Finite Group Databasebeta

← #36356804

Subgroups — #36356804

Order 22144.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#19149641C212yesmaximal, complemented
2C2×#19149641C211yesmaximal, characteristic, complemented
3C2×#19149637C211yesmaximal, characteristic, complemented
4#26011417C211yesmaximal, characteristic, complemented
5#26011413C212yesmaximal, complemented
6#26011409C212yesmaximal, complemented
7#26011408C211yesmaximal, characteristic, complemented
8#26011401C211yesmaximal, characteristic
9#26011342C211yesmaximal, characteristic, complemented
10#26011338C212yesmaximal, complemented
11#398×C173C211yesmaximal, characteristic, complemented
12C22×D692C2211yescharacteristic
13C22×D692C2212yes—
14C22×C4×D173C2212yes—
15C22×C4×D173C2211yescharacteristic
16#19149641—24no—
17#19149641—28no—
18#19149637—24no—
19C2×C4⋊Dic173C2211yescharacteristic
20C2×#14583805C2212yes—
21C2×#14583803C2212yes—
22C2×#14583803C2211yescharacteristic
23C2×#14583803C2212yes—
24C2×#14583803C2212yes—
25C2×#14583803C2211yescharacteristic
26C2×#14583803C2212yes—
27C2×#14583803C2212yes—
28C2×C4×Dic173C2211yescharacteristic
29C2×C4×Dic173C2212yes—
30C2×C4×Dic173C2211yescharacteristic
31#19149586C2211yescharacteristic
32#19149586C2212yes—
33C2×C4⋊C4×C173C2211yescharacteristic
34C2×C4⋊C4×C173C2212yes—
35C2×C4⋊C4×C173C2211yescharacteristic
36#95×C173C2211yescharacteristic
37#95×C173C2212yes—
38C23×D173C2311yescharacteristic
39C23×D173C2312yes—
40C2×D692—24no—
41C2×D692—24no—
42C2×D692—24no—
43C2×D692—22no—
44C2×D692—24no—
45C22×Dic173C2312yes—
46C22×Dic173C2311yescharacteristic
47C22×Dic173C2311yescharacteristic
48C22×Dic173C2311yescharacteristic
49C2×C4×D173—22no—
50C2×C4×D173—22no—
51C2×C4×D173—24no—
52C2×C4×D173—24no—
53C2×C4×D173—24no—
54C2×C4×D173D414yescomplemented
55C4⋊Dic173—22no—
56#14583805—24no—
57#14583803—24no—
58#14583803—24no—
59#14583803—22no—
60#14583803—24no—
61#14583803—22no—
62#14583803—24no—
63#14583803—24no—
64C4×Dic173—22no—
65C4×Dic173—24no—
66C4×Dic173D414yescomplemented
67C22×C4×C173C2312yes—
68C22×C4×C173C2311yescharacteristic
69C22×C4×C173C2311yescharacteristic
70C22×C4×C173C2311yescharacteristic
71C22×C4×C173C2312yes—
72C4⋊C4×C173—22no—
73C4⋊C4×C173—24no—
74C4⋊C4×C173—22no—
75C22×D173—22no—
76C22×D173—22no—
77C22×D173—21no—
78C22×D173—21no—
79C22×D173—22no—
80C22×D173—22no—
81C22×D173—22no—
82C22×D173—22no—
83C22×D173—22no—
84C22×D173—22no—
85C22×D173C2×D412yes—
86C22×D173#4114yes—
87D692—28no—
88D692—216no—
89C2×Dic173—22no—
90C2×Dic173—22no—
91C2×Dic173—21no—
92C2×Dic173—22no—
93C2×Dic173—22no—
94C2×Dic173—22no—
95C2×Dic173C2×D412yes—
96C2×Dic173C2×D412yes—
97C2×Dic173C2×D412yes—
98C2×Dic173#4112yes—
99C4×D173—28no—
100C4×D173—48no—
101C23×C173C2411yescharacteristic
102C2×C4×C173—21no—
103C2×C4×C173—22no—
104C2×C4×C173—21no—
105C2×C4×C173—22no—
106C2×C4×C173—22no—
107C2×C4×C173—22no—
108C2×C4×C173—22no—
109C2×C4×C173—21no—
110C2×C4×C173—22no—
111C2×C4×C173—22no—
112C2×C4×C173C2×D412yes—
113C2×C4×C173C2×D412yes—
114C2×C4×C173#4114yes—
115C22×C173C22×D411yescharacteristic
116C22×C173C22×D411yescharacteristic
117C22×C173C2×#4111yescharacteristic
118C22×C173C2×#4112yes—
119C22×C173C2×#4112yes—
120C2×D173—22no—
121C2×D173—24no—
122C2×D173—24no—
123C2×D173—24no—
124C2×D173—24no—
125C2×D173—42no—
126C2×D173—41no—
127C2×D173—42no—
128C2×D173—42no—
129C2×D173—42no—
130C2×D173—42no—
131C2×D173—41no—
132C4×C173—22no—
133C4×C173—22no—
134C4×C173—24no—
135C4×C173—42no—
136C4×C173—41no—
137Dic173—22no—
138Dic173—42no—
139Dic173—41no—
140Dic173#12114yes—
141C2×C173C2×#12111yescharacteristic
