Finite Group Databasebeta

← #36356487

Subgroups — #36356487

Order 22144.

660 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149641C211yesmaximal, characteristic
2C2×#19149636C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D173C211yesmaximal, characteristic, complemented
4#26011383C211yesmaximal, characteristic, complemented
5#26011381C211yesmaximal, characteristic, complemented
6#26011377C211yesmaximal, characteristic, complemented
7#26011369C211yesmaximal, characteristic
8#26011369C211yesmaximal, characteristic
9#26011355C211yesmaximal, characteristic, complemented
10#26011350C211yesmaximal, characteristic, complemented
11#26011350C211yesmaximal, characteristic, complemented
12#26011342C211yesmaximal, characteristic, complemented
13C4×#14583803C211yesmaximal, characteristic, complemented
14#26011338C211yesmaximal, characteristic, complemented
15#387×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C22×C4×D173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21#19149641C418yescomplemented
22#19149636—24no—
23C2×C4⋊Dic173C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C4⋊C4×D173—24no—
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×C4×Dic173C2211yescharacteristic
38C2×C4×Dic173C2211yescharacteristic
39C2×C4×Dic173C2211yescharacteristic
40C2×C4×Dic173C2211yescharacteristic
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
55C23×D173C2311yescharacteristic
56C2×D692—22no—
57C2×D692—22no—
58C2×D692C2×C414yes—
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C22×Dic173C2311yescharacteristic
64C22×Dic173C2311yescharacteristic
65C2×C4×D173—22no—
66C2×C4×D173—22no—
67C2×C4×D173—22no—
68C2×C4×D173—22no—
69C2×C4×D173—22no—
70C2×C4×D173—22no—
71C2×C4×D173—22no—
72C2×C4×D173—22no—
73C2×C4×D173—41no—
74C2×C4×D173—41no—
75C2×C4×D173C2×C414yes—
76C2×C4×D173C2×C414yescomplemented
77C4⋊Dic173—22no—
78#14583805—22no—
79#14583805—22no—
80#14583805—22no—
81#14583805D414yescomplemented
82#14583803—22no—
83#14583803—22no—
84#14583803—22no—
85#14583803—22no—
86#14583803—41no—
87#14583803—41no—
88#14583803C2×C414yes—
89#14583803C2×C414yescomplemented
90C4×Dic173—22no—
91C4×Dic173—41no—
92C4×Dic173—41no—
93C4×Dic173C2×C414yescomplemented
94C22×C4×C173C2311yescharacteristic
95C22×C4×C173C2311yescharacteristic
96C22×C4×C173C2311yescharacteristic
97C22×C4×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C4⋊C4×C173—22no—
102C4⋊C4×C173C2×C414yescomplemented
103C42×C173—22no—
104C22×D173—21no—
105C22×D173—21no—
106C22×D173—21no—
107C22×D173—21no—
108C22×D173—21no—
109C22×D173—21no—
110C22×D173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173C22×C412yes—
115C22×D173C22×C412yes—
116C22×D173#4112yes—
117C22×D173#4112yes—
118D692—28no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173C22×C412yes—
133C2×Dic173C22×C412yes—
134C2×Dic173C2×D412yes—
135C2×Dic173C2×D412yes—
136C2×Dic173#4112yes—
137C4×D173—28no—
138C4×D173—28no—
139C4×D173—44no—
140C4×D173—44no—
141C23×C173C2411yescharacteristic
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173C22×C412yes—
160C2×C4×C173C22×C412yes—
161C2×C4×C173C22×C412yes—
162C2×C4×C173C2×D412yes—
163C22×C173C23×C411yescharacteristic
164C22×C173C22×D411yescharacteristic
165C22×C173C2×#4111yescharacteristic
166C22×C173C2×#4111yescharacteristic
167C22×C173C2×#4111yescharacteristic
168C22×C173H(5,𝔽2)11yescharacteristic
169C22×C1732−1+411yescharacteristic
170C2×D173—22no—
171C2×D173—22no—
172C2×D173—22no—
173C2×D173—22no—
174C2×D173—22no—
175C2×D173—22no—
176C2×D173—22no—
177C2×D173—22no—
178C2×D173—22no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173#11714yescomplemented
184C4×C173—22no—
185C4×C173—22no—
186C4×C173—22no—
187C4×C173—22no—
188C4×C173—41no—
189C4×C173—41no—
190C4×C173—41no—
191Dic173—22no—
192Dic173—22no—
193Dic173—41no—
194Dic173—41no—
195Dic173—41no—
196Dic173C4×D414yes—
197C2×C173C2×#11711yescharacteristic
198C2×C173C2×C4×D411yescharacteristic
199C2×C173#52011yescharacteristic
200C2×C173#54611yescharacteristic
201C2×C173#54811yescharacteristic
202C2×C173#55211yescharacteristic
203C2×C173#55311yescharacteristic
204D173—24no—
205D173—42no—
206C173#235211yescharacteristic, complemented, Sylow
207#2352—1731nomaximal, Sylow
208C2×C4×D4—1731no—
209C2×#117—1731no—
210C22×C4⋊C4—1731no—
211#388—1731no—
212#388—1731no—
213#387—1731no—
214#387D17311yescharacteristic, complemented
215#386—1731no—
216#386—1731no—
