Finite Group Databasebeta

← #27174457

Subgroups — #27174457

Order 12928.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×C22⋊D101C211yesmaximal, characteristic
2C2×#15692237C212yesmaximal, complemented
3C2×C22⋊C4×D101C211yesmaximal, characteristic, complemented
4#20265093C212yesmaximal, complemented
5#20265092C212yesmaximal, complemented
6#20265053C212yesmaximal, complemented
7#20265052C211yesmaximal, characteristic, complemented
8#20265049C211yesmaximal, characteristic, complemented
9#20265033C211yesmaximal, characteristic, complemented
10#20265032C211yesmaximal, characteristic, complemented
11#379×C101C211yesmaximal, characteristic, complemented
12C24×D101C2211yescharacteristic
13C22×C22⋊D101—24no—
14C22×C22⋊D101—28no—
15C22×C22⋊D101C2212yes—
16C22×C22⋊D101C2212yes—
17C23×Dic101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic, complemented
19C22×C4×D101C2211yescharacteristic, complemented
20#15692237—28no—
21C2×C4⋊Dic101C2212yescomplemented
22C2×#13108053C2212yescomplemented
23C2×C22⋊Dic101C2211yescharacteristic
24C2×C22⋊Dic101C2211yescharacteristic
25C2×C22⋊Dic101C2212yescomplemented
26C2×C22⋊Dic101C2212yescomplemented
27C2×#13108051C2212yescomplemented
28C2×#13108051C2211yescharacteristic, complemented
29C2×#13108051C2211yescharacteristic, complemented
30C2×#13108051C2212yescomplemented
31C22⋊C4×D101—24no—
32#15692162C2212yescomplemented
33#15692162C2212yescomplemented
34C25×C101C2211yescharacteristic
35C2×C22⋊C4×C101C2211yescharacteristic
36C2×C22⋊C4×C101C2211yescharacteristic
37C2×C22⋊C4×C101C2212yescomplemented
38C2×C22⋊C4×C101C2212yescomplemented
39C23×D101—22no—
40C23×D101—22no—
41C23×D101—24no—
42C23×D101—42no—
43C23×D101D414yes—
44C23×D101C2311yescharacteristic
45C23×D101C2311yescharacteristic
46C2×C22⋊D101—232no—
47C2×C22⋊D101—24no—
48C2×C22⋊D101—24no—
49C2×C22⋊D101—24no—
50C2×C22⋊D101—24no—
51C2×C22⋊D101—24no—
52C2×C22⋊D101D418yes—
53C22×Dic101—22no—
54C22×Dic101—24no—
55C22×Dic101C2311yescharacteristic
56C22×Dic101C2311yescharacteristic
57C22×Dic101C2312yes—
58C22×Dic101C2312yes—
59C2×C4×D101—22no—
60C2×C4×D101—22no—
61C2×C4×D101—24no—
62C2×C4×D101—42no—
63C4⋊Dic101—24no—
64#13108053—24no—
65C22⋊Dic101—22no—
66C22⋊Dic101—24no—
67C22⋊Dic101—22no—
68C22⋊Dic101—42no—
69#13108051—22no—
70#13108051—22no—
71#13108051—24no—
72#13108051—42no—
73C24×C101—24no—
74C24×C101—22no—
75C24×C101—22no—
76C24×C101—22no—
77C24×C101—42no—
78C24×C101C2312yes—
79C24×C101C2311yescharacteristic
80C22×C4×C101C2311yescharacteristic
81C22×C4×C101C2311yescharacteristic
82C22×C4×C101C2312yes—
83C22⋊C4×C101—22no—
84C22⋊C4×C101—24no—
85C22⋊C4×C101—22no—
86C22⋊C4×C101—42no—
87C23×C101—22no—
88C23×C101—22no—
89C23×C101—28no—
90C23×C101—22no—
91C23×C101—22no—
92C23×C101—22no—
93C23×C101—22no—
94C23×C101—22no—
95C23×C101—22no—
96C23×C101—48no—
97C23×C101—48no—
98C23×C101—44no—
99C23×C101—44no—
100C23×C101C2×D414yes—
101C23×C101C2×D412yes—
102C23×C101#4112yes—
103C23×C101#4112yes—
104C23×C101C2411yescharacteristic
105C22×D101—21no—
106C22×D101—21no—
107C22×D101—22no—
108C22×D101—28no—