142C2×C173#52511yescharacteristic
143C2×C173#52512yes—
144C2×C173#52912yes—
145C2×C173#53211yescharacteristic
146D173—42no—
147D173—44no—
148C173#239211yescharacteristic, complemented, Sylow
149#2392—1731nomaximal, Sylow
150C2×C4×D4—1732no—
151C2×C4×D4—1731no—
152C2×#117—1731no—
153#398D17311yescharacteristic, complemented
154#397—1731no—
155#396—1732no—
156#394—1731no—
157#394—1732no—
158#392—1731no—
159C4×C4⋊C4—1731no—
160C4×C22⋊C4—1732no—
161C22×D4—1731no—
162C22×D4—1732no—
163C23×C4—1732no—
164C23×C4—1731no—
165C4×D4—3468no—
166C4×D4—3464no—
167#117—3464no—
168C2×C4⋊C4—1731no—
169C2×C4⋊C4—1732no—
170C2×C4⋊C4C2×D17311yescharacteristic
171C2×C4⋊C4C2×D17312yes—
172C2×C4⋊C4C2×D17311yescharacteristic
173C2×C22⋊C4—1731no—
174C2×C22⋊C4—1731no—
175C2×C22⋊C4—1732no—
176C2×C22⋊C4—1732no—
177C2×C22⋊C4—1732no—
178C2×C22⋊C4—1732no—
179C2×C22⋊C4—1732no—
180C2×C42—1731no—
181C2×C42—1732no—
182C2×C42—1731no—
183#95—1732no—
184#95—1731no—
185#95C2×D17312yes—
186#95C2×D17311yescharacteristic
187C24—1731no—
188C24—1732no—
189C2×D4—3464no—
190C2×D4—3464no—
191C2×D4—3464no—
192C2×D4—3462no—
193C2×D4—3464no—
194C22×C4—3462no—
195C22×C4—3462no—
196C22×C4—3464no—
197C22×C4—3464no—
198C22×C4—3464no—
199C22×C4—1732no—
200C22×C4—1731no—
201C22×C4—1731no—
202C22×C4—1734no—
203C22×C4—1731no—
204C22×C4C22×D17312yes—
205C22×C4C22×D17311yescharacteristic
206C22×C4C22×D17311yescharacteristic
207C22×C4C22×D17311yescharacteristic
208C22×C4C22×D17312yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—3462no—
213C4⋊C4—3464no—
214C22⋊C4—3464no—
215C22⋊C4—3464no—
216C22⋊C4—3462no—
217C22⋊C4—3464no—
218C22⋊C4—3462no—
219C22⋊C4—3464no—
220C22⋊C4—3464no—
221C42—3462no—
222C42—3464no—
223C42—1734no—
224C23—3461no—
225C23—3462no—
226C23—3462no—
227C23—3461no—
228C23—3462no—
229C23—3462no—
230C23—3462no—
231C23—3462no—
232C23—3462no—
233C23—3462no—
234C23—1732no—
235C23—1734no—
236C23C23×D17311yescharacteristic
237D4—3468no—
238D4—34616no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—3468no—
250C2×C4—3462no—
251C2×C4—3462no—
252C2×C4—3461no—
253C2×C4—3462no—
254C2×C4—3462no—
255C2×C4—3462no—
256C2×C4—1732no—
257C2×C4—1732no—
258C2×C4—1732no—
259C2×C4—1732no—
260C2×C4—6928no—
261C2×C4D4×D17312yes—
262C2×C4D4×D17312yes—
263C2×C4#1458382714yes—
264C22—3462no—
265C22—3464no—
266C22—3464no—
267C22—3464no—
268C22—3464no—
269C22—6922no—
270C22—6921no—
271C22—6922no—
272C22—6922no—
273C22—6922no—
274C22—6921no—
275C22—6922no—
276C22C2×D4×D17311yescharacteristic
277C22C2×D4×D17311yescharacteristic
278C22#41×D17311yescharacteristic
279C22#41×D17312yes—
280C22C2×#1458382712yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—3462no—
287C4—1734no—
288C4—6922no—
289C4—6921no—
290C2—6922no—
291C2—6924no—
292C2#121×D17311yescharacteristic
293C2#2601199911yescharacteristic
294C2#2601200412yes—
295C2#2601202612yes—
296C2#2601204211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#398×C173C2
prime radical#26011401C2
hypercenter#398D173
hypocenterC173#2392
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#19149641C2semidirect
2. C2×#19149641C2semidirect
3. C2×#19149637C2semidirect
4. #26011417C2semidirect
5. #26011413C2semidirect
6. #26011409C2semidirect
7. #26011408C2semidirect
9. #26011342C2semidirect
10. #26011338C2semidirect
11. #398×C173C2semidirect
54. C2×C4×D173D4semidirect
66. C4×Dic173D4semidirect
148. C173#2392semidirect
153. #398D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496412
C2×#191496372
#260114172
#260114132
#260114092
#260114082
#260114012
#260113422
#260113382
#398×C1732
#2392173