Finite Group Databasebeta

← #37576203

Subgroups — #37576203

Order 24448.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259300C211yesmaximal, characteristic
2C2×#20259279C211yesmaximal, characteristic
3#27132601C211yesmaximal, characteristic, complemented
4#27132594C211yesmaximal, characteristic, complemented
5#27132593C211yesmaximal, characteristic, complemented
6#27132583C211yesmaximal, characteristic, complemented
7#27132567C211yesmaximal, characteristic, complemented
8#27132547C211yesmaximal, characteristic, complemented
9#27132546C211yesmaximal, characteristic, complemented
10#27132534C211yesmaximal, characteristic
11#27132534C211yesmaximal, characteristic
12#27132525C211yesmaximal, characteristic
13#27132520C211yesmaximal, characteristic
14#95×D191C211yesmaximal, characteristic
15#389×C191C211yesmaximal, characteristic, complemented
16C22×Dic382C2211yescharacteristic
17C22×C4×D191C2211yescharacteristic
18C22×C4×D191C2211yescharacteristic
19C22×C4×D191C2211yescharacteristic
20#20259300—24no—
21#20259279—24no—
22C2×C4⋊Dic191C2211yescharacteristic
23C2×Dic191⋊C4C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×Dic191⋊C4C2211yescharacteristic
27C2×Dic191⋊C4C2211yescharacteristic
28C2×Dic191⋊C4C2211yescharacteristic
29C2×(C2×D191)⋊C4C2211yescharacteristic
30C2×(C2×D191)⋊C4C2211yescharacteristic
31C2×(C2×D191)⋊C4C2211yescharacteristic
32C2×(C2×D191)⋊C4C2211yescharacteristic
33C2×C4×Dic191C2211yescharacteristic
34C191⋊#95C2211yescharacteristic
35C191⋊#95C2211yescharacteristic
36C191⋊#95C2211yescharacteristic
37C191⋊#95C2211yescharacteristic
38C191⋊#95C2211yescharacteristic
39C191⋊#95C2211yescharacteristic
40C191⋊#95C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C191⋊#95C2211yescharacteristic
46C2×C4⋊C4×C191C2211yescharacteristic
47C2×C4⋊C4×C191C2211yescharacteristic
48C2×C4⋊C4×C191C2211yescharacteristic
49C2×C4⋊C4×C191C2211yescharacteristic
50C2×C42×C191C2211yescharacteristic
51#95×C191C2211yescharacteristic
52#95×C191C2211yescharacteristic
53C23×D191C2311yescharacteristic
54C2×Dic382—22no—
55C2×Dic382—22no—
56C2×Dic382—41no—
57C22×Dic191C2311yescharacteristic
58C22×Dic191C2311yescharacteristic
59C22×Dic191C2311yescharacteristic
60C22×Dic191C2311yescharacteristic
61C22×Dic191C2311yescharacteristic
62C22×Dic191C2311yescharacteristic
63C22×Dic191C2311yescharacteristic
64C2×C4×D191—22no—
65C2×C4×D191—22no—
66C2×C4×D191—22no—
67C2×C4×D191—22no—
68C2×C4×D191—22no—
69C2×C4×D191—22no—
70C2×C4×D191—41no—
71C2×C4×D191—41no—
72C2×C4×D191Q814yescomplemented
73C4⋊Dic191—22no—
74Dic191⋊C4—22no—
75Dic191⋊C4—22no—
76Dic191⋊C4—22no—
77Dic191⋊C4—22no—
78Dic191⋊C4—22no—
79Dic191⋊C4—41no—
80(C2×D191)⋊C4—22no—
81(C2×D191)⋊C4—22no—
82(C2×D191)⋊C4—22no—
83(C2×D191)⋊C4D414yes—
84C4×Dic191—22no—
85C22×C4×C191C2311yescharacteristic
86C22×C4×C191C2311yescharacteristic
87C22×C4×C191C2311yescharacteristic
88C22×C4×C191C2311yescharacteristic
89C22×C4×C191C2311yescharacteristic
90C22×C4×C191C2311yescharacteristic
91C22×C4×C191C2311yescharacteristic
92C4⋊C4×C191—22no—
93C4⋊C4×C191—22no—