217#382—1731no—
218#380—1731no—
219#380—1731no—
220C4×C4⋊C4—1731no—
221C4×C22⋊C4—1731no—
222C4×C22⋊C4—1731no—
223C22×D4—1731no—
224C23×C4—1731no—
225C23×C4—1731no—
226C23×C4—1731no—
227C23×C4—1731no—
228C4×D4—1738no—
229#117—3464no—
230C2×C4⋊C4—3464no—
231C2×C4⋊C4—1731no—
232C2×C4⋊C4—1731no—
233C2×C4⋊C4—1731no—
234C2×C4⋊C4—1731no—
235C2×C4⋊C4—1731no—
236C2×C4⋊C4C2×D17311yescharacteristic
237C2×C4⋊C4C2×D17311yescharacteristic
238C2×C22⋊C4—1731no—
239C2×C22⋊C4—1731no—
240C2×C22⋊C4—1731no—
241C2×C22⋊C4—1731no—
242C2×C22⋊C4—1731no—
243C2×C22⋊C4—1731no—
244C2×C22⋊C4—1731no—
245C2×C22⋊C4—1731no—
246C2×C42—1731no—
247C2×C42—1731no—
248C2×C42—1731no—
249C2×C42—1731no—
250C2×C42C2×D17311yescharacteristic
251#95—1731no—
252#95—1731no—
253#95—1731no—
254#95—1731no—
255#95—1731no—
256#95—1731no—
257#95C2×D17311yescharacteristic
258#95C2×D17311yescharacteristic
259#95C2×D17311yescharacteristic
260#95C2×D17311yescharacteristic
261C24—1731no—
262C24—1731no—
263C2×D4—3462no—
264C2×D4—3462no—
265C2×D4—1734no—
266C22×C4—3462no—
267C22×C4—3462no—
268C22×C4—3462no—
269C22×C4—3462no—
270C22×C4—3462no—
271C22×C4—3462no—
272C22×C4—3462no—
273C22×C4—3462no—
274C22×C4—1731no—
275C22×C4—1731no—
276C22×C4—1731no—
277C22×C4—1731no—
278C22×C4—1731no—
279C22×C4—1734no—
280C22×C4—1734no—
281C22×C4—1731no—
282C22×C4—6921no—
283C22×C4—6921no—
284C22×C4C22×D17311yescharacteristic
285C22×C4C22×D17311yescharacteristic
286C22×C4C22×D17311yescharacteristic
287C22×C4C22×D17311yescharacteristic
288C22×C4C22×D17311yescharacteristic
289C22×C4C22×D17311yescharacteristic
290C22×C4C22×D17311yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—3462no—
293C4⋊C4—3462no—
294C4⋊C4—3462no—
295C4⋊C4—3462no—
296C4⋊C4—1734no—
297C4⋊C4C4×D17314yescomplemented
298C22⋊C4—3462no—
299C22⋊C4—3462no—
300C22⋊C4—3462no—
301C22⋊C4—3462no—
302C22⋊C4—1734no—
303C22⋊C4—1734no—
304C22⋊C4—6921no—
305C22⋊C4—6921no—
306C42—22no—
307C42—3462no—
308C42—1734no—
309C42—6921no—
310C42—6921no—
311C23—3461no—
312C23—3461no—
313C23—3461no—
314C23—3461no—
315C23—3461no—
316C23—3461no—
317C23—3461no—
318C23—3461no—
319C23—3461no—
320C23—3461no—
321C23—1732no—
322C23—1732no—
323C23—1732no—
324C23—1732no—
325C23C23×D17311yescharacteristic
326D4—3468no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3468no—
345C2×C4—3468no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—1732no—
360C2×C4—1732no—
361C2×C4—1732no—
362C2×C4—1732no—
363C2×C4—1732no—
364C2×C4—6924no—
365C2×C4—6924no—
366C2×C4C2×C4×D17312yes—
367C2×C4C2×C4×D17312yes—
368C2×C4C2×C4×D17312yes—
369C2×C4D4×D17312yes—
370C22—3462no—
371C22—3462no—
372C22—3462no—
373C22—3462no—
374C22—3462no—
375C22—3462no—
376C22—3462no—
377C22—3462no—
378C22—3462no—
379C22—1734no—
380C22—6921no—
381C22—6921no—
382C22—6921no—
383C22—6921no—
384C22C22×C4×D17311yescharacteristic
385C22C2×D4×D17311yescharacteristic
386C22#41×D17311yescharacteristic
387C22#41×D17311yescharacteristic
388C22#41×D17311yescharacteristic
389C22#1914972511yescharacteristic
390C22#1914972711yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—3462no—
399C4—3462no—
400C4—1734no—
401C4—6921no—
402C4—6921no—
403C4—6921no—
404C2—3464no—
405C2—6922no—
406C2#117×D17311yescharacteristic
407C2C4×D4×D17311yescharacteristic
408C2#2601197311yescharacteristic
409C2#2601213311yescharacteristic
410C2#2601215111yescharacteristic
411C2#2601217311yescharacteristic
412C2#2601219211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#387×C173C2
socle, cosocleC23×C173C24
prime radicalC22×D692C22
hypercenter#387D173
hypocenterC173#2352
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×#19149636C2semidirect
3. C2×C4⋊C4×D173C2semidirect
4. #26011383C2semidirect
5. #26011381C2semidirect
6. #26011377C2semidirect
9. #26011355C2semidirect
10. #26011350C2semidirect
11. #26011350C2semidirect
12. #26011342C2semidirect
13. C4×#14583803C2semidirect
14. #26011338C2semidirect
15. #387×C173C2semidirect
21. #19149641C4semidirect
76. C2×C4×D173C2×C4semidirect
81. #14583805D4semidirect
89. #14583803C2×C4semidirect
93. C4×Dic173C2×C4semidirect
102. C4⋊C4×C173C2×C4semidirect
183. C2×D173#117semidirect
206. C173#2352semidirect
214. #387D173semidirect
297. C4⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496412
C2×#191496362
C2×C4⋊C4×D1732
#260113832
#260113812
#260113772
#260113692
#260113552
#260113502
#260113422
C4×#145838032
#260113382
#387×C1732
#2352173