Finite Group Databasebeta

← #37549745

Subgroups — #37549745

Order 23168.

378 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 70 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×#19153895C211yesmaximal, characteristic, complemented
2C2×#19153875C212yesmaximal
3#26042222C211yesmaximal, characteristic, complemented
4#26042157C212yesmaximal, complemented
5C2×#19153801C211yesmaximal, characteristic
6#19153895—24no—
7C2×C4⋊#10624382—21no—
8C2×C22⋊#10624382C2212yes—
9C2×C22⋊#10624382C2212yes—
10C2×C4×#10624382—21no—
11#19153875—24no—
12(C2×#10624382)⋊C4—21no—
13(C2×#10624382)⋊C4—21no—
14C22×D724C2211yescharacteristic
15C22×C4×D181C2211yescharacteristic
16#19153801—21no—
17#19153801—21no—
18#19153801—21no—
19#19153801—21no—
20C2×#15668428C411yescharacteristic, complemented
21C2×#15668426C411yescharacteristic, complemented
22C2×#15668426C411yescharacteristic, complemented
23C2×#15668426C2211yescharacteristic
24C2×C4⋊C4×C181C411yescharacteristic, complemented
25C22×#10624382—22no—
26C22×#10624382—22no—
27C4⋊#10624382—24no—
28C22⋊#10624382—22no—
29C22⋊#10624382—22no—
30C22⋊#10624382—22no—
31C22⋊#10624382D414yes—
32C4×#10624382—24no—
33C23×D181C2×C411yescharacteristic
34C23×D181C2311yescharacteristic
35C2×D724—21no—
36C2×D724—21no—
37C2×D724—22no—
38C2×D724—22no—
39C22×Dic181C2×C411yescharacteristic
40C22×Dic181C2×C411yescharacteristic
41C2×C4×D181—22no—
42C2×C4×D181—22no—
43C2×C4×D181D411yescharacteristic
44C2×C4×D181D411yescharacteristic
45C2×C4×D181D411yescharacteristic
46C2×C4×D181D411yescharacteristic
47#15668428—21no—
48#15668428—21no—
49#15668426—21no—
50#15668426—21no—
51#15668426—21no—
52#15668426—21no—
53#15668426—21no—
54#15668426—21no—
55C22×C4×C181C2×C411yescharacteristic
56C22×C4×C181C2×C411yescharacteristic
57C22×C4×C181C2×C411yescharacteristic
58C4⋊C4×C181—21no—
59C4⋊C4×C181—21no—
60C2×#10624382—24no—
61C2×#10624382—42no—
62C2×#10624382—42no—
63C2×#10624382—42no—
64C2×#10624382—42no—
65C2×#10624382—42no—
66C22×D181—21no—
67C22×D181—22no—
68C22×D181—41no—
69C22×D181—41no—
70C22×D181—41no—
71C22×D181—41no—
72C22×D181C2×D411yescharacteristic
73C22×D181C2×D411yescharacteristic
74C22×D181C2×D412yes—
75C22×D181#4112yes—
76D724—41no—
77D724—42no—
78D724—41no—
79C2×Dic181—22no—
80C2×Dic181—21no—
81C2×Dic181—41no—
82C2×Dic181C22⋊C411yescharacteristic
83C2×Dic181C22⋊C411yescharacteristic
84C4×D181—21no—
85C4×D181—21no—
86C4×D181—22no—
87C4×D181—22no—
88C4×D181—21no—
89C4×D181—21no—
90C23×C181C22×C411yescharacteristic
91C2×C4×C181—22no—
92C2×C4×C181—21no—
93C2×C4×C181—21no—
94C2×C4×C181—22no—
95C2×C4×C181—41no—
96C2×C4×C181C22⋊C411yescharacteristic
97C2×C4×C181C22⋊C411yescharacteristic
98C22×C181—21no—
99C22×C181—21no—
100C22×C181C2×C22⋊C411yescharacteristic
101C22×C181C4×D412yes—
102C2×D181—21no—
103C2×D181—21no—
104C2×D181—21no—
105C2×D181—21no—
106C2×D181—42no—
107C2×D181—42no—
108C2×D181—41no—
109C2×D181—41no—
110C2×D181—41no—
111C2×D181—82no—
112C2×D181#12011yescharacteristic
113C2×D181#12112yes—
114C2×D181#12311yescharacteristic
115#10624382—44no—
116#10624382—82no—
117C4×C181—21no—
118C4×C181—21no—
119C4×C181—41no—
120C4×C181—41no—
121Dic181—21no—
122Dic181—21no—
123Dic181—41no—
124C2×C181—22no—
