Finite Group Databasebeta

← #27174396

Subgroups — #27174396

Order 12928.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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
2C2×#15692213C211yesmaximal, characteristic, complemented
3#20265116C211yesmaximal, characteristic, complemented
4#20265114C211yesmaximal, characteristic, complemented
5#20265104C212yesmaximal, complemented
6#20265103C211yesmaximal, characteristic, complemented
7#20265094C212yesmaximal
8C4⋊C4×Dic101C212yesmaximal, complemented
9#20265024C212yesmaximal, complemented
10#20265024C211yesmaximal, characteristic, complemented
11#398×C101C211yesmaximal, characteristic, complemented
12C22×D404C2211yescharacteristic
13C22×C4×D101C2211yescharacteristic
14C22×C4×D101C2211yescharacteristic
15#15692217—24no—
16#15692213—24no—
17C2×C4⋊Dic101C2212yes—
18C2×C4⋊Dic101C2211yescharacteristic
19C2×#13108051C2212yes—
20C2×#13108051C2212yes—
21C2×#13108051C2212yes—
22C2×#13108051C2212yes—
23C2×#13108051C2211yescharacteristic
24C2×#13108051C2211yescharacteristic
25C2×C4×Dic101C2212yes—
26C2×C4×Dic101C2212yes—
27C2×C4×Dic101C2211yescharacteristic
28#15692162C2211yescharacteristic
29#15692162C2212yes—
30#15692162C2212yes—
31#15692162C2212yes—
32C2×C4⋊C4×C101C2212yes—
33C2×C4⋊C4×C101C2211yescharacteristic
34C2×C4⋊C4×C101C2211yescharacteristic
35#95×C101C2212yes—
36#95×C101C2211yescharacteristic
37C23×D101C2311yescharacteristic
38C23×D101C2311yescharacteristic
39C2×D404—22no—
40C2×D404—22no—
41C2×D404—22no—
42C22×Dic101C2312yes—
43C22×Dic101C2311yescharacteristic
44C22×Dic101C2312yes—
45C22×Dic101C2311yescharacteristic
46C2×C4×D101—24no—
47C2×C4×D101—22no—
48C2×C4×D101—24no—
49C2×C4×D101D414yes—
50C4⋊Dic101—22no—
51C4⋊Dic101—24no—
52#13108051—24no—
53#13108051—22no—
54#13108051—24no—
55#13108051—22no—
56#13108051—24no—
57#13108051—24no—
58C4×Dic101—24no—
59C4×Dic101—22no—
60C4×Dic101—24no—
61C22×C4×C101C2311yescharacteristic
62C22×C4×C101C2312yes—
63C22×C4×C101C2311yescharacteristic
64C22×C4×C101C2311yescharacteristic
65C22×C4×C101C2312yes—
66C4⋊C4×C101—22no—
67C4⋊C4×C101—22no—
68C4⋊C4×C101—24no—
69C23×C101C2411yescharacteristic
70C22×D101—21no—
71C22×D101—22no—
72C22×D101—22no—
73C22×D101—22no—
74C22×D101—22no—
75C22×D101—21no—
76C22×D101—21no—
77C22×D101—21no—
78C22×D101C2×D412yes—
79C22×D101#4112yes—
80C2×C4×C101—21no—
81C2×C4×C101—22no—
82C2×C4×C101—22no—
83C2×C4×C101—22no—
84C2×C4×C101—21no—
85C2×C4×C101—22no—
86C2×C4×C101—21no—
87C2×C4×C101—22no—
88C2×C4×C101—22no—
89C2×C4×C101—22no—
90C2×C4×C101C2×D412yes—
91C2×C4×C101#4112yes—
92C2×C4×C101#4114yes—
93C2×Dic101—21no—
94C2×Dic101—22no—
95C2×Dic101—22no—
96C2×Dic101—22no—
97C2×Dic101—22no—
98C2×Dic101—22no—
99C2×Dic101—22no—
100C2×Dic101—22no—
101C2×Dic101C2×D412yes—
102C2×Dic101#4114yes—
103D404—28no—
104C4×D101—28no—
105C4×D101—44no—
106C22×C101C22×D411yescharacteristic
107C22×C101C2×#4112yes—
108C22×C101C2×#4111yescharacteristic
109C22×C101C2×#4111yescharacteristic
110C22×C101C2×#4112yes—
111C2×D101—22no—
112C2×D101—22no—
113C2×D101—22no—
114C2×D101—24no—
115C2×D101—22no—
116C2×D101—41no—
117C2×D101—42no—
118C2×D101—42no—
119C2×D101—42no—
120C2×D101—41no—
121C4×C101—22no—
122C4×C101—22no—
123C4×C101—24no—
124C4×C101—41no—
125C4×C101—42no—
