Finite Group Databasebeta

← #36362277

Subgroups — #36362277

Order 22144.

484 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149687C211yesmaximal, characteristic, complemented
2C2×#19149685C211yesmaximal, characteristic, complemented
3#26012038C211yesmaximal, characteristic
4#26011565C211yesmaximal, characteristic, complemented
5#26011564C211yesmaximal, characteristic, complemented
6#26011563C211yesmaximal, characteristic, complemented
7#26011562C211yesmaximal, characteristic, complemented
8#26011561C211yesmaximal, characteristic, complemented
9#26011560C211yesmaximal, characteristic, complemented
10#26011559C211yesmaximal, characteristic, complemented
11#26011513C211yesmaximal, characteristic, complemented
12#26011456C211yesmaximal, characteristic, complemented
13#26011456C211yesmaximal, characteristic, complemented
14#97×D173C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18#19149687—24no—
19#19149685—24no—
20C2×#14583810C2211yescharacteristic
21C2×#14583809C2211yescharacteristic
22C2×#14583809C2211yescharacteristic
23C2×#14583809C2211yescharacteristic
24C2×#14583809C2211yescharacteristic
25C22×#13063436C2211yescharacteristic
26C2×C8×D173C2211yescharacteristic
27C2×C8×D173C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29C2×C8×D173C2211yescharacteristic
30#19149678C411yescharacteristic
31#19149675C411yescharacteristic
32#19149665C411yescharacteristic
33#19149665C411yescharacteristic
34#19149654C411yescharacteristic
35#19149652C411yescharacteristic
36#19149652C411yescharacteristic
37#19149650C411yescharacteristic
38C4×D692C2211yescharacteristic
39C4×C22⋊D173C2211yescharacteristic
40C4×C22⋊D173C2211yescharacteristic
41C42×D173C2211yescharacteristic
42C4⋊#13063436C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149608C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47#19149591C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic
52#19149589C2211yescharacteristic
53#19149589C2211yescharacteristic
54C2×M16×C173C2211yescharacteristic
55C22×C8×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic
57C4⋊C8×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C22⋊C8×C173C2211yescharacteristic
60#97×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—22no—
63#14583826—22no—
64#14583826—22no—
65C2×Dic346C2×C411yescharacteristic
66C2×Dic346C2×C411yescharacteristic
67C2×D692C2×C411yescharacteristic
68C2×D692C2×C411yescharacteristic
69C2×D692C2×C411yescharacteristic
70C2×C22⋊D173C2×C411yescharacteristic
71C2×C22⋊D173C2×C411yescharacteristic
72C2×C22⋊D173C2×C411yescharacteristic
73C2×C22⋊D173C2×C411yescharacteristic
74C2×C22⋊D173C2×C411yescharacteristic
75C2×C22⋊D173C2×C411yescharacteristic
76C2×C4×D173C2×C411yescharacteristic
77C2×C4×D173C2×C411yescharacteristic
78C2×C4×D173C2311yescharacteristic
79C2×C4×D173C2311yescharacteristic
80C2×C4×D173C2311yescharacteristic
81#14583810—22no—
82#14583809—22no—
83#14583809—22no—
84#14583809—22no—
85#14583809—22no—
86C2×#13063436—22no—
87C2×#13063436C2311yescharacteristic
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C8×D173—22no—
92C8×D173—22no—
93C8×D173—22no—
94C8×D173D414yescomplemented
95C4⋊Dic173C2×C411yescharacteristic
96#14583805C2×C411yescharacteristic
97#14583805C2×C411yescharacteristic
98C22⋊Dic173C2×C411yescharacteristic
99C22⋊Dic173C2×C411yescharacteristic
100#14583803C2×C411yescharacteristic
101#14583803C2×C411yescharacteristic
102#14583803C2×C411yescharacteristic
103#14583803C2×C411yescharacteristic
