Finite Group Databasebeta

← #37575757

Subgroups — #37575757

Order 24448.

696 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 151 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×(C22⋊D191)⋊C4C212yesmaximal, complemented
3C2×#20259278C211yesmaximal, characteristic, complemented
4#27132594C211yesmaximal, characteristic, complemented
5#27132587C212yesmaximal, complemented
6#27132581C211yesmaximal, characteristic, complemented
7#27132578C212yesmaximal, complemented
8#27132572C211yesmaximal, characteristic
9#27132514C211yesmaximal, characteristic, complemented
10#27132510C212yesmaximal, complemented
11#397×C191C211yesmaximal, characteristic, complemented
12C22×D764C2211yescharacteristic
13C22×C22⋊D191C2212yes—
14C23×Dic191C2211yescharacteristic
15C22×C4×D191C2212yes—
16#20259284—24no—
17(C22⋊D191)⋊C4—28no—
18#20259278—24no—
19C2×Dic191⋊C4C2212yes—
20C2×Dic191⋊C4C2211yescharacteristic
21C2×Dic191⋊C4C2211yescharacteristic
22C2×Dic191⋊C4C2212yes—
23C2×C22⋊Dic191C2211yescharacteristic
24C2×(C2×D191)⋊C4C2212yes—
25C2×(C2×D191)⋊C4C2212yes—
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×C22⋊C4×C191C2211yescharacteristic
35C2×C22⋊C4×C191C2212yes—
36#95×C191C2212yes—
37#95×C191C2211yescharacteristic
38C23×D191C2312yes—
39C2×D764—24no—
40C2×D764—22no—
41C2×C22⋊D191—24no—
42C2×C22⋊D191—24no—
43C2×C22⋊D191—24no—
44C22×Dic191—22no—
45C22×Dic191—22no—
46C22×Dic191D414yescomplemented
47C22×Dic191C2311yescharacteristic
48C22×Dic191C2311yescharacteristic
49C22×Dic191C2311yescharacteristic
50C22×Dic191C2312yes—
51C22×Dic191C2311yescharacteristic
52C2×C4×D191—24no—
53C2×C4×D191—24no—
54C2×C4×D191—24no—
55Dic191⋊C4—22no—
56Dic191⋊C4—24no—
57Dic191⋊C4—24no—
58Dic191⋊C4—22no—
59C22⋊Dic191—22no—
60(C2×D191)⋊C4—24no—
61(C2×D191)⋊C4—24no—
62(C2×D191)⋊C4—24no—
63(C2×D191)⋊C4—24no—
64C4×Dic191—22no—
65C4×Dic191—24no—
66C4×Dic191D414yescomplemented
67C24×C191C2311yescharacteristic
68C22×C4×C191C2312yes—
69C22×C4×C191C2312yes—
70C22×C4×C191C2311yescharacteristic
71C22×C4×C191C2311yescharacteristic
72C4⋊C4×C191—22no—
73C22⋊C4×C191—24no—
74C22⋊C4×C191—22no—
75C22×D191—22no—
76C22×D191—22no—
77C22×D191—22no—
78C22×D191—22no—
79C22×D191—22no—
80C22×D191—22no—
81C22×D191#4114yes—
82D764—28no—
83C22⋊D191—216no—
84C2×Dic191—22no—
85C2×Dic191—21no—
86C2×Dic191—22no—
87C2×Dic191—21no—
88C2×Dic191—22no—
89C2×Dic191—28no—
90C2×Dic191—21no—
91C2×Dic191—22no—
92C2×Dic191C2×D412yes—
93C2×Dic191C2×D412yes—
94C2×Dic191C2×D412yes—
95C2×Dic191C2×D412yes—
96C2×Dic191#4114yes—
97C2×Dic191#4112yes—
98C4×D191—48no—
99C23×C191—21no—
100C23×C191—21no—
101C23×C191—22no—
102C23×C191—22no—
103C23×C191C2×D412yes—
104C23×C191C2411yescharacteristic
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×C191—22no—
113C2×C4×C191—22no—
114C2×C4×C191C2×D412yes—
115C22×C191—22no—
116C22×C191—24no—
117C22×C191—41no—
118C22×C191—42no—
119C22×C191—41no—
120C22×C191C22×D411yescharacteristic
121C22×C191C22×D411yescharacteristic
122C22×C191C2×#4111yescharacteristic
123C22×C191C2×#4112yes—
124C22×C191C2×#4112yes—
125C2×D191—24no—
126C2×D191—24no—
127C2×D191—24no—
128C2×D191—42no—
129C2×D191—42no—
130C2×D191—42no—
131C2×D191—42no—
132C4×C191—22no—
133C4×C191—42no—
134C4×C191—42no—
135C4×C191—41no—
136Dic191—22no—
137Dic191—24no—
138Dic191—22no—
139Dic191—41no—
140Dic191#12114yes—
141C2×C191—42no—
142C2×C191C2×#12111yescharacteristic
143C2×C191#52512yes—
144C2×C191#52511yescharacteristic
145C2×C191#52912yes—
146C2×C191#53211yescharacteristic
