Finite Group Databasebeta

← #27145021

Subgroups — #27145021

Order 12416.

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×#15690775C211yesmaximal, characteristic, complemented
2C2×#15690770C211yesmaximal, characteristic
3#20260676C212yesmaximal, complemented
4#20260661C211yesmaximal, characteristic, complemented
5#20260650C212yesmaximal
6#20260648C211yesmaximal, characteristic, complemented
7#20260647C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic97C212yesmaximal, complemented
9#20260582C212yesmaximal, complemented
10#20260580C211yesmaximal, characteristic, complemented
11#397×C97C211yesmaximal, characteristic, complemented
12C22×Dic194C2211yescharacteristic
13C22×C22⋊D97C2211yescharacteristic
14C23×Dic97C2211yescharacteristic
15C22×C4×D97C2211yescharacteristic
16#15690775—24no—
17#15690770—24no—
18C2×#13095490C2212yes—
19C2×#13095490C2211yescharacteristic
20C2×C22⋊Dic97C2212yes—
21C2×#13095488C2212yes—
22C2×#13095488C2211yescharacteristic
23C2×#13095488C2212yes—
24C2×C4×Dic97C2212yes—
25C2×C4×Dic97C2212yes—
26C2×C4×Dic97C2211yescharacteristic
27#15690717C2211yescharacteristic
28#15690717C2212yes—
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32C2×C4⋊C4×C97C2211yescharacteristic
33C2×C22⋊C4×C97C2211yescharacteristic
34C2×C22⋊C4×C97C2212yes—
35#95×C97C2211yescharacteristic
36#95×C97C2212yes—
37C23×D97C2311yescharacteristic
38C2×Dic194—22no—
39C2×Dic194—24no—
40C2×C22⋊D97—22no—
41C2×C22⋊D97—22no—
42C2×C22⋊D97—22no—
43C22×Dic97—22no—
44C22×Dic97—24no—
45C22×Dic97C2312yes—
46C22×Dic97C2311yescharacteristic
47C22×Dic97C2311yescharacteristic
48C22×Dic97C2312yes—
49C22×Dic97C2311yescharacteristic
50C2×C4×D97—22no—
51C2×C4×D97—24no—
52#13095490—22no—
53#13095490—24no—
54C22⋊Dic97—24no—
55#13095488—24no—
56#13095488—22no—
57#13095488—24no—
58C4×Dic97—24no—
59C4×Dic97—24no—
60C4×Dic97D414yes—
61C24×C97C2311yescharacteristic
62C22×C4×C97C2312yes—
63C22×C4×C97C2312yes—
64C22×C4×C97C2311yescharacteristic
65C22×C4×C97C2311yescharacteristic
66C4⋊C4×C97—22no—
67C22⋊C4×C97—24no—
68C22⋊C4×C97—22no—
69C23×C97—21no—
70C23×C97—22no—
71C23×C97—22no—
72C23×C97—21no—
73C23×C97#4112yes—
74C23×C97C2411yescharacteristic
75C22×D97—21no—
76C22×D97—22no—
77C22×D97—22no—
78C22×D97—21no—
79C22×D97#4112yes—
80C2×C4×C97—22no—
81C2×C4×C97—21no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97—22no—
89C2×C4×C97C2×D412yes—
90C22⋊D97—28no—
91C2×Dic97—22no—
92C2×Dic97—21no—
93C2×Dic97—21no—
94C2×Dic97—22no—
95C2×Dic97—21no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97—22no—
100C2×Dic97—44no—
101C2×Dic97C2×D412yes—
102C2×Dic97C2×D412yes—
103C2×Dic97#4114yes—
104C2×Dic97#4114yes—
105C4×D97—44no—
106Dic194—28no—
107C22×C97—22no—
108C22×C97—24no—
109C22×C97—41no—
110C22×C97—41no—
111C22×C97—42no—
112C22×C97C22×D411yescharacteristic
113C22×C97C2×#4111yescharacteristic
114C22×C97C2×#4112yes—
115C22×C97C2×#4112yes—
116C22×C97C2×#4111yescharacteristic
117C2×D97—22no—
118C2×D97—24no—
119C2×D97—41no—
120C2×D97—41no—
121C2×D97—42no—
122C4×C97—22no—
123C4×C97—42no—
124C4×C97—41no—
125C4×C97—42no—
126Dic97—24no—
127Dic97—22no—
128Dic97—41no—
129Dic97—42no—
130Dic97#12414yes—
131C2×C97—42no—
132C2×C97#52511yescharacteristic
133C2×C97#52511yescharacteristic
