Finite Group Databasebeta

← #36356113

Subgroups — #36356113

Order 22144.

516 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149674C211yesmaximal, characteristic
2#26011383C211yesmaximal, characteristic
3#26011382C211yesmaximal, characteristic
4#26011381C211yesmaximal, characteristic
5#26011378C211yesmaximal, characteristic, complemented
6#26011376C211yesmaximal, characteristic, complemented
7#26011358C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic173C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic173C211yesmaximal, characteristic, complemented
10#26011342C211yesmaximal, characteristic, complemented
11#26011338C211yesmaximal, characteristic
12#26011338C211yesmaximal, characteristic
13#26011338C211yesmaximal, characteristic
14C42×Dic173C211yesmaximal, characteristic, complemented
15#387×C173C211yesmaximal, characteristic, complemented
16C22×C4×D173C2211yescharacteristic
17#19149674C418yescomplemented
18C2×C4⋊Dic173C2211yescharacteristic
19C2×C4⋊Dic173C2211yescharacteristic
20C2×#14583805C2211yescharacteristic
21C2×#14583805C2211yescharacteristic
22C2×#14583803C2211yescharacteristic
23C2×#14583803C2211yescharacteristic
24C2×#14583803C2211yescharacteristic
25C2×#14583803C2211yescharacteristic
26C2×#14583803C2211yescharacteristic
27C2×#14583803C2211yescharacteristic
28C2×C4×Dic173C2211yescharacteristic
29C2×C4×Dic173C2211yescharacteristic
30C2×C4×Dic173C2211yescharacteristic
31C2×C4×Dic173C2211yescharacteristic
32C2×C4×Dic173C2211yescharacteristic
33C2×C4×Dic173C2211yescharacteristic
34C2×C4×Dic173C2211yescharacteristic
35C2×C4×Dic173C2211yescharacteristic
36C2×C4×Dic173C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45C2×C4⋊C4×C173C2211yescharacteristic
46C2×C4⋊C4×C173C2211yescharacteristic
47C2×C42×C173C2211yescharacteristic
48#95×C173C2211yescharacteristic
49#95×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51#95×C173C2211yescharacteristic
52C23×D173C2311yescharacteristic
53C22×Dic173C2311yescharacteristic
54C22×Dic173C2311yescharacteristic
55C22×Dic173C2311yescharacteristic
56C22×Dic173C2311yescharacteristic
57C22×Dic173C2311yescharacteristic
58C22×Dic173C2311yescharacteristic
59C22×Dic173C2311yescharacteristic
60C2×C4×D173—22no—
61C2×C4×D173—22no—
62C2×C4×D173—22no—
63C4⋊Dic173—22no—
64C4⋊Dic173C2×C414yescomplemented
65#14583805—22no—
66#14583805C2×C414yescomplemented
67#14583803—22no—
68#14583803—22no—
69#14583803—22no—
70#14583803C2×C414yes—
71#14583803C2×C414yes—
72#14583803C2×C414yes—
73C4×Dic173—22no—
74C4×Dic173—22no—
75C4×Dic173—22no—
76C4×Dic173—22no—
77C4×Dic173—22no—
78C4×Dic173—22no—
79C4×Dic173—22no—
80C4×Dic173—22no—
81C4×Dic173C2×C414yescomplemented
82C22×C4×C173C2311yescharacteristic
83C22×C4×C173C2311yescharacteristic
84C22×C4×C173C2311yescharacteristic
85C22×C4×C173C2311yescharacteristic
86C22×C4×C173C2311yescharacteristic
87C22×C4×C173C2311yescharacteristic
88C22×C4×C173C2311yescharacteristic
89C4⋊C4×C173—22no—
