Finite Group Databasebeta

← #36356429

Subgroups — #36356429

Order 22144.

604 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149637C211yesmaximal, characteristic
2C2×#19149636C211yesmaximal, characteristic
3C2×C4⋊C4×D173C211yesmaximal, characteristic
4#26011388C211yesmaximal, characteristic, complemented
5#26011386C211yesmaximal, characteristic, complemented
6#26011377C211yesmaximal, characteristic, complemented
7#26011376C211yesmaximal, characteristic, complemented
8#26011374C211yesmaximal, characteristic, complemented
9#26011353C211yesmaximal, characteristic
10#26011353C211yesmaximal, characteristic
11#26011350C211yesmaximal, characteristic
12C4×#14583805C211yesmaximal, characteristic, complemented
13#26011342C211yesmaximal, characteristic, complemented
14#26011338C211yesmaximal, characteristic
15#387×C173C211yesmaximal, characteristic, complemented
16C22×C4×D173C2211yescharacteristic
17C22×C4×D173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19#19149637—24no—
20#19149636—24no—
21C2×C4⋊Dic173C2211yescharacteristic
22C2×C4⋊Dic173C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×#14583805C2211yescharacteristic
29C4⋊C4×D173C418yescomplemented
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×C4×Dic173C2211yescharacteristic
35C2×C4×Dic173C2211yescharacteristic
36C2×C4×Dic173C2211yescharacteristic
37C2×C4×Dic173C2211yescharacteristic
38C2×C4×Dic173C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45#19149586C2211yescharacteristic
46#19149586C2211yescharacteristic
47C2×C4⋊C4×C173C2211yescharacteristic
48C2×C4⋊C4×C173C2211yescharacteristic
49C2×C42×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51#95×C173C2211yescharacteristic
52#95×C173C2211yescharacteristic
53#95×C173C2211yescharacteristic
54C23×D173C2311yescharacteristic
55C22×Dic173C2311yescharacteristic
56C22×Dic173C2311yescharacteristic
57C22×Dic173C2311yescharacteristic
58C22×Dic173C2311yescharacteristic
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C2×C4×D173—22no—
63C2×C4×D173—22no—
64C2×C4×D173—22no—
65C2×C4×D173—22no—
66C2×C4×D173—22no—
67C2×C4×D173—22no—
68C2×C4×D173C2×C414yes—
69C2×C4×D173C2×C414yes—
70C2×C4×D173C2×C414yes—
71C4⋊Dic173—22no—
72C4⋊Dic173C2×C414yes—
73#14583805—22no—
74#14583805—22no—
75#14583805—22no—
76#14583805—22no—
77#14583805C2×C414yes—
78#14583805C2×C414yes—
79#14583803—22no—
80#14583803—22no—
81#14583803—22no—
82#14583803D414yes—
83C4×Dic173—22no—
84C4×Dic173—22no—
85C4×Dic173—22no—
86C4×Dic173—22no—
87C4×Dic173—22no—
88C22×C4×C173C2311yescharacteristic
89C22×C4×C173C2311yescharacteristic
90C22×C4×C173C2311yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C22×C4×C173C2311yescharacteristic
94C22×C4×C173C2311yescharacteristic
95C4⋊C4×C173—22no—
96C4⋊C4×C173C2×C414yes—
97C42×C173—22no—
98C22×D173—21no—
99C22×D173—21no—
100C22×D173—21no—
101C22×D173—21no—
102C22×D173C22×C412yes—
103C22×D173C2×D412yes—
104C22×D173#4112yes—
105C2×Dic173—21no—
106C2×Dic173—21no—
107C2×Dic173—21no—
108C2×Dic173—21no—
109C2×Dic173—21no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173C22×C412yes—
121C2×Dic173C22×C412yes—
122C2×Dic173C22×C412yes—
123C2×Dic173C2×D412yes—
124C2×Dic173#4112yes—
125C2×Dic173#4112yes—
126C4×D173—28no—
127C4×D173—28no—
128C4×D173—28no—
129C23×C173C2411yescharacteristic
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173C22×C412yes—
148C2×C4×C173C22×C412yes—
149C2×C4×C173C22×C412yes—
150C2×C4×C173C2×D412yes—
151C22×C173C23×C411yescharacteristic
152C22×C173C22×D411yescharacteristic
153C22×C173C2×#4111yescharacteristic
154C22×C173C2×#4111yescharacteristic
155C22×C173C2×#4111yescharacteristic
156C22×C1732−1+411yescharacteristic
157C22×C1732−1+411yescharacteristic
158C2×D173—22no—
159C2×D173—22no—
160C2×D173—22no—
161C2×D173—22no—
162C2×D173—22no—
163C2×D173—22no—
164C2×D173C4×D414yes—
165C4×C173—22no—
166C4×C173—22no—
167C4×C173—22no—
168C4×C173—22no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173—41no—
172Dic173—22no—
173Dic173—22no—
174Dic173—22no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178Dic173#11714yescomplemented
179C2×C173C2×#11711yescharacteristic
180C2×C173C2×C4×D411yescharacteristic
181C2×C173#51911yescharacteristic
