Finite Group Databasebeta

← #36356777

Subgroups — #36356777

Order 22144.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C22×#14583825C211yesmaximal, characteristic, complemented
2C2×#19149641C211yesmaximal, characteristic, complemented
3C2×#19149639C212yesmaximal, complemented
4#26011411C211yesmaximal, characteristic, complemented
5#26011406C212yesmaximal, complemented
6#26011404C211yesmaximal, characteristic, complemented
7#26011400C212yesmaximal, complemented
8#26011399C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic173C212yesmaximal, complemented
10#26011338C211yesmaximal, characteristic, complemented
11#394×C173C211yesmaximal, characteristic, complemented
12C2×#14583825—28no—
13C2×#14583825—24no—
14C22×Dic346C2211yescharacteristic
15C22×D692C2211yescharacteristic
16C22×C22⋊D173C2212yes—
17C22×C22⋊D173C2212yes—
18C23×Dic173C2212yes—
19C22×C4×D173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21#19149641—24no—
22#19149639—28no—
23C2×#14583805C2212yes—
24C2×C22⋊Dic173C2212yes—
25C2×#14583803C2211yescharacteristic
26C2×#14583803C2212yes—
27C2×#14583803C2212yes—
28C2×#14583803C2211yescharacteristic
29C2×C4×Dic173C2212yes—
30C2×C4×Dic173C2211yescharacteristic
31#19149586C2211yescharacteristic
32#19149586C2212yes—
33#19149586C2212yes—
34C22×D4×C173C2211yescharacteristic
35C2×C4⋊C4×C173C2211yescharacteristic
36C2×C22⋊C4×C173C2212yes—
37C2×C22⋊C4×C173C2212yes—
38#95×C173C2211yescharacteristic
39C23×D173C2311yescharacteristic
40C23×D173C2311yescharacteristic
41#14583825—232no—
42C2×Dic346—24no—
43C2×Dic346—22no—
44C2×D692—22no—
45C2×D692—22no—
46C2×D692—22no—
47C2×C22⋊D173—24no—
48C2×C22⋊D173—24no—
49C2×C22⋊D173—24no—
50C2×C22⋊D173—24no—
51C2×C22⋊D173—24no—
52C2×C22⋊D173—24no—
53C22×Dic173—24no—
54C22×Dic173—24no—
55C22×Dic173D418yescomplemented
56C22×Dic173C2311yescharacteristic
57C22×Dic173C2312yes—
58C22×Dic173C2312yes—
59C22×Dic173C2311yescharacteristic
60C2×C4×D173—24no—
61C2×C4×D173—24no—
62C2×C4×D173—22no—
63C2×C4×D173D414yescomplemented
64#14583805—24no—
65C22⋊Dic173—24no—
66#14583803—22no—
67#14583803—24no—
68#14583803—24no—
69#14583803—22no—
70C4×Dic173—24no—
71C4×Dic173—22no—
72C24×C173C2312yes—
73C2×D4×C173—24no—
74C2×D4×C173—22no—
75C22×C4×C173C2312yes—
76C22×C4×C173C2311yescharacteristic
77C22×C4×C173C2311yescharacteristic
78C22×C4×C173C2311yescharacteristic
79C4⋊C4×C173—22no—
80C22⋊C4×C173—24no—
81C22⋊C4×C173—24no—
82C22×D173—22no—
83C22×D173—22no—
84C22×D173—21no—
85C22×D173—22no—
86C22×D173—21no—
87C22×D173—21no—
88C22×D173—21no—
89C22×D173—22no—
90C22×D173C2×D412yes—
91C22×D173#4112yescomplemented
92Dic346—28no—
93D692—28no—
94C22⋊D173—216no—
95C22⋊D173—216no—
96C2×Dic173—22no—
97C2×Dic173—22no—
98C2×Dic173—22no—
99C2×Dic173—22no—
100C2×Dic173—216no—
101C2×Dic173—21no—
102C2×Dic173—22no—
103C2×Dic173C2×D414yes—
104C2×Dic173C2×D414yes—
105C2×Dic173C2×D412yes—
106C2×Dic173#4114yescomplemented
107C4×D173—28no—
108C4×D173—44no—
109C23×C173—22no—
110C23×C173—22no—
111C23×C173—22no—
112C23×C173—22no—
113C23×C173—22no—
114C23×C173—22no—
115C23×C173C2×D414yes—
116C23×C173C2411yescharacteristic
117D4×C173—28no—
118C2×C4×C173—22no—
119C2×C4×C173—21no—
120C2×C4×C173—21no—
121C2×C4×C173—22no—
122C2×C4×C173—21no—
123C2×C4×C173—22no—
124C2×C4×C173—22no—
125C2×C4×C173—22no—
126C2×C4×C173C2×D412yes—
127C2×C4×C173#4112yescomplemented
128C22×C173—24no—
129C22×C173—24no—
130C22×C173—24no—
131C22×C173—42no—
132C22×C173—42no—
133C22×C173—42no—
134C22×C173—42no—
135C22×C173C22×D412yes—
136C22×C173C22×D411yescharacteristic
137C22×C173C2×#4111yescharacteristic
138C22×C173C2×#4112yes—
139C22×C173C2×#4111yescharacteristic
140C2×D173—22no—
141C2×D173—24no—
142C2×D173—22no—
143C2×D173—22no—
144C2×D173—22no—
145C2×D173—41no—
146C2×D173—42no—
147C2×D173—42no—
148C2×D173—42no—
149C2×D173—41no—
150C4×C173—22no—
151C4×C173—22no—
152C4×C173—41no—
153C4×C173—42no—
154Dic173—24no—
155Dic173—24no—
156Dic173—41no—
157Dic173#12014yes—
158C2×C173—44no—
159C2×C173C2×#12011yescharacteristic
160C2×C173#52512yes—