90C4⋊C4×C173C2×C414yescomplemented
91C42×C173—22no—
92C22×D173—21no—
93C22×D173—21no—
94C22×D173—21no—
95C22×D173—21no—
96C22×D173—21no—
97C22×D173—21no—
98C22×D173C22×C412yes—
99C2×Dic173—21no—
100C2×Dic173—21no—
101C2×Dic173—21no—
102C2×Dic173—21no—
103C2×Dic173—21no—
104C2×Dic173—21no—
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×Dic173C22×C412yes—
114C2×Dic173C22×C412yes—
115C2×Dic173C22×C412yes—
116C2×Dic173#4112yes—
117C2×Dic173#4112yes—
118C2×Dic173#4112yes—
119C2×Dic173#4112yes—
120C4×D173—44no—
121C23×C173C2411yescharacteristic
122C2×C4×C173—21no—
123C2×C4×C173—21no—
124C2×C4×C173—21no—
125C2×C4×C173—21no—
126C2×C4×C173—21no—
127C2×C4×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
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×C173C22×C412yes—
139C2×C4×C173C22×C412yes—
140C2×C4×C173C22×C412yes—
141C2×C4×C173#4112yes—
142C2×C4×C173#4112yes—
143C22×C173C23×C411yescharacteristic
144C22×C173C2×#4111yescharacteristic
145C22×C173C2×#4111yescharacteristic
146C22×C173C2×#4111yescharacteristic
147C22×C173C2×#4111yescharacteristic
148C22×C173C2×#4111yescharacteristic
149C22×C173C2×#4111yescharacteristic
150C2×D173—22no—
151C2×D173—22no—
152C2×D173—22no—
153C2×D173—41no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C4×C173—22no—
158C4×C173—22no—
159C4×C173—22no—
160C4×C173—22no—
161C4×C173—22no—
162C4×C173—41no—
163C4×C173—41no—
164Dic173—22no—
165Dic173—22no—
166Dic173—22no—
167Dic173—22no—
168Dic173—41no—
169Dic173—41no—
170Dic173#12614yescomplemented
171C2×C173C4×#4111yescharacteristic
172C2×C173C4×#4111yescharacteristic
173C2×C173C4×#4111yescharacteristic
174C2×C173C2×#12611yescharacteristic
175C2×C173#52911yescharacteristic
176C2×C173#52911yescharacteristic
177C2×C173#52911yescharacteristic
178D173—42no—
179C173C4×#12611yescharacteristic, complemented, Sylow
180C4×#126—1731nomaximal, Sylow
181C2×#126—1731no—
182#388—1731no—
183#388—1731no—
184#388—1731no—
185#387—1731no—
186#387—1731no—
187#387D17311yescharacteristic, complemented
188#383—1731no—
189C4×C4⋊C4—1731no—
190C4×C4⋊C4—1731no—
191C4×C4⋊C4—1731no—
192C4×C22⋊C4—1731no—
193C4×C22⋊C4—1731no—
194C4×C22⋊C4—1731no—
195C43—1731no—
196C23×C4—1731no—
197#126—1738no—
198C2×C4⋊C4—1731no—
199C2×C4⋊C4—1731no—
200C2×C4⋊C4—1731no—
201C2×C4⋊C4—1731no—
202C2×C4⋊C4C2×D17311yescharacteristic
203C2×C4⋊C4C2×D17311yescharacteristic
204C2×C22⋊C4—1731no—
205C2×C22⋊C4—1731no—
206C2×C22⋊C4—1731no—
207C2×C22⋊C4—1731no—
208C2×C22⋊C4—1731no—
209C2×C22⋊C4—1731no—
210C2×C42—1731no—
211C2×C42—1731no—
212C2×C42—1731no—
213C2×C42—1731no—
214C2×C42—1731no—
215C2×C42—1731no—
216C2×C42—1731no—
217C2×C42—1731no—
218C2×C42—1731no—
219C2×C42C2×D17311yescharacteristic
220#95—1731no—
221#95—1731no—
222#95—1731no—
223#95—1731no—
224#95—1731no—
225#95—1731no—
226#95—1731no—
227#95—1731no—
228#95C2×D17311yescharacteristic
229#95C2×D17311yescharacteristic
230#95C2×D17311yescharacteristic
231#95C2×D17311yescharacteristic