125C2×C181#38611yescharacteristic
126C2×C181#41011yescharacteristic
127C2×C181#41011yescharacteristic
128D181—21no—
129D181—22no—
130D181—21no—
131D181—81no—
132C181#189911yescharacteristic, complemented, Sylow
133#1899—1811nomaximal, Sylow
134C2×#123—1811no—
135C2×#117—1811no—
136C2×#99—1812no—
137#381—1811no—
138#342—1812no—
139C22×D4—1811no—
140C23×C4—1811no—
141#123—3621no—
142#123—3621no—
143#123—3621no—
144#123—3621no—
145#117—3624no—
146C2×C4⋊C4—3621no—
147C2×C4⋊C4—1811no—
148C2×C4⋊C4#1062438211yescharacteristic, complemented
149C2×C22⋊C4—1812no—
150C2×C22⋊C4—1811no—
151C2×C22⋊C4—1811no—
152C2×C22⋊C4—1811no—
153C2×C22⋊C4—1812no—
154C2×C42—3621no—
155#99—3624no—
156#95—3621no—
157#95—3621no—
158C24—1811no—
159C24—1811no—
160C2×D4—3622no—
161C2×D4—3621no—
162C2×D4—3621no—
163C2×D4—3622no—
164C22×C4—3622no—
165C22×C4—3622no—
166C22×C4—3622no—
167C22×C4—3622no—
168C22×C4—1811no—
169C22×C4—1811no—
170C22×C4—1811no—
171C22×C4—1811no—
172C22×C4—1811no—
173C22×C4—1811no—
174C22×C4C2×#1062438211yescharacteristic
175C22×C4C2×#1062438211yescharacteristic
176C22×C4C2×#1062438211yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—3621no—
180C4⋊C4—3621no—
181C4⋊C4—3624no—
182C22⋊C4—3621no—
183C22⋊C4—3621no—
184C22⋊C4—3621no—
185C22⋊C4—3621no—
186C22⋊C4—3622no—
187C22⋊C4—3621no—
188C22⋊C4—3621no—
189C22⋊C4—3622no—
190C22⋊C4—3622no—
191C22⋊C4—1814no—
192C42—3624no—
193C23—3622no—
194C23—3621no—
195C23—1811no—
196C23—1811no—
197C23—1812no—
198C23—1812no—
199C23—7241no—
200C23—7241no—
201C23—7241no—
202C23—7241no—
203C23C22×#1062438211yescharacteristic
204D4—7241no—
205D4—7242no—
206D4—7241no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—3621no—
213C2×C4—3621no—
214C2×C4—3622no—
215C2×C4—3622no—
216C2×C4—3621no—
217C2×C4—3621no—
218C2×C4—3622no—
219C2×C4—3624no—
220C2×C4—3621no—
221C2×C4—1811no—
222C2×C4—1811no—
223C2×C4—7242no—
224C2×C4—7242no—
225C2×C4—7242no—
226C2×C4—7242no—
227C2×C4—7242no—
228C2×C4—7241no—
229C2×C4C22⋊#1062438211yescharacteristic
230C2×C4C22⋊#1062438211yescharacteristic
231C22—21no—
232C22—21no—
233C22—3621no—
234C22—3621no—
235C22—3621no—
236C22—3621no—
237C22—14482no—
238C22—1811no—
239C22—1812no—
240C22—1811no—
241C22—7242no—
242C22—7242no—
243C22—7241no—
244C22—7241no—
245C22—7241no—
246C22C2×C22⋊#1062438211yescharacteristic
247C22D4×#1062438212yes—
248C4—21no—
249C4—21no—
250C4—41no—
251C4—41no—
252C4—3621no—
253C4—3621no—
254C4—14482no—
255C4—7244no—
256C4—7241no—
257C2—22no—
258C2—3622no—
259C2—3621no—
260C2—3621no—
261C2—14481no—
262C2#2604222711yescharacteristic
263C2#2604225811yescharacteristic
264C2#2604226011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
FittingC2×C4⋊C4×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC22×D724C22
cosocleC23×D181C23
hypercenterC2×C4⋊C4#10624382
hypocenterC181#1899
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19153895C2semidirect
3. #26042222C2semidirect
4. #26042157C2semidirect
20. C2×#15668428C4semidirect
21. C2×#15668426C4semidirect
22. C2×#15668426C4semidirect
24. C2×C4⋊C4×C181C4semidirect
132. C181#1899semidirect
148. C2×C4⋊C4#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538952
C2×#191538752
#260422222
#260421572
C2×#191538012
#1899181