Finite Group Databasebeta

← #37575710

Subgroups — #37575710

Order 24448.

456 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259317C211yesmaximal, characteristic
2#27132562C211yesmaximal, characteristic, complemented
3#27132561C211yesmaximal, characteristic, complemented
4#27132560C211yesmaximal, characteristic, complemented
5#27132555C211yesmaximal, characteristic
6#27132554C211yesmaximal, characteristic
7#27132552C211yesmaximal, characteristic
8#27132546C211yesmaximal, characteristic, complemented
9#27132545C211yesmaximal, characteristic, complemented
10#27132537C211yesmaximal, characteristic
11#27132537C211yesmaximal, characteristic
12#27132534C211yesmaximal, characteristic
13#27132532C211yesmaximal, characteristic, complemented
14#27132530C211yesmaximal, characteristic, complemented
15#387×C191C211yesmaximal, characteristic, complemented
16C22×C4×D191C2211yescharacteristic
17#20259317C418yescomplemented
18C2×C4⋊Dic191C2211yescharacteristic
19C2×C4⋊Dic191C2211yescharacteristic
20C2×C4⋊Dic191C2211yescharacteristic
21C2×C4⋊Dic191C2211yescharacteristic
22C2×Dic191⋊C4C2211yescharacteristic
23C2×Dic191⋊C4C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×Dic191⋊C4C2211yescharacteristic
27C2×Dic191⋊C4C2211yescharacteristic
28C2×(C2×D191)⋊C4C2211yescharacteristic
29C2×(C2×D191)⋊C4C2211yescharacteristic
30C2×(C2×D191)⋊C4C2211yescharacteristic
31C2×(C2×D191)⋊C4C2211yescharacteristic
32C2×(C2×D191)⋊C4C2211yescharacteristic
33C2×(C2×D191)⋊C4C2211yescharacteristic
34C2×C4×Dic191C2211yescharacteristic
35C2×C4×Dic191C2211yescharacteristic
36C2×C4×Dic191C2211yescharacteristic
37C191⋊#95C2211yescharacteristic
38C191⋊#95C2211yescharacteristic
39C191⋊#95C2211yescharacteristic
40C191⋊#95C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C2×C4⋊C4×C191C2211yescharacteristic
46C2×C4⋊C4×C191C2211yescharacteristic
47C2×C42×C191C2211yescharacteristic
48#95×C191C2211yescharacteristic
49#95×C191C2211yescharacteristic
50#95×C191C2211yescharacteristic
51#95×C191C2211yescharacteristic
52C23×D191C2311yescharacteristic
53C22×Dic191C2311yescharacteristic
54C22×Dic191C2311yescharacteristic
55C22×Dic191C2311yescharacteristic
56C22×Dic191C2311yescharacteristic
57C22×Dic191C2311yescharacteristic
58C22×Dic191C2311yescharacteristic
59C22×Dic191C2311yescharacteristic
60C2×C4×D191—41no—
61C2×C4×D191—41no—
62C2×C4×D191—41no—
63C4⋊Dic191—41no—
64C4⋊Dic191—41no—
65C4⋊Dic191—41no—
66C4⋊Dic191C2×C414yescomplemented
67Dic191⋊C4—41no—
68Dic191⋊C4—41no—
69Dic191⋊C4—41no—
70Dic191⋊C4—41no—
71Dic191⋊C4—41no—
72Dic191⋊C4C2×C414yescomplemented
73(C2×D191)⋊C4—41no—
74(C2×D191)⋊C4—41no—
75(C2×D191)⋊C4—41no—
76(C2×D191)⋊C4C2×C414yes—
77(C2×D191)⋊C4C2×C414yes—
78(C2×D191)⋊C4C2×C414yes—
79C4×Dic191—41no—
80C4×Dic191—41no—
81C4×Dic191C2×C414yescomplemented
82C22×C4×C191C2311yescharacteristic
83C22×C4×C191C2311yescharacteristic
84C22×C4×C191C2311yescharacteristic
85C22×C4×C191C2311yescharacteristic
