Finite Group Databasebeta

← #27174452

Subgroups — #27174452

Order 12928.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108075C211yesmaximal, characteristic, complemented
2C2×#15692225C212yesmaximal, complemented
3C2×C22⋊C4×D101C211yesmaximal, characteristic, complemented
4#20265088C211yesmaximal, characteristic, complemented
5#20265087C211yesmaximal, characteristic, complemented
6#20265086C211yesmaximal, characteristic, complemented
7#20265055C212yesmaximal, complemented
8#20265055C212yesmaximal, complemented
9#20265053C212yesmaximal, complemented
10#20265039C211yesmaximal, characteristic, complemented
11#393×C101C211yesmaximal, characteristic, complemented
12C24×D101C2211yescharacteristic
13C2×#13108075—28no—
14C2×#13108075—24no—
15C22×D404C2212yes—
16C22×D404C2212yes—
17C22×D404C2211yescharacteristic, complemented
18C22×C22⋊D101C2212yes—
19C22×C4×D101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2212yescomplemented
22#15692225—28no—
23C2×C22⋊Dic101C2211yescharacteristic
24C2×#13108051C2212yes—
25C2×#13108051C2211yescharacteristic
26C2×#13108051C2212yes—
27C2×#13108051C2211yescharacteristic
28C2×#13108051C2212yescomplemented
29C2×#13108051C2212yescomplemented
30C22⋊C4×D101—24no—
31#15692162C2212yescomplemented
32#15692162C2211yescharacteristic, complemented
33#15692162C2212yescomplemented
34C22×Q8×C101C2211yescharacteristic, complemented
35C2×C22⋊C4×C101C2212yes—
36C2×C22⋊C4×C101C2211yescharacteristic
37#95×C101C2211yescharacteristic, complemented
38#95×C101C2212yescomplemented
39C23×D101—22no—
40C23×D101—22no—
41C23×D101—22no—
42C23×D101—24no—
43C23×D101—42no—
44C23×D101C2312yes—
45C23×D101C2311yescharacteristic
46C23×D101C2311yescharacteristic
47#13108075—232no—
48C2×D404—24no—
49C2×D404—24no—
50C2×D404—24no—
51C2×D404—22no—
52C2×D404—24no—
53C2×D404—24no—
54C2×D404—42no—
55C2×D404D418yescomplemented
56C2×C22⋊D101—24no—
57C2×C22⋊D101—24no—
58C2×C22⋊D101—42no—
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2312yes—
61C22×Dic101C2311yescharacteristic
62C2×C4×D101—24no—
63C2×C4×D101—24no—
64C2×C4×D101—24no—
65C2×C4×D101—24no—
66C2×C4×D101—22no—
67C2×C4×D101—42no—
68C2×C4×D101D414yescomplemented
69C22⋊Dic101—22no—
70#13108051—24no—
71#13108051—22no—
72#13108051—24no—
73#13108051—22no—
74#13108051—42no—
75#13108051—42no—
76C24×C101C2311yescharacteristic
77C2×Q8×C101—24no—
78C2×Q8×C101—22no—
79C22×C4×C101C2311yescharacteristic
80C22×C4×C101C2312yes—
81C22×C4×C101C2311yescharacteristic
82C22×C4×C101C2312yes—
83C22⋊C4×C101—22no—
84C22⋊C4×C101—24no—
85C23×C101—21no—
86C23×C101—22no—
87C23×C101—21no—
88C23×C101—22no—
89C23×C101#4112yescomplemented
90C23×C101C2411yescharacteristic
91C22×D101—21no—
92C22×D101—21no—
93C22×D101—22no—
94C22×D101—28no—
95C22×D101—21no—
96C22×D101—22no—
97C22×D101—22no—
98C22×D101—22no—
99C22×D101—22no—
100C22×D101—22no—
101C22×D101—21no—
102C22×D101—22no—
103C22×D101—22no—
104C22×D101—22no—
105C22×D101—48no—
106C22×D101—48no—
107C22×D101—44no—
108C22×D101—44no—
109C22×D101C2×D414yes—
