Finite Group Databasebeta

← #27174373

Subgroups — #27174373

Order 12928.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C2×#15692217C211yesmaximal, characteristic, complemented
2C2×#15692217C212yesmaximal, complemented
3C2×#15692213C211yesmaximal, characteristic, complemented
4#20265103C211yesmaximal, characteristic, complemented
5#20265099C212yesmaximal, complemented
6#20265095C212yesmaximal, complemented
7#20265094C211yesmaximal, characteristic, complemented
8#20265087C211yesmaximal, characteristic
9#20265028C211yesmaximal, characteristic, complemented
10#20265024C212yesmaximal, complemented
11#398×C101C211yesmaximal, characteristic, complemented
12C22×D404C2211yescharacteristic
13C22×D404C2212yes—
14C22×C4×D101C2212yes—
15C22×C4×D101C2211yescharacteristic
16#15692217—28no—
17#15692217—24no—
18#15692213—24no—
19C2×C4⋊Dic101C2211yescharacteristic
20C2×#13108053C2212yes—
21C2×#13108051C2212yes—
22C2×#13108051C2212yes—
23C2×#13108051C2211yescharacteristic
24C2×#13108051C2212yes—
25C2×#13108051C2211yescharacteristic
26C2×#13108051C2212yes—
27C2×#13108051C2212yes—
28C2×C4×Dic101C2211yescharacteristic
29C2×C4×Dic101C2212yes—
30C2×C4×Dic101C2211yescharacteristic
31#15692162C2212yes—
32#15692162C2211yescharacteristic
33C2×C4⋊C4×C101C2211yescharacteristic
34C2×C4⋊C4×C101C2212yes—
35C2×C4⋊C4×C101C2211yescharacteristic
36#95×C101C2212yes—
37#95×C101C2211yescharacteristic
38C23×D101C2311yescharacteristic
39C23×D101C2312yes—
40C2×D404—24no—
41C2×D404—24no—
42C2×D404—24no—
43C2×D404—22no—
44C2×D404—24no—
45C22×Dic101C2312yes—
46C22×Dic101C2311yescharacteristic
47C22×Dic101C2311yescharacteristic
48C22×Dic101C2311yescharacteristic
49C2×C4×D101—22no—
50C2×C4×D101—22no—
51C2×C4×D101—24no—
52C2×C4×D101—24no—
53C2×C4×D101—24no—
54C2×C4×D101D414yescomplemented
55C4⋊Dic101—22no—
56#13108053—24no—
57#13108051—24no—
58#13108051—24no—
59#13108051—22no—
60#13108051—24no—
61#13108051—24no—
62#13108051—22no—
63#13108051—24no—
64C4×Dic101—22no—
65C4×Dic101—24no—
66C4×Dic101D414yescomplemented
67C22×C4×C101C2312yes—
68C22×C4×C101C2311yescharacteristic
69C22×C4×C101C2311yescharacteristic
70C22×C4×C101C2311yescharacteristic
71C22×C4×C101C2312yes—
72C4⋊C4×C101—22no—
73C4⋊C4×C101—24no—
74C4⋊C4×C101—22no—
75C23×C101C2411yescharacteristic
76C22×D101—21no—
77C22×D101—22no—
78C22×D101—22no—
79C22×D101—22no—
80C22×D101—21no—
81C22×D101—22no—
82C22×D101—22no—
83C22×D101—22no—
84C22×D101—22no—
85C22×D101—22no—
86C22×D101C2×D412yes—
87C22×D101#4114yes—
88C2×C4×C101—22no—
89C2×C4×C101—21no—
90C2×C4×C101—21no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101—21no—
94C2×C4×C101—22no—
95C2×C4×C101—22no—
96C2×C4×C101—22no—
97C2×C4×C101—22no—
98C2×C4×C101C2×D412yes—
99C2×C4×C101C2×D412yes—
100C2×C4×C101#4114yes—
101C2×Dic101—22no—
102C2×Dic101—22no—
103C2×Dic101—21no—
104C2×Dic101—22no—
105C2×Dic101—22no—
106C2×Dic101—22no—
107C2×Dic101C2×D412yes—
108C2×Dic101C2×D412yes—
109C2×Dic101C2×D412yes—
110C2×Dic101#4112yes—
111D404—28no—
112D404—216no—
113C4×D101—28no—
114C4×D101—48no—
115C22×C101C22×D411yescharacteristic
116C22×C101C22×D411yescharacteristic
117C22×C101C2×#4111yescharacteristic
118C22×C101C2×#4112yes—
119C22×C101C2×#4112yes—
120C2×D101—22no—
121C2×D101—24no—
122C2×D101—24no—
123C2×D101—24no—
124C2×D101—24no—
125C2×D101—42no—
126C2×D101—41no—
127C2×D101—42no—
128C2×D101—42no—
129C2×D101—42no—
130C2×D101—41no—
131C2×D101—42no—
132C4×C101—22no—
133C4×C101—22no—
134C4×C101—24no—
135C4×C101—42no—
136C4×C101—41no—
137Dic101—22no—
138Dic101—42no—
139Dic101—41no—
140Dic101#12114yes—
141C2×C101C2×#12111yescharacteristic
