Finite Group Databasebeta

← #37534765

Subgroups — #37534765

Order 23168.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153844C212yesmaximal, complemented
2#26041860C211yesmaximal, characteristic, complemented
3C2×#19153742C212yesmaximal, complemented
4#26041180C211yesmaximal, characteristic, complemented
5#343×C181C211yesmaximal, characteristic, complemented
6C2×#15668449C2212yes—
7C22×D724C2211yescharacteristic
8#19153844—24no—
9#19153844—24no—
10C2×#15668438—22no—
11C2×D1448—22no—
12#19153796—22no—
13#19153791—22no—
14#19153790—22no—
15C2×#15668426—22no—
16C2×C4×Dic181C2211yescharacteristic
17#19153742—24no—
18C2×M16×C181C2212yes—
19C2×C4⋊C4×C181C2211yescharacteristic
20C23×D181—21no—
21#15668449—22no—
22#15668449—22no—
23#15668449—22no—
24#15668449—22no—
25C2×Dic362D412yes—
26C2×D724—22no—
27C2×D724—22no—
28C2×D724—22no—
29C2×D724—21no—
30C2×D724D412yes—
31C2×D724D412yescomplemented
32C2×C22⋊D181—22no—
33C22×Dic181—21no—
34C2×C4×D181—22no—
35#15668438—28no—
36D1448—28no—
37C4⋊Dic181—22no—
38#15668426—24no—
39#15668426—44no—
40C4×Dic181—22no—
41C22×C4×C181D412yes—
42C22×C4×C181C2311yescharacteristic
43M16×C181—22no—
44M16×C181D414yescomplemented
45C2×C8×C181—22no—
46C4⋊C4×C181—22no—
47C22×D181—22no—
48C22×D181—22no—
49C22×D181—42no—
50C22×D181—42no—
51C22×D181—41no—
52C22×D181—41no—
53Dic362—22no—
54Dic362—22no—
55D724—22no—
56D724—24no—
57D724—22no—
58D724—42no—
59D724—82no—
60C22⋊D181—42no—
61C22⋊D181—42no—
62C2×Dic181—22no—
63C2×Dic181—22no—
64C2×Dic181—41no—
65C2×Dic181—41no—
66C4×D181—42no—
67C4×D181—42no—
68C23×C181C2×D411yescharacteristic
69C2×C4×C181—22no—
70C2×C4×C181—42no—
71C2×C4×C181C2×D411yescharacteristic
72C2×C4×C181C2×D412yes—
73C2×C4×C181C2×D412yes—
74C2×C4×C181#4111yescharacteristic
75C8×C181—24no—
76C22×C181—21no—
77C22×C181—21no—
78C22×C181#12011yescharacteristic
79C22×C181#12111yescharacteristic
80C22×C181#12711yescharacteristic
81C2×D181—44no—
82C2×D181—42no—
83C2×D181—42no—
84C2×D181—42no—
85C2×D181—42no—
86C2×D181—82no—
87C2×D181—81no—
88C2×D181—81no—
89C4×C181—42no—
90C4×C181#12012yes—
91C4×C181#12112yes—
92Dic181—42no—
93Dic181—42no—
94C2×C181—21no—
95C2×C181—21no—
96C2×C181#39211yescharacteristic
97C2×C181#45311yescharacteristic
98C2×C181#45411yescharacteristic
99D181—82no—
100D181—82no—
101C181#200611yescharacteristic, complemented, Sylow
102#2006—1811nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1812no—
104#537—1811no—
105C2×#104—1812no—
106#390—1811no—
107#343D18111yescharacteristic, complemented
108C2×#41—1812no—
109C22×D4—1811no—
110AGL(1,ℤ/8)—3624no—
111AGL(1,ℤ/8)—3624no—
112C2×SD16—3622no—
113C2×D8—3622no—
114C2×M16C2×D18112yes—
115C22⋊Q8—3622no—
116#121—3622no—
117#121—3622no—
118C2×C4⋊C4C2×D18111yescharacteristic
119C2×C22⋊C4—3622no—
120C2×C42—1811no—
121#104—3624no—
122C24—3621no—
123#41—3622no—
124#41—3622no—
125#41—3622no—
126#41—3622no—
127C2×Q8—1812no—
128C2×D4—3622no—
129C2×D4—3622no—
130C2×D4—3622no—
131C2×D4—3622no—
132C2×D4—3621no—
133C2×D4—1812no—
134C2×D4—1812no—
135C22×C4—3621no—
136C22×C4—3622no—
137C22×C4D72412yes—
138C22×C4C22×D18111yescharacteristic
139SD16—3628no—
140D8—3628no—
141M16—22no—
142M16D72414yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—3622no—
146C22⋊C4—3624no—
147C22⋊C4—7244no—
148C42—3622no—
149C23—3622no—
150C23—3622no—
151C23—7242no—
152C23—7242no—
153C23—7241no—
154C23—7241no—
155C23C2×D72411yescharacteristic
156Q8—3622no—
157Q8—3622no—
158D4—3622no—
159D4—3624no—
160D4—3622no—
161D4—14482no—
162D4—7242no—
163D4—7242no—
164D4—7242no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—3622no—
168C2×C4—3622no—
169C2×C4—7241no—
170C2×C4—7241no—
171C2×C4—7242no—
172C2×C4—7242no—
173C2×C4D4×D18111yescharacteristic
174C2×C4D4×D18112yes—
175C2×C4C2×D72412yes—
176C2×C4#1566845011yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—14482no—
181C22—14481no—
182C22—14481no—
183C22—7244no—
184C22—7242no—
185C22—7242no—
186C22—7242no—
187C22—7242no—
188C22#1915378611yescharacteristic
189C22#1915379111yescharacteristic
190C22#1915381311yescharacteristic
191C4—42no—
192C4—7242no—
193C4—7242no—
194C4#1915378612yes—
195C4#1915379112yes—
196C2—21no—
197C2—21no—
198C2—14482no—
199C2—14482no—
200C2#2604118911yescharacteristic
201C2#2604144911yescharacteristic
202C2#2604145411yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#343×C181C2
socleC22×C181#120
hypercenter#343D181
hypocenterC181#2006
LCSG ▷ C22×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19153844C2semidirect
2. #26041860C2semidirect
3. C2×#19153742C2semidirect
4. #26041180C2semidirect
5. #343×C181C2semidirect
31. C2×D724D4semidirect
44. M16×C181D4semidirect
101. C181#2006semidirect
107. #343D181semidirect
142. M16D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
#260418602
C2×#191537422
#260411802
#343×C1812
#2006181