147D191—44no—
148C191#239211yescharacteristic, complemented, Sylow
149#2392—1911nomaximal, Sylow
150C2×C4×D4—1911no—
151C2×C4×D4—1912no—
152C2×#117—1911no—
153#398—1911no—
154#397D19111yescharacteristic, complemented
155#396—1912no—
156#394—1912no—
157#394—1911no—
158#392—1911no—
159C4×C4⋊C4—1911no—
160C4×C22⋊C4—1912no—
161C22×D4—1912no—
162C22×D4—1911no—
163C23×C4—1912no—
164C23×C4—1911no—
165C4×D4—3828no—
166C4×D4—3824no—
167#117—3824no—
168C2×C4⋊C4—1911no—
169C2×C4⋊C4—1911no—
170C2×C4⋊C4—1912no—
171C2×C4⋊C4—1912no—
172C2×C4⋊C4C2×D19111yescharacteristic
173C2×C22⋊C4—1911no—
174C2×C22⋊C4—1912no—
175C2×C22⋊C4—1912no—
176C2×C22⋊C4—1912no—
177C2×C22⋊C4—1912no—
178C2×C22⋊C4C2×D19111yescharacteristic
179C2×C22⋊C4C2×D19112yes—
180C2×C42—1911no—
181C2×C42—1912no—
182C2×C42—1911no—
183#95—1912no—
184#95—1911no—
185#95C2×D19112yes—
186#95C2×D19111yescharacteristic
187C24—1912no—
188C24C22×D19111yescharacteristic
189C2×D4—3824no—
190C2×D4—3824no—
191C2×D4—3824no—
192C2×D4—3824no—
193C2×D4—3822no—
194C22×C4—3824no—
195C22×C4—3824no—
196C22×C4—3824no—
197C22×C4—3822no—
198C22×C4—3822no—
199C22×C4—1911no—
200C22×C4—1914no—
201C22×C4—1912no—
202C22×C4—1911no—
203C22×C4—1911no—
204C22×C4—1911no—
205C22×C4C22×D19112yes—
206C22×C4C22×D19112yes—
207C22×C4C22×D19111yescharacteristic
208C22×C4C22×D19111yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—3824no—
211C4⋊C4—3822no—
212C4⋊C4—3824no—
213C4⋊C4—3822no—
214C22⋊C4—24no—
215C22⋊C4—22no—
216C22⋊C4—3824no—
217C22⋊C4—3824no—
218C22⋊C4—3824no—
219C22⋊C4—3824no—
220C22⋊C4—3822no—
221C42—3822no—
222C42—3824no—
223C42—1914no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—3822no—
229C23—3822no—
230C23—3822no—
231C23—3822no—
232C23—3822no—
233C23—3822no—
234C23—1914no—
235C23D4×D19112yes—
236C23C23×D19111yescharacteristic
237D4—3828no—
238D4—38216no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—3822no—
249C2×C4—3821no—
250C2×C4—3822no—
251C2×C4—3821no—
252C2×C4—3822no—
253C2×C4—3828no—
254C2×C4—3821no—
255C2×C4—3822no—
256C2×C4—1912no—
257C2×C4—1912no—
258C2×C4—1914no—
259C2×C4—1912no—
260C2×C4—1912no—
261C2×C4—1912no—
262C2×C4—7648no—
263C2×C4D4×D19112yes—
264C22—22no—
265C22—24no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—3824no—
270C22—3824no—
271C22—3824no—
272C22—7642no—
273C22—7642no—
274C22—7642no—
275C22—7642no—
276C22C2×D4×D19111yescharacteristic
277C22C2×D4×D19111yescharacteristic
278C22#41×D19111yescharacteristic
279C22#41×D19112yes—
280C22#41×D19112yes—
281C4—22no—
282C4—41no—
283C4—42no—
284C4—42no—
285C4—3822no—
286C4—3822no—
287C4—3824no—
288C4—1914no—
289C4—7641no—
290C2—42no—
291C2—7644no—
292C2#121×D19111yescharacteristic
293C2#2713317111yescharacteristic
294C2#2713317112yes—
295C2#2713319612yes—
296C2#2713321211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#397×C191C2
prime radical#27132572C2
hypercenter#397D191
hypocenterC191#2392
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#20259284C2semidirect
2. C2×(C22⋊D191)⋊C4C2semidirect
3. C2×#20259278C2semidirect
4. #27132594C2semidirect
5. #27132587C2semidirect
6. #27132581C2semidirect
7. #27132578C2semidirect
9. #27132514C2semidirect
10. #27132510C2semidirect
11. #397×C191C2semidirect
46. C22×Dic191D4semidirect
66. C4×Dic191D4semidirect
148. C191#2392semidirect
154. #397D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592842
C2×(C22⋊D191)⋊C42
C2×#202592782
#271325942
#271325872
#271325812
#271325782
#271325722
#271325142
#271325102
#397×C1912
#2392191