Finite Group Databasebeta

← #27174421

Subgroups — #27174421

Order 12928.

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

normal subgroups only

This group has 568 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×#15692220C211yesmaximal, characteristic, complemented
2C2×#15692217C211yesmaximal, characteristic
3#20265115C211yesmaximal, characteristic, complemented
4#20265106C212yesmaximal, complemented
5#20265095C211yesmaximal, characteristic
6#20265094C211yesmaximal, characteristic
7#20265090C212yesmaximal, complemented
8#20265024C212yesmaximal, complemented
9#20265024C212yesmaximal, complemented
10#20265022C211yesmaximal, characteristic, complemented
11#400×C101C211yesmaximal, characteristic, complemented
12C22×Dic202C2211yescharacteristic
13C22×D404C2211yescharacteristic
14C22×C4×D101C2212yes—
15#15692220—24no—
16#15692217—24no—
17C2×#13108053C2211yescharacteristic
18C2×#13108053C2211yescharacteristic
19C2×#13108051C2212yes—
20C2×#13108051C2212yes—
21C2×#13108051C2212yes—
22C2×#13108051C2212yes—
23C2×#13108051C2212yes—
24C2×C4×Dic101C2212yes—
25C2×C4×Dic101C2211yescharacteristic
26C2×C4×Dic101C2211yescharacteristic
27C2×C4×Dic101C2211yescharacteristic
28#15692162C2212yes—
29#15692162C2211yescharacteristic
30#15692162C2212yes—
31#15692162C2212yes—
32C2×C4⋊C4×C101C2211yescharacteristic
33C2×C4⋊C4×C101C2211yescharacteristic
34#95×C101C2211yescharacteristic
35#95×C101C2212yes—
36#95×C101C2212yes—
37C23×D101C2312yes—
38C2×Dic202—22no—
39C2×Dic202—22no—
40C2×Dic202—22no—
41C2×D404—24no—
42C2×D404—22no—
43C22×Dic101C2312yes—
44C22×Dic101C2311yescharacteristic
45C22×Dic101C2311yescharacteristic
46C22×Dic101C2311yescharacteristic
47C22×Dic101C2311yescharacteristic
48C2×C4×D101—24no—
49C2×C4×D101—24no—
50C2×C4×D101—24no—
51#13108053—22no—
52#13108053—22no—
53#13108051—24no—
54#13108051—24no—
55#13108051—24no—
56#13108051—24no—
57#13108051—24no—
58C4×Dic101—22no—
59C4×Dic101—22no—
60C4×Dic101—24no—
61C4×Dic101D414yes—
62C22×C4×C101C2312yes—
63C22×C4×C101C2312yes—
64C22×C4×C101C2311yescharacteristic
65C22×C4×C101C2311yescharacteristic
66C22×C4×C101C2311yescharacteristic
67C4⋊C4×C101—22no—
68C4⋊C4×C101—22no—
69C23×C101C2411yescharacteristic
70C22×D101—22no—
71C22×D101—22no—
72C22×D101—22no—
73C22×D101—22no—
74C22×D101—22no—
75C22×D101—22no—
76C22×D101#4114yes—
77C2×C4×C101—21no—
78C2×C4×C101—22no—
79C2×C4×C101—21no—
80C2×C4×C101—21no—
81C2×C4×C101—22no—
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×C101C2×D412yes—
89C2×C4×C101#4112yes—
90C2×Dic101—21no—
91C2×Dic101—21no—
92C2×Dic101—21no—
93C2×Dic101—21no—
94C2×Dic101—21no—
95C2×Dic101—22no—
96C2×Dic101—22no—
97C2×Dic101—22no—
98C2×Dic101—22no—
99C2×Dic101C2×D412yes—
100C2×Dic101C2×D412yes—
101C2×Dic101#4112yes—
102C2×Dic101#4114yes—
103D404—28no—
104C4×D101—48no—
105Dic202—28no—
106C22×C101C22×D411yescharacteristic
107C22×C101C2×#4112yes—
108C22×C101C2×#4111yescharacteristic
109C22×C101C2×#4112yes—
110C22×C101C2×#4111yescharacteristic
111C2×D101—24no—
112C2×D101—24no—
113C2×D101—24no—
114C2×D101—42no—
115C2×D101—42no—
116C2×D101—42no—
117C2×D101—42no—
118C4×C101—22no—
119C4×C101—22no—
120C4×C101—41no—
121C4×C101—42no—
122C4×C101—42no—
123Dic101—22no—
124Dic101—22no—
125Dic101—41no—
126Dic101—42no—