142C2×C101#52511yescharacteristic
143C2×C101#52512yes—
144C2×C101#52912yes—
145C2×C101#53211yescharacteristic
146D101—42no—
147D101—44no—
148#2392—1011nomaximal, Sylow
149C101#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—1011no—
151C2×C4×D4—1012no—
152C2×#117—1011no—
153#398D10111yescharacteristic, complemented
154#397—1011no—
155#396—1012no—
156#394—1011no—
157#394—1012no—
158#392—1011no—
159C4×C4⋊C4—1011no—
160C4×C22⋊C4—1012no—
161C22×D4—1011no—
162C22×D4—1012no—
163C23×C4—1011no—
164C23×C4—1012no—
165C4×D4—2024no—
166C4×D4—2028no—
167#117—2024no—
168C2×C4⋊C4—1011no—
169C2×C4⋊C4—1012no—
170C2×C4⋊C4C2×D10111yescharacteristic
171C2×C4⋊C4C2×D10112yes—
172C2×C4⋊C4C2×D10111yescharacteristic
173C2×C22⋊C4—1011no—
174C2×C22⋊C4—1011no—
175C2×C22⋊C4—1012no—
176C2×C22⋊C4—1012no—
177C2×C22⋊C4—1012no—
178C2×C22⋊C4—1012no—
179C2×C22⋊C4—1012no—
180C2×C42—1011no—
181C2×C42—1011no—
182C2×C42—1012no—
183#95—1012no—
184#95—1011no—
185#95C2×D10112yes—
186#95C2×D10111yescharacteristic
187C24—1011no—
188C24—1012no—
189C2×D4—2024no—
190C2×D4—2024no—
191C2×D4—2024no—
192C2×D4—2022no—
193C2×D4—2024no—
194C22×C4—1011no—
195C22×C4—1011no—
196C22×C4—1014no—
197C22×C4—1012no—
198C22×C4—1011no—
199C22×C4—2024no—
200C22×C4—2024no—
201C22×C4—2024no—
202C22×C4—2022no—
203C22×C4—2022no—
204C22×C4C22×D10112yes—
205C22×C4C22×D10111yescharacteristic
206C22×C4C22×D10111yescharacteristic
207C22×C4C22×D10111yescharacteristic
208C22×C4C22×D10112yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—2022no—
213C4⋊C4—2024no—
214C22⋊C4—2022no—
215C22⋊C4—2024no—
216C22⋊C4—2024no—
217C22⋊C4—2024no—
218C22⋊C4—2022no—
219C22⋊C4—2024no—
220C22⋊C4—2024no—
221C42—1014no—
222C42—2022no—
223C42—2024no—
224C23—1012no—
225C23—1014no—
226C23—2021no—
227C23—2022no—
228C23—2022no—
229C23—2022no—
230C23—2021no—
231C23—2022no—
232C23—2022no—
233C23—2022no—
234C23—2022no—
235C23—2022no—
236C23C23×D10111yescharacteristic
237D4—2028no—
238D4—20216no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—1012no—
250C2×C4—1012no—
251C2×C4—1012no—
252C2×C4—1012no—
253C2×C4—4048no—
254C2×C4—2028no—
255C2×C4—2022no—
256C2×C4—2021no—
257C2×C4—2022no—
258C2×C4—2022no—
259C2×C4—2022no—
260C2×C4—2022no—
261C2×C4D4×D10112yes—
262C2×C4D4×D10112yes—
263C2×C4#1310807514yes—
264C22—4041no—
265C22—4042no—
266C22—4042no—
267C22—4041no—
268C22—4042no—
269C22—4042no—
270C22—4042no—
271C22—2022no—
272C22—2024no—
273C22—2024no—
274C22—2024no—
275C22—2024no—
276C22C2×D4×D10111yescharacteristic
277C22C2×D4×D10111yescharacteristic
278C22#41×D10111yescharacteristic
279C22#41×D10112yes—
280C22C2×#1310807512yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—1014no—
287C4—4041no—
288C4—4042no—
289C4—2022no—
290C2—4044no—
291C2—4042no—
292C2#121×D10111yescharacteristic
293C2#2026568511yescharacteristic
294C2#2026569012yes—
295C2#2026571212yes—
296C2#2026572811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#398×C101C2
prime radical#20265087C2
hypercenter#398D101
hypocenterC101#2392
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#15692217C2semidirect
2. C2×#15692217C2semidirect
3. C2×#15692213C2semidirect
4. #20265103C2semidirect
5. #20265099C2semidirect
6. #20265095C2semidirect
7. #20265094C2semidirect
9. #20265028C2semidirect
10. #20265024C2semidirect
11. #398×C101C2semidirect
54. C2×C4×D101D4semidirect
66. C4×Dic101D4semidirect
149. C101#2392semidirect
153. #398D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922172
C2×#156922132
#202651032
#202650992
#202650952
#202650942
#202650872
#202650282
#202650242
#398×C1012
#2392101