Finite Group Databasebeta

← #36356469

Subgroups — #36356469

Order 22144.

704 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D173C211yesmaximal, characteristic
2C2×#19149639C211yesmaximal, characteristic
3C2×#19149639C211yesmaximal, characteristic
4#26011388C211yesmaximal, characteristic, complemented
5#26011386C211yesmaximal, characteristic, complemented
6#26011366C211yesmaximal, characteristic, complemented
7#26011365C211yesmaximal, characteristic, complemented
8#26011362C211yesmaximal, characteristic, complemented
9#26011355C211yesmaximal, characteristic, complemented
10#26011349C211yesmaximal, characteristic, complemented
11#26011348C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic173C211yesmaximal, characteristic, complemented
13C2×#19149586C211yesmaximal, characteristic, complemented
14#95×D173C211yesmaximal, characteristic, complemented
15#385×C173C211yesmaximal, characteristic, complemented
16C22×C22⋊D173C2211yescharacteristic
17C23×Dic173C2211yescharacteristic
18C23×Dic173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21C22×C4×D173C2211yescharacteristic
22C4×C22⋊D173—24no—
23#19149639—24no—
24#19149639C418yescomplemented
25C2×C4⋊Dic173C2211yescharacteristic
26C2×C4⋊Dic173C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×#14583805C2211yescharacteristic
29C2×#14583805C2211yescharacteristic
30C2×#14583805C2211yescharacteristic
31C2×#14583805C2211yescharacteristic
32C2×C22⋊Dic173C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
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
47C23×C4×C173C2211yescharacteristic
48C2×C4⋊C4×C173C2211yescharacteristic
49C2×C4⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C22⋊C4×C173C2211yescharacteristic
52#95×C173C2211yescharacteristic
53#95×C173C2211yescharacteristic
54C23×D173C2311yescharacteristic
55C2×C22⋊D173C2×C414yes—
56C2×C22⋊D173D414yes—
57C2×C22⋊D173Q814yes—
58C22×Dic173—22no—
59C22×Dic173—22no—
60C22×Dic173—41no—
61C22×Dic173—41no—
62C22×Dic173—41no—
63C22×Dic173C2×C414yescomplemented
64C22×Dic173C2311yescharacteristic
65C22×Dic173C2311yescharacteristic
66C22×Dic173C2311yescharacteristic
67C22×Dic173C2311yescharacteristic
68C22×Dic173C2311yescharacteristic
69C22×Dic173C2311yescharacteristic
70C22×Dic173C2311yescharacteristic
71C2×C4×D173—22no—
72C2×C4×D173—22no—
73C2×C4×D173—22no—
74C2×C4×D173—22no—
75C2×C4×D173—41no—
76C2×C4×D173—41no—
77C2×C4×D173—41no—
78C2×C4×D173—41no—
79C2×C4×D173C2×C414yescomplemented
80C4⋊Dic173—22no—
81C4⋊Dic173—41no—
82#14583805—22no—
83#14583805—22no—
84#14583805—41no—
85#14583805—41no—
86#14583805C2×C414yescomplemented
87C22⋊Dic173—22no—
88#14583803—22no—
89#14583803—22no—
90#14583803C2×C414yescomplemented
91C4×Dic173—22no—
92C4×Dic173—22no—
93C4×Dic173C2×C414yescomplemented
94C24×C173C2311yescharacteristic
95C22×C4×C173—22no—
96C22×C4×C173—22no—
97C22×C4×C173—41no—
98C22×C4×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C4⋊C4×C173—22no—
105C4⋊C4×C173—41no—
106C22⋊C4×C173—22no—
107C22⋊C4×C173C2×C414yescomplemented
108C22×D173—21no—
109C22×D173—21no—
110C22×D173—21no—
111C22×D173—21no—
112C22×D173C22×C412yes—
113C22×D173C2×D412yes—
114C22×D173C2×Q812yes—
115C22⋊D173C4⋊C4116yescomplemented
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
