Finite Group Databasebeta

← #37575878

Subgroups — #37575878

Order 24448.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D191C211yesmaximal, characteristic, complemented
2C2×#20259299C212yesmaximal, complemented
3C2×(C22⋊D191)⋊C4C211yesmaximal, characteristic
4#27132584C211yesmaximal, characteristic, complemented
5#27132582C211yesmaximal, characteristic, complemented
6#27132575C211yesmaximal, characteristic, complemented
7#27132567C212yesmaximal, complemented
8#27132539C212yesmaximal
9#27132537C212yesmaximal, complemented
10#27132522C211yesmaximal, characteristic, complemented
11#393×C191C211yesmaximal, characteristic, complemented
12C22×Dic382C2211yescharacteristic
13C22×Dic382C2212yes—
14C2×Q8×D191—28no—
15C2×Q8×D191—24no—
16C22×C22⋊D191C2211yescharacteristic
17C23×Dic191C2211yescharacteristic
18C22×C4×D191C2212yes—
19C22×C4×D191C2211yescharacteristic
20#20259299—28no—
21(C22⋊D191)⋊C4—24no—
22C2×C4⋊Dic191C2212yes—
23C2×Dic191⋊C4C2212yes—
24C2×Dic191⋊C4C2212yes—
25C2×Dic191⋊C4C2211yescharacteristic
26C2×C22⋊Dic191C2212yes—
27C2×(C2×D191)⋊C4C2212yes—
28C2×(C2×D191)⋊C4C2211yescharacteristic
29C2×C4×Dic191C2211yescharacteristic
30C191⋊#95C2211yescharacteristic
31C191⋊#95C2212yes—
32C191⋊#95C2212yes—
33C191⋊#95C2212yes—
34C22×Q8×C191C2211yescharacteristic
35C2×C22⋊C4×C191C2212yes—
36C2×C22⋊C4×C191C2211yescharacteristic
37#95×C191C2211yescharacteristic
38#95×C191C2212yes—
39C23×D191C2311yescharacteristic
40C2×Dic382—24no—
41C2×Dic382—24no—
42C2×Dic382—24no—
43C2×Dic382—22no—
44C2×Dic382D418yescomplemented
45Q8×D191—232no—
46C2×C22⋊D191—22no—
47C2×C22⋊D191—22no—
48C2×C22⋊D191Q814yescomplemented
49C22×Dic191—22no—
50C22×Dic191—24no—
51C22×Dic191C2311yescharacteristic
52C22×Dic191C2312yes—
53C22×Dic191C2312yes—
54C22×Dic191C2311yescharacteristic
55C22×Dic191C2311yescharacteristic
56C2×C4×D191—24no—
57C2×C4×D191—24no—
58C2×C4×D191—24no—
59C2×C4×D191—24no—
60C2×C4×D191D414yes—
61C4⋊Dic191—24no—
62Dic191⋊C4—24no—
63Dic191⋊C4—22no—
64Dic191⋊C4—24no—
65C22⋊Dic191—24no—
66(C2×D191)⋊C4—22no—
67(C2×D191)⋊C4—24no—
68C4×Dic191—22no—
69C24×C191C2311yescharacteristic
70C2×Q8×C191—24no—
71C2×Q8×C191—22no—
72C22×C4×C191C2311yescharacteristic
73C22×C4×C191C2312yes—
74C22×C4×C191C2312yes—
75C22×C4×C191C2311yescharacteristic
76C22⋊C4×C191—22no—
77C22⋊C4×C191—24no—
78C22×D191—22no—
79C22×D191—22no—
80C22×D191C2×D412yes—
81C22×D191C2×Q812yes—
82C22×D191#4112yes—
83Dic382—28no—
84Dic382—216no—
85C22⋊D191—28no—
86C2×Dic191—21no—
87C2×Dic191—22no—
88C2×Dic191—22no—
89C2×Dic191—21no—
90C2×Dic191—21no—
91C2×Dic191—22no—
92C2×Dic191—22no—
93C2×Dic191—22no—
94C2×Dic191—22no—
95C2×Dic191—44no—
96C2×Dic191C2×D414yes—
97C2×Dic191C2×D412yes—
98C2×Dic191C2×D414yes—
99C2×Dic191C2×Q812yescomplemented
100C4×D191—216no—
101C4×D191—28no—
102C23×C191—22no—
103C23×C191—21no—
104C23×C191—21no—
105C23×C191—22no—
106C23×C191C2×Q812yescomplemented
107C23×C191C2411yescharacteristic
108Q8×C191—28no—
109C2×C4×C191—22no—
110C2×C4×C191—22no—
111C2×C4×C191—22no—
112C2×C4×C191—21no—
113C2×C4×C191—22no—
114C2×C4×C191—22no—
115C2×C4×C191—22no—
116C2×C4×C191—22no—
117C2×C4×C191C2×D414yes—
118C2×C4×C191C2×D412yes—
119C22×C191—22no—
120C22×C191—24no—
121C22×C191—41no—
122C22×C191—41no—
123C22×C191—42no—
124C22×C191C22×D411yescharacteristic
125C22×C191C22×D412yes—
126C22×C191C22×Q811yescharacteristic
127C22×C191C2×#4111yescharacteristic
128C22×C1912−1+412yes—
129C2×D191—22no—
130C2×D191—22no—
131C2×D191—24no—
132C2×D191—24no—
133C2×D191C22⋊Q814yescomplemented
134C4×C191—24no—
135C4×C191—22no—
136C4×C191—41no—
137C4×C191—42no—
138Dic191—24no—
139Dic191—22no—
140Dic191—41no—
141Dic191—42no—
142Dic191#12014yes—
143C2×C191—42no—
144C2×C191C2×#12011yescharacteristic
145C2×C191C2×C22⋊Q811yescharacteristic
