Finite Group Databasebeta

← #27179425

Subgroups — #27179425

Order 12928.

1102 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 181 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×C4×D101C211yesmaximal, characteristic, complemented
2C2×#15692239C211yesmaximal, characteristic, complemented
3C2×#15692238C211yesmaximal, characteristic, complemented
4C2×#15692225C211yesmaximal, characteristic
5#20265055C214yesmaximal, complemented
6#20265051C214yesmaximal, complemented
7#20265038C211yesmaximal, characteristic, complemented
8#20265032C211yesmaximal, characteristic, complemented
9#381×C101C211yesmaximal, characteristic, complemented
10C24×D101C2211yescharacteristic
11C22×D404C2212yes—
12C22×C22⋊D101C2211yescharacteristic
13C23×Dic101C2211yescharacteristic
14C22×C4×D101—24no—
15C22×C4×D101—24no—
16C22×C4×D101—24no—
17C22×C4×D101C2214yes—
18#15692239—24no—
19#15692238—24no—
20#15692225—24no—
21C2×C4⋊Dic101C2212yescomplemented
22C2×#13108053C2212yescomplemented
23C2×C22⋊Dic101C2211yescharacteristic
24C2×#13108051C2214yes—
25C2×#13108051C2212yes—
26C2×#13108051C2212yescomplemented
27C2×#13108051C2212yescomplemented
28#15692162C2214yescomplemented
29C23×C4×C101C2211yescharacteristic
30C2×C22⋊C4×C101C2211yescharacteristic
31C2×C22⋊C4×C101C2211yescharacteristic
32#95×C101C2214yescomplemented
33C23×D101—22no—
34C23×D101—22no—
35C23×D101—28no—
36C23×D101D414yes—
37C23×D101C2311yescharacteristic
38C23×D101C2312yes—
39C2×D404—24no—
40C2×D404—28no—
41C2×C22⋊D101—22no—
42C2×C22⋊D101—22no—
43C2×C22⋊D101—22no—
44C22×Dic101—22no—
45C22×Dic101—22no—
46C22×Dic101D414yescomplemented
47C22×Dic101C2311yescharacteristic
48C22×Dic101C2312yes—
49C22×Dic101C2312yes—
50C2×C4×D101—232no—
51C2×C4×D101—28no—
52C2×C4×D101—28no—
53C2×C4×D101—28no—
54C4⋊Dic101—24no—
55#13108053—24no—
56C22⋊Dic101—22no—
57#13108051—24no—
58#13108051—28no—
59#13108051—24no—
60#13108051—24no—
61C24×C101C2311yescharacteristic
62C22×C4×C101—22no—
63C22×C4×C101—22no—
64C22×C4×C101D414yescomplemented
65C22×C4×C101C2312yes—
66C22×C4×C101C2312yes—
67C22×C4×C101C2312yes—
68C22⋊C4×C101—22no—
69C22⋊C4×C101—22no—
70C23×C101—24no—
71C23×C101C2×D412yes—
72C23×C101C2×D412yes—
73C23×C101C2×D412yes—
74C23×C101C2411yescharacteristic
75C22×D101—216no—
76C22×D101—216no—
77C22×D101—28no—
78C22×D101—22no—
79C22×D101—22no—
80C22×D101—24no—
81C22×D101—216no—
82C22×D101—21no—
83C22×D101—21no—
84C22×D101—21no—
85C22×D101—24no—
86C22×D101C2×D414yes—
87C22×D101#4118yes—
88C2×C4×C101—28no—
89C2×C4×C101—22no—
90C2×C4×C101—22no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101—24no—
94C2×C4×C101—24no—
95C2×C4×C101C2×D414yes—
96C22⋊D101—44no—
97C2×Dic101—28no—
98C2×Dic101—22no—
99C2×Dic101—22no—
100C2×Dic101—21no—
101C2×Dic101—22no—
102C2×Dic101—21no—
103C2×Dic101—21no—
104C2×Dic101—24no—
105C2×Dic101C2×D414yes—
106D404—48no—
107C4×D101—232no—
108C22×C101—24no—
109C22×C101—24no—
110C22×C101—24no—
111C22×C101#12014yes—
112C22×C101C22×D411yescharacteristic
113C22×C101C22×D411yescharacteristic
114C22×C101C22×D411yescharacteristic
115C22×C101C2×#4114yes—
116C2×D101—232no—
117C2×D101—24no—
118C2×D101—24no—
119C2×D101—28no—
120C2×D101—28no—
121C2×D101—41no—
122C2×D101—42no—
123C2×D101—42no—
124C2×D101—42no—
125C2×D101#12318yescomplemented
126C4×C101—24no—
127C4×C101—42no—
128C4×C101—42no—
129Dic101—24no—
130Dic101—41no—
131Dic101—42no—
132C2×C101—24no—
133C2×C101C2×#12011yescharacteristic
134C2×C101C2×#12312yes—
135C2×C101#52512yes—
136C2×C101#52512yes—
137D101—28no—
138D101—81no—
139#2849—1011nomaximal, Sylow