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—44no—
132C2×Dic173—44no—
133C2×Dic173C22×C412yes—
134C2×Dic173C22×C412yes—
135C2×Dic173C22×C412yes—
136C2×Dic173C2×D412yes—
137C2×Dic173C2×Q812yes—
138C2×Dic173#4112yes—
139C4×D173—44no—
140C4×D173—44no—
141C4×D173—44no—
142C23×C173—21no—
143C23×C173—21no—
144C23×C173—21no—
145C23×C173—21no—
146C23×C173C22×C412yes—
147C23×C173C2×D412yes—
148C23×C173C2×Q812yes—
149C23×C173C2411yescharacteristic
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×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—44no—
166C2×C4×C173C22×C412yes—
167C2×C4×C173C22×C412yes—
168C2×C4×C173#4112yes—
169C22×C173—22no—
170C22×C173—22no—
171C22×C173—22no—
172C22×C173—22no—
173C22×C173—22no—
174C22×C173—22no—
175C22×C173C2×C4⋊C414yes—
176C22×C173C23×C411yescharacteristic
177C22×C173C22×D411yescharacteristic
178C22×C173C22×Q811yescharacteristic
179C22×C173C2×#4111yescharacteristic
180C22×C173C2×#4111yescharacteristic
181C22×C173H(5,𝔽2)11yescharacteristic
182C22×C173H(5,𝔽2)11yescharacteristic
183C2×D173—22no—
184C2×D173—22no—
185C2×D173—22no—
186C2×D173—22no—
187C2×D173—22no—
188C2×D173—22no—
189C2×D173C2×C4⋊C414yes—
190C4×C173—22no—
191C4×C173—22no—
192C4×C173—22no—
193C4×C173—41no—
194C4×C173—41no—
195C4×C173—41no—
196Dic173—22no—
197Dic173—22no—
198Dic173—22no—
199Dic173—41no—
200Dic173—41no—
201Dic173—41no—
202Dic173C2×C4⋊C414yes—
203C2×C173—24no—
204C2×C173C22×C4⋊C411yescharacteristic
205C2×C173C4×#4111yescharacteristic
206C2×C173#51811yescharacteristic
207C2×C173#54611yescharacteristic
208C2×C173#54611yescharacteristic
209C2×C173#55411yescharacteristic
210C2×C173#55411yescharacteristic
211D173—24no—
212C173#234911yescharacteristic, complemented, Sylow
213#2349—1731nomaximal, Sylow
214C2×C4×D4—1731no—
215C2×C4×D4—1731no—
216C2×C4×D4—1731no—
217#389—1731no—
218#389—1731no—
219#385—1731no—
220#385—1731no—
221#385—1731no—
222#385D17311yescharacteristic, complemented
223#382—1731no—
224#380—1731no—
225#380—1731no—
226C4×C4⋊C4—1731no—
227C2×#95—1731no—
228C2×#95—1731no—
229C22×D4—1731no—
230C23×C4—1731no—
231C23×C4—1731no—
232C23×C4—1731no—
233C23×C4—1731no—
234C23×C4—1731no—
235C23×C4C2×D17311yescharacteristic
236C4×D4—3464no—
237C4×D4—3464no—
238C4×D4—1738no—
239C2×C4⋊C4—1731no—
240C2×C4⋊C4—1731no—
241C2×C4⋊C4—1731no—
242C2×C4⋊C4—1731no—
243C2×C4⋊C4—1731no—
244C2×C4⋊C4—1731no—
245C2×C4⋊C4—1731no—
246C2×C4⋊C4C2×D17311yescharacteristic
247C2×C4⋊C4C2×D17311yescharacteristic
248C2×C22⋊C4—1731no—
249C2×C22⋊C4—1731no—
250C2×C22⋊C4—1731no—
251C2×C22⋊C4—1731no—
252C2×C22⋊C4C2×D17311yescharacteristic
253C2×C22⋊C4C2×D17311yescharacteristic
254C2×C42—1731no—
255C2×C42—1731no—
256C2×C42—1731no—
257#95—1731no—
258#95—1731no—
259#95—1731no—
260#95—1731no—
261#95—1731no—
262#95—1731no—
263#95—1731no—
264#95—1731no—
265#95C2×D17311yescharacteristic
266#95C2×D17311yescharacteristic
267C24—1731no—
268C24C22×D17311yescharacteristic
269C2×D4—1734no—
270C2×D4—1734no—
271C2×D4—1734no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—3462no—
276C22×C4—3462no—
277C22×C4—3462no—
278C22×C4—3462no—
279C22×C4—3462no—
280C22×C4—3462no—
281C22×C4—1731no—
282C22×C4—1731no—
283C22×C4—1731no—
284C22×C4—1731no—