109C22×D101—22no—
110C22×D101—21no—
111C22×D101—21no—
112C22×D101—22no—
113C22×D101—28no—
114C22×D101—44no—
115C22×D101—48no—
116C22×D101—48no—
117C22×D101C2×D412yes—
118C22×D101C2×D414yes—
119C22×D101C2×D412yes—
120C2×C4×C101—22no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—22no—
124C2×C4×C101—22no—
125C2×C4×C101—22no—
126C2×C4×C101—22no—
127C22⋊D101—216no—
128C22⋊D101—216no—
129C2×Dic101—28no—
130C2×Dic101—22no—
131C2×Dic101—22no—
132C2×Dic101—22no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—22no—
136C2×Dic101—22no—
137C2×Dic101—22no—
138C2×Dic101—22no—
139C2×Dic101C2×D414yes—
140C4×D101—44no—
141C4×D101—44no—
142C22×C101—22no—
143C22×C101—22no—
144C22×C101—24no—
145C22×C101—24no—
146C22×C101—24no—
147C22×C101—24no—
148C22×C101—24no—
149C22×C101—416no—
150C22×C101—42no—
151C22×C101—42no—
152C22×C101—42no—
153C22×C101—42no—
154C22×C101#12314yescomplemented
155C22×C101C22×D411yescharacteristic
156C22×C101C22×D412yes—
157C22×C101C2×#4111yescharacteristic
158C22×C101C2×#4111yescharacteristic
159C22×C101H(5,𝔽2)12yes—
160C2×D101—22no—
161C2×D101—22no—
162C2×D101—24no—
163C2×D101—24no—
164C2×D101—22no—
165C2×D101—22no—
166C2×D101—22no—
167C2×D101—416no—
168C2×D101—42no—
169C2×D101—42no—
170C2×D101#12014yes—
171C4×C101—41no—
172C4×C101—41no—
173C4×C101—42no—
174Dic101—24no—
175Dic101—41no—
176Dic101—41no—
177Dic101—42no—
178C2×C101—24no—
179C2×C101—44no—
180C2×C101C2×#12011yescharacteristic
181C2×C101C2×#12311yescharacteristic
182C2×C101#53411yescharacteristic
183C2×C101#54612yes—
184C2×C101#54612yes—
185D101—24no—
186D101—42no—
187#2405—1011nomaximal, Sylow
188C101#240511yescharacteristic, complemented, Sylow
189C23×D4—1011no—
190C2×#123—1012no—
191C22×C22⋊C4—1011no—
192#394—1012no—
193#394—1012no—
194#386—1012no—
195#385—1011no—
196#385—1011no—
197#379—1011no—
198#379—1011no—
199#379D10111yescharacteristic, complemented
200C25—1011no—
201C25C2×D10111yescharacteristic
202C22×D4—1012no—
203C22×D4—1012no—
204C22×D4—2028no—
205C22×D4—2024no—
206C23×C4—1011no—
207C23×C4—1011no—
208C23×C4—1011no—
209#123—2028no—
210C2×C4⋊C4—1012no—
211C2×C4⋊C4—1012no—
212C2×C22⋊C4—1012no—
213C2×C22⋊C4—1012no—
214C2×C22⋊C4—1011no—
215C2×C22⋊C4—1012no—
216C2×C22⋊C4—1011no—
217C2×C22⋊C4—1011no—
218C2×C22⋊C4—1011no—
219C2×C22⋊C4—1012no—
220C2×C22⋊C4—2024no—
221C2×C22⋊C4C2×D10111yescharacteristic
222C2×C22⋊C4C2×D10111yescharacteristic
223C2×C22⋊C4C2×D10112yescomplemented
224C2×C22⋊C4C2×D10112yescomplemented
225#95—1012no—
226#95—1012no—
227C24—24no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—42no—
232C24—1011no—
233C24—1014no—
234C24—1011no—
235C24—4042no—
236C24—2024no—
237C24—2022no—
238C24—2022no—
239C24C22×D10112yes—
240C24C22×D10111yescharacteristic
241C2×D4—1018no—
242C2×D4—2024no—
243C2×D4—2024no—
244C2×D4—2024no—
245C2×D4—2024no—
246C2×D4—20232no—
247C2×D4—2024no—
248C22×C4—1012no—
249C22×C4—1011no—
250C22×C4—1012no—
251C22×C4—1011no—
252C22×C4—4042no—
253C22×C4—2022no—
254C22×C4—2022no—
255C22×C4—2024no—
256C22×C4—2024no—
257C22×C4—2022no—
258C22×C4C22×D10111yescharacteristic
