteristic
70	C₂×C₇₃	#2169	1	1	yes	characteristic
71	C₂×C₇₃	#2170	1	1	yes	characteristic
72	#2183	—	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	#12054	1	1	yes	characteristic, complemented, Sylow
78	C₈×Q₈	—	73	2	no	—
79	C₄×Q₁₆	#64408	1	1	yes	characteristic
80	C₄⋊C₁₆	—	73	2	no	—
81	C₄⋊C₁₆	—	146	2	no	—
82	#346	—	146	2	no	—
83	C₄×C₁₆	—	73	1	no	—
84	C₄×C₁₆	—	73	1	no	—
85	#335	—	73	1	no	—
86	Q₈⋊C₈	—	73	2	no	—
87	C₈²	—	73	1	no	—
88	C₂×Q₁₆	#10596833	1	1	yes	characteristic
89	C₄×Q₈	C₂×#64408	1	2	yes	—
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	#10596833	1	1	yes	characteristic
95	C₄⋊C₈	—	73	2	no	—
96	C₄⋊C₈	—	146	2	no	—
97	#103	#10596833	1	2	yes	—
98	C₄×C₈	—	73	1	no	—
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	—	146	2	no	—
101	C₄×C₈	C₂×#64408	1	1	yes	characteristic
102	C₂×Q₈	C₂×#10596833	1	2	yes	—
103	Q₁₆	—	2	2	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	146	2	no	—
107	C₂×C₈	—	146	1	no	—
108	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
109	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂×#10596833	1	2	yes	—
112	C₄²	—	2	2	no	—
113	C₄²	C₂²×#64408	1	1	yes	characteristic
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	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	D₄×#64408	1	1	yes	characteristic
123	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
124	C₂×C₄	#14556678	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	292	2	no	—
127	C₈	—	73	1	no	—
128	C₈	—	73	1	no	—
129	C₈	—	146	2	no	—
130	C₈	—	146	1	no	—
131	C₈	#14556665	1	2	yes	—
132	C₂²	D₄×#10596833	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	2	no	—
136	C₄	Q₁₆×#64408	1	1	yes	characteristic
137	C₄	#17989631	1	1	yes	characteristic
138	C₄	C₂×#14556665	1	1	yes	characteristic
139	C₄	#17989645	1	1	yes	characteristic
140	C₂	Q₁₆×#10596833	1	1	yes	characteristic
141	C₂	#22626315	1	1	yes	characteristic
142	C₂	#22626318	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626346

C₂

yes

maximal

#22624960

C₂

yes

maximal, characteristic

Q₈⋊#11924031

C₂

yes

maximal

C₈×#11924031

C₂

yes

maximal, characteristic

Q₁₆×#10596831

C₂

yes

maximal, characteristic

C₄⋊#11924031

—

C₄⋊#11924031

C₂²

yes

—

#17989475

—

C₄×#11924031

C₂²

yes

characteristic

C₄×#11924031

C₂²

yes

characteristic

Q₈×#10596831

C₂²

yes

—

#17988340

C₄

yes

characteristic

Q₈⋊#10596831

C₄

yes

—

C₈×#10596831

C₂²

yes

characteristic

C₄×Q₁₆×C₇₃

C₄

yes

—

—

D₄

yes

characteristic

C₂×#11924031

D₄

yes

characteristic

C₄⋊#10596831

—

C₄⋊#10596831

C₂×C₄

yes

—

C₄×#10596831

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂³

yes

characteristic

C₂×Q₁₆×C₇₃

C₈

yes

characteristic

C₄×Q₈×C₇₃

C₂×C₄

yes

—

#107×C₇₃

C₈

yes

characteristic

#103×C₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

Q₁₆

yes

—

—

C₂×D₄

yes

yes

C₂×Q₈×C₇₃

C₂×C₈

yes

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

#131

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

M₃₂

yes

—

—

—

—

C₂×Q₁₆

yes

yes

C₂²×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×Q₁₆

yes

characteristic

C₄×C₇₃

#443

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

C₄×C₇₃

yes

—

C₂×C₇₃

C₈×Q₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₈×Q₁₆

—

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₇₃

#12054

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

#64408

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

#346

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×Q₁₆

#10596833

yes

characteristic

C₄×Q₈

C₂×#64408

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

#107

#10596833

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

146

—

#103

#10596833

yes

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

146

—

101

C₄×C₈

C₂×#64408

yes

characteristic

102

C₂×Q₈

C₂×#10596833

yes

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

146

—

107

C₂×C₈

—

146

—

108

C₂×C₈

C₂×#10596833

yes

characteristic

109

C₂×C₈

C₂×#10596833

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×#10596833

yes

—

112

C₄²

—

113

C₄²

C₂²×#64408

yes

characteristic

114

C₁₆

—

292

—

115

C₁₆

—

116

C₁₆

—

146

—

117

C₁₆

—

146

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

D₄×#64408

yes

characteristic

123

C₂×C₄

C₂²×#10596833

yes

characteristic

124

C₂×C₄

#14556678

yes

characteristic

125

C₈

—

126

C₈

—

292

—

127

C₈

—

128

C₈

—

129

C₈

—

146

—

130

C₈

—

146

—

131

C₈

#14556665

yes

—

132

C₂²

D₄×#10596833

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

Q₁₆×#64408

yes

characteristic

137

C₄

#17989631

yes

characteristic

138

C₄

C₂×#14556665

yes

characteristic

139

C₄

#17989645

yes

characteristic

140

C₂

Q₁₆×#10596833

yes

characteristic

141

C₂

#22626315

yes

characteristic

142

C₂

#22626318

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₄²	C₂²×#64408
Fitting	C₄×Q₁₆×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₈×D₄
cosocle	C₄×#10596831	C₂³
hypercenter	C₄×Q₁₆	#64408
hypocenter	C₇₃	#12054
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×#64408

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₄²

C₂²×#64408

Fitting

C₄×Q₁₆×C₇₃

C₄

socle, prime radical

C₂²×C₇₃

C₈×D₄

C₂³

C₄×Q₁₆

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
33. #11924031	Q₁₆	semidirect
77. C₇₃	#12054	semidirect

H ≅ G/N

type

33. #11924031

Q₁₆

semidirect

77. C₇₃

#12054

semidirect

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

subgroup	index
#22626346	2
#22624960	2
Q₈⋊#11924031	2
C₈×#11924031	2
Q₁₆×#10596831	2
#12054	73

subgroup

index

Q₁₆×#10596831

Subgroups — #34011905

Conjugacy classes of subgroups

Complements

Maximal subgroups