Finite Group Databasebeta

← #24915158

Subgroups — #24915158

Order 10624.

472 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19127642C211yesmaximal, characteristic, complemented
2#19127625C211yesmaximal, characteristic, complemented
3#19127624C211yesmaximal, characteristic, complemented
4#19127620C211yesmaximal, characteristic, complemented
5#19127612C211yesmaximal, characteristic, complemented
6#19127601C211yesmaximal, characteristic
7#19127586C211yesmaximal, characteristic, complemented
8#19127583C211yesmaximal, characteristic, complemented
9#19127578C211yesmaximal, characteristic
10#19127577C211yesmaximal, characteristic
11#19127575C211yesmaximal, characteristic, complemented
12#19127573C211yesmaximal, characteristic, complemented
13#19127560C211yesmaximal, characteristic, complemented
14#19127559C211yesmaximal, characteristic, complemented
15#393×C83C211yesmaximal, characteristic, complemented
16C22×C22⋊D83C2211yescharacteristic
17C23×Dic83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C2×C4⋊Dic83C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×C4×Dic83C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C22×Q8×C83C2211yescharacteristic
45C2×C22⋊C4×C83C2211yescharacteristic
46C2×C22⋊C4×C83C2211yescharacteristic
47C2×C22⋊C4×C83C2211yescharacteristic
48#95×C83C2211yescharacteristic
49#95×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51C23×D83C2311yescharacteristic
52C2×C22⋊D83—41no—
53C2×C22⋊D83—41no—
54C2×C22⋊D83Q814yescomplemented
55C22×Dic83—22no—
56C22×Dic83—41no—
57C22×Dic83—41no—
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C2×C4×D83—22no—
66C2×C4×D83—41no—
67C2×C4×D83—41no—
68C4⋊Dic83—22no—
69Dic83⋊C4—22no—
70Dic83⋊C4—41no—
71Dic83⋊C4—41no—
72Dic83⋊C4—41no—
73Dic83⋊C4—41no—
74Dic83⋊C4—41no—
75Dic83⋊C4—41no—
76C22⋊Dic83—22no—
77C22⋊Dic83—41no—
78(C2×D83)⋊C4—22no—
79(C2×D83)⋊C4—41no—
80(C2×D83)⋊C4—41no—
81(C2×D83)⋊C4—41no—
82(C2×D83)⋊C4—41no—
83C4×Dic83—41no—
84C24×C83C2311yescharacteristic
85C2×Q8×C83—41no—
86C2×Q8×C83—41no—
87C2×Q8×C83—41no—
88C22×C4×C83C2311yescharacteristic
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22⋊C4×C83—41no—
95C22⋊C4×C83—41no—
96C22⋊C4×C83—41no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83C2×Q812yes—
104C23×C83C2411yescharacteristic
105C22×D83—21no—
106C22×D83—21no—
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83C2×Q812yes—
112Q8×C83—44no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C22⋊D83—44no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—44no—
151C2×Dic83C2×Q812yes—
152C2×Dic83#4112yes—
153C2×Dic83#4112yes—
154C4×D83—44no—
155C22×C83—22no—
156C22×C83—22no—
157C22×C83—22no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83C22×Q811yescharacteristic
163C22×C83C2×#4111yescharacteristic
164C22×C83C2×#4111yescharacteristic
165C22×C83H(5,𝔽2)11yescharacteristic
166C22×C83H(5,𝔽2)11yescharacteristic
167C22×C83H(5,𝔽2)11yescharacteristic
168C22×C83H(5,𝔽2)11yescharacteristic
169C2×D83—22no—
170C2×D83—22no—
171C2×D83—22no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83—41no—
182Dic83—22no—
183Dic83—22no—
184Dic83—22no—
185Dic83—41no—
186Dic83—41no—
187Dic83—41no—
188Dic83—41no—
189C2×C83—42no—
190C2×C83#53811yescharacteristic
191C2×C83#53811yescharacteristic
192C2×C83#54311yescharacteristic
193C2×C83#55411yescharacteristic
194C2×C83#55411yescharacteristic
195C2×C83#55511yescharacteristic
196C2×C83#56011yescharacteristic
197D83—42no—
198#2784—831nomaximal, Sylow
199C83#278411yescharacteristic, complemented, Sylow
200#400—831no—
201#396—831no—
202#396—831no—
203#395—831no—
204#393—831no—
205#393D8311yescharacteristic, complemented
206#390—831no—
207#387—831no—
208#387—831no—
209#386—831no—
210#386—831no—
211#385—831no—
212#385—831no—
213#380—831no—
214#380—831no—
215C22×Q8C2×D8311yescharacteristic