86C22×C4×C191C2311yescharacteristic
87C22×C4×C191C2311yescharacteristic
88C22×C4×C191C2311yescharacteristic
89C4⋊C4×C191—41no—
90C4⋊C4×C191C2×C414yescomplemented
91C42×C191—41no—
92C22×D191—21no—
93C22×D191—21no—
94C22×D191—21no—
95C22×D191—21no—
96C22×D191—21no—
97C22×D191—21no—
98C22×D191C22×C412yes—
99C2×Dic191—21no—
100C2×Dic191—21no—
101C2×Dic191—21no—
102C2×Dic191—21no—
103C2×Dic191—21no—
104C2×Dic191—21no—
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×Dic191C22×C412yes—
118C2×Dic191C22×C412yes—
119C2×Dic191C22×C412yes—
120C4×D191—44no—
121C23×C191C2411yescharacteristic
122C2×C4×C191—21no—
123C2×C4×C191—21no—
124C2×C4×C191—21no—
125C2×C4×C191—21no—
126C2×C4×C191—21no—
127C2×C4×C191—21no—
128C2×C4×C191—21no—
129C2×C4×C191—21no—
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×C191C22×C412yes—
141C2×C4×C191C22×C412yes—
142C2×C4×C191C22×C412yes—
143C22×C191C23×C411yescharacteristic
144C22×C191H(5,𝔽2)11yescharacteristic
145C22×C191H(5,𝔽2)11yescharacteristic
146C22×C191H(5,𝔽2)11yescharacteristic
147C22×C1912−1+411yescharacteristic
148C22×C1912−1+411yescharacteristic
149C22×C1912−1+411yescharacteristic
150C2×D191—22no—
151C2×D191—22no—
152C2×D191—22no—
153C2×D191—41no—
154C2×D191—41no—
155C2×D191—41no—
156C2×D191—41no—
157C4×C191—22no—
158C4×C191—22no—
159C4×C191—22no—
160C4×C191—41no—
161C4×C191—41no—
162C4×C191—41no—
163C4×C191—41no—
164Dic191—22no—
165Dic191—22no—
166Dic191—22no—
167Dic191—41no—
168Dic191—41no—
169Dic191—41no—
170Dic191—41no—
171C2×C191#52011yescharacteristic
172C2×C191#52011yescharacteristic
173C2×C191#52011yescharacteristic
174C2×C191#56011yescharacteristic
175C2×C191#56311yescharacteristic
176C2×C191#56311yescharacteristic
177C2×C191#56311yescharacteristic
178D191—42no—
179C191#238311yescharacteristic, complemented, Sylow
180#2383—1911nomaximal, Sylow
181C2×#126—1911no—
182#389—1911no—
183#389—1911no—
184#389—1911no—
185#388—1911no—
186#388—1911no—
187#388—1911no—
188#387—1911no—
189#387—1911no—
190#387D19111yescharacteristic, complemented
191#385—1911no—
192#385—1911no—
193#385—1911no—
194#384—1911no—
195#383—1911no—
196C23×C4—1911no—
197#126—1918no—
198C2×C4⋊C4—1911no—
199C2×C4⋊C4—1911no—
200C2×C4⋊C4—1911no—
201C2×C4⋊C4—1911no—
202C2×C4⋊C4—1911no—
203C2×C4⋊C4—1911no—
204C2×C4⋊C4—1911no—
205C2×C4⋊C4—1911no—
206C2×C4⋊C4—1911no—
207C2×C4⋊C4—1911no—
208C2×C4⋊C4C2×D19111yescharacteristic
209C2×C4⋊C4C2×D19111yescharacteristic
210C2×C22⋊C4—1911no—
211C2×C22⋊C4—1911no—
212C2×C22⋊C4—1911no—
213C2×C22⋊C4—1911no—
214C2×C22⋊C4—1911no—
215C2×C22⋊C4—1911no—
216C2×C42—1911no—
217C2×C42—1911no—
218C2×C42—1911no—
219C2×C42C2×D19111yescharacteristic
220#95—1911no—
221#95—1911no—
222#95—1911no—
223#95—1911no—
224#95—1911no—
225#95—1911no—
226#95—1911no—
227#95—1911no—
228#95C2×D19111yescharacteristic
229#95C2×D19111yescharacteristic
230#95C2×D19111yescharacteristic
231#95C2×D19111yescharacteristic
232C24—1911no—
233C22×C4—1911no—
234C22×C4—1911no—
235C22×C4—1911no—
