Finite Group Databasebeta

← #27174369

Subgroups — #27174369

Order 12928.

696 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 151 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×#15692215C212yesmaximal, complemented
3C2×#15692211C211yesmaximal, characteristic, complemented
4#20265108C211yesmaximal, characteristic, complemented
5#20265101C212yesmaximal, complemented
6#20265095C211yesmaximal, characteristic, complemented
7#20265092C212yesmaximal, complemented
8#20265086C211yesmaximal, characteristic
9#20265028C211yesmaximal, characteristic, complemented
10#20265024C212yesmaximal, complemented
11#397×C101C211yesmaximal, characteristic, complemented
12C22×D404C2211yescharacteristic
13C22×C22⋊D101C2212yes—
14C23×Dic101C2211yescharacteristic
15C22×C4×D101C2212yes—
16#15692217—24no—
17#15692215—28no—
18#15692211—24no—
19C2×#13108053C2212yes—
20C2×#13108053C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×#13108053C2212yes—
23C2×C22⋊Dic101C2211yescharacteristic
24C2×#13108051C2212yes—
25C2×#13108051C2212yes—
26C2×#13108051C2212yes—
27C2×#13108051C2212yes—
28C2×C4×Dic101C2211yescharacteristic
29C2×C4×Dic101C2212yes—
30C2×C4×Dic101C2211yescharacteristic
31#15692162C2212yes—
32#15692162C2211yescharacteristic
33C2×C4⋊C4×C101C2211yescharacteristic
34C2×C22⋊C4×C101C2211yescharacteristic
35C2×C22⋊C4×C101C2212yes—
36#95×C101C2212yes—
37#95×C101C2211yescharacteristic
38C23×D101C2312yes—
39C2×D404—24no—
40C2×D404—22no—
41C2×C22⋊D101—24no—
42C2×C22⋊D101—24no—
43C2×C22⋊D101—24no—
44C22×Dic101—22no—
45C22×Dic101—22no—
46C22×Dic101D414yescomplemented
47C22×Dic101C2311yescharacteristic
48C22×Dic101C2311yescharacteristic
49C22×Dic101C2311yescharacteristic
50C22×Dic101C2312yes—
51C22×Dic101C2311yescharacteristic
52C2×C4×D101—24no—
53C2×C4×D101—24no—
54C2×C4×D101—24no—
55#13108053—22no—
56#13108053—22no—
57#13108053—24no—
58#13108053—24no—
59C22⋊Dic101—22no—
60#13108051—24no—
61#13108051—24no—
62#13108051—24no—
63#13108051—24no—
64C4×Dic101—22no—
65C4×Dic101—24no—
66C4×Dic101D414yescomplemented
67C24×C101C2311yescharacteristic
68C22×C4×C101C2312yes—
69C22×C4×C101C2312yes—
70C22×C4×C101C2311yescharacteristic
71C22×C4×C101C2311yescharacteristic
72C4⋊C4×C101—22no—
73C22⋊C4×C101—24no—
74C22⋊C4×C101—22no—
75C23×C101—21no—
76C23×C101—21no—
77C23×C101—22no—
78C23×C101—22no—
79C23×C101C2×D412yes—
80C23×C101C2411yescharacteristic
81C22×D101—22no—
82C22×D101—22no—
83C22×D101—22no—
84C22×D101—22no—
85C22×D101—22no—
86C22×D101—22no—
87C22×D101#4114yes—
88C2×C4×C101—22no—
89C2×C4×C101—22no—
90C2×C4×C101—21no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101—22no—
94C2×C4×C101—22no—
95C2×C4×C101—22no—
96C2×C4×C101—22no—
97C2×C4×C101C2×D412yes—
98C22⋊D101—216no—
99C2×Dic101—22no—
100C2×Dic101—21no—
101C2×Dic101—22no—
102C2×Dic101—21no—
103C2×Dic101—22no—
104C2×Dic101—28no—
105C2×Dic101—21no—
106C2×Dic101—22no—
107C2×Dic101C2×D412yes—
108C2×Dic101C2×D412yes—
109C2×Dic101C2×D412yes—
110C2×Dic101C2×D412yes—
111C2×Dic101#4114yes—
112C2×Dic101#4112yes—
113D404—28no—
114C4×D101—48no—
115C22×C101—22no—
116C22×C101—24no—
117C22×C101—41no—
118C22×C101—42no—
119C22×C101—41no—
120C22×C101C22×D411yescharacteristic
121C22×C101C22×D411yescharacteristic
122C22×C101C2×#4111yescharacteristic
123C22×C101C2×#4112yes—
124C22×C101C2×#4112yes—
125C2×D101—24no—
126C2×D101—24no—
127C2×D101—24no—
128C2×D101—42no—
129C2×D101—42no—
130C2×D101—42no—
131C2×D101—42no—
132C4×C101—22no—
133C4×C101—42no—
134C4×C101—42no—
135C4×C101—41no—
136Dic101—22no—
137Dic101—24no—
138Dic101—22no—
139Dic101—41no—
140Dic101#12114yes—
141C2×C101—42no—
