Finite Group Databasebeta

← #27174394

Subgroups — #27174394

Order 12928.

560 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 143 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×#15692215C211yesmaximal, characteristic
2C2×#15692213C211yesmaximal, characteristic, complemented
3#20265117C211yesmaximal, characteristic, complemented
4#20265113C211yesmaximal, characteristic, complemented
5#20265108C211yesmaximal, characteristic, complemented
6#20265106C212yesmaximal, complemented
7#20265092C212yesmaximal
8#20265028C212yesmaximal, complemented
9C22⋊C4×Dic101C212yesmaximal, complemented
10#20265024C211yesmaximal, characteristic, complemented
11#397×C101C211yesmaximal, characteristic, complemented
12C22×C22⋊D101C2211yescharacteristic
13C23×Dic101C2211yescharacteristic
14C22×C4×D101C2211yescharacteristic
15#15692215—24no—
16#15692213—24no—
17C2×C4⋊Dic101C2211yescharacteristic
18C2×#13108053C2212yes—
19C2×#13108053C2212yes—
20C2×#13108053C2211yescharacteristic
21C2×C22⋊Dic101C2212yes—
22C2×#13108051C2212yes—
23C2×#13108051C2212yes—
24C2×#13108051C2211yescharacteristic
25C2×C4×Dic101C2212yes—
26C2×C4×Dic101C2212yes—
27C2×C4×Dic101C2211yescharacteristic
28#15692162C2211yescharacteristic
29#15692162C2212yes—
30#15692162C2212yes—
31#15692162C2212yes—
32C2×C4⋊C4×C101C2211yescharacteristic
33C2×C22⋊C4×C101C2212yes—
34C2×C22⋊C4×C101C2211yescharacteristic
35#95×C101C2211yescharacteristic
36#95×C101C2212yes—
37C23×D101C2311yescharacteristic
38C2×C22⋊D101—22no—
39C2×C22⋊D101—22no—
40C2×C22⋊D101—22no—
41C22×Dic101—22no—
42C22×Dic101—24no—
43C22×Dic101C2311yescharacteristic
44C22×Dic101C2312yes—
45C22×Dic101C2311yescharacteristic
46C22×Dic101C2312yes—
47C22×Dic101C2311yescharacteristic
48C2×C4×D101—24no—
49C2×C4×D101D414yes—
50C4⋊Dic101—22no—
51#13108053—22no—
52#13108053—24no—
53#13108053—24no—
54C22⋊Dic101—24no—
55#13108051—24no—
56#13108051—22no—
57#13108051—24no—
58C4×Dic101—24no—
59C4×Dic101—22no—
60C4×Dic101—24no—
61C24×C101C2311yescharacteristic
62C22×C4×C101C2312yes—
63C22×C4×C101C2311yescharacteristic
64C22×C4×C101C2312yes—
65C22×C4×C101C2311yescharacteristic
66C4⋊C4×C101—22no—
67C22⋊C4×C101—24no—
68C22⋊C4×C101—22no—
69C23×C101—21no—
70C23×C101—22no—
71C23×C101—22no—
72C23×C101—21no—
73C23×C101#4112yes—
74C23×C101C2411yescharacteristic
75C22×D101—22no—
76C22×D101—22no—
77C22×D101—21no—
78C22×D101—21no—
79C22×D101C2×D412yes—
80C2×C4×C101—22no—
81C2×C4×C101—21no—
82C2×C4×C101—22no—
83C2×C4×C101—22no—
84C2×C4×C101—22no—
85C2×C4×C101—22no—
86C2×C4×C101—22no—
87C2×C4×C101—22no—
88C2×C4×C101—22no—
89C2×C4×C101C2×D412yes—
90C22⋊D101—28no—
91C2×Dic101—21no—
92C2×Dic101—22no—
93C2×Dic101—22no—
94C2×Dic101—21no—
95C2×Dic101—22no—
96C2×Dic101—21no—
97C2×Dic101—22no—
98C2×Dic101—22no—
99C2×Dic101—22no—
100C2×Dic101—44no—
101C2×Dic101C2×D412yes—
102C2×Dic101#4114yes—
103C2×Dic101#4112yes—
104C2×Dic101#4114yes—
105C4×D101—28no—
106C22×C101—22no—
107C22×C101—24no—
108C22×C101—41no—
109C22×C101—42no—
110C22×C101—41no—
111C22×C101C22×D411yescharacteristic
112C22×C101C2×#4112yes—
113C22×C101C2×#4111yescharacteristic
114C22×C101C2×#4112yes—
115C22×C101C2×#4111yescharacteristic
116C2×D101—22no—
117C2×D101—22no—
118C2×D101—22no—
119C2×D101—42no—
120C2×D101—42no—
121C4×C101—22no—
122C4×C101—42no—
123C4×C101—42no—
124C4×C101—41no—
125Dic101—22no—
126Dic101—24no—
127Dic101—41no—
128Dic101—42no—
129Dic101#12314yes—
