Finite Group Databasebeta

← #37576168

Subgroups — #37576168

Order 24448.

1088 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 173 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×D4×D191C211yesmaximal, characteristic
2C2×#20259311C211yesmaximal, characteristic, complemented
3C2×#20259298C211yesmaximal, characteristic, complemented
4#27132578C214yesmaximal, complemented
5#27132566C211yesmaximal, characteristic, complemented
6#27132565C211yesmaximal, characteristic, complemented
7#27132518C214yesmaximal, complemented
8C4×(C2×D191)⋊C4C211yesmaximal, characteristic, complemented
9C2×#124×C191C211yesmaximal, characteristic, complemented
10C24×D191C2212yes—
11C22×D764C2211yescharacteristic
12C22×C22⋊D191C2212yes—
13C2×D4×D191—24no—
14C2×D4×D191—24no—
15C2×D4×D191—24no—
16C22×C4×D191C2211yescharacteristic
17#20259311—24no—
18#20259298—24no—
19C2×C4⋊Dic191C2211yescharacteristic, complemented
20C2×Dic191⋊C4C2212yescomplemented
21C2×C22⋊Dic191C2214yescomplemented
22C2×(C2×D191)⋊C4C2212yes—
23C2×(C2×D191)⋊C4C2212yes—
24C2×(C2×D191)⋊C4C2214yescomplemented
25C2×(C2×D191)⋊C4C2214yescomplemented
26C2×C4×Dic191C2211yescharacteristic, complemented
27C191⋊#95C2212yescomplemented
28C22×Q8×C191C2211yescharacteristic, complemented
29C22×D4×C191C2211yescharacteristic
30#124×C191—24no—
31C2×C22⋊C4×C191C2214yescomplemented
32C2×C42×C191C2211yescharacteristic, complemented
33C23×D191—24no—
34C23×D191—24no—
35C23×D191—28no—
36C23×D191—24no—
37C23×D191—44no—
38C23×D191C2312yes—
39C23×D191C2311yescharacteristic
40C2×D764—22no—
41C2×D764—22no—
42C2×D764D414yes—
43C2×C22⋊D191—24no—
44C2×C22⋊D191—24no—
45C2×C22⋊D191—24no—
46D4×D191—232no—
47C22×Dic191C2312yes—
48C22×Dic191C2312yes—
49C22×Dic191C2311yescharacteristic
50C2×C4×D191—22no—
51C2×C4×D191—22no—
52C2×C4×D191D414yes—
53C4⋊Dic191—22no—
54Dic191⋊C4—24no—
55C22⋊Dic191—44no—
56(C2×D191)⋊C4—24no—
57(C2×D191)⋊C4—28no—
58(C2×D191)⋊C4—24no—
59(C2×D191)⋊C4—44no—
60C4×Dic191—22no—
61C24×C191C2312yes—
62C2×Q8×C191—22no—
63C2×Q8×C191—22no—
64C2×Q8×C191—22no—
65C2×D4×C191—22no—
66C2×D4×C191—22no—
67C2×D4×C191D414yes—
68C22×C4×C191C2312yes—
69C22×C4×C191C2312yes—
70C22×C4×C191C2311yescharacteristic
71C22⋊C4×C191—28no—
72C42×C191—22no—
73C22×D191—21no—
74C22×D191—21no—
75C22×D191—22no—
76C22×D191—216no—
77C22×D191—22no—
78C22×D191—22no—
79C22×D191—24no—
80C22×D191—24no—
81C22×D191—416no—
82C22×D191—416no—
83C22×D191—48no—
84C22×D191—48no—
85C22×D191C2×D414yes—
86C22×D191C2×D412yes—
87C22×D191#4114yes—
88D764—28no—
89C22⋊D191—216no—
90C2×Dic191—21no—
91C2×Dic191—21no—
92C2×Dic191—22no—
93C2×Dic191—22no—
94C2×Dic191—22no—
95C2×Dic191—24no—
96C2×Dic191—22no—
97C2×Dic191C2×D412yes—
98C4×D191—28no—
99C23×C191—22no—
100C23×C191—22no—
101C23×C191—24no—
102C23×C191—24no—
103C23×C191C2×D414yes—
104C23×C191C2411yescharacteristic
105Q8×C191—44no—
106D4×C191—28no—
107C2×C4×C191—22no—
108C2×C4×C191—22no—
109C2×C4×C191—22no—
110C2×C4×C191—22no—
111C2×C4×C191—22no—
112C2×C4×C191—22no—
113C2×C4×C191C2×D412yes—
114C2×C4×C191C2×D412yes—
115C2×C4×C191C2×D412yes—
116C22×C191—24no—
117C22×C191—28no—
118C22×C191—42no—
119C22×C191—42no—
120C22×C191—44no—
121C22×C191C22×D411yescharacteristic
122C22×C191C22×D411yescharacteristic
123C22×C191C22×D411yescharacteristic
124C22×C191C2×#4112yes—
125C22×C191H(5,𝔽2)12yes—
126C2×D191—22no—
127C2×D191—22no—
128C2×D191—24no—
129C2×D191—24no—
130C2×D191—24no—
131C2×D191—28no—
132C2×D191—432no—
133C2×D191—42no—
134C2×D191—42no—
135C2×D191—44no—
136C2×D191#12414yescomplemented
137C4×C191—42no—
138C4×C191—42no—
139C4×C191#12014yes—
140Dic191—22no—
141Dic191—42no—
142Dic191—42no—
143C2×C191—44no—
144C2×C191C2×#12011yescharacteristic
145C2×C191C2×#12411yescharacteristic
146C2×C191#53511yescharacteristic
147C2×C191#54612yes—
148C2×C191#54612yes—
149D191—24no—
150D191—44no—