232C24—1731no—
233C22×C4—3462no—
234C22×C4—3462no—
235C22×C4—3462no—
236C22×C4—1731no—
237C22×C4—1731no—
238C22×C4—1731no—
239C22×C4—1731no—
240C22×C4—1731no—
241C22×C4—1731no—
242C22×C4—1731no—
243C22×C4C22×D17311yescharacteristic
244C22×C4C22×D17311yescharacteristic
245C22×C4C22×D17311yescharacteristic
246C22×C4C22×D17311yescharacteristic
247C22×C4C22×D17311yescharacteristic
248C22×C4C22×D17311yescharacteristic
249C22×C4C22×D17311yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—3462no—
252C4⋊C4—3462no—
253C4⋊C4—1734no—
254C4⋊C4—1734no—
255C4⋊C4C4×D17314yescomplemented
256C22⋊C4—3462no—
257C22⋊C4—3462no—
258C22⋊C4—3462no—
259C22⋊C4—1734no—
260C22⋊C4—1734no—
261C22⋊C4—1734no—
262C42—22no—
263C42—3462no—
264C42—3462no—
265C42—3462no—
266C42—3462no—
267C42—3462no—
268C42—3462no—
269C42—3462no—
270C42—3462no—
271C42—1734no—
272C23—3461no—
273C23—3461no—
274C23—3461no—
275C23—3461no—
276C23—3461no—
277C23—3461no—
278C23—1732no—
279C23C23×D17311yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—3461no—
301C2×C4—3461no—
302C2×C4—3461no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—1732no—
311C2×C4—1732no—
312C2×C4—1732no—
313C2×C4—1732no—
314C2×C4—1732no—
315C2×C4—1732no—
316C2×C4—1732no—
317C2×C4—6924no—
318C2×C4C2×C4×D17312yes—
319C2×C4C2×C4×D17312yes—
320C2×C4C2×C4×D17312yes—
321C2×C4#1458382512yes—
322C2×C4#1458382712yes—
323C22—3462no—
324C22—3462no—
325C22—3462no—
326C22—6921no—
327C22—6921no—
328C22—6921no—
329C22—6921no—
330C22C22×C4×D17311yescharacteristic
331C22#41×D17311yescharacteristic
332C22#41×D17311yescharacteristic
333C22#41×D17311yescharacteristic
334C22#41×D17311yescharacteristic
335C22C2×#1458382511yescharacteristic
336C22C2×#1458382711yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—3462no—
345C4—3462no—
346C4—3462no—
347C4—3462no—
348C4—1734no—
349C4—6921no—
350C4—6921no—
351C2—6922no—
352C2C4×#1458382511yescharacteristic
353C2#2601195911yescharacteristic
354C2C4×#1458382711yescharacteristic
355C2#126×D17311yescharacteristic
356C2#2601201811yescharacteristic
357C2#2601202611yescharacteristic
358C2#2601202711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#387×C173C2
socle, cosocleC23×C173C24
prime radicalC23×D173C23
hypercenter#387D173
hypocenterC173C4×#126
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
5. #26011378C2semidirect
6. #26011376C2semidirect
7. #26011358C2semidirect
8. C4⋊C4×Dic173C2semidirect
9. C4⋊C4×Dic173C2semidirect
10. #26011342C2semidirect
14. C42×Dic173C2semidirect
15. #387×C173C2semidirect
17. #19149674C4semidirect
64. C4⋊Dic173C2×C4semidirect
66. #14583805C2×C4semidirect
81. C4×Dic173C2×C4semidirect
90. C4⋊C4×C173C2×C4semidirect
170. Dic173#126semidirect
179. C173C4×#126semidirect
187. #387D173semidirect
255. C4⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496742
#260113832
#260113822
#260113812
#260113782
#260113762
#260113582
C4⋊C4×Dic1732
#260113422
#260113382
C42×Dic1732
#387×C1732
C4×#126173