130C2×C101—42no—
131C2×C101C2×#12311yescharacteristic
132C2×C101#52511yescharacteristic
133C2×C101#52912yes—
134C2×C101#52912yes—
135C2×C101#53211yescharacteristic
136D101—42no—
137#2395—1011nomaximal, Sylow
138C101#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—1011no—
140C2×#117—1011no—
141#400—1011no—
142#399—1011no—
143#398—1011no—
144#397—1012no—
145#397D10111yescharacteristic, complemented
146#394—1012no—
147C4×C4⋊C4—1012no—
148C4×C22⋊C4—1012no—
149C4×C22⋊C4—1011no—
150C22×D4—1011no—
151C23×C4—1011no—
152C23×C4—1011no—
153C4×D4—2024no—
154#117—2024no—
155C2×C4⋊C4—1011no—
156C2×C4⋊C4—1011no—
157C2×C4⋊C4—1012no—
158C2×C4⋊C4—1012no—
159C2×C4⋊C4C2×D10111yescharacteristic
160C2×C22⋊C4—1012no—
161C2×C22⋊C4—1012no—
162C2×C22⋊C4—1011no—
163C2×C22⋊C4—1012no—
164C2×C22⋊C4C2×D10111yescharacteristic
165C2×C22⋊C4C2×D10112yes—
166C2×C42—1012no—
167C2×C42—1011no—
168C2×C42—1012no—
169#95—1012no—
170#95—1012no—
171#95—1011no—
172#95—1012no—
173#95C2×D10111yescharacteristic
174#95C2×D10112yes—
175C24—1011no—
176C24C22×D10111yescharacteristic
177C2×D4—2022no—
178C2×D4—2022no—
179C2×D4—2022no—
180C22×C4—1011no—
181C22×C4—1012no—
182C22×C4—1011no—
183C22×C4—1014no—
184C22×C4—1011no—
185C22×C4—1012no—
186C22×C4—2024no—
187C22×C4—2022no—
188C22×C4—2024no—
189C22×C4C22×D10112yes—
190C22×C4C22×D10112yes—
191C22×C4C22×D10111yescharacteristic
192C22×C4C22×D10111yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—2024no—
195C4⋊C4—2024no—
196C4⋊C4—2022no—
197C4⋊C4—2022no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—2024no—
201C22⋊C4—2024no—
202C22⋊C4—2022no—
203C22⋊C4—2024no—
204C42—2024no—
205C42—2024no—
206C42—2022no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—21no—
211C23—1012no—
212C23—2022no—
213C23—2022no—
214C23—2021no—
215C23—2021no—
216C23#1310807312yes—
217C23C23×D10111yescharacteristic
218D4—2028no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—1012no—
229C2×C4—1014no—
230C2×C4—1014no—
231C2×C4—1012no—
232C2×C4—4044no—
233C2×C4—2028no—
234C2×C4—2022no—
235C2×C4—2021no—
236C2×C4—2022no—
237C2×C4—2022no—
238C2×C4—2022no—
239C2×C4—2021no—
240C2×C4—2022no—
241C2×C4—2022no—
242C2×C4—2021no—
243C2×C4D4×D10112yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—4042no—
250C22—4042no—
251C22—2022no—
252C22—2022no—
253C22—2022no—
254C22C2×D4×D10111yescharacteristic
255C22#41×D10112yes—
256C22#41×D10111yescharacteristic
257C22#41×D10112yes—
258C22C2×#1310807311yescharacteristic
259C4—22no—
260C4—42no—
261C4—42no—
262C4—41no—
263C4—1014no—
264C4—4041no—
265C4—4042no—
266C4—2022no—
267C4—2024no—
268C2—42no—
269C2—4042no—
270C2#123×D10111yescharacteristic
271C2#2026568711yescharacteristic
272C2#2026570412yes—
273C2#2026571012yes—
274C2#2026572611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#397×C101C2
prime radicalC22×C22⋊D101C22
hypercenter#397D101
hypocenterC101#2395
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#15692213C2semidirect
3. #20265117C2semidirect
4. #20265113C2semidirect
5. #20265108C2semidirect
6. #20265106C2semidirect
8. #20265028C2semidirect
9. C22⋊C4×Dic101C2semidirect
10. #20265024C2semidirect
11. #397×C101C2semidirect
138. C101#2395semidirect
145. #397D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922152
C2×#156922132
#202651172
#202651132
#202651082
#202651062
#202650922
#202650282
C22⋊C4×Dic1012
#202650242
#397×C1012
#2395101