Finite Group Databasebeta

← #27179457

Subgroups — #27179457

Order 12928.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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×#15692239C211yesmaximal, characteristic
2C22×C4×Dic101C211yesmaximal, characteristic, complemented
3#20265067C212yesmaximal, complemented
4#20265066C212yesmaximal, complemented
5#20265064C212yesmaximal, complemented
6#20265063C212yesmaximal, complemented
7#20265056C211yesmaximal, characteristic
8#20265054C211yesmaximal, characteristic
9#20265049C211yesmaximal, characteristic, complemented
10#20265038C211yesmaximal, characteristic, complemented
11#385×C101C211yesmaximal, characteristic, complemented
12C22×C22⋊D101C2211yescharacteristic
13C23×Dic101C2211yescharacteristic
14C23×Dic101C2211yescharacteristic
15#15692239—24no—
16C2×C4⋊Dic101C2211yescharacteristic
17C2×C4⋊Dic101C2211yescharacteristic
18C2×#13108053C2212yes—
19C2×C22⋊Dic101C2211yescharacteristic
20C2×#13108051C2212yes—
21C2×#13108051C2212yes—
22C2×#13108051C2211yescharacteristic
23C2×#13108051C2211yescharacteristic
24C2×C4×Dic101—24no—
25C2×C4×Dic101C2212yes—
26C2×C4×Dic101C2212yes—
27#15692162C2212yes—
28#15692162C2212yes—
29#15692162C2212yes—
30#15692162C2212yes—
31#15692162C2212yes—
32C23×C4×C101C2211yescharacteristic
33C2×C4⋊C4×C101C2212yes—
34C2×C22⋊C4×C101C2211yescharacteristic
35C2×C22⋊C4×C101C2211yescharacteristic
36#95×C101C2212yes—
37C23×D101C2311yescharacteristic
38C2×C22⋊D101—22no—
39C2×C22⋊D101—22no—
40C2×C22⋊D101—22no—
41C22×Dic101—22no—
42C22×Dic101—22no—
43C22×Dic101—22no—
44C22×Dic101—24no—
45C22×Dic101D414yes—
46C22×Dic101C2311yescharacteristic
47C22×Dic101C2312yes—
48C22×Dic101C2311yescharacteristic
49C22×Dic101C2311yescharacteristic
50C22×Dic101C2312yes—
51C4⋊Dic101—22no—
52C4⋊Dic101—22no—
53#13108053—24no—
54C22⋊Dic101—22no—
55#13108051—24no—
56#13108051—22no—
57#13108051—22no—
58#13108051—24no—
59C4×Dic101—24no—
60C4×Dic101—24no—
61C24×C101C2311yescharacteristic
62C22×C4×C101—22no—
63C22×C4×C101—24no—
64C22×C4×C101C2311yescharacteristic
65C22×C4×C101C2312yes—
66C22×C4×C101C2311yescharacteristic
67C22×C4×C101C2312yes—
68C4⋊C4×C101—24no—
69C22⋊C4×C101—22no—
70C22⋊C4×C101—22no—
71C23×C101—22no—
72C23×C101—22no—
73C23×C101C2×D412yes—
74C23×C101#4112yes—
75C23×C101#4112yes—
76C23×C101C2411yescharacteristic
77C22×D101—21no—
78C22×D101—22no—
79C22×D101—22no—
80C22×D101—21no—
81C22×D101—21no—
82C2×C4×C101—28no—
83C2×C4×C101—22no—
84C2×C4×C101—22no—
85C2×C4×C101—21no—
86C2×C4×C101—21no—
87C2×C4×C101—22no—
88C2×C4×C101—22no—
89C2×C4×C101—22no—
90C2×C4×C101—22no—
91C2×C4×C101—22no—
92C2×C4×C101#4114yes—
93C22⋊D101—44no—
94C2×Dic101—28no—
95C2×Dic101—28no—
96C2×Dic101—22no—
97C2×Dic101—22no—
98C2×Dic101—21no—
99C2×Dic101—21no—
100C2×Dic101—21no—
101C2×Dic101—21no—
102C2×Dic101—21no—
103C2×Dic101—22no—
104C2×Dic101—22no—
105C2×Dic101—22no—
106C2×Dic101—21no—
107C2×Dic101—21no—
108C2×Dic101C2×D412yes—
109C2×Dic101C2×D412yes—
110C2×Dic101#4114yes—
111C22×C101—22no—
112C22×C101—22no—
113C22×C101—24no—
114C22×C101—24no—
115C22×C101#12314yes—
116C22×C101C22×D411yescharacteristic
117C22×C101C2×#4112yes—
118C22×C101C2×#4112yes—
119C22×C101C2×#4111yescharacteristic
120C22×C101C2×#4111yescharacteristic
121C2×D101—41no—
122C2×D101—41no—
123C2×D101—41no—
124C2×D101—42no—
125C2×D101—42no—
126C4×C101—24no—
127C4×C101—41no—
128C4×C101—41no—
129C4×C101—42no—
130Dic101—22no—
131Dic101—22no—
132Dic101—24no—
133Dic101—41no—
134Dic101—42no—
