Finite Group Databasebeta

← #37575761

Subgroups — #37575761

Order 24448.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#20259284C211yesmaximal, characteristic, complemented
2C2×#20259284C212yesmaximal, complemented
3C2×#20259280C211yesmaximal, characteristic, complemented
4#27132589C211yesmaximal, characteristic, complemented
5#27132585C212yesmaximal, complemented
6#27132581C212yesmaximal, complemented
7#27132580C211yesmaximal, characteristic, complemented
8#27132573C211yesmaximal, characteristic
9#27132514C211yesmaximal, characteristic, complemented
10#27132510C212yesmaximal, complemented
11#398×C191C211yesmaximal, characteristic, complemented
12C22×D764C2211yescharacteristic
13C22×D764C2212yes—
14C22×C4×D191C2212yes—
15C22×C4×D191C2211yescharacteristic
16#20259284—28no—
17#20259284—24no—
18#20259280—24no—
19C2×C4⋊Dic191C2211yescharacteristic
20C2×Dic191⋊C4C2212yes—
21C2×(C2×D191)⋊C4C2212yes—
22C2×(C2×D191)⋊C4C2212yes—
23C2×(C2×D191)⋊C4C2211yescharacteristic
24C2×(C2×D191)⋊C4C2212yes—
25C2×(C2×D191)⋊C4C2211yescharacteristic
26C2×(C2×D191)⋊C4C2212yes—
27C2×(C2×D191)⋊C4C2212yes—
28C2×C4×Dic191C2211yescharacteristic
29C2×C4×Dic191C2212yes—
30C2×C4×Dic191C2211yescharacteristic
31C191⋊#95C2212yes—
32C191⋊#95C2211yescharacteristic
33C2×C4⋊C4×C191C2211yescharacteristic
34C2×C4⋊C4×C191C2212yes—
35C2×C4⋊C4×C191C2211yescharacteristic
36#95×C191C2212yes—
37#95×C191C2211yescharacteristic
38C23×D191C2311yescharacteristic
39C23×D191C2312yes—
40C2×D764—24no—
41C2×D764—24no—
42C2×D764—24no—
43C2×D764—22no—
44C2×D764—24no—
45C22×Dic191C2312yes—
46C22×Dic191C2311yescharacteristic
47C22×Dic191C2311yescharacteristic
48C22×Dic191C2311yescharacteristic
49C2×C4×D191—22no—
50C2×C4×D191—22no—
51C2×C4×D191—24no—
52C2×C4×D191—24no—
53C2×C4×D191—24no—
54C2×C4×D191D414yescomplemented
55C4⋊Dic191—22no—
56Dic191⋊C4—24no—
57(C2×D191)⋊C4—24no—
58(C2×D191)⋊C4—24no—
59(C2×D191)⋊C4—22no—
60(C2×D191)⋊C4—24no—
61(C2×D191)⋊C4—22no—
62(C2×D191)⋊C4—24no—
63(C2×D191)⋊C4—24no—
64C4×Dic191—22no—
65C4×Dic191—24no—
66C4×Dic191D414yescomplemented
67C22×C4×C191C2312yes—
68C22×C4×C191C2311yescharacteristic
69C22×C4×C191C2311yescharacteristic
70C22×C4×C191C2311yescharacteristic
71C22×C4×C191C2312yes—
72C4⋊C4×C191—22no—
73C4⋊C4×C191—24no—
74C4⋊C4×C191—22no—
75C22×D191—21no—
76C22×D191—22no—
77C22×D191—22no—
78C22×D191—22no—
79C22×D191—21no—
80C22×D191—22no—
81C22×D191—22no—
82C22×D191—22no—
83C22×D191—22no—
84C22×D191—22no—
85C22×D191C2×D412yes—
86C22×D191#4114yes—
87D764—28no—
88D764—216no—
89C2×Dic191—22no—
90C2×Dic191—22no—
91C2×Dic191—21no—
92C2×Dic191—22no—
93C2×Dic191—22no—
94C2×Dic191—22no—
95C2×Dic191C2×D412yes—
96C2×Dic191C2×D412yes—
97C2×Dic191C2×D412yes—
98C2×Dic191#4112yes—
99C4×D191—28no—
100C4×D191—48no—
101C23×C191C2411yescharacteristic
102C2×C4×C191—22no—
103C2×C4×C191—21no—
104C2×C4×C191—21no—
105C2×C4×C191—22no—
106C2×C4×C191—22no—
107C2×C4×C191—21no—
108C2×C4×C191—22no—
109C2×C4×C191—22no—
110C2×C4×C191—22no—
111C2×C4×C191—22no—
112C2×C4×C191C2×D412yes—
113C2×C4×C191C2×D412yes—
114C2×C4×C191#4114yes—
115C22×C191C22×D411yescharacteristic
116C22×C191C22×D411yescharacteristic
117C22×C191C2×#4111yescharacteristic
118C22×C191C2×#4112yes—
119C22×C191C2×#4112yes—
120C2×D191—22no—
121C2×D191—24no—
122C2×D191—24no—
123C2×D191—24no—
124C2×D191—24no—
125C2×D191—42no—
126C2×D191—41no—
127C2×D191—42no—
128C2×D191—42no—
129C2×D191—42no—
130C2×D191—41no—
131C2×D191—42no—
132C4×C191—22no—
133C4×C191—22no—
134C4×C191—24no—
135C4×C191—42no—
136C4×C191—41no—
137Dic191—22no—
138Dic191—42no—
139Dic191—41no—
140Dic191#12114yes—