236C22×C4—1911no—
237C22×C4—1911no—
238C22×C4—1911no—
239C22×C4—1911no—
240C22×C4—7641no—
241C22×C4—7641no—
242C22×C4—7641no—
243C22×C4C22×D19111yescharacteristic
244C22×C4C22×D19111yescharacteristic
245C22×C4C22×D19111yescharacteristic
246C22×C4C22×D19111yescharacteristic
247C22×C4C22×D19111yescharacteristic
248C22×C4C22×D19111yescharacteristic
249C22×C4C22×D19111yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—1914no—
252C4⋊C4—1914no—
253C4⋊C4—7641no—
254C4⋊C4—7641no—
255C4⋊C4—7641no—
256C4⋊C4—7641no—
257C4⋊C4—7641no—
258C4⋊C4—7641no—
259C4⋊C4—7641no—
260C4⋊C4—7641no—
261C4⋊C4C4×D19114yescomplemented
262C22⋊C4—1914no—
263C22⋊C4—1914no—
264C22⋊C4—1914no—
265C22⋊C4—7641no—
266C22⋊C4—7641no—
267C22⋊C4—7641no—
268C42—41no—
269C42—1914no—
270C42—7641no—
271C42—7641no—
272C23—3821no—
273C23—3821no—
274C23—3821no—
275C23—3821no—
276C23—3821no—
277C23—3821no—
278C23—1912no—
279C23C23×D19111yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—3821no—
299C2×C4—3821no—
300C2×C4—3821no—
301C2×C4—3821no—
302C2×C4—3821no—
303C2×C4—3821no—
304C2×C4—3821no—
305C2×C4—3821no—
306C2×C4—3821no—
307C2×C4—3821no—
308C2×C4—3821no—
309C2×C4—3821no—
310C2×C4—3821no—
311C2×C4—3821no—
312C2×C4—3821no—
313C2×C4—3821no—
314C2×C4—3821no—
315C2×C4—3821no—
316C2×C4—1912no—
317C2×C4—1912no—
318C2×C4—1912no—
319C2×C4—7644no—
320C2×C4C2×C4×D19112yes—
321C2×C4C2×C4×D19112yes—
322C2×C4C2×C4×D19112yes—
323C22—3822no—
324C22—3822no—
325C22—3822no—
326C22—7641no—
327C22—7641no—
328C22—7641no—
329C22—7641no—
330C22C22×C4×D19111yescharacteristic
331C22#2025936711yescharacteristic
332C22#2025936811yescharacteristic
333C22#2025936811yescharacteristic
334C22#2025936911yescharacteristic
335C22#2025936911yescharacteristic
336C22#2025937011yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—3822no—
345C4—3822no—
346C4—3822no—
347C4—7641no—
348C4—7641no—
349C4—7641no—
350C4—7641no—
351C2—7642no—
352C2#2713314211yescharacteristic
353C2#2713314311yescharacteristic
354C2#2713314511yescharacteristic
355C2#2713340311yescharacteristic
356C2#2713341411yescharacteristic
357C2#2713341511yescharacteristic
358C2#2713341711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#387×C191C2
socle, cosocleC23×C191C24
prime radicalC23×D191C23
hypercenter#387D191
hypocenterC191#2383
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #27132562C2semidirect
3. #27132561C2semidirect
4. #27132560C2semidirect
8. #27132546C2semidirect
9. #27132545C2semidirect
13. #27132532C2semidirect
14. #27132530C2semidirect
15. #387×C191C2semidirect
17. #20259317C4semidirect
66. C4⋊Dic191C2×C4semidirect
72. Dic191⋊C4C2×C4semidirect
81. C4×Dic191C2×C4semidirect
90. C4⋊C4×C191C2×C4semidirect
179. C191#2383semidirect
190. #387D191semidirect
261. C4⋊C4C4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593172
#271325622
#271325612
#271325602
#271325552
#271325542
#271325522
#271325462
#271325452
#271325372
#271325342
#271325322
#271325302
#387×C1912
#2383191