161C2×C173#52511yescharacteristic
162C2×C173#53212yes—
163C2×C173#53211yescharacteristic
164D173—42no—
165D173—42no—
166C173#238811yescharacteristic, complemented, Sylow
167#2388—1731nomaximal, Sylow
168C22×#41—1731no—
169C2×C4×D4—1731no—
170C2×C4×D4—1732no—
171#395—1731no—
172#394—1732no—
173#394D17311yescharacteristic, complemented
174#393—1731no—
175#392—1731no—
176#392—1732no—
177C4×C22⋊C4—1732no—
178C4×C22⋊C4—1731no—
179C2×#41—3468no—
180C2×#41—3464no—
181C22×Q8—1731no—
182C22×D4—1732no—
183C22×D4—1732no—
184C22×D4—1731no—
185C22×D4C2×D17311yescharacteristic
186C23×C4—1731no—
187C23×C4—1732no—
188C23×C4—1731no—
189C4×D4—3468no—
190C4×D4—3464no—
191C2×C4⋊C4—1732no—
192C2×C4⋊C4C2×D17311yescharacteristic
193C2×C22⋊C4—1732no—
194C2×C22⋊C4—1732no—
195C2×C22⋊C4—1731no—
196C2×C22⋊C4—1731no—
197C2×C22⋊C4—1732no—
198C2×C22⋊C4C2×D17312yes—
199C2×C22⋊C4C2×D17312yes—
200C2×C42—1731no—
201C2×C42—1732no—
202#95—1731no—
203#95—1732no—
204#95—1732no—
205#95C2×D17311yescharacteristic
206C24—1731no—
207C24—1731no—
208C24C22×D17312yes—
209#41—34632no—
210C2×Q8—3464no—
211C2×Q8—3462no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—3462no—
215C2×D4—3462no—
216C2×D4—3464no—
217C2×D4—3464no—
218C2×D4—3464no—
219C2×D4—3464no—
220C2×D4—3464no—
221C2×D4—3464no—
222C2×D4—3462no—
223C22×C4—3464no—
224C22×C4—3464no—
225C22×C4—3462no—
226C22×C4—3464no—
227C22×C4—3464no—
228C22×C4—1731no—
229C22×C4—1738no—
230C22×C4—1732no—
231C22×C4—1732no—
232C22×C4—1734no—
233C22×C4—1731no—
234C22×C4C22×D17312yes—
235C22×C4C22×D17311yescharacteristic
236C22×C4C22×D17311yescharacteristic
237C22×C4C22×D17311yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—3464no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—3462no—
243C22⋊C4—3464no—
244C22⋊C4—3464no—
245C22⋊C4—3464no—
246C22⋊C4—3462no—
247C42—3462no—
248C42—3464no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—3462no—
256C23—3461no—
257C23—3462no—
258C23—3462no—
259C23—3462no—
260C23—3461no—
261C23—3461no—
262C23—3461no—
263C23—1732no—
264C23—1732no—
265C23D4×D17314yes—
266C23C23×D17311yescharacteristic
267Q8—3468no—
268D4—28no—
269D4—3468no—
270D4—34616no—
271D4—34616no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—3468no—
281C2×C4—3462no—
282C2×C4—3462no—
283C2×C4—3462no—
284C2×C4—3462no—
285C2×C4—34616no—
286C2×C4—3461no—
287C2×C4—3462no—
288C2×C4—1734no—
289C2×C4—1734no—
290C2×C4—1734no—
291C2×C4—1732no—
292C2×C4—6924no—
293C2×C4D4×D17312yes—
294C2×C4#1458382512yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—3462no—
303C22—3464no—
304C22—3462no—
305C22—3462no—
306C22—3462no—
307C22—6921no—
308C22—6922no—
309C22—6922no—
310C22—6922no—
311C22—6921no—
312C22C2×D4×D17312yes—
313C22C2×D4×D17311yescharacteristic
314C22#41×D17311yescharacteristic
315C22#41×D17312yes—
316C22C2×#1458382511yescharacteristic
317C4—22no—
318C4—22no—
319C4—41no—
320C4—42no—
321C4—3464no—
322C4—3464no—
323C4—1734no—
324C4—6921no—
325C2—44no—
326C2—6922no—
327C2—6922no—
328C2#120×D17311yescharacteristic
329C2#2601199912yes—
330C2#2601200111yescharacteristic
331C2#2601203612yes—
332C2#2601204011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#394×C173C2
hypercenter#394D173
hypocenterC173#2388
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#14583825C2semidirect
2. C2×#19149641C2semidirect
3. C2×#19149639C2semidirect
4. #26011411C2semidirect
5. #26011406C2semidirect
6. #26011404C2semidirect
7. #26011400C2semidirect
8. #26011399C2semidirect
9. C22⋊C4×Dic173C2semidirect
10. #26011338C2semidirect
11. #394×C173C2semidirect
55. C22×Dic173D4semidirect
63. C2×C4×D173D4semidirect
91. C22×D173#41semidirect
106. C2×Dic173#41semidirect
127. C2×C4×C173#41semidirect
166. C173#2388semidirect
173. #394D173semidirect
294. C2×C4#14583825semidirect

Maximal subgroups

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

subgroupindex
C22×#145838252
C2×#191496412
C2×#191496392
#260114112
#260114062
#260114042
#260114002
#260113992
C22⋊C4×Dic1732
#260113382
#394×C1732
#2388173