141C2×C191C2×#12111yescharacteristic
142C2×C191#52511yescharacteristic
143C2×C191#52512yes—
144C2×C191#52912yes—
145C2×C191#53211yescharacteristic
146D191—42no—
147D191—44no—
148C191#239211yescharacteristic, complemented, Sylow
149#2392—1911nomaximal, Sylow
150C2×C4×D4—1912no—
151C2×C4×D4—1911no—
152C2×#117—1911no—
153#398D19111yescharacteristic, complemented
154#397—1911no—
155#396—1912no—
156#394—1911no—
157#394—1912no—
158#392—1911no—
159C4×C4⋊C4—1911no—
160C4×C22⋊C4—1912no—
161C22×D4—1911no—
162C22×D4—1912no—
163C23×C4—1912no—
164C23×C4—1911no—
165C4×D4—3828no—
166C4×D4—3824no—
167#117—3824no—
168C2×C4⋊C4—1911no—
169C2×C4⋊C4—1912no—
170C2×C4⋊C4C2×D19111yescharacteristic
171C2×C4⋊C4C2×D19112yes—
172C2×C4⋊C4C2×D19111yescharacteristic
173C2×C22⋊C4—1911no—
174C2×C22⋊C4—1911no—
175C2×C22⋊C4—1912no—
176C2×C22⋊C4—1912no—
177C2×C22⋊C4—1912no—
178C2×C22⋊C4—1912no—
179C2×C22⋊C4—1912no—
180C2×C42—1911no—
181C2×C42—1912no—
182C2×C42—1911no—
183#95—1912no—
184#95—1911no—
185#95C2×D19112yes—
186#95C2×D19111yescharacteristic
187C24—1911no—
188C24—1912no—
189C2×D4—3824no—
190C2×D4—3824no—
191C2×D4—3824no—
192C2×D4—3822no—
193C2×D4—3824no—
194C22×C4—3822no—
195C22×C4—3822no—
196C22×C4—3824no—
197C22×C4—3824no—
198C22×C4—3824no—
199C22×C4—1912no—
200C22×C4—1911no—
201C22×C4—1911no—
202C22×C4—1914no—
203C22×C4—1911no—
204C22×C4C22×D19112yes—
205C22×C4C22×D19111yescharacteristic
206C22×C4C22×D19111yescharacteristic
207C22×C4C22×D19111yescharacteristic
208C22×C4C22×D19112yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—3822no—
213C4⋊C4—3824no—
214C22⋊C4—3824no—
215C22⋊C4—3824no—
216C22⋊C4—3822no—
217C22⋊C4—3824no—
218C22⋊C4—3822no—
219C22⋊C4—3824no—
220C22⋊C4—3824no—
221C42—3822no—
222C42—3824no—
223C42—1914no—
224C23—3821no—
225C23—3822no—
226C23—3822no—
227C23—3822no—
228C23—3821no—
229C23—3822no—
230C23—3822no—
231C23—3822no—
232C23—3822no—
233C23—3822no—
234C23—1912no—
235C23—1914no—
236C23C23×D19111yescharacteristic
237D4—3828no—
238D4—38216no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—3828no—
250C2×C4—3822no—
251C2×C4—3822no—
252C2×C4—3821no—
253C2×C4—3822no—
254C2×C4—3822no—
255C2×C4—3822no—
256C2×C4—1912no—
257C2×C4—1912no—
258C2×C4—1912no—
259C2×C4—1912no—
260C2×C4—7648no—
261C2×C4D4×D19112yes—
262C2×C4D4×D19112yes—
263C2×C4#1569028814yes—
264C22—3824no—
265C22—3822no—
266C22—3824no—
267C22—3824no—
268C22—3824no—
269C22—7642no—
270C22—7642no—
271C22—7641no—
272C22—7642no—
273C22—7642no—
274C22—7641no—
275C22—7642no—
276C22C2×D4×D19111yescharacteristic
277C22C2×D4×D19111yescharacteristic
278C22#41×D19111yescharacteristic
279C22#41×D19112yes—
280C22C2×#1569028812yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—3822no—
287C4—1914no—
288C4—7641no—
289C4—7642no—
290C2—7644no—
291C2—7642no—
292C2#121×D19111yescharacteristic
293C2#2713317111yescharacteristic
294C2#2713317612yes—
295C2#2713319812yes—
296C2#2713321411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#398×C191C2
prime radical#27132573C2
hypercenter#398D191
hypocenterC191#2392
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#20259284C2semidirect
2. C2×#20259284C2semidirect
3. C2×#20259280C2semidirect
4. #27132589C2semidirect
5. #27132585C2semidirect
6. #27132581C2semidirect
7. #27132580C2semidirect
9. #27132514C2semidirect
10. #27132510C2semidirect
11. #398×C191C2semidirect
54. C2×C4×D191D4semidirect
66. C4×Dic191D4semidirect
148. C191#2392semidirect
153. #398D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592842
C2×#202592802
#271325892
#271325852
#271325812
#271325802
#271325732
#271325142
#271325102
#398×C1912
#2392191