104C4×Dic173C2×C411yescharacteristic
105C4×Dic173C2×C411yescharacteristic
106C4×Dic173C2311yescharacteristic
107C2×D4×C173C2×C411yescharacteristic
108C22×C4×C173C2311yescharacteristic
109C22×C4×C173C2311yescharacteristic
110M16×C173—22no—
111C2×C8×C173—22no—
112C2×C8×C173C2311yescharacteristic
113C2×C8×C173C2311yescharacteristic
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C4⋊C4×C173C2×C411yescharacteristic
117C22⋊C4×C173C2×C411yescharacteristic
118C22⋊C4×C173C2×C411yescharacteristic
119C42×C173C2311yescharacteristic
120C22×D173C22×C411yescharacteristic
121C22×D173C22×C411yescharacteristic
122C22×D173C22×C411yescharacteristic
123Dic346—22no—
124Dic346—22no—
125D692—22no—
126D692—22no—
127D692—41no—
128C22⋊D173—22no—
129C22⋊D173—22no—
130C22⋊D173—22no—
131C22⋊D173—22no—
132C22⋊D173—41no—
133C22⋊D173—41no—
134C2×Dic173C22×C411yescharacteristic
135C2×Dic173C22×C411yescharacteristic
136C2×Dic173C22×C411yescharacteristic
137C2×Dic173C22×C411yescharacteristic
138C2×Dic173C22×C411yescharacteristic
139C4×D173—21no—
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—41no—
144C4×D173—41no—
145C4×D173C2×D412yes—
146C4×D173#4112yes—
147#13063436—21no—
148#13063436—21no—
149#13063436—21no—
150#13063436C2×D412yes—
151C23×C173C22×C411yescharacteristic
152C23×C173C22×C411yescharacteristic
153D4×C173—41no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173C22×C411yescharacteristic
159C2×C4×C173C22×C411yescharacteristic
160C2×C4×C173C22×C411yescharacteristic
161C2×C4×C173C22×C411yescharacteristic
162C2×C4×C173C2411yescharacteristic
163C8×C173—21no—
164C8×C173—21no—
165C8×C173—21no—
166C8×C173C2×D412yes—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173C23×C411yescharacteristic
174C2×D173—21no—
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—21no—
179C2×D173—21no—
180C2×D173—21no—
181C2×D173—21no—
182C2×D173C4×D412yes—
183C4×C173—21no—
184C4×C173—21no—
185C4×C173—21no—
186C4×C173—21no—
187C4×C173C22×D411yescharacteristic
188C4×C173C2×#4111yescharacteristic
189Dic173—21no—
190Dic173—21no—
191Dic173—21no—
192Dic173—21no—
193Dic173C4×D412yes—
194C2×C173—41no—
195C2×C173—41no—
196C2×C173C2×C4×D411yescharacteristic
197C2×C173#56811yescharacteristic
198C2×C173#56811yescharacteristic
199D173—22no—
200D173—41no—
201D173—41no—
202C173#293211yescharacteristic, complemented, Sylow
203#2932—1731nomaximal, Sylow
204C2×#131—1731no—
205C2×#131—1731no—
206#532—1731no—
207#435—1731no—
208#435—1731no—
209#435—1731no—
210#435—1731no—
211#435—1731no—
212#435—1731no—
213#435—1731no—
214#435D17311yescharacteristic, complemented
215#423—1731no—
216#408—1731no—
217#408—1731no—
218C2×#97—1731no—
219C2×#41—1731no—
220C2×#41—1731no—
221#131—3464no—
222#131—3464no—
223C2×M16—1731no—
224C2×M16—1731no—
225C2×M16—1731no—
226C2×M16—1731no—
227C2×M16—1731no—
228C2×M16C2×D17311yescharacteristic
229C22×C8—1731no—
230C22×C8—1731no—
231C22×C8—1731no—
232C22×C8—1731no—
233C22×C8—1731no—
234C22×C8C2×D17311yescharacteristic
235C4⋊Q8—1731no—
236#127—1731no—
237#124—1731no—
238#124—1731no—
239#121—1731no—
240#121—1731no—
241#121—1731no—
242#121—1731no—
243C4×D4—1731no—
244C4×D4—1731no—
245C4×D4—1731no—
246C4×D4C2×D17311yescharacteristic
247C2×C42—1731no—
248C4⋊C8—1731no—
249C4⋊C8—1731no—
250C4⋊C8—1731no—
251C4⋊C8C2×D17311yescharacteristic
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256C22⋊C8—1731no—
257C22⋊C8—1731no—
258C22⋊C8C2×D17311yescharacteristic
259C22⋊C8C2×D17311yescharacteristic