94C4⋊C4×C191—41no—
95C4⋊C4×C191—41no—
96C42×C191—22no—
97C22×D191—21no—
98C22×D191—21no—
99C22×D191—21no—
100C22×D191—21no—
101C22×D191C2×D412yes—
102C22×D191C2×Q812yes—
103C22×D191#4112yes—
104Dic382—44no—
105C2×Dic191—21no—
106C2×Dic191—21no—
107C2×Dic191—21no—
108C2×Dic191—21no—
109C2×Dic191—21no—
110C2×Dic191—21no—
111C2×Dic191—21no—
112C2×Dic191—21no—
113C2×Dic191—21no—
114C2×Dic191—21no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—21no—
118C2×Dic191—21no—
119C2×Dic191—21no—
120C2×Dic191—21no—
121C2×Dic191—21no—
122C2×Dic191—21no—
123C2×Dic191C2×D412yes—
124C2×Dic191C2×Q812yes—
125C2×Dic191#4112yes—
126C4×D191—28no—
127C4×D191—44no—
128C4×D191—44no—
129C23×C191C2411yescharacteristic
130C2×C4×C191—21no—
131C2×C4×C191—21no—
132C2×C4×C191—21no—
133C2×C4×C191—21no—
134C2×C4×C191—21no—
135C2×C4×C191—21no—
136C2×C4×C191—21no—
137C2×C4×C191—21no—
138C2×C4×C191—21no—
139C2×C4×C191—21no—
140C2×C4×C191—21no—
141C2×C4×C191—21no—
142C2×C4×C191—21no—
143C2×C4×C191—21no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191—21no—
147C2×C4×C191—21no—
148C2×C4×C191C2×D412yes—
149C2×C4×C191C2×Q812yes—
150C2×C4×C191#4112yes—
151C22×C191C22×D411yescharacteristic
152C22×C191C22×Q811yescharacteristic
153C22×C191C2×#4111yescharacteristic
154C22×C191C2×#4111yescharacteristic
155C22×C191C2×#4111yescharacteristic
156C22×C191H(5,𝔽2)11yescharacteristic
157C22×C1912−1+411yescharacteristic
158C2×D191—22no—
159C2×D191—22no—
160C2×D191—22no—
161C2×D191—22no—
162C2×D191—22no—
163C2×D191—22no—
164C2×D191C22⋊Q814yes—
165C4×C191—22no—
166C4×C191—22no—
167C4×C191—22no—
168C4×C191—41no—
169C4×C191—41no—
170C4×C191—41no—
171C4×C191—41no—
172Dic191—22no—
173Dic191—22no—
174Dic191—22no—
175Dic191—41no—
176Dic191—41no—
177Dic191—41no—
178Dic191—41no—
179C2×C191C2×C22⋊Q811yescharacteristic
180C2×C191#53311yescharacteristic
181C2×C191#54411yescharacteristic
182C2×C191#54611yescharacteristic
183C2×C191#54711yescharacteristic
184C2×C191#55111yescharacteristic
185C2×C191#55211yescharacteristic
186D191—24no—
187C191#243811yescharacteristic, complemented, Sylow
188#2438—1911nomaximal, Sylow
189C2×C22⋊Q8—1911no—
190C2×#117—1911no—
191#400—1911no—
192#398—1911no—
193#398—1911no—
194#395—1911no—
195#391—1911no—
196#389D19111yescharacteristic, complemented
197#387—1911no—
198#387—1911no—
199#385—1911no—
200#385—1911no—
201#381—1911no—
202#380—1911no—
203C2×#95—1911no—
204C22×Q8—1911no—
205C23×C4—1911no—
206C23×C4—1911no—
207C23×C4—1911no—
208C22⋊Q8—3824no—
209#117—3824no—
210C2×C4⋊C4—1911no—
211C2×C4⋊C4—1911no—
212C2×C4⋊C4—1911no—
213C2×C4⋊C4—1911no—
214C2×C4⋊C4—1911no—
215C2×C4⋊C4—1911no—
216C2×C4⋊C4—1911no—
217C2×C4⋊C4C2×D19111yescharacteristic
218C2×C4⋊C4C2×D19111yescharacteristic
219C2×C4⋊C4C2×D19111yescharacteristic
220C2×C4⋊C4C2×D19111yescharacteristic
221C2×C22⋊C4—1911no—
222C2×C22⋊C4—1911no—
223C2×C22⋊C4—1911no—
224C2×C22⋊C4—1911no—
225C2×C42—1911no—
226C2×C42C2×D19111yescharacteristic
227#95—1911no—
228#95—1911no—
229#95—1911no—
230#95—1911no—
231#95—1911no—
232#95—1911no—
233#95—1911no—
234#95—1911no—