135C2×C101—24no—
136C2×C101C2×#12311yescharacteristic
137C2×C101#52511yescharacteristic
138C2×C101#52511yescharacteristic
139C2×C101#52912yes—
140C2×C101#52912yes—
141D101—81no—
142#2853—1011nomaximal, Sylow
143C101#285311yescharacteristic, complemented, Sylow
144C2×#123—1011no—
145C22×C42—1011no—
146#388—1012no—
147#388—1012no—
148#387—1012no—
149#387—1012no—
150#386—1011no—
151#386—1011no—
152#385—1011no—
153#385D10111yescharacteristic, complemented
154#381—1011no—
155C22×D4—1011no—
156C23×C4—1011no—
157C23×C4—1011no—
158C23×C4C2×D10111yescharacteristic
159#123—2024no—
160C2×C4⋊C4—1011no—
161C2×C4⋊C4—1011no—
162C2×C4⋊C4—1012no—
163C2×C4⋊C4C2×D10112yes—
164C2×C22⋊C4—1011no—
165C2×C22⋊C4—1011no—
166C2×C22⋊C4—1011no—
167C2×C22⋊C4—1012no—
168C2×C22⋊C4—1012no—
169C2×C22⋊C4C2×D10111yescharacteristic
170C2×C22⋊C4C2×D10111yescharacteristic
171C2×C42—1012no—
172C2×C42—1012no—
173C2×C42—2024no—
174#95—1012no—
175#95—1012no—
176#95—1012no—
177#95—1012no—
178#95—1012no—
179#95C2×D10112yes—
180C24—1011no—
181C24C22×D10111yescharacteristic
182C2×D4—2022no—
183C2×D4—2022no—
184C2×D4—2022no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—1014no—
188C22×C4—1011no—
189C22×C4—1011no—
190C22×C4—1012no—
191C22×C4—1012no—
192C22×C4—1011no—
193C22×C4—2024no—
194C22×C4—2022no—
195C22×C4—2022no—
196C22×C4—2022no—
197C22×C4C22×D10112yes—
198C22×C4C22×D10111yescharacteristic
199C22×C4C22×D10111yescharacteristic
200C22×C4C22×D10112yes—
201C4⋊C4—24no—
202C4⋊C4—2024no—
203C4⋊C4—2022no—
204C4⋊C4—2022no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2024no—
208C22⋊C4—2022no—
209C22⋊C4—2022no—
210C22⋊C4—2022no—
211C22⋊C4—2024no—
212C42—2024no—
213C42—2024no—
214C23—22no—
215C23—22no—
216C23—2021no—
217C23—2022no—
218C23—2022no—
219C23—2021no—
220C23—2021no—
221C23D4×D10112yes—
222C23#1310807412yes—
223C23#1310807512yes—
224C23C23×D10111yescharacteristic
225D4—4044no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—1012no—
237C2×C4—1012no—
238C2×C4—1014no—
239C2×C4—2021no—
240C2×C4—2022no—
241C2×C4—2022no—
242C2×C4—2021no—
243C2×C4—2022no—
244C2×C4—2021no—
245C2×C4—2028no—
246C2×C4—2022no—
247C2×C4—2022no—
248C2×C4—2021no—
249C2×C4—2021no—
250C2×C4—2021no—
251C2×C4—2021no—
252C2×C4—2028no—
253C2×C4#1310807314yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—4041no—
259C22—4041no—
260C22—4041no—
261C22—4042no—
262C22—4042no—
263C22#1569224014yes—
264C22C2×D4×D10111yescharacteristic
265C22#41×D10112yes—
266C22C2×#1310807312yes—
267C22C2×#1310807411yescharacteristic
268C22C2×#1310807511yescharacteristic
269C4—24no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—4041no—
274C4—4042no—
275C4—2022no—
276C4—2022no—
277C4—2024no—
278C2—24no—
279C2—8081no—
280C2C2×#1569224011yescharacteristic
281C2#2026568311yescharacteristic
282C2#2026568511yescharacteristic
283C2#2026570712yes—
284C2#2026571312yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#385×C101C2
prime radicalC22×C22⋊D101C22
hypercenter#385D101
hypocenterC101#2853
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic101C2semidirect
3. #20265067C2semidirect
4. #20265066C2semidirect
5. #20265064C2semidirect
6. #20265063C2semidirect
9. #20265049C2semidirect
10. #20265038C2semidirect
11. #385×C101C2semidirect
143. C101#2853semidirect
153. #385D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922392
C22×C4×Dic1012
#202650672
#202650662
#202650642
#202650632
#202650562
#202650542
#202650492
#202650382
#385×C1012
#2853101