285C22×C4—1731no—
286C22×C4—1731no—
287C22×C4—1731no—
288C22×C4—1734no—
289C22×C4—1734no—
290C22×C4—6921no—
291C22×C4—6921no—
292C22×C4—6921no—
293C22×C4—6921no—
294C22×C4—6921no—
295C22×C4—6921no—
296C22×C4—6921no—
297C22×C4C22×D17311yescharacteristic
298C22×C4C22×D17311yescharacteristic
299C22×C4C22×D17311yescharacteristic
300C22×C4C22×D17311yescharacteristic
301C22×C4C22×D17311yescharacteristic
302C22×C4C22×D17311yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—3462no—
306C4⋊C4—3462no—
307C4⋊C4—3462no—
308C4⋊C4—1734no—
309C4⋊C4—6921no—
310C4⋊C4—6921no—
311C4⋊C4—6921no—
312C22⋊C4—22no—
313C22⋊C4—3462no—
314C22⋊C4—3462no—
315C22⋊C4—3462no—
316C22⋊C4—1734no—
317C22⋊C4C4×D17314yescomplemented
318C42—3462no—
319C42—3462no—
320C42—1734no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—3461no—
326C23—3461no—
327C23—3461no—
328C23—3461no—
329C23—1732no—
330C23—1732no—
331C23—1732no—
332C23C2×C4×D17312yes—
333C23D4×D17312yes—
334C23Q8×D17312yes—
335C23C23×D17311yescharacteristic
336D4—17316no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—44no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—1732no—
369C2×C4—1732no—
370C2×C4—1732no—
371C2×C4—1732no—
372C2×C4—1732no—
373C2×C4—1732no—
374C2×C4—6924no—
375C2×C4—6924no—
376C2×C4—6924no—
377C2×C4—6924no—
378C2×C4—6924no—
379C2×C4C2×C4×D17312yes—
380C2×C4C2×C4×D17312yes—
381C2×C4#1458382512yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—3462no—
389C22—3462no—
390C22—3462no—
391C22—3462no—
392C22—3462no—
393C22—3462no—
394C22—1734no—
395C22C4⋊C4×D17314yes—
396C22C22×C4×D17311yescharacteristic
397C22C2×D4×D17311yescharacteristic
398C22C2×Q8×D17311yescharacteristic
399C22#41×D17311yescharacteristic
400C22C2×#1458382511yescharacteristic
401C22#1914972411yescharacteristic
402C22#1914972511yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—3462no—
410C4—3462no—
411C4—3462no—
412C4—1734no—
413C4—6921no—
414C4—6921no—
415C4—6921no—
416C2—24no—
417C2—3464no—
418C2C2×C4⋊C4×D17311yescharacteristic
419C2C4×#1458382511yescharacteristic
420C2#2601196311yescharacteristic
421C2#2601212811yescharacteristic
422C2#2601213311yescharacteristic
423C2#2601219511yescharacteristic
424C2#2601219911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#385×C173C2
socle, cosocleC23×C173C24
prime radicalC22×C22⋊D173C22
hypercenter#385D173
hypocenterC173#2349
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #26011388C2semidirect
5. #26011386C2semidirect
6. #26011366C2semidirect
7. #26011365C2semidirect
8. #26011362C2semidirect
9. #26011355C2semidirect
10. #26011349C2semidirect
11. #26011348C2semidirect
12. C4⋊C4×Dic173C2semidirect
13. C2×#19149586C2semidirect
14. #95×D173C2semidirect
15. #385×C173C2semidirect
24. #19149639C4semidirect
63. C22×Dic173C2×C4semidirect
79. C2×C4×D173C2×C4semidirect
86. #14583805C2×C4semidirect
90. #14583803C2×C4semidirect
93. C4×Dic173C2×C4semidirect
107. C22⋊C4×C173C2×C4semidirect
115. C22⋊D173C4⋊C4semidirect
212. C173#2349semidirect
222. #385D173semidirect
317. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1732
C2×#191496392
#260113882
#260113862
#260113662
#260113652
#260113622
#260113552
#260113492
#260113482
C4⋊C4×Dic1732
C2×#191495862
#95×D1732
#385×C1732
#2349173