140C101#284911yescharacteristic, complemented, Sylow
141C24×C4—1011no—
142C2×#123—1011no—
143C2×#123—1011no—
144C2×#120—1011no—
145#386—1014no—
146#385—1014no—
147#381—1011no—
148#381D10111yescharacteristic, complemented
149#379—1011no—
150C25—1011no—
151C22×D4—1011no—
152C22×D4—1012no—
153C23×C4—1011no—
154C23×C4—1014no—
155C23×C4—2024no—
156C23×C4—2024no—
157C23×C4—2024no—
158C23×C4C2×D10111yescharacteristic
159#123—2024no—
160#123—2024no—
161#120—2024no—
162C2×C4⋊C4—1012no—
163C2×C4⋊C4—1012no—
164C2×C22⋊C4—1011no—
165C2×C22⋊C4—1012no—
166C2×C22⋊C4—1012no—
167C2×C22⋊C4—1012no—
168C2×C22⋊C4—1014no—
169C2×C22⋊C4C2×D10111yescharacteristic
170C2×C22⋊C4C2×D10111yescharacteristic
171#95—1014no—
172#95C2×D10114yescomplemented
173C24—1012no—
174C24—1014no—
175C24—1011no—
176C24—2028no—
177C24—2022no—
178C24—2022no—
179C24C22×D10111yescharacteristic
180C2×D4—2024no—
181C2×D4—2028no—
182C2×D4—2022no—
183C2×D4—2022no—
184C2×D4—2022no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—1014no—
188C22×C4—1012no—
189C22×C4—1012no—
190C22×C4—1011no—
191C22×C4—2028no—
192C22×C4—2028no—
193C22×C4—20232no—
194C22×C4—2022no—
195C22×C4—2022no—
196C22×C4—2028no—
197C22×C4D40414yescomplemented
198C22×C4C22×D10112yes—
199C22×C4C22×D10112yes—
200C22×C4C22×D10112yes—
201C4⋊C4—2024no—
202C4⋊C4—2024no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—2024no—
206C22⋊C4—2028no—
207C22⋊C4—2024no—
208C22⋊C4—2022no—
209C22⋊C4—2024no—
210C23—24no—
211C23—1014no—
212C23—1018no—
213C23—20216no—
214C23—20216no—
215C23—20216no—
216C23—2021no—
217C23—2021no—
218C23—2021no—
219C23—2024no—
220C23—2022no—
221C23—2022no—
222C23—2024no—
223C23—2028no—
224C23D4×D10112yes—
225C23D4×D10112yes—
226C23C2×D40412yes—
227C23C23×D10111yescharacteristic
228D4—4048no—
229D4—4044no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—28no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—1014no—
238C2×C4—20232no—
239C2×C4—2021no—
240C2×C4—2021no—
241C2×C4—2022no—
242C2×C4—2021no—
243C2×C4—2024no—
244C2×C4—2022no—
245C2×C4—2022no—
246C2×C4—2028no—
247C2×C4C2×D40414yes—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—1018no—
252C22—4041no—
253C22—4042no—
254C22—4042no—
255C22—4042no—
256C22—20232no—
257C22—2024no—
258C22—2024no—
259C22—2028no—
260C22—2028no—
261C22#1569222514yes—
262C22C2×D4×D10111yescharacteristic
263C22C2×D4×D10111yescharacteristic
264C22C22×D40411yescharacteristic
265C22#41×D10114yes—
266C4—24no—
267C4—42no—
268C4—42no—
269C4—4041no—
270C4—4042no—
271C4—2024no—
272C2—24no—
273C2—8081no—
274C2—2028no—
275C2C2×#1569222511yescharacteristic
276C2#123×D10112yes—
277C2#2026568512yes—
278C2#2026568912yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#381×C101C2
prime radicalC2×#15692225C2
hypercenter#381D101
hypocenterC101#2849
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C23×C4×D101C2semidirect
2. C2×#15692239C2semidirect
3. C2×#15692238C2semidirect
5. #20265055C2semidirect
6. #20265051C2semidirect
7. #20265038C2semidirect
8. #20265032C2semidirect
9. #381×C101C2semidirect
21. C2×C4⋊Dic101C22semidirect
22. C2×#13108053C22semidirect
26. C2×#13108051C22semidirect
27. C2×#13108051C22semidirect
28. #15692162C22semidirect
32. #95×C101C22semidirect
46. C22×Dic101D4semidirect
64. C22×C4×C101D4semidirect
125. C2×D101#123semidirect
140. C101#2849semidirect
148. #381D101semidirect
172. #95C2×D101semidirect
197. C22×C4D404semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1012
C2×#156922392
C2×#156922382
C2×#156922252
#202650552
#202650512
#202650382
#202650322
#381×C1012
#2849101