Group #34011900 — subgroups

Subgroups — #34011900

Order 18688.

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

normal subgroups only

This group has 218 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	#22626345	C₂	1	2	yes	maximal
2	#22624960	C₂	1	1	yes	maximal, characteristic
3	#22624898	C₂	1	2	yes	maximal
4	C₈×#11924031	C₂	1	1	yes	maximal, characteristic
5	#22608737	C₂	1	1	yes	maximal, characteristic
6	#17989508	—	2	2	no	—
7	C₄⋊#11924031	C₂²	1	2	yes	—
8	#17989473	—	2	2	no	—
9	C₄×#11924031	C₂²	1	1	yes	characteristic
10	C₄×#11924031	C₂²	1	1	yes	characteristic
11	#17988711	C₂²	1	2	yes	—
12	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
13	#17988317	C₄	1	2	yes	—
14	C₈×#10596831	C₂²	1	1	yes	characteristic
15	#335×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	C₂×Dic₂₉₂	C₈	1	1	yes	characteristic
21	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
22	#14556525	C₈	1	1	yes	characteristic
23	#14556520	—	2	2	no	—
24	#14556516	C₈	1	2	yes	—
25	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₄×#10596831	C₂³	1	1	yes	characteristic
27	C₈×Dic₇₃	—	2	2	no	—
28	C₄⋊C₈×C₇₃	C₂×C₄	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×Dic₁₄₆	C₂×C₈	1	2	yes	—
35	Dic₂₉₂	C₁₆	1	4	yes	complemented
36	C₂×#10596831	—	2	1	no	—
37	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
38	C₂×#10596831	#41	1	1	yes	characteristic
39	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
40	#11924007	—	2	2	no	—
41	C₄×Dic₇₃	—	2	2	no	—
42	C₂×C₈×C₇₃	—	2	2	no	—
43	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
49	C₂×Dic₇₃	—	2	2	no	—
50	C₂×Dic₇₃	—	2	2	no	—
51	Dic₁₄₆	—	2	2	no	—
52	Dic₁₄₆	C₂×C₁₆	1	4	yes	—
53	C₈×C₇₃	—	2	1	no	—
54	C₈×C₇₃	—	4	2	no	—
55	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
56	#10596831	—	2	2	no	—
57	#10596831	—	2	1	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	C₄×Q₁₆	1	1	yes	characteristic
63	C₄×C₇₃	#443	1	1	yes	characteristic
64	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×C₇₃	#504	1	1	yes	characteristic
66	Dic₇₃	—	2	4	no	—
67	Dic₇₃	—	4	2	no	—
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	#64408	1	1	yes	characteristic
86	Q₈⋊C₈	—	73	2	no	—
87	C₈²	—	73	1	no	—
88	C₂×Q₁₆	—	73	1	no	—
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	—	146	2	no	—
96	C₄⋊C₈	C₂×#64408	1	2	yes	—
97	#103	—	73	2	no	—
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	—	73	2	no	—
103	Q₁₆	—	73	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	146	1	no	—
108	C₂×C₈	#11924036	1	1	yes	characteristic
109	C₂×C₈	#11924036	1	1	yes	characteristic
110	C₄⋊C₄	—	73	2	no	—
111	C₄⋊C₄	—	146	2	no	—
112	C₄²	—	146	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	—	73	4	no	—
119	Q₈	—	146	2	no	—
120	C₂×C₄	—	146	2	no	—
121	C₂×C₄	—	146	2	no	—
122	C₂×C₄	D₄×#64408	1	1	yes	characteristic
123	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
124	C₂×C₄	#14556679	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	4	2	no	—
127	C₈	—	73	1	no	—
128	C₈	—	73	1	no	—
129	C₈	—	146	2	no	—
130	C₈	—	146	1	no	—
131	C₈	#14556664	1	2	yes	—
132	C₂²	#17989615	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	292	2	no	—
135	C₄	—	146	4	no	—
136	C₄	Q₁₆×#64408	1	1	yes	characteristic
137	C₄	#17989629	1	1	yes	characteristic
138	C₄	C₂×#14556664	1	1	yes	characteristic
139	C₄	#17989646	1	1	yes	characteristic
140	C₂	#22625519	1	1	yes	characteristic
141	C₂	#22626310	1	1	yes	characteristic
142	C₂	#22626320	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626345

