Finite Group Databasebeta

← #37575927

Subgroups — #37575927

Order 24448.

610 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 165 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×#20259305C211yesmaximal, characteristic
2C2×#20259305C211yesmaximal, characteristic
3C2×#20259281C211yesmaximal, characteristic, complemented
4#27132596C212yesmaximal, complemented
5#27132585C212yesmaximal, complemented
6#27132558C212yesmaximal, complemented
7#27132553C212yesmaximal, complemented
8#27132542C211yesmaximal, characteristic
9#27132536C211yesmaximal, characteristic, complemented
10#27132526C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C191C211yesmaximal, characteristic, complemented
12C22×C22⋊D191C2211yescharacteristic
13#20259305—24no—
14#20259305—24no—
15#20259281—24no—
16C2×C4⋊Dic191C2212yes—
17C2×Dic191⋊C4C2212yes—
18C2×Dic191⋊C4C2211yescharacteristic
19C2×Dic191⋊C4C2212yes—
20C2×Dic191⋊C4C2211yescharacteristic
21C2×C22⋊Dic191C2211yescharacteristic
22C2×C22⋊Dic191C2211yescharacteristic
23C2×C22⋊Dic191C2211yescharacteristic
24C2×(C2×D191)⋊C4C2212yes—
25C2×(C2×D191)⋊C4C2211yescharacteristic
26C2×(C2×D191)⋊C4C2212yes—
27C2×(C2×D191)⋊C4C2211yescharacteristic
28C2×C4×Dic191C2212yes—
29C191⋊#95C2212yes—
30C191⋊#95C2212yes—
31C191⋊#95C2212yes—
32C191⋊#95C2212yes—
33C23×C4×C191C2211yescharacteristic
34C23×C4×C191C2211yescharacteristic
35C23×C4×C191C2211yescharacteristic
36C2×C4⋊C4×C191—24no—
37C2×C4⋊C4×C191C2212yes—
38C2×C4⋊C4×C191C2212yes—
39C23×D191C2311yescharacteristic
40C2×C22⋊D191—22no—
41C2×C22⋊D191—22no—
42C2×C22⋊D191—22no—
43C22×Dic191C2311yescharacteristic
44C22×Dic191C2311yescharacteristic
45C22×Dic191C2312yes—
46C22×Dic191C2311yescharacteristic
47C22×Dic191C2312yes—
48C4⋊Dic191—24no—
49Dic191⋊C4—24no—
50Dic191⋊C4—22no—
51Dic191⋊C4—22no—
52Dic191⋊C4—24no—
53C22⋊Dic191—22no—
54C22⋊Dic191—22no—
55C22⋊Dic191D414yes—
56(C2×D191)⋊C4—24no—
57(C2×D191)⋊C4—22no—
58(C2×D191)⋊C4—22no—
59(C2×D191)⋊C4—24no—
60C4×Dic191—24no—
61C24×C191C2311yescharacteristic
62C22×C4×C191—22no—
63C22×C4×C191—22no—
64C22×C4×C191—22no—
65C22×C4×C191—24no—
66C22×C4×C191—22no—
67C22×C4×C191—22no—
68C22×C4×C191—22no—
69C22×C4×C191D414yes—
70C22×C4×C191C2312yes—
71C22×C4×C191C2311yescharacteristic
72C22×C4×C191C2311yescharacteristic
73C22×C4×C191C2312yes—
74C4⋊C4×C191—24no—
75C4⋊C4×C191—24no—
76C22×D191—21no—
77C22×D191—21no—
78C22×D191—22no—
79C22×D191—21no—
80C22×D191—22no—
81C22⋊D191—44no—
82C2×Dic191—22no—
83C2×Dic191—22no—
84C2×Dic191—22no—
85C2×Dic191—21no—
86C2×Dic191—21no—
87C2×Dic191—21no—
88C2×Dic191—21no—
89C2×Dic191—22no—
90C2×Dic191—22no—
91C2×Dic191—21no—
92C2×Dic191—22no—
93C2×Dic191—22no—
94C2×Dic191C2×D414yes—
95C23×C191—22no—
96C23×C191—22no—
97C23×C191C2×D412yes—
98C23×C191C2×D412yes—
99C23×C191#4112yes—
100C23×C191C2411yescharacteristic
101C2×C4×C191—28no—
102C2×C4×C191—28no—
103C2×C4×C191—22no—
104C2×C4×C191—21no—
105C2×C4×C191—21no—
106C2×C4×C191—21no—
107C2×C4×C191—21no—
108C2×C4×C191—22no—
109C2×C4×C191—22no—
110C2×C4×C191—22no—
111C2×C4×C191—22no—
112C2×C4×C191—44no—
113C2×C4×C191C2×D412yes—
114C2×C4×C191C2×D412yes—
115C2×C4×C191#4114yes—
116C22×C191—22no—
117C22×C191—22no—
118C22×C191—24no—
119C22×C191—24no—
120C22×C191#12114yes—
121C22×C191C22×D411yescharacteristic
122C22×C191C22×D411yescharacteristic
123C22×C191C2×#4111yescharacteristic
124C22×C191C2×#4112yes—
125C22×C1912−1+412yes—
126C2×D191—41no—
127C2×D191—41no—
128C2×D191—41no—
129C2×D191—42no—
130C2×D191—42no—
131C4×C191—22no—
132C4×C191—22no—
133C4×C191—24no—
134C4×C191—42no—
135Dic191—24no—
136Dic191—41no—
137Dic191—41no—