110C22×D101C2×D414yes—
111C22×D101C2×D412yes—
112C22×D101#4112yescomplemented
113Q8×C101—28no—
114C2×C4×C101—22no—
115C2×C4×C101—22no—
116C2×C4×C101—22no—
117C2×C4×C101—21no—
118C2×C4×C101—22no—
119C2×C4×C101—22no—
120C2×C4×C101—22no—
121C2×C4×C101—22no—
122C2×C4×C101C2×D414yes—
123C2×C4×C101C2×D412yes—
124C22⋊D101—48no—
125C2×Dic101—22no—
126C2×Dic101—22no—
127C2×Dic101—21no—
128C2×Dic101—22no—
129C2×Dic101—22no—
130C2×Dic101—22no—
131C2×Dic101C2×D412yes—
132D404—28no—
133D404—216no—
134D404—48no—
135C4×D101—216no—
136C4×D101—28no—
137C4×D101—44no—
138C22×C101—22no—
139C22×C101—24no—
140C22×C101—41no—
141C22×C101—41no—
142C22×C101—42no—
143C22×C101C22×D411yescharacteristic
144C22×C101C22×D412yes—
145C22×C101C2×#4111yescharacteristic
146C22×C101C2×#4111yescharacteristic
147C22×C101H(5,𝔽2)12yes—
148C2×D101—22no—
149C2×D101—22no—
150C2×D101—24no—
151C2×D101—24no—
152C2×D101—24no—
153C2×D101—22no—
154C2×D101—24no—
155C2×D101—24no—
156C2×D101—24no—
157C2×D101—416no—
158C2×D101—41no—
159C2×D101—42no—
160C2×D101—42no—
161C2×D101—41no—
162C2×D101—42no—
163C2×D101—42no—
164C2×D101—42no—
165C2×D101#12014yes—
166C4×C101—24no—
167C4×C101—22no—
168C4×C101—41no—
169C4×C101—42no—
170Dic101—22no—
171Dic101—41no—
172Dic101—42no—
173C2×C101—42no—
174C2×C101C2×#12011yescharacteristic
175C2×C101#52511yescharacteristic
176C2×C101#53511yescharacteristic
177C2×C101#54612yes—
178C2×C101#54612yes—
179D101—24no—
180D101—44no—
181D101—42no—
182#2404—1011nomaximal, Sylow
183C101#240411yescharacteristic, complemented, Sylow
184C22×#41—1011no—
185C2×#120—1012no—
186C22×C22⋊C4—1011no—
187#393—1011no—
188#393D10111yescharacteristic, complemented
189#392—1011no—
190#392—1011no—
191#386—1012no—
192#386—1012no—
193#386—1012no—
194#381—1011no—
195C25—1011no—
196C2×#41—2024no—
197C2×#41—2028no—
198C22×Q8C2×D10111yescharacteristic, complemented
199C22×D4—1012no—
200C22×D4—1012no—
201C22×D4—1012no—
202C22×D4—1011no—
203C23×C4—1011no—
204C23×C4—1011no—
205C23×C4—1012no—
206#120—2028no—
207C2×C22⋊C4—1012no—
208C2×C22⋊C4—1012no—
209C2×C22⋊C4—1011no—
210C2×C22⋊C4—1011no—
211C2×C22⋊C4—1011no—
212C2×C22⋊C4—1012no—
213C2×C22⋊C4—1012no—
214C2×C22⋊C4—2024no—
215C2×C22⋊C4C2×D10111yescharacteristic
216C2×C22⋊C4C2×D10112yes—
217#95—1012no—
218#95—1011no—
219#95—1012no—
220#95C2×D10111yescharacteristic, complemented
221#95C2×D10112yescomplemented
222C24—1011no—
223C24—1011no—
224C24—1012no—
225C24—4042no—
226C24—2024no—
227C24—2022no—
228C24—2022no—
229C24—2022no—
230C24C22×D10111yescharacteristic
231#41—20232no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×D4—1018no—
235C2×D4—4042no—
236C2×D4—4042no—
237C2×D4—2024no—
238C2×D4—2024no—
239C2×D4—2024no—
240C2×D4—2024no—
241C2×D4—2024no—
242C2×D4—2024no—
243C2×D4—2024no—
244C2×D4—2022no—
245C22×C4—1011no—
246C22×C4—1014no—
247C22×C4—1012no—
248C22×C4—1011no—
249C22×C4—4042no—
250C22×C4—2024no—
251C22×C4—2022no—
252C22×C4—2024no—
253C22×C4—2024no—
254C22×C4—2024no—
255C22×C4C22×D10112yes—
256C22×C4C22×D10111yescharacteristic