260#97—1731no—
261#97—1731no—
262#97—1731no—
263#97C2×D17311yescharacteristic
264#41—3462no—
265#41—3462no—
266#41—3462no—
267#41—3462no—
268C2×Q8—1731no—
269C2×Q8—1731no—
270C2×D4—1731no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4—1731no—
274C2×D4—1731no—
275C2×D4—1731no—
276C2×D4—1731no—
277C2×D4—1731no—
278C2×D4—1731no—
279C2×D4C4×D17311yescharacteristic
280C22×C4—1731no—
281C22×C4—1731no—
282C22×C4—1731no—
283C22×C4—1731no—
284C22×C4—1731no—
285C22×C4C22×D17311yescharacteristic
286C22×C4C22×D17311yescharacteristic
287M16—22no—
288M16—3462no—
289M16—3462no—
290M16—3462no—
291M16—3462no—
292M16—3462no—
293C2×C8—22no—
294C2×C8—3462no—
295C2×C8—3462no—
296C2×C8—3462no—
297C2×C8—3462no—
298C2×C8—1731no—
299C2×C8—1731no—
300C2×C8—1731no—
301C2×C8—1731no—
302C2×C8—1734no—
303C2×C8C22×D17311yescharacteristic
304C2×C8C22×D17311yescharacteristic
305C2×C8C22×D17311yescharacteristic
306C2×C8C22×D17311yescharacteristic
307C4⋊C4—1731no—
308C4⋊C4—1731no—
309C4⋊C4—1731no—
310C4⋊C4C4×D17311yescharacteristic
311C22⋊C4—1731no—
312C22⋊C4—1731no—
313C22⋊C4—1731no—
314C22⋊C4—1731no—
315C22⋊C4—1731no—
316C22⋊C4—1731no—
317C22⋊C4C4×D17311yescharacteristic
318C22⋊C4C4×D17311yescharacteristic
319C42—1731no—
320C42—1731no—
321C42—1731no—
322C42C22×D17311yescharacteristic
323C23—1731no—
324C23—1731no—
325C23—1731no—
326C23C2×C4×D17311yescharacteristic
327C23C2×C4×D17311yescharacteristic
328Q8—3462no—
329Q8—3462no—
330D4—41no—
331D4—3462no—
332D4—3462no—
333D4—3462no—
334D4—3462no—
335D4—3462no—
336D4—3462no—
337D4—6921no—
338D4—6921no—
339D4—6921no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—1731no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—1731no—
352C2×C4—1731no—
353C2×C4—1732no—
354C2×C4—1732no—
355C2×C4—6921no—
356C2×C4—6921no—
357C2×C4C2×C4×D17311yescharacteristic
358C2×C4C2×C4×D17311yescharacteristic
359C2×C4C2×C4×D17311yescharacteristic
360C2×C4C2×C4×D17311yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8—3461no—
366C8—3461no—
367C8—3461no—
368C8—1732no—
369C8D4×D17312yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—3461no—
377C22—3461no—
378C22—3461no—
379C22—3461no—
380C22—3461no—
381C22—3461no—
382C22—3461no—
383C22—3461no—
384C22—1732no—
385C22C22×C4×D17311yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—3461no—
391C4—3461no—
392C4—3461no—
393C4—3461no—
394C4—1732no—
395C4C2×D4×D17311yescharacteristic
396C4#41×D17311yescharacteristic
397C2—41no—
398C2—41no—
399C2—3462no—
400C2—6921no—
401C2—6921no—
402C2C4×D4×D17311yescharacteristic
403C2#2601225611yescharacteristic
404C2#2601225711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D173
commutator, socleC22×C173C23×C4
Fitting#435×C173C2
prime radical#26012038C2
cosocleC2×C4×C173C24
hypercenter#435D173
hypocenterC173#2932
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19149687C2semidirect
2. C2×#19149685C2semidirect
4. #26011565C2semidirect
5. #26011564C2semidirect
6. #26011563C2semidirect
7. #26011562C2semidirect
8. #26011561C2semidirect
9. #26011560C2semidirect
10. #26011559C2semidirect
11. #26011513C2semidirect
12. #26011456C2semidirect
13. #26011456C2semidirect
14. #97×D173C2semidirect
15. #435×C173C2semidirect
94. C8×D173D4semidirect
202. C173#2932semidirect
214. #435D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496872
C2×#191496852
#260120382
#260115652
#260115642
#260115632
#260115622
#260115612
#260115602
#260115592
#260115132
#260114562
#97×D1732
#435×C1732
#2932173