126Dic101—42no—
127Dic101—41no—
128Dic101—42no—
129Dic101#12314yes—
130C2×C101C2×#12311yescharacteristic
131C2×C101#52511yescharacteristic
132C2×C101#52912yes—
133C2×C101#52912yes—
134C2×C101#53211yescharacteristic
135D101—42no—
136D101—42no—
137#2395—1011nomaximal, Sylow
138C101#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—1011no—
140C2×#117—1011no—
141#400—1011no—
142#399—1011no—
143#398D10111yescharacteristic, complemented
144#397—1011no—
145#397—1012no—
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—1012no—
157C2×C4⋊C4C2×D10111yescharacteristic
158C2×C4⋊C4C2×D10111yescharacteristic
159C2×C4⋊C4C2×D10112yes—
160C2×C22⋊C4—1012no—
161C2×C22⋊C4—1011no—
162C2×C22⋊C4—1011no—
163C2×C22⋊C4—1012no—
164C2×C22⋊C4—1012no—
165C2×C22⋊C4—1012no—
166C2×C42—1012no—
167C2×C42—1011no—
168C2×C42—1012no—
169#95—1011no—
170#95—1012no—
171#95—1012no—
172#95—1012no—
173#95C2×D10112yes—
174#95C2×D10111yescharacteristic
175C24—1011no—
176C24—1011no—
177C2×D4—2022no—
178C2×D4—2022no—
179C2×D4—2022no—
180C22×C4—1011no—
181C22×C4—1014no—
182C22×C4—1012no—
183C22×C4—1011no—
184C22×C4—1012no—
185C22×C4—2022no—
186C22×C4—2024no—
187C22×C4—2024no—
188C22×C4C22×D10111yescharacteristic
189C22×C4C22×D10112yes—
190C22×C4C22×D10111yescharacteristic
191C22×C4C22×D10111yescharacteristic
192C22×C4C22×D10112yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—2022no—
197C4⋊C4—2024no—
198C22⋊C4—2024no—
199C22⋊C4—2024no—
200C22⋊C4—2022no—
201C22⋊C4—2022no—
202C22⋊C4—2024no—
203C22⋊C4—2024no—
204C42—2024no—
205C42—2024no—
206C42—2022no—
207C23—1012no—
208C23—1012no—
209C23—2021no—
210C23—2022no—
211C23—2022no—
212C23—2021no—
213C23—2021no—
214C23—2021no—
215C23—2022no—
216C23—2022no—
217C23C23×D10111yescharacteristic
218D4—2028no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—1012no—
230C2×C4—1014no—
231C2×C4—4044no—
232C2×C4—2028no—
233C2×C4—2021no—
234C2×C4—2022no—
235C2×C4—2022no—
236C2×C4—2022no—
237C2×C4—2022no—
238C2×C4—2022no—
239C2×C4—2022no—
240C2×C4—2022no—
241C2×C4D4×D10112yes—
242C2×C4#1310807312yes—
243C2×C4#1310807514yes—
244C22—4041no—
245C22—4041no—
246C22—4042no—
247C22—4042no—
248C22—4042no—
249C22—2022no—
250C22—2022no—
251C22—2022no—
252C22—2022no—
253C22—2024no—
254C22C2×D4×D10111yescharacteristic
255C22#41×D10112yes—
256C22#41×D10111yescharacteristic
257C22C2×#1310807311yescharacteristic
258C22C2×#1310807512yes—
259C4—22no—
260C4—22no—
261C4—24no—
262C4—41no—
263C4—42no—
264C4—1014no—
265C4—4041no—
266C4—4042no—
267C4—4042no—
268C2—4042no—
269C2—4042no—
270C2#123×D10111yescharacteristic
271C2#2026568711yescharacteristic
272C2#2026571212yes—
273C2#2026571312yes—
274C2#2026572811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#398×C101C2
prime radicalC22×D404C22
hypercenter#398D101
hypocenterC101#2395
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#15692213C2semidirect
3. #20265116C2semidirect
4. #20265114C2semidirect
5. #20265104C2semidirect
6. #20265103C2semidirect
8. C4⋊C4×Dic101C2semidirect
9. #20265024C2semidirect
10. #20265024C2semidirect
11. #398×C101C2semidirect
138. C101#2395semidirect
143. #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
#202651162
#202651142
#202651042
#202651032
#202650942
C4⋊C4×Dic1012
#202650242
#398×C1012
#2395101