259C22×C4C22×D10112yes—
260C22×C4C22×D10111yescharacteristic
261C4⋊C4—2024no—
262C4⋊C4—2024no—
263C22⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—4042no—
268C22⋊C4—4042no—
269C22⋊C4—2022no—
270C22⋊C4—2022no—
271C22⋊C4—2024no—
272C22⋊C4—2022no—
273C22⋊C4—2024no—
274C22⋊C4—2022no—
275C23—22no—
276C23—22no—
277C23—28no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—48no—
285C23—48no—
286C23—44no—
287C23—44no—
288C23—1014no—
289C23—1012no—
290C23—1012no—
291C23—4048no—
292C23—4048no—
293C23—4044no—
294C23—2028no—
295C23—2022no—
296C23—2021no—
297C23—2021no—
298C23—2022no—
299C23—2028no—
300C23—2021no—
301C23—2021no—
302C23—2022no—
303C23D4×D10114yes—
304C23D4×D10112yes—
305C23#1310807312yes—
306C23#1310807412yes—
307C23C23×D10111yescharacteristic
308D4—20216no—
309D4—20216no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—1014no—
318C2×C4—4044no—
319C2×C4—4044no—
320C2×C4—2022no—
321C2×C4—2021no—
322C2×C4—2021no—
323C2×C4—2022no—
324C2×C4—2022no—
325C2×C4—2022no—
326C2×C4—2022no—
327C2×C4—2028no—
328C2×C4—2022no—
329C2×C4—2022no—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—416no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—1014no—
343C22—40416no—
344C22—4042no—
345C22—4042no—
346C22—2022no—
347C22—2022no—
348C22—2022no—
349C22—2022no—
350C22—2024no—
351C22—2024no—
352C22—2022no—
353C22#1569223714yescomplemented
354C22C2×D4×D10111yescharacteristic
355C22C2×D4×D10112yes—
356C22C2×#1310807311yescharacteristic
357C22C2×#1310807411yescharacteristic
358C22#1569230012yes—
359C4—41no—
360C4—42no—
361C4—41no—
362C4—4041no—
363C4—4042no—
364C4—4041no—
365C4—2024no—
366C2—24no—
367C2—44no—
368C2—4042no—
369C2—2024no—
370C2#120×D10111yescharacteristic
371C2C2×#1569223711yescharacteristic
372C2#2026573711yescharacteristic
373C2#2026581412yes—
374C2#2026581712yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#379×C101C2
prime radicalC23×C22⋊D101C2
hypercenter#379D101
hypocenterC101#2405
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#15692237C2semidirect
3. C2×C22⋊C4×D101C2semidirect
4. #20265093C2semidirect
5. #20265092C2semidirect
6. #20265053C2semidirect
7. #20265052C2semidirect
8. #20265049C2semidirect
9. #20265033C2semidirect
10. #20265032C2semidirect
11. #379×C101C2semidirect
18. C22×C4×D101C22semidirect
19. C22×C4×D101C22semidirect
21. C2×C4⋊Dic101C22semidirect
22. C2×#13108053C22semidirect
25. C2×C22⋊Dic101C22semidirect
26. C2×C22⋊Dic101C22semidirect
27. C2×#13108051C22semidirect
28. C2×#13108051C22semidirect
29. C2×#13108051C22semidirect
30. C2×#13108051C22semidirect
32. #15692162C22semidirect
33. #15692162C22semidirect
37. C2×C22⋊C4×C101C22semidirect
38. C2×C22⋊C4×C101C22semidirect
154. C22×C101#123semidirect
188. C101#2405semidirect
199. #379D101semidirect
223. C2×C22⋊C4C2×D101semidirect
224. C2×C22⋊C4C2×D101semidirect
353. C22#15692237semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1012
C2×#156922372
C2×C22⋊C4×D1012
#202650932
#202650922
#202650532
#202650522
#202650492
#202650332
#202650322
#379×C1012
#2405101