182C2×C173#54711yescharacteristic
183C2×C173#54711yescharacteristic
184C2×C173#55211yescharacteristic
185C2×C173#55211yescharacteristic
186D173—24no—
187C173#234411yescharacteristic, complemented, Sylow
188#2344—1731nomaximal, Sylow
189C2×#117—1731no—
190C2×#117—1731no—
191C22×C4⋊C4—1731no—
192#389—1731no—
193#389—1731no—
194#387—1731no—
195#387—1731no—
196#387—1731no—
197#387D17311yescharacteristic, complemented
198#381—1731no—
199#381—1731no—
200#380—1731no—
201C4×C4⋊C4—1731no—
202C4×C4⋊C4—1731no—
203C4×C22⋊C4—1731no—
204C23×C4—1731no—
205C23×C4—1731no—
206C23×C4—1731no—
207#117—3464no—
208#117—3464no—
209C2×C4⋊C4—1731no—
210C2×C4⋊C4—1731no—
211C2×C4⋊C4—1731no—
212C2×C4⋊C4—1731no—
213C2×C4⋊C4—1731no—
214C2×C4⋊C4—1731no—
215C2×C4⋊C4—1731no—
216C2×C4⋊C4—1731no—
217C2×C4⋊C4—1738no—
218C2×C4⋊C4C2×D17311yescharacteristic
219C2×C4⋊C4C2×D17311yescharacteristic
220C2×C22⋊C4—1731no—
221C2×C22⋊C4—1731no—
222C2×C22⋊C4—1731no—
223C2×C22⋊C4—1731no—
224C2×C42—1731no—
225C2×C42—1731no—
226C2×C42—1731no—
227C2×C42—1731no—
228C2×C42—1731no—
229C2×C42C2×D17311yescharacteristic
230#95—1731no—
231#95—1731no—
232#95—1731no—
233#95—1731no—
234#95—1731no—
235#95—1731no—
236#95—1731no—
237#95—1731no—
238#95C2×D17311yescharacteristic
239#95C2×D17311yescharacteristic
240#95C2×D17311yescharacteristic
241#95C2×D17311yescharacteristic
242C24—1731no—
243C22×C4—3462no—
244C22×C4—3462no—
245C22×C4—3462no—
246C22×C4—3462no—
247C22×C4—3462no—
248C22×C4—3462no—
249C22×C4—1731no—
250C22×C4—1731no—
251C22×C4—1731no—
252C22×C4—1731no—
253C22×C4—1731no—
254C22×C4—1731no—
255C22×C4—1734no—
256C22×C4—1734no—
257C22×C4—1734no—
258C22×C4—1731no—
259C22×C4C22×D17311yescharacteristic
260C22×C4C22×D17311yescharacteristic
261C22×C4C22×D17311yescharacteristic
262C22×C4C22×D17311yescharacteristic
263C22×C4C22×D17311yescharacteristic
264C22×C4C22×D17311yescharacteristic
265C22×C4C22×D17311yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—3462no—
268C4⋊C4—3462no—
269C4⋊C4—3462no—
270C4⋊C4—3462no—
271C4⋊C4—3462no—
272C4⋊C4—1734no—
273C4⋊C4—1734no—
274C4⋊C4—1734no—
275C4⋊C4C4×D17314yes—
276C22⋊C4—3462no—
277C22⋊C4—3462no—
278C22⋊C4—3462no—
279C22⋊C4—1734no—
280C42—22no—
281C42—3462no—
282C42—3462no—
283C42—3462no—
284C42—3462no—
285C42—3462no—
286C23—3461no—
287C23—3461no—
288C23—3461no—
289C23—3461no—
290C23—1732no—
291C23—1732no—
292C23—1732no—
293C23C23×D17311yescharacteristic
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—3468no—
312C2×C4—3468no—
313C2×C4—3468no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—1732no—
330C2×C4—1732no—
331C2×C4—1732no—
332C2×C4—1732no—
333C2×C4—1732no—
334C2×C4—1732no—
335C2×C4C2×C4×D17312yes—
336C2×C4C2×C4×D17312yes—
337C2×C4C2×C4×D17312yes—
338C2×C4D4×D17312yes—
339C22—3462no—
340C22—3462no—
341C22—3462no—
342C22—3462no—
343C22—3462no—
344C22—3462no—
345C22—1734no—
346C22C22×C4×D17311yescharacteristic
347C22C2×D4×D17311yescharacteristic
348C22#41×D17311yescharacteristic
349C22#41×D17311yescharacteristic
350C22#41×D17311yescharacteristic
351C22#1914972611yescharacteristic
352C22#1914972711yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3462no—
361C4—3462no—
362C4—3462no—
363C4—1734no—
364C4—6921no—
365C4—6921no—
366C4—6921no—
367C2—3464no—
368C2#117×D17311yescharacteristic
369C2C4×D4×D17311yescharacteristic
370C2#2601196711yescharacteristic
371C2#2601213911yescharacteristic
372C2#2601214311yescharacteristic
373C2#2601217311yescharacteristic
374C2#2601217711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#387×C173C2
socle, cosocleC23×C173C24
prime radicalC23×D173C23
hypercenter#387D173
hypocenterC173#2344
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
4. #26011388C2semidirect
5. #26011386C2semidirect
6. #26011377C2semidirect
7. #26011376C2semidirect
8. #26011374C2semidirect
12. C4×#14583805C2semidirect
13. #26011342C2semidirect
15. #387×C173C2semidirect
29. C4⋊C4×D173C4semidirect
178. Dic173#117semidirect
187. C173#2344semidirect
197. #387D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496372
C2×#191496362
C2×C4⋊C4×D1732
#260113882
#260113862
#260113772
#260113762
#260113742
#260113532
#260113502
C4×#145838052
#260113422
#260113382
#387×C1732
#2344173