127Dic101#12414yes—
128C2×C101#52511yescharacteristic
129C2×C101#52511yescharacteristic
130C2×C101C2×#12411yescharacteristic
131C2×C101#52912yes—
132C2×C101#52912yes—
133D101—44no—
134#2398—1011nomaximal, Sylow
135C101#239811yescharacteristic, complemented, Sylow
136C2×C4×Q8—1011no—
137C2×C4×D4—1011no—
138#400—1011no—
139#400D10111yescharacteristic, complemented
140#397—1012no—
141#394—1011no—
142#394—1011no—
143#393—1012no—
144C4×C22⋊C4—1012no—
145C4×C22⋊C4—1012no—
146#376—1011no—
147C22×Q8—1011no—
148C22×D4—1011no—
149C23×C4—1012no—
150C4×Q8—2024no—
151C4×D4—2024no—
152C2×C4⋊C4—1011no—
153C2×C4⋊C4—1011no—
154C2×C4⋊C4C2×D10111yescharacteristic
155C2×C4⋊C4C2×D10111yescharacteristic
156C2×C22⋊C4—1012no—
157C2×C22⋊C4—1012no—
158C2×C22⋊C4—1012no—
159C2×C22⋊C4—1012no—
160C2×C22⋊C4—1012no—
161C2×C42—1011no—
162C2×C42—1011no—
163C2×C42—1011no—
164C2×C42—1012no—
165#95—1012no—
166#95—1011no—
167#95—1012no—
168#95—1012no—
169#95C2×D10111yescharacteristic
170#95C2×D10112yes—
171#95C2×D10112yes—
172C24—1012no—
173C2×Q8—2022no—
174C2×Q8—2022no—
175C2×Q8—2022no—
176C2×D4—2024no—
177C2×D4—2022no—
178C22×C4—1011no—
179C22×C4—1011no—
180C22×C4—1011no—
181C22×C4—1012no—
182C22×C4—1011no—
183C22×C4—2024no—
184C22×C4—2024no—
185C22×C4—2024no—
186C22×C4C22×D10112yes—
187C22×C4C22×D10112yes—
188C22×C4C22×D10111yescharacteristic
189C22×C4C22×D10111yescharacteristic
190C22×C4C22×D10111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—2022no—
194C4⋊C4—2022no—
195C22⋊C4—2024no—
196C22⋊C4—2024no—
197C22⋊C4—2024no—
198C22⋊C4—2024no—
199C22⋊C4—2024no—
200C42—1014no—
201C42—2024no—
202C42—2022no—
203C42—2022no—
204C23—1014no—
205C23—2022no—
206C23—2022no—
207C23—2022no—
208C23—2022no—
209C23—2022no—
210C23—2022no—
211C23C23×D10111yescharacteristic
212Q8—2028no—
213D4—2028no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—1012no—
226C2×C4—1014no—
227C2×C4—1012no—
228C2×C4—1012no—
229C2×C4—4048no—
230C2×C4—2021no—
231C2×C4—2022no—
232C2×C4—2022no—
233C2×C4—2022no—
234C2×C4—2022no—
235C2×C4—2021no—
236C2×C4—2021no—
237C2×C4—2021no—
238C2×C4—2021no—
239C2×C4D4×D10112yes—
240C2×C4#1310807512yes—
241C22—4042no—
242C22—4042no—
243C22—4042no—
244C22—4042no—
245C22—2024no—
246C22—2024no—
247C22—2024no—
248C22C2×D4×D10111yescharacteristic
249C22#41×D10112yes—
250C22#41×D10111yescharacteristic
251C22#41×D10112yes—
252C22C2×#1310807511yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—1014no—
259C4—4041no—
260C4—4042no—
261C4—2022no—
262C4—2022no—
263C2—4044no—
264C2#2026568511yescharacteristic
265C2#2026569011yescharacteristic
266C2#124×D10111yescharacteristic
267C2#2026571012yes—
268C2#2026571212yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#400×C101C2
prime radicalC22×D404C22
hypercenter#400D101
hypocenterC101#2398
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#15692220C2semidirect
3. #20265115C2semidirect
4. #20265106C2semidirect
7. #20265090C2semidirect
8. #20265024C2semidirect
9. #20265024C2semidirect
10. #20265022C2semidirect
11. #400×C101C2semidirect
135. C101#2398semidirect
139. #400D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922202
C2×#156922172
#202651152
#202651062
#202650952
#202650942
#202650902
#202650242
#202650222
#400×C1012
#2398101