134C2×C97C2×#12411yescharacteristic
135C2×C97#52912yes—
136C2×C97#52912yes—
137D97—42no—
138#2398—971nomaximal, Sylow
139C97#239811yescharacteristic, complemented, Sylow
140C2×C4×Q8—971no—
141C2×C4×D4—971no—
142#400—972no—
143#397—971no—
144#397D9711yescharacteristic, complemented
145#394—972no—
146#393—971no—
147#393—971no—
148C4×C22⋊C4—972no—
149C4×C22⋊C4—972no—
150#376—971no—
151C22×Q8—971no—
152C22×D4—971no—
153C23×C4—971no—
154C23×C4—971no—
155C4×Q8—1944no—
156C4×D4—1944no—
157C2×C4⋊C4—971no—
158C2×C4⋊C4—972no—
159C2×C4⋊C4C2×D9711yescharacteristic
160C2×C22⋊C4—972no—
161C2×C22⋊C4—971no—
162C2×C22⋊C4—972no—
163C2×C22⋊C4—972no—
164C2×C22⋊C4C2×D9711yescharacteristic
165C2×C22⋊C4C2×D9712yes—
166C2×C42—971no—
167C2×C42—972no—
168C2×C42—972no—
169#95—971no—
170#95—972no—
171#95—972no—
172#95—972no—
173#95—972no—
174#95C2×D9711yescharacteristic
175#95C2×D9712yes—
176C24—971no—
177C24C22×D9711yescharacteristic
178C2×Q8—1944no—
179C2×Q8—1942no—
180C2×D4—1942no—
181C2×D4—1942no—
182C2×D4—1942no—
183C22×C4—971no—
184C22×C4—972no—
185C22×C4—971no—
186C22×C4—972no—
187C22×C4—971no—
188C22×C4—1942no—
189C22×C4—1944no—
190C22×C4—1942no—
191C22×C4—1944no—
192C22×C4C22×D9712yes—
193C22×C4C22×D9711yescharacteristic
194C22×C4C22×D9712yes—
195C22×C4C22×D9711yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—1942no—
198C4⋊C4—1944no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—1944no—
202C22⋊C4—1944no—
203C22⋊C4—1944no—
204C22⋊C4—1942no—
205C42—974no—
206C42—1944no—
207C42—1944no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—972no—
213C23—1941no—
214C23—1942no—
215C23—1942no—
216C23—1941no—
217C23#1309551012yes—
218C23C23×D9711yescharacteristic
219Q8—1948no—
220D4—1948no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—972no—
231C2×C4—974no—
232C2×C4—972no—
233C2×C4—974no—
234C2×C4—3884no—
235C2×C4—3884no—
236C2×C4—1941no—
237C2×C4—1942no—
238C2×C4—1942no—
239C2×C4—1942no—
240C2×C4—1942no—
241C2×C4—1942no—
242C2×C4—1941no—
243C2×C4—1942no—
244C2×C4—1941no—
245C2×C4D4×D9712yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—3881no—
252C22—3881no—
253C22—3882no—
254C22—1942no—
255C22—1944no—
256C22C2×D4×D9711yescharacteristic
257C22#41×D9712yes—
258C22#41×D9711yescharacteristic
259C22#41×D9712yes—
260C22C2×#1309551011yescharacteristic
261C4—22no—
262C4—42no—
263C4—41no—
264C4—42no—
265C4—974no—
266C4—3881no—
267C4—3882no—
268C4—1944no—
269C4—1942no—
270C2—42no—
271C2—3882no—
272C2#2026124311yescharacteristic
273C2#2026124511yescharacteristic
274C2#124×D9711yescharacteristic
275C2#2026126212yes—
276C2#2026126812yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#397×C97C2
prime radicalC22×C22⋊D97C22
hypercenter#397D97
hypocenterC97#2398
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#15690775C2semidirect
3. #20260676C2semidirect
4. #20260661C2semidirect
6. #20260648C2semidirect
7. #20260647C2semidirect
8. C22⋊C4×Dic97C2semidirect
9. #20260582C2semidirect
10. #20260580C2semidirect
11. #397×C97C2semidirect
139. C97#2398semidirect
144. #397D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907752
C2×#156907702
#202606762
#202606612
#202606502
#202606482
#202606472
C22⋊C4×Dic972
#202605822
#202605802
#397×C972
#239897