Finite Group Databasebeta

← #27174489

Subgroups — #27174489

Order 12928.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 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×D404C211yesmaximal, characteristic, complemented
2C2×#15692230C212yesmaximal, complemented
3C2×#15692225C212yesmaximal, complemented
4C2×C4×C22⋊D101C211yesmaximal, characteristic, complemented
5#20265095C212yesmaximal, complemented
6#20265087C212yesmaximal, complemented
7#20265078C212yesmaximal, complemented
8#20265032C212yesmaximal, complemented
9#380×C101C211yesmaximal, characteristic, complemented
10C24×D101C2212yes—
11C22×D404—28no—
12C22×D404—24no—
13C22×D404C2212yes—
14C22×D404C2212yes—
15C22×D404C2214yescomplemented
16C22×C22⋊D101C2211yescharacteristic
17C22×C4×D101C2211yescharacteristic, complemented
18#15692230—28no—
19#15692225—28no—
20C4×C22⋊D101—24no—
21C2×#13108053C2211yescharacteristic, complemented
22C2×C22⋊Dic101C2211yescharacteristic
23C2×#13108051C2214yescomplemented
24C2×#13108051C2214yescomplemented
25C2×#13108051C2211yescharacteristic, complemented
26C2×#13108051C2214yescomplemented
27C2×C4×Dic101C2211yescharacteristic, complemented
28C23×C4×C101C2211yescharacteristic
29C2×C4⋊C4×C101C2212yescomplemented
30C2×C22⋊C4×C101C2212yes—
31#95×C101C2212yescomplemented
32C23×D101—24no—
33C23×D101—28no—
34C23×D101—24no—
35C23×D101—24no—
36C23×D101—44no—
37C23×D101C2314yes—
38C23×D101C2311yescharacteristic
39C2×D404—232no—
40C2×D404—28no—
41C2×D404—24no—
42C2×D404—24no—
43C2×D404—24no—
44C2×D404—28no—
45C2×D404—28no—
46C2×D404—44no—
47C2×D404D418yescomplemented
48C2×C22⋊D101—22no—
49C2×C22⋊D101—22no—
50C2×C22⋊D101D414yescomplemented
51C22×Dic101C2311yescharacteristic
52C22×Dic101C2311yescharacteristic
53C22×Dic101C2311yescharacteristic
54C2×C4×D101—22no—
55C2×C4×D101—24no—
56#13108053—22no—
57C22⋊Dic101—22no—
58#13108051—28no—
59#13108051—22no—
60#13108051—28no—
61#13108051—44no—
62C4×Dic101—22no—
63C24×C101C2311yescharacteristic
64C22×C4×C101—24no—
65C22×C4×C101D414yescomplemented
66C22×C4×C101C2314yes—
67C22×C4×C101C2311yescharacteristic
68C22×C4×C101C2311yescharacteristic
69C4⋊C4×C101—24no—
70C22⋊C4×C101—24no—
71C23×C101—22no—
72C23×C101—22no—
73C23×C101C2×D412yes—
74C23×C101C2×D412yes—
75C23×C101#4112yes—
76C23×C101C2411yescharacteristic
77C22×D101—216no—
78C22×D101—21no—
79C22×D101—24no—
80C22×D101—22no—
81C22×D101—24no—
82C22×D101—24no—
83C22×D101—21no—
84C22×D101—24no—
85C22×D101—24no—
86C22×D101—22no—
87C22×D101—24no—
88C22×D101—416no—
89C22×D101—416no—
90C22×D101—48no—
91C22×D101—48no—
92C22×D101C2×D418yes—
93C22×D101C2×D412yescomplemented
94C2×C4×C101—22no—
95C2×C4×C101—28no—
96C2×C4×C101—24no—
97C2×C4×C101—24no—
98C2×C4×C101—24no—
99C2×C4×C101C2×D414yes—
100C2×C4×C101C2×D412yes—
101C2×C4×C101C2×D412yes—
102C22⋊D101—28no—
103C2×Dic101—21no—
104C2×Dic101—21no—
105C2×Dic101—21no—
106C2×Dic101—22no—
107C2×Dic101—21no—
108C2×Dic101—22no—
109C2×Dic101C2×D412yescomplemented
110D404—216no—
111D404—216no—
112D404—416no—
113C4×D101—44no—
114C22×C101—22no—
115C22×C101—22no—
116C22×C101—24no—
117C22×C101—24no—
118C22×C101#12114yescomplemented
119C22×C101C22×D412yes—
120C22×C101C22×D411yescharacteristic
121C22×C101C22×D411yescharacteristic
122C22×C101C2×#4111yescharacteristic
123C22×C101H(5,𝔽2)12yes—
124C2×D101—28no—
125C2×D101—22no—
126C2×D101—24no—
127C2×D101—28no—
128C2×D101—28no—
129C2×D101—432no—
130C2×D101—41no—
131C2×D101—44no—
132C2×D101—41no—
133C2×D101—44no—
134C2×D101—44no—
135C2×D101—42no—
136C2×D101—44no—
137C4×C101—22no—
138C4×C101—44no—
139C4×C101#12014yescomplemented
140Dic101—22no—
141Dic101—41no—
142Dic101—41no—
143C2×C101—24no—
144C2×C101C2×#12011yescharacteristic
145C2×C101C2×#12111yescharacteristic
146C2×C101#53511yescharacteristic
147C2×C101D4212yes—
148C2×C101#54612yes—
149D101—42no—
150D101—48no—
151#2408—1011nomaximal, Sylow
152C101#240811yescharacteristic, complemented, Sylow
153C23×D4—1011no—
154C2×#121—1012no—
155C2×#120—1012no—
156C2×C4×D4—1011no—
157#394—1012no—