C₂

yes

maximal

#22624960

C₂

yes

maximal, characteristic

#22624898

C₂

yes

maximal

C₈×#11924031

C₂

yes

maximal, characteristic

#22608737

C₂

yes

maximal, characteristic

#17989508

—

C₄⋊#11924031

C₂²

yes

—

#17989473

—

C₄×#11924031

C₂²

yes

characteristic

C₄×#11924031

C₂²

yes

characteristic

#17988711

C₂²

yes

—

C₄×Dic₂₉₂

C₄

yes

characteristic

#17988317

C₄

yes

—

C₈×#10596831

C₂²

yes

characteristic

#335×C₇₃

C₄

yes

—

—

D₄

yes

characteristic

C₂×#11924031

D₄

yes

characteristic

C₂×Dic₂₉₂

C₈

yes

characteristic

C₄×Dic₁₄₆

C₂×C₄

yes

—

#14556525

C₈

yes

characteristic

#14556520

—

#14556516

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂³

yes

characteristic

C₈×Dic₇₃

—

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

Q₁₆

yes

complemented

C₂×Dic₁₄₆

C₂×C₈

yes

—

Dic₂₉₂

C₁₆

yes

complemented

C₂×#10596831

—

C₂×#10596831

C₂×D₄

yes

yes

C₄⋊Dic₇₃

C₂×C₈

yes

—

#11924007

—

C₄×Dic₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

#131

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

C₂×C₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

—

#10596831

—

#10596831

—

#10596831

C₂×Q₁₆

yes

yes

C₂²×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄×Q₁₆

yes

characteristic

C₄×C₇₃

#443

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

Dic₇₃

—

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

#64408

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

#107

—

C₄⋊C₈

—

146

—

C₄⋊C₈

C₂×#64408

yes

—

#103

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

146

—

101

C₄×C₈

C₂×#64408

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₂×C₈

#11924036

yes

characteristic

109

C₂×C₈

#11924036

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

146

—

112

C₄²

—

146

—

113

C₄²

C₂²×#64408

yes

characteristic

114

C₁₆

—

292

—

115

C₁₆

—

116

C₁₆

—

146

—

117

C₁₆

—

146

—

118

Q₈

—

119

Q₈

—

146

—

120

C₂×C₄

—

146

—

121

C₂×C₄

—

146

—

122

C₂×C₄

D₄×#64408

yes

characteristic

123

C₂×C₄

C₂×#11924036

yes

characteristic

124

C₂×C₄

#14556679

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

146

—

130

C₈

—

146

—

131

C₈

#14556664

yes

—

132

C₂²

#17989615

yes

characteristic

133

C₄

—

134

C₄

—

292

—

135

C₄

—

146

—

136

C₄

Q₁₆×#64408

yes

characteristic

137

C₄

#17989629

yes

characteristic

138

C₄

C₂×#14556664

yes

characteristic

139

C₄

#17989646

yes

characteristic

140

C₂

#22625519

yes

characteristic

141

C₂

#22626310

yes

characteristic

142

C₂

#22626320

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	#335×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₈×D₄
cosocle	C₄×#10596831	C₂³
hypercenter	#335	#64408
hypocenter	C₇₃	C₁₆×Q₁₆
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#64408

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₄²

C₂²×#64408

Fitting

#335×C₇₃

C₄

socle, prime radical

C₂²×C₇₃

C₈×D₄

C₂³

C₇₃

C₁₆×Q₁₆

LCS

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
33. #11924031	Q₁₆	semidirect
35. Dic₂₉₂	C₁₆	semidirect
77. C₇₃	C₁₆×Q₁₆	semidirect

H ≅ G/N

type

33. #11924031

Q₁₆

semidirect

35. Dic₂₉₂

C₁₆

semidirect

77. C₇₃

C₁₆×Q₁₆

semidirect

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

subgroup	index
#22626345	2
#22624960	2
#22624898	2
C₈×#11924031	2
#22608737	2
C₁₆×Q₁₆	73

subgroup

index

C₁₆×Q₁₆

Subgroups — #34011900

Conjugacy classes of subgroups

Complements

Maximal subgroups