146C2×C191#53611yescharacteristic
147C2×C191#54812yes—
148C2×C191D4×Q812yes—
149D191—24no—
150C191#240911yescharacteristic, complemented, Sylow
151#2409—1911nomaximal, Sylow
152C23×Q8—1911no—
153C2×C22⋊Q8—1912no—
154C2×C4×D4—1911no—
155#395—1911no—
156#395—1911no—
157#393—1911no—
158#393D19111yescharacteristic, complemented
159#391—1912no—
160#386—1912no—
161#385—1912no—
162#380—1911no—
163C22×Q8—3828no—
164C22×Q8—3824no—
165C22×Q8—1911no—
166C22×Q8—1912no—
167C22×Q8C2×D19111yescharacteristic
168C22×D4—1911no—
169C23×C4—1911no—
170C23×C4—1912no—
171C23×C4—1911no—
172C22⋊Q8—3828no—
173C4×D4—3824no—
174C2×C4⋊C4—1911no—
175C2×C4⋊C4—1912no—
176C2×C4⋊C4—1912no—
177C2×C4⋊C4—1912no—
178C2×C22⋊C4—1912no—
179C2×C22⋊C4—1911no—
180C2×C22⋊C4—1912no—
181C2×C22⋊C4C2×D19111yescharacteristic
182C2×C22⋊C4C2×D19112yes—
183C2×C42—1911no—
184#95—1911no—
185#95—1912no—
186#95—1912no—
187#95—1912no—
188#95C2×D19111yescharacteristic
189#95C2×D19112yes—
190C24—1911no—
191C24C22×D19111yescharacteristic
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—3824no—
195C2×Q8—3824no—
196C2×Q8—3824no—
197C2×Q8—38232no—
198C2×Q8—3822no—
199C2×Q8—1918no—
200C2×D4—3822no—
201C2×D4—3822no—
202C2×D4—1914no—
203C22×C4—3824no—
204C22×C4—3824no—
205C22×C4—3824no—
206C22×C4—3824no—
207C22×C4—3822no—
208C22×C4—3824no—
209C22×C4—1912no—
210C22×C4—1911no—
211C22×C4—1911no—
212C22×C4—1912no—
213C22×C4—1914no—
214C22×C4—1911no—
215C22×C4C22×D19111yescharacteristic
216C22×C4C22×D19112yes—
217C22×C4C22×D19112yes—
218C22×C4C22×D19111yescharacteristic
219C4⋊C4—3824no—
220C4⋊C4—3824no—
221C4⋊C4—3822no—
222C4⋊C4—3824no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—3824no—
226C22⋊C4—3822no—
227C22⋊C4—3824no—
228C42—3822no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—3822no—
234C23—3822no—
235C23—1912no—
236C23—1912no—
237C23—1912no—
238C23Q8×D19112yescomplemented
239C23C23×D19111yescharacteristic
240Q8—28no—
241Q8—3828no—
242Q8—38216no—
243D4—3828no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—38216no—
253C2×C4—3828no—
254C2×C4—3821no—
255C2×C4—3822no—
256C2×C4—3822no—
257C2×C4—3821no—
258C2×C4—3822no—
259C2×C4—3821no—
260C2×C4—3822no—
261C2×C4—3822no—
262C2×C4—3822no—
263C2×C4—1914no—
264C2×C4—1912no—
265C2×C4—1912no—
266C2×C4—1914no—
267C2×C4—7644no—
268C2×C4D4×D19114yes—
269C2×C4D4×D19112yes—
270C22—24no—
271C22—22no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—3822no—
276C22—3824no—
277C22—3824no—
278C22—3822no—
279C22—1914no—
280C22C2×D4×D19111yescharacteristic
281C22C2×D4×D19112yes—
282C22C2×Q8×D19111yescharacteristic
283C22#41×D19111yescharacteristic
284C22#2025936912yes—
285C4—24no—
286C4—22no—
287C4—41no—
288C4—42no—
289C4—3824no—
290C4—3822no—
291C4—1914no—
292C4—7642no—
293C4—7641no—
294C2—42no—
295C2—3824no—
296C2#120×D19111yescharacteristic
297C2C22⋊Q8×D19111yescharacteristic
298C2#2713323311yescharacteristic
299C2#2713332112yes—
300C2#2713332612yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#393×C191C2
prime radicalC22×C22⋊D191C22
hypercenter#393D191
hypocenterC191#2409
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D191C2semidirect
2. C2×#20259299C2semidirect
4. #27132584C2semidirect
5. #27132582C2semidirect
6. #27132575C2semidirect
7. #27132567C2semidirect
9. #27132537C2semidirect
10. #27132522C2semidirect
11. #393×C191C2semidirect
44. C2×Dic382D4semidirect
48. C2×C22⋊D191Q8semidirect
99. C2×Dic191C2×Q8semidirect
106. C23×C191C2×Q8semidirect
133. C2×D191C22⋊Q8semidirect
150. C191#2409semidirect
158. #393D191semidirect
238. C23Q8×D191semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1912
C2×#202592992
C2×(C22⋊D191)⋊C42
#271325842
#271325822
#271325752
#271325672
#271325392
#271325372
#271325222
#393×C1912
#2409191