158#392—1012no—
159#390—1012no—
160#380D10111yescharacteristic, complemented
161#379—1012no—
162C25—1012no—
163C22×D4—1011no—
164C22×D4—1012no—
165C22×D4—1014no—
166C22×D4—1012no—
167C22×D4—2028no—
168C22×D4—2024no—
169C23×C4—1011no—
170C23×C4C2×D10111yescharacteristic
171#121—2028no—
172#120—2028no—
173C4×D4—2024no—
174C2×C4⋊C4—1011no—
175C2×C4⋊C4C2×D10112yescomplemented
176C2×C22⋊C4—1014no—
177C2×C22⋊C4—1011no—
178C2×C22⋊C4—1011no—
179C2×C22⋊C4—1014no—
180C2×C22⋊C4—1014no—
181C2×C22⋊C4C2×D10112yes—
182C2×C42—1011no—
183#95C2×D10112yescomplemented
184C24—1014no—
185C24—1011no—
186C24—4044no—
187C24—2028no—
188C24—2024no—
189C24—2024no—
190C24—2024no—
191C24C22×D10111yescharacteristic
192C2×D4—1014no—
193C2×D4—1018no—
194C2×D4—4044no—
195C2×D4—2028no—
196C2×D4—2028no—
197C2×D4—2028no—
198C2×D4—20232no—
199C2×D4—2022no—
200C2×D4—2022no—
201C2×D4—2024no—
202C2×D4—2024no—
203C2×D4—2024no—
204C22×C4—24no—
205C22×C4—1011no—
206C22×C4—1011no—
207C22×C4—1011no—
208C22×C4—2024no—
209C22×C4—2022no—
210C22×C4D40414yescomplemented
211C22×C4C22×D10114yes—
212C22×C4C22×D10111yescharacteristic
213C22×C4C22×D10111yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—2022no—
216C22⋊C4—24no—
217C22⋊C4—4044no—
218C22⋊C4—2028no—
219C22⋊C4—2028no—
220C22⋊C4—2022no—
221C22⋊C4—2022no—
222C42—2022no—
223C23—22no—
224C23—22no—
225C23—1018no—
226C23—1012no—
227C23—4048no—
228C23—40416no—
229C23—40416no—
230C23—4048no—
231C23—20216no—
232C23—2024no—
233C23—2022no—
234C23—2024no—
235C23—2021no—
236C23—2024no—
237C23—2021no—
238C23—2024no—
239C23—2024no—
240C23—2022no—
241C23—2024no—
242C23D4×D10112yes—
243C23C2×D40412yes—
244C23#1310807512yes—
245C23C23×D10111yescharacteristic
246D4—40416no—
247D4—20216no—
248D4—2028no—
249D4—20216no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—1012no—
256C2×C4—4044no—
257C2×C4—2022no—
258C2×C4—2021no—
259C2×C4—2021no—
260C2×C4—2022no—
261C2×C4—2021no—
262C2×C4—2021no—
263C2×C4D4×D10114yes—
264C2×C4C2×D40412yes—
265C2×C4C2×D40412yes—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—40432no—
271C22—4044no—
272C22—4041no—
273C22—4044no—
274C22—4044no—
275C22—4042no—
276C22—4041no—
277C22—4044no—
278C22—2024no—
279C22—2028no—
280C22—2028no—
281C22—2028no—
282C22—2022no—
283C22#1569223014yescomplemented
284C22C2×D4×D10111yescharacteristic
285C22C2×D4×D10112yes—
286C22C22×D40411yescharacteristic
287C22C2×#1310807511yescharacteristic
288C22#1569230112yes—
289C4—22no—
290C4—44no—
291C4—4041no—
292C4—4041no—
293C4—2022no—
294C4#1569222514yescomplemented
295C2—24no—
296C2—4048no—
297C2—4042no—
298C2C2×#1569222511yescharacteristic
299C2C2×#1569223011yescharacteristic
300C2#2026574611yescharacteristic
301C2#2026581112yes—
302C2#2026581812yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#380×C101C2
hypercenter#380D101
hypocenterC101#2408
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D404C2semidirect
2. C2×#15692230C2semidirect
3. C2×#15692225C2semidirect
4. C2×C4×C22⋊D101C2semidirect
5. #20265095C2semidirect
6. #20265087C2semidirect
7. #20265078C2semidirect
8. #20265032C2semidirect
9. #380×C101C2semidirect
15. C22×D404C22semidirect
17. C22×C4×D101C22semidirect
21. C2×#13108053C22semidirect
23. C2×#13108051C22semidirect
24. C2×#13108051C22semidirect
25. C2×#13108051C22semidirect
26. C2×#13108051C22semidirect
27. C2×C4×Dic101C22semidirect
29. C2×C4⋊C4×C101C22semidirect
31. #95×C101C22semidirect
47. C2×D404D4semidirect
50. C2×C22⋊D101D4semidirect
65. C22×C4×C101D4semidirect
93. C22×D101C2×D4semidirect
109. C2×Dic101C2×D4semidirect
118. C22×C101#121semidirect
139. C4×C101#120semidirect
152. C101#2408semidirect
160. #380D101semidirect
175. C2×C4⋊C4C2×D101semidirect
183. #95C2×D101semidirect
210. C22×C4D404semidirect
283. C22#15692230semidirect
294. C4#15692225semidirect

Maximal subgroups

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

subgroupindex
C23×D4042
C2×#156922302
C2×#156922252
C2×C4×C22⋊D1012
#202650952
#202650872
#202650782
#202650322
#380×C1012
#2408101