142C2×C101C2×#12111yescharacteristic
143C2×C101#52512yes—
144C2×C101#52511yescharacteristic
145C2×C101#52912yes—
146C2×C101#53211yescharacteristic
147D101—44no—
148#2392—1011nomaximal, Sylow
149C101#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—1011no—
151C2×C4×D4—1012no—
152C2×#117—1011no—
153#398—1011no—
154#397D10111yescharacteristic, complemented
155#396—1012no—
156#394—1012no—
157#394—1011no—
158#392—1011no—
159C4×C4⋊C4—1011no—
160C4×C22⋊C4—1012no—
161C22×D4—1012no—
162C22×D4—1011no—
163C23×C4—1011no—
164C23×C4—1012no—
165C4×D4—2024no—
166C4×D4—2028no—
167#117—2024no—
168C2×C4⋊C4—1011no—
169C2×C4⋊C4—1011no—
170C2×C4⋊C4—1012no—
171C2×C4⋊C4—1012no—
172C2×C4⋊C4C2×D10111yescharacteristic
173C2×C22⋊C4—1011no—
174C2×C22⋊C4—1012no—
175C2×C22⋊C4—1012no—
176C2×C22⋊C4—1012no—
177C2×C22⋊C4—1012no—
178C2×C22⋊C4C2×D10111yescharacteristic
179C2×C22⋊C4C2×D10112yes—
180C2×C42—1011no—
181C2×C42—1011no—
182C2×C42—1012no—
183#95—1012no—
184#95—1011no—
185#95C2×D10112yes—
186#95C2×D10111yescharacteristic
187C24—1012no—
188C24C22×D10111yescharacteristic
189C2×D4—2024no—
190C2×D4—2024no—
191C2×D4—2024no—
192C2×D4—2024no—
193C2×D4—2022no—
194C22×C4—1011no—
195C22×C4—1014no—
196C22×C4—1012no—
197C22×C4—1011no—
198C22×C4—1011no—
199C22×C4—1011no—
200C22×C4—2024no—
201C22×C4—2024no—
202C22×C4—2024no—
203C22×C4—2022no—
204C22×C4—2022no—
205C22×C4C22×D10112yes—
206C22×C4C22×D10112yes—
207C22×C4C22×D10111yescharacteristic
208C22×C4C22×D10111yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—2024no—
211C4⋊C4—2022no—
212C4⋊C4—2024no—
213C4⋊C4—2022no—
214C22⋊C4—24no—
215C22⋊C4—22no—
216C22⋊C4—2024no—
217C22⋊C4—2024no—
218C22⋊C4—2024no—
219C22⋊C4—2024no—
220C22⋊C4—2022no—
221C42—1014no—
222C42—2022no—
223C42—2024no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—1014no—
229C23—2022no—
230C23—2022no—
231C23—2022no—
232C23—2022no—
233C23—2022no—
234C23—2022no—
235C23D4×D10112yes—
236C23C23×D10111yescharacteristic
237D4—2028no—
238D4—20216no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—1012no—
249C2×C4—1012no—
250C2×C4—1012no—
251C2×C4—1012no—
252C2×C4—1012no—
253C2×C4—1014no—
254C2×C4—4048no—
255C2×C4—2028no—
256C2×C4—2021no—
257C2×C4—2022no—
258C2×C4—2022no—
259C2×C4—2021no—
260C2×C4—2022no—
261C2×C4—2021no—
262C2×C4—2022no—
263C2×C4D4×D10112yes—
264C22—22no—
265C22—24no—
266C22—41no—
267C22—42no—
268C22—41no—
269C22—4042no—
270C22—4042no—
271C22—4042no—
272C22—4042no—
273C22—2024no—
274C22—2024no—
275C22—2024no—
276C22C2×D4×D10111yescharacteristic
277C22C2×D4×D10111yescharacteristic
278C22#41×D10111yescharacteristic
279C22#41×D10112yes—
280C22#41×D10112yes—
281C4—22no—
282C4—42no—
283C4—42no—
284C4—41no—
285C4—1014no—
286C4—4041no—
287C4—2022no—
288C4—2024no—
289C4—2022no—
290C2—42no—
291C2—4044no—
292C2#121×D10111yescharacteristic
293C2#2026568511yescharacteristic
294C2#2026568512yes—
295C2#2026571012yes—
296C2#2026572611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#397×C101C2
prime radical#20265086C2
hypercenter#397D101
hypocenterC101#2392
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#15692217C2semidirect
2. C2×#15692215C2semidirect
3. C2×#15692211C2semidirect
4. #20265108C2semidirect
5. #20265101C2semidirect
6. #20265095C2semidirect
7. #20265092C2semidirect
9. #20265028C2semidirect
10. #20265024C2semidirect
11. #397×C101C2semidirect
46. C22×Dic101D4semidirect
66. C4×Dic101D4semidirect
149. C101#2392semidirect
154. #397D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922172
C2×#156922152
C2×#156922112
#202651082
#202651012
#202650952
#202650922
#202650862
#202650282
#202650242
#397×C1012
#2392101