Finite Group Databasebeta

← #36356802

Subgroups — #36356802

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×D4×Dic173C211yesmaximal, characteristic, complemented
2C2×#19149639C212yesmaximal, complemented
3C2×#19149637C211yesmaximal, characteristic, complemented
4#26011421C211yesmaximal, characteristic, complemented
5#26011418C211yesmaximal, characteristic, complemented
6#26011414C212yesmaximal, complemented
7#26011406C212yesmaximal, complemented
8#26011399C211yesmaximal, characteristic
9#26011342C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic173C212yesmaximal, complemented
11#394×C173C211yesmaximal, characteristic, complemented
12C22×C22⋊D173C2212yes—
13C23×Dic173C2212yes—
14C22×C4×D173C2211yescharacteristic
15D4×Dic173—24no—
16#19149639—28no—
17#19149637—24no—
18C2×C4⋊Dic173C2211yescharacteristic
19C2×C4⋊Dic173C2211yescharacteristic
20C2×#14583805C2212yes—
21C2×#14583805C2212yes—
22C2×C22⋊Dic173C2212yes—
23C2×C22⋊Dic173C2212yes—
24C2×#14583803C2211yescharacteristic
25C2×#14583803C2211yescharacteristic
26C2×#14583803C2212yes—
27C2×C4×Dic173C2211yescharacteristic
28C2×C4×Dic173C2212yes—
29C2×C4×Dic173C2211yescharacteristic
30#19149586C2211yescharacteristic
31#19149586C2212yes—
32#19149586C2212yes—
33C22×D4×C173C2211yescharacteristic
34C2×C4⋊C4×C173C2211yescharacteristic
35C2×C22⋊C4×C173C2212yes—
36C2×C22⋊C4×C173C2212yes—
37#95×C173C2211yescharacteristic
38C23×D173C2311yescharacteristic
39C2×C22⋊D173—24no—
40C2×C22⋊D173—24no—
41C2×C22⋊D173—24no—
42C22×Dic173—24no—
43C22×Dic173—24no—
44C22×Dic173—24no—
45C22×Dic173C2311yescharacteristic
46C22×Dic173C2312yes—
47C22×Dic173C2311yescharacteristic
48C22×Dic173C2312yes—
49C22×Dic173C2311yescharacteristic
50C2×C4×D173—22no—
51C2×C4×D173—22no—
52C2×C4×D173D414yescomplemented
53C4⋊Dic173—22no—
54C4⋊Dic173—22no—
55#14583805—24no—
56#14583805—24no—
57C22⋊Dic173—24no—
58C22⋊Dic173—24no—
59#14583803—24no—
60#14583803—22no—
61#14583803—22no—
62C4×Dic173—22no—
63C4×Dic173—24no—
64C4×Dic173D414yescomplemented
65C24×C173C2312yes—
66C2×D4×C173—24no—
67C2×D4×C173—22no—
68C22×C4×C173C2311yescharacteristic
69C22×C4×C173C2312yes—
70C22×C4×C173C2311yescharacteristic
71C22×C4×C173C2311yescharacteristic
72C4⋊C4×C173—22no—
73C22⋊C4×C173—24no—
74C22⋊C4×C173—24no—
75C22×D173—21no—
76C22×D173—21no—
77C22×D173—22no—
78C22×D173—22no—
79C22×D173C2×D412yes—
80C22⋊D173—216no—
81C2×Dic173—22no—
82C2×Dic173—22no—
83C2×Dic173—22no—
84C2×Dic173—22no—
85C2×Dic173—22no—
86C2×Dic173—21no—
87C2×Dic173—22no—
88C2×Dic173—22no—
89C2×Dic173—48no—
90C2×Dic173C2×D412yes—
91C2×Dic173C2×D412yes—
92C2×Dic173C2×D412yes—
93C2×Dic173#4114yes—
94C2×Dic173#4112yes—
95C4×D173—28no—
96C23×C173—22no—
97C23×C173—22no—
98C23×C173—22no—
99C23×C173—22no—
100C23×C173—22no—
101C23×C173—22no—
102C23×C173#4114yes—
103C23×C173C2411yescharacteristic
104D4×C173—28no—
105C2×C4×C173—21no—
106C2×C4×C173—21no—
107C2×C4×C173—22no—
108C2×C4×C173—22no—
109C2×C4×C173—21no—
110C2×C4×C173—22no—
111C2×C4×C173—22no—
112C2×C4×C173—22no—
113C2×C4×C173C2×D412yes—
114C2×C4×C173C2×D412yes—
115C22×C173—24no—
116C22×C173—24no—
117C22×C173—24no—
118C22×C173—42no—
119C22×C173—42no—
120C22×C173—42no—
121C22×C173—42no—
122C22×C173C22×D411yescharacteristic
123C22×C173C22×D411yescharacteristic
124C22×C173C2×#4111yescharacteristic
125C22×C173C2×#4112yes—
126C22×C173C2×#4112yes—
127C2×D173—22no—
128C2×D173—24no—
129C2×D173—41no—
130C2×D173—42no—
131C2×D173—41no—
132C4×C173—22no—
133C4×C173—22no—
134C4×C173—41no—
135C4×C173—42no—
136Dic173—24no—
137Dic173—22no—
138Dic173—42no—
139Dic173—41no—