257C22×C4C22×D10111yescharacteristic
258C22×C4C22×D10112yes—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—4042no—
262C22⋊C4—4042no—
263C22⋊C4—2022no—
264C22⋊C4—2024no—
265C22⋊C4—2022no—
266C22⋊C4—2022no—
267C22⋊C4—2024no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—1014no—
273C23—1012no—
274C23—1014no—
275C23—1012no—
276C23—4048no—
277C23—4044no—
278C23—4044no—
279C23—4048no—
280C23—2028no—
281C23—2021no—
282C23—2022no—
283C23—2022no—
284C23—2022no—
285C23—2022no—
286C23—2022no—
287C23—2021no—
288C23—2022no—
289C23—2022no—
290C23—2022no—
291C23—2021no—
292C23—2021no—
293C23—2022no—
294C23#1310807512yescomplemented
295C23C23×D10111yescharacteristic
296Q8—28no—
297D4—4048no—
298D4—4048no—
299D4—20216no—
300D4—2028no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—1012no—
310C2×C4—4044no—
311C2×C4—2028no—
312C2×C4—20216no—
313C2×C4—2021no—
314C2×C4—2022no—
315C2×C4—2022no—
316C2×C4—2022no—
317C2×C4—2022no—
318C2×C4—2022no—
319C2×C4D4×D10112yes—
320C2×C4D4×D10114yes—
321C22—24no—
322C22—22no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—1014no—
327C22—40416no—
328C22—4041no—
329C22—4042no—
330C22—4042no—
331C22—4042no—
332C22—4042no—
333C22—4041no—
334C22—4042no—
335C22—2024no—
336C22—2024no—
337C22—2024no—
338C22—2024no—
339C22—2022no—
340C22—2022no—
341C22—2024no—
342C22—2024no—
343C22—2022no—
344C22C2×D4×D10111yescharacteristic
345C22C2×D4×D10112yes—
346C22#41×D10111yescharacteristic
347C22C2×#1310807511yescharacteristic
348C22#1569230112yes—
349C4—22no—
350C4—24no—
351C4—41no—
352C4—42no—
353C4—4041no—
354C4—4042no—
355C4—2022no—
356C2—42no—
357C2—4042no—
358C2—4044no—
359C2—2024no—
360C2#120×D10111yescharacteristic
361C2#2026569011yescharacteristic
362C2#2026574511yescharacteristic
363C2#2026581812yes—
364C2#2026582012yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#393×C101C2
hypercenter#393D101
hypocenterC101#2404
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#13108075C2semidirect
2. C2×#15692225C2semidirect
3. C2×C22⋊C4×D101C2semidirect
4. #20265088C2semidirect
5. #20265087C2semidirect
6. #20265086C2semidirect
7. #20265055C2semidirect
8. #20265055C2semidirect
9. #20265053C2semidirect
10. #20265039C2semidirect
11. #393×C101C2semidirect
17. C22×D404C22semidirect
21. C22×C4×D101C22semidirect
28. C2×#13108051C22semidirect
29. C2×#13108051C22semidirect
31. #15692162C22semidirect
32. #15692162C22semidirect
33. #15692162C22semidirect
34. C22×Q8×C101C22semidirect
37. #95×C101C22semidirect
38. #95×C101C22semidirect
55. C2×D404D4semidirect
68. C2×C4×D101D4semidirect
89. C23×C101#41semidirect
112. C22×D101#41semidirect
183. C101#2404semidirect
188. #393D101semidirect
198. C22×Q8C2×D101semidirect
220. #95C2×D101semidirect
221. #95C2×D101semidirect
294. C23#13108075semidirect

Maximal subgroups

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

subgroupindex
C22×#131080752
C2×#156922252
C2×C22⋊C4×D1012
#202650882
#202650872
#202650862
#202650552
#202650532
#202650392
#393×C1012
#2404101