151C191#243511yescharacteristic, complemented, Sylow
152#2435—1911nomaximal, Sylow
153C23×D4—1911no—
154C2×#124—1911no—
155C2×#124D19111yescharacteristic, complemented
156C2×C22⋊Q8—1911no—
157#394—1914no—
158#390—1911no—
159#390—1911no—
160#379—1914no—
161C4×C22⋊C4—1911no—
162C25—1912no—
163C22×Q8C2×D19111yescharacteristic, complemented
164C22×D4—3824no—
165C22×D4—3824no—
166C22×D4—3824no—
167C22×D4—1912no—
168C22×D4—1911no—
169C22×D4C2×D19111yescharacteristic
170C23×C4—1911no—
171#124—24no—
172#124—3824no—
173C22⋊Q8—3824no—
174C2×C4⋊C4—1911no—
175C2×C4⋊C4—1912no—
176C2×C22⋊C4—1912no—
177C2×C22⋊C4—1914no—
178C2×C22⋊C4—1912no—
179C2×C22⋊C4—1914no—
180C2×C22⋊C4—1914no—
181C2×C22⋊C4C2×D19114yescomplemented
182C2×C42—1911no—
183C2×C42C2×D19111yescharacteristic, complemented
184#95—1912no—
185C24—3824no—
186C24—3824no—
187C24—3828no—
188C24—3824no—
189C24—1912no—
190C24—1911no—
191C24—7644no—
192C24C22×D19112yes—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—38232no—
199C2×D4—3824no—
200C2×D4—3824no—
201C2×D4—3824no—
202C2×D4—3822no—
203C2×D4—3822no—
204C2×D4—1914no—
205C2×D4C22⋊D19114yes—
206C22×C4—3822no—
207C22×C4—3822no—
208C22×C4—1912no—
209C22×C4—1912no—
210C22×C4—1911no—
211C22×C4—1914no—
212C22×C4C22×D19112yes—
213C22×C4C22×D19112yes—
214C22×C4C22×D19111yescharacteristic
215C4⋊C4—3824no—
216C4⋊C4—3822no—
217C22⋊C4—28no—
218C22⋊C4—3824no—
219C22⋊C4—3824no—
220C22⋊C4—3828no—
221C22⋊C4—7644no—
222C22⋊C4—7644no—
223C42—22no—
224C42—3822no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—38216no—
230C23—3821no—
231C23—3821no—
232C23—3822no—
233C23—3822no—
234C23—3822no—
235C23—3824no—
236C23—3824no—
237C23—1914no—
238C23—1912no—
239C23—1914no—
240C23—76416no—
241C23—76416no—
242C23—7648no—
243C23—7648no—
244C23C2×C22⋊D19114yes—
245C23C23×D19111yescharacteristic
246Q8—44no—
247D4—28no—
248D4—38216no—
249D4—3828no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—3821no—
257C2×C4—3821no—
258C2×C4—3828no—
259C2×C4—3822no—
260C2×C4—3822no—
261C2×C4—3822no—
262C2×C4—3824no—
263C2×C4—3822no—
264C2×C4—1912no—
265C2×C4D4×D19112yes—
266C2×C4D4×D19112yes—
267C2×C4C2×C22⋊D19112yes—
268C22—28no—
269C22—24no—
270C22—42no—
271C22—44no—
272C22—42no—
273C22—3822no—
274C22—3824no—
275C22—3824no—
276C22—3822no—
277C22—3828no—
278C22—3824no—
279C22—1914no—
280C22—76432no—
281C22—7642no—
282C22—7644no—
283C22—7642no—
284C22C2×D4×D19111yescharacteristic
285C22C2×D4×D19111yescharacteristic
286C22C22×C22⋊D19111yescharacteristic
287C22#41×D19112yes—
288C22#2025936812yes—
289C4—42no—
290C4—42no—
291C4—3822no—
292C4—7642no—
293C4—7642no—
294C4#2025929014yes—
295C2—44no—
296C2—3824no—
297C2—7644no—
298C2C2×#2025929011yescharacteristic
299C2#124×D19111yescharacteristic
300C2#2713323111yescharacteristic
301C2#2713330112yes—
302C2#2713330612yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
FittingC2×#124×C191C2
prime radicalC22×D4×D191C2
hypercenterC2×#124D191
hypocenterC191#2435
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#20259311C2semidirect
3. C2×#20259298C2semidirect
4. #27132578C2semidirect
5. #27132566C2semidirect
6. #27132565C2semidirect
7. #27132518C2semidirect
8. C4×(C2×D191)⋊C4C2semidirect
9. C2×#124×C191C2semidirect
19. C2×C4⋊Dic191C22semidirect
20. C2×Dic191⋊C4C22semidirect
21. C2×C22⋊Dic191C22semidirect
24. C2×(C2×D191)⋊C4C22semidirect
25. C2×(C2×D191)⋊C4C22semidirect
26. C2×C4×Dic191C22semidirect
27. C191⋊#95C22semidirect
28. C22×Q8×C191C22semidirect
31. C2×C22⋊C4×C191C22semidirect
32. C2×C42×C191C22semidirect
136. C2×D191#124semidirect
151. C191#2435semidirect
155. C2×#124D191semidirect
163. C22×Q8C2×D191semidirect
181. C2×C22⋊C4C2×D191semidirect
183. C2×C42C2×D191semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1912
C2×#202593112
C2×#202592982
#271325782
#271325662
#271325652
#271325182
C4×(C2×D191)⋊C42
C2×#124×C1912
#2435191