140Dic173#12114yes—
141C2×C173—44no—
142C2×C173C2×#12111yescharacteristic
143C2×C173#52511yescharacteristic
144C2×C173#52512yes—
145C2×C173#52912yes—
146C2×C173#53211yescharacteristic
147D173—42no—
148C173#239211yescharacteristic, complemented, Sylow
149#2392—1731nomaximal, Sylow
150C2×C4×D4—1731no—
151C2×C4×D4—1732no—
152C2×#117—1731no—
153#398—1731no—
154#397—1731no—
155#396—1732no—
156#394—1732no—
157#394D17311yescharacteristic, complemented
158#392—1731no—
159C4×C4⋊C4—1731no—
160C4×C22⋊C4—1732no—
161C22×D4—1732no—
162C22×D4C2×D17311yescharacteristic
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⋊C4—1731no—
171C2×C4⋊C4—1732no—
172C2×C4⋊C4C2×D17311yescharacteristic
173C2×C22⋊C4—1732no—
174C2×C22⋊C4—1732no—
175C2×C22⋊C4—1731no—
176C2×C22⋊C4—1731no—
177C2×C22⋊C4—1732no—
178C2×C22⋊C4C2×D17312yes—
179C2×C22⋊C4C2×D17312yes—
180C2×C42—1731no—
181C2×C42—1732no—
182C2×C42—1731no—
183#95—1732no—
184#95—1732no—
185#95—1731no—
186#95C2×D17311yescharacteristic
187C24—1731no—
188C24C22×D17312yes—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—3464no—
192C2×D4—3464no—
193C2×D4—3464no—
194C22×C4—3462no—
195C22×C4—3462no—
196C22×C4—3464no—
197C22×C4—3464no—
198C22×C4—3464no—
199C22×C4—1731no—
200C22×C4—1732no—
201C22×C4—1732no—
202C22×C4—1731no—
203C22×C4—1734no—
204C22×C4—1731no—
205C22×C4C22×D17311yescharacteristic
206C22×C4C22×D17312yes—
207C22×C4C22×D17311yescharacteristic
208C22×C4C22×D17311yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—3462no—
211C4⋊C4—3464no—
212C4⋊C4—3464no—
213C4⋊C4—3462no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—3464no—
217C22⋊C4—3462no—
218C22⋊C4—3464no—
219C22⋊C4—3464no—
220C22⋊C4—3462no—
221C42—3462no—
222C42—3464no—
223C42—1734no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—3461no—
231C23—3461no—
232C23—3462no—
233C23—3462no—
234C23—1732no—
235C23#1458382514yes—
236C23C23×D17311yescharacteristic
237D4—28no—
238D4—34616no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—3468no—
248C2×C4—3462no—
249C2×C4—3462no—
250C2×C4—3462no—
251C2×C4—3461no—
252C2×C4—3462no—
253C2×C4—3462no—
254C2×C4—3462no—
255C2×C4—3462no—
256C2×C4—1732no—
257C2×C4—1734no—
258C2×C4—1732no—
259C2×C4—1732no—
260C2×C4—1732no—
261C2×C4—6928no—
262C2×C4D4×D17312yes—
263C2×C4D4×D17312yes—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—3462no—
272C22—3464no—
273C22—6921no—
274C22—6922no—
275C22—6921no—
276C22C2×D4×D17311yescharacteristic
277C22C2×D4×D17311yescharacteristic
278C22#41×D17311yescharacteristic
279C22#41×D17312yes—
280C22C2×#1458382512yes—
281C4—22no—
282C4—22no—
283C4—41no—
284C4—42no—
285C4—3464no—
286C4—3462no—
287C4—1734no—
288C4—6922no—
289C4—6921no—
290C2—44no—
291C2—6922no—
292C2#121×D17311yescharacteristic
293C2#2601199711yescharacteristic
294C2#2601200112yes—
295C2#2601201812yes—
296C2#2601204011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#394×C173C2
prime radical#26011399C2
hypercenter#394D173
hypocenterC173#2392
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic173C2semidirect
2. C2×#19149639C2semidirect
3. C2×#19149637C2semidirect
4. #26011421C2semidirect
5. #26011418C2semidirect
6. #26011414C2semidirect
7. #26011406C2semidirect
9. #26011342C2semidirect
10. C22⋊C4×Dic173C2semidirect
11. #394×C173C2semidirect
52. C2×C4×D173D4semidirect
64. C4×Dic173D4semidirect
148. C173#2392semidirect
157. #394D173semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1732
C2×#191496392
C2×#191496372
#260114212
#260114182
#260114142
#260114062
#260113992
#260113422
C22⋊C4×Dic1732
#394×C1732
#2392173