138Dic191—41no—
139Dic191—42no—
140C2×C191—24no—
141C2×C191C2×#12111yescharacteristic
142C2×C191#52511yescharacteristic
143C2×C191#53611yescharacteristic
144C2×C191#54712yes—
145C2×C191#55212yes—
146D191—81no—
147C191#241311yescharacteristic, complemented, Sylow
148#2413—1911nomaximal, Sylow
149C2×#123—1911no—
150C2×#123—1911no—
151C2×#117—1911no—
152C22×C4⋊C4D19111yescharacteristic, complemented
153#398—1912no—
154#396—1912no—
155#389—1912no—
156#388—1912no—
157#386—1911no—
158#385—1911no—
159#381—1911no—
160C22×D4—1911no—
161C23×C4C2×D19111yescharacteristic
162C23×C4C2×D19111yescharacteristic
163C23×C4C2×D19111yescharacteristic
164#123—3824no—
165#123—3824no—
166#117—3824no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—1912no—
169C2×C4⋊C4—1911no—
170C2×C4⋊C4—1912no—
171C2×C4⋊C4—1911no—
172C2×C4⋊C4—1912no—
173C2×C4⋊C4C2×D19112yes—
174C2×C4⋊C4C2×D19112yes—
175C2×C22⋊C4—1911no—
176C2×C22⋊C4—1911no—
177C2×C22⋊C4—1911no—
178C2×C22⋊C4—1912no—
179C2×C22⋊C4—1911no—
180C2×C22⋊C4—1912no—
181C2×C22⋊C4—1911no—
182C2×C42—1912no—
183#95—1912no—
184#95—1912no—
185#95—1912no—
186#95—1912no—
187C24—1911no—
188C24C22×D19111yescharacteristic
189C2×D4—3822no—
190C2×D4—3822no—
191C2×D4—3822no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—1911no—
200C22×C4—1912no—
201C22×C4—1911no—
202C22×C4—1911no—
203C22×C4—1912no—
204C22×C4C22⋊D19114yes—
205C22×C4C22×D19112yes—
206C22×C4C22×D19111yescharacteristic
207C22×C4C22×D19111yescharacteristic
208C22×C4C22×D19112yes—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—3824no—
212C4⋊C4—3824no—
213C4⋊C4—3822no—
214C4⋊C4—3822no—
215C4⋊C4—3824no—
216C22⋊C4—3824no—
217C22⋊C4—3822no—
218C22⋊C4—3822no—
219C22⋊C4—3822no—
220C22⋊C4—3822no—
221C22⋊C4—3824no—
222C22⋊C4—1914no—
223C42—3824no—
224C23—22no—
225C23—22no—
226C23—3821no—
227C23—3821no—
228C23—3821no—
229C23—3822no—
230C23—3822no—
231C23D4×D19112yes—
232C23C2×C22⋊D19112yes—
233C23#1569028612yes—
234C23C23×D19111yescharacteristic
235D4—7644no—
236C2×C4—28no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—44no—
248C2×C4—3822no—
249C2×C4—3822no—
250C2×C4—3822no—
251C2×C4—3821no—
252C2×C4—3821no—
253C2×C4—3821no—
254C2×C4—3821no—
255C2×C4—3821no—
256C2×C4—3822no—
257C2×C4—3822no—
258C2×C4—3822no—
259C2×C4—3822no—
260C2×C4—1914no—
261C2×C4C2×C22⋊D19112yes—
262C2×C4C2×C22⋊D19112yes—
263C2×C4#1569028714yes—
264C22—22no—
265C22—24no—
266C22—24no—
267C22—22no—
268C22—7641no—
269C22—7642no—
270C22—7642no—
271C22—7641no—
272C22—7641no—
273C22#2025929414yes—
274C22C2×D4×D19111yescharacteristic
275C22C22×C22⋊D19111yescharacteristic
276C22C2×#1569028611yescharacteristic
277C22C2×#1569028712yes—
278C22#2025937012yes—
279C4—22no—
280C4—22no—
281C4—24no—
282C4—42no—
283C4—3824no—
284C4—7641no—
285C4—7642no—
286C4—7641no—
287C4—7641no—
288C2—24no—
289C2—15281no—
290C2C2×#2025929411yescharacteristic
291C2#2713317411yescharacteristic
292C2#2713323711yescharacteristic
293C2#2713331412yes—
294C2#2713334812yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
FittingC22×C4⋊C4×C191C2
prime radicalC22×C22⋊D191C22
hypercenterC22×C4⋊C4D191
hypocenterC191#2413
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#20259281C2semidirect
4. #27132596C2semidirect
5. #27132585C2semidirect
6. #27132558C2semidirect
7. #27132553C2semidirect
9. #27132536C2semidirect
10. #27132526C2semidirect
11. C22×C4⋊C4×C191C2semidirect
147. C191#2413semidirect
152. C22×C4⋊C4D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593052
C2×#202592812
#271325962
#271325852
#271325582
#271325532
#271325422
#271325362
#271325262
C22×C4⋊C4×C1912
#2413191