216C22×D4—831no—
217C23×C4—831no—
218C23×C4—831no—
219C2×C4⋊C4—831no—
220C2×C4⋊C4—831no—
221C2×C4⋊C4—831no—
222C2×C4⋊C4—831no—
223C2×C4⋊C4—831no—
224C2×C4⋊C4—831no—
225C2×C4⋊C4—831no—
226C2×C4⋊C4—831no—
227C2×C22⋊C4—831no—
228C2×C22⋊C4—831no—
229C2×C22⋊C4—831no—
230C2×C22⋊C4—831no—
231C2×C22⋊C4—831no—
232C2×C22⋊C4—831no—
233C2×C22⋊C4—831no—
234C2×C22⋊C4C2×D8311yescharacteristic
235C2×C22⋊C4C2×D8311yescharacteristic
236C2×C22⋊C4C2×D8311yescharacteristic
237C2×C42—831no—
238#95—831no—
239#95—831no—
240#95—831no—
241#95—831no—
242#95—831no—
243#95—831no—
244#95—831no—
245#95—831no—
246#95—831no—
247#95C2×D8311yescharacteristic
248#95C2×D8311yescharacteristic
249#95C2×D8311yescharacteristic
250C24—831no—
251C24C22×D8311yescharacteristic
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×D4—3321no—
256C2×D4—3321no—
257C2×D4—834no—
258C22×C4—3321no—
259C22×C4—3321no—
260C22×C4—3321no—
261C22×C4—3321no—
262C22×C4—831no—
263C22×C4—831no—
264C22×C4—831no—
265C22×C4—831no—
266C22×C4—831no—
267C22×C4—831no—
268C22×C4—831no—
269C22×C4—1662no—
270C22×C4—1662no—
271C22×C4C22×D8311yescharacteristic
272C22×C4C22×D8311yescharacteristic
273C22×C4C22×D8311yescharacteristic
274C22×C4C22×D8311yescharacteristic
275C22×C4C22×D8311yescharacteristic
276C22×C4C22×D8311yescharacteristic
277C4⋊C4—3321no—
278C4⋊C4—3321no—
279C4⋊C4—3321no—
280C4⋊C4—3321no—
281C4⋊C4—3321no—
282C4⋊C4—3321no—
283C4⋊C4—1662no—
284C4⋊C4—1662no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—3321no—
289C22⋊C4—3321no—
290C22⋊C4—3321no—
291C22⋊C4—3321no—
292C22⋊C4—3321no—
293C22⋊C4—1662no—
294C22⋊C4—1662no—
295C42—3321no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—832no—
303C23—1661no—
304C23—1661no—
305C23—1661no—
306C23—1661no—
307C23—1661no—
308C23—1661no—
309C23Q8×D8312yes—
310C23C23×D8311yescharacteristic
311Q8—44no—
312D4—3324no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—3324no—
332C2×C4—3324no—
333C2×C4—832no—
334C2×C4—832no—
335C2×C4—832no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4—1661no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4—1661no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4—1661no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3321no—
362C22—3321no—
363C22—3321no—
364C22—3321no—
365C22—1662no—
366C22—1662no—
367C22—1662no—
368C22C2×Q8×D8311yescharacteristic
369C22#41×D8311yescharacteristic
370C22#41×D8311yescharacteristic
371C22#1458183711yescharacteristic
372C22#1458183711yescharacteristic
373C22#1458183711yescharacteristic
374C22#1458183811yescharacteristic
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3321no—
382C4—3321no—
383C4—3321no—
384C4—3321no—
385C4—1662no—
386C4—1662no—
387C4—1662no—
388C2—42no—
389C2—3322no—
390C2#1912828511yescharacteristic
391C2#1912828511yescharacteristic
392C2#1912832611yescharacteristic
393C2#1912840511yescharacteristic
394C2#1912840511yescharacteristic
395C2#1912841311yescharacteristic
396C2#1912844011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#393×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#393D83
hypocenterC83#2784
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #19127642C2semidirect
2. #19127625C2semidirect
3. #19127624C2semidirect
4. #19127620C2semidirect
5. #19127612C2semidirect
7. #19127586C2semidirect
8. #19127583C2semidirect
11. #19127575C2semidirect
12. #19127573C2semidirect
13. #19127560C2semidirect
14. #19127559C2semidirect
15. #393×C83C2semidirect
54. C2×C22⋊D83Q8semidirect
199. C83#2784semidirect
205. #393D83semidirect

Maximal subgroups

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

subgroupindex
#191276422
#191276252
#191276242
#191276202
#191276122
#191276012
#191275862
#191275832
#191275782
#191275772
#191275752
#191275732
#191275602
#191275592
#393×C832
#278483