235#95—1911no—
236#95—1911no—
237#95—1911no—
238#95—1911no—
239#95C2×D19111yescharacteristic
240#95C2×D19111yescharacteristic
241C24—1911no—
242C2×Q8—3822no—
243C2×Q8—3822no—
244C2×Q8—7641no—
245C22×C4—3822no—
246C22×C4—3822no—
247C22×C4—3822no—
248C22×C4—3822no—
249C22×C4—3822no—
250C22×C4—3822no—
251C22×C4—1911no—
252C22×C4—1911no—
253C22×C4—1911no—
254C22×C4—1911no—
255C22×C4—1911no—
256C22×C4—1911no—
257C22×C4—1911no—
258C22×C4—1914no—
259C22×C4—7641no—
260C22×C4—7641no—
261C22×C4C22×D19111yescharacteristic
262C22×C4C22×D19111yescharacteristic
263C22×C4C22×D19111yescharacteristic
264C22×C4C22×D19111yescharacteristic
265C22×C4C22×D19111yescharacteristic
266C22×C4C22×D19111yescharacteristic
267C22×C4C22×D19111yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—3822no—
273C4⋊C4—3822no—
274C4⋊C4—3822no—
275C4⋊C4—3822no—
276C4⋊C4—3822no—
277C4⋊C4—3822no—
278C4⋊C4—7641no—
279C22⋊C4—3822no—
280C22⋊C4—3822no—
281C22⋊C4—3822no—
282C22⋊C4—1914no—
283C42—22no—
284C42—3822no—
285C23—3821no—
286C23—3821no—
287C23—3821no—
288C23—3821no—
289C23—1912no—
290C23—1912no—
291C23—1912no—
292C23C23×D19111yescharacteristic
293Q8—7644no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—3828no—
313C2×C4—3821no—
314C2×C4—3821no—
315C2×C4—3821no—
316C2×C4—3821no—
317C2×C4—3821no—
318C2×C4—3821no—
319C2×C4—3821no—
320C2×C4—3821no—
321C2×C4—3821no—
322C2×C4—3821no—
323C2×C4—3821no—
324C2×C4—3821no—
325C2×C4—3821no—
326C2×C4—3821no—
327C2×C4—3821no—
328C2×C4—3821no—
329C2×C4—3821no—
330C2×C4—3821no—
331C2×C4—1912no—
332C2×C4—1912no—
333C2×C4—1912no—
334C2×C4—7644no—
335C2×C4—7644no—
336C2×C4D4×D19112yes—
337C2×C4Q8×D19112yes—
338C2×C4#1569028712yes—
339C22—3822no—
340C22—3822no—
341C22—3822no—
342C22—3822no—
343C22—3822no—
344C22—3822no—
345C22—1914no—
346C22C2×D4×D19111yescharacteristic
347C22C2×Q8×D19111yescharacteristic
348C22#41×D19111yescharacteristic
349C22#41×D19111yescharacteristic
350C22C2×#1569028711yescharacteristic
351C22#2025936711yescharacteristic
352C22#2025937011yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3822no—
361C4—3822no—
362C4—3822no—
363C4—7641no—
364C4—7641no—
365C4—7641no—
366C4—7641no—
367C2—3824no—
368C2C22⋊Q8×D19111yescharacteristic
369C2#2713321711yescharacteristic
370C2#2713329011yescharacteristic
371C2#2713331011yescharacteristic
372C2#2713331411yescharacteristic
373C2#2713334211yescharacteristic
374C2#2713334511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#389×C191C2
prime radicalC23×D191C23
hypercenter#389D191
hypocenterC191#2438
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #27132601C2semidirect
4. #27132594C2semidirect
5. #27132593C2semidirect
6. #27132583C2semidirect
7. #27132567C2semidirect
8. #27132547C2semidirect
9. #27132546C2semidirect
15. #389×C191C2semidirect
72. C2×C4×D191Q8semidirect
187. C191#2438semidirect
196. #389D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593002
C2×#202592792
#271326012
#271325942
#271325932
#271325832
#271325672
#271325472
#271325462
#271325342
#271325252
#271325202
#95×D1912
#389×C1912
#2438191