Finite Group Databasebeta

← #37575999

Subgroups — #37575999

Order 24448.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259306C211yesmaximal, characteristic
2C2×C4×C22⋊D191C211yesmaximal, characteristic
3#27132605C211yesmaximal, characteristic, complemented
4#27132596C211yesmaximal, characteristic, complemented
5#27132592C211yesmaximal, characteristic, complemented
6#27132589C211yesmaximal, characteristic, complemented
7#27132550C211yesmaximal, characteristic, complemented
8#27132549C211yesmaximal, characteristic, complemented
9#27132540C211yesmaximal, characteristic
10#27132534C211yesmaximal, characteristic, complemented
11#27132525C211yesmaximal, characteristic, complemented
12#27132524C211yesmaximal, characteristic, complemented
13#27132521C211yesmaximal, characteristic, complemented
14C2×C191⋊#95C211yesmaximal, characteristic, complemented
15#385×C191C211yesmaximal, characteristic, complemented
16C22×C22⋊D191C2211yescharacteristic
17C23×Dic191C2211yescharacteristic
18C23×Dic191C2211yescharacteristic
19C22×C4×D191C2211yescharacteristic
20#20259306—24no—
21C4×C22⋊D191—24no—
22C2×C4⋊Dic191C2211yescharacteristic
23C2×C4⋊Dic191C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×Dic191⋊C4C2211yescharacteristic
27C2×C22⋊Dic191C2211yescharacteristic
28C2×(C2×D191)⋊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
46C23×C4×C191C2211yescharacteristic
47C2×C4⋊C4×C191C2211yescharacteristic
48C2×C4⋊C4×C191C2211yescharacteristic
49C2×C22⋊C4×C191C2211yescharacteristic
50C2×C22⋊C4×C191C2211yescharacteristic
51#95×C191C2211yescharacteristic
52#95×C191C2211yescharacteristic
53C23×D191C2311yescharacteristic
54C2×C22⋊D191—22no—
55C2×C22⋊D191—22no—
56C2×C22⋊D191D414yes—
57C22×Dic191—22no—
58C22×Dic191—22no—
59C22×Dic191—22no—
60C22×Dic191—22no—
61C22×Dic191—22no—
62C22×Dic191—41no—
63C22×Dic191C2311yescharacteristic
64C22×Dic191C2311yescharacteristic
65C22×Dic191C2311yescharacteristic
66C22×Dic191C2311yescharacteristic
67C22×Dic191C2311yescharacteristic
68C22×Dic191C2311yescharacteristic
69C22×Dic191C2311yescharacteristic
70C2×C4×D191—22no—
71C2×C4×D191—22no—
72C2×C4×D191—41no—
73C4⋊Dic191—22no—
74C4⋊Dic191—22no—
75Dic191⋊C4—22no—
76Dic191⋊C4—22no—
77Dic191⋊C4—41no—
78C22⋊Dic191—22no—
79(C2×D191)⋊C4—22no—
80(C2×D191)⋊C4—22no—
81(C2×D191)⋊C4—22no—
82(C2×D191)⋊C4—22no—
83(C2×D191)⋊C4—41no—
84C4×Dic191—22no—
85C24×C191C2311yescharacteristic
86C22×C4×C191—22no—
87C22×C4×C191—22no—
88C22×C4×C191—41no—
89C22×C4×C191C2311yescharacteristic
90C22×C4×C191C2311yescharacteristic
91C22×C4×C191C2311yescharacteristic
92C22×C4×C191C2311yescharacteristic
93C22×C4×C191C2311yescharacteristic
94C22×C4×C191C2311yescharacteristic
95C4⋊C4×C191—22no—
96C4⋊C4×C191—41no—
97C22⋊C4×C191—22no—
98C22⋊C4×C191—22no—
99C22×D191—21no—
100C22×D191—21no—
101C22×D191—21no—
102C22×D191—21no—
103C22×D191—21no—
104C22×D191—21no—
105C22×D191C2×D412yes—
106C22⋊D191—28no—
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×Dic191—21no—
124C2×Dic191—21no—
125C2×Dic191—21no—
126C2×Dic191—44no—
127C2×Dic191—44no—
128C2×Dic191C2×D412yes—
129C2×Dic191#4112yes—
130C4×D191—44no—
131C23×C191—21no—
132C23×C191—21no—
133C23×C191—21no—
134C23×C191—21no—
135C23×C191C2×D412yes—
136C23×C191#4112yes—
137C23×C191#4112yes—
138C23×C191C2411yescharacteristic
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×C191—21no—
149C2×C4×C191—21no—
150C2×C4×C191—21no—
151C2×C4×C191—21no—
152C2×C4×C191—21no—
153C2×C4×C191—21no—
154C2×C4×C191—21no—
155C2×C4×C191—21no—
156C2×C4×C191—44no—
157C2×C4×C191#4112yes—
158C22×C191—22no—
159C22×C191—22no—
160C22×C191—22no—
161C22×C191—22no—
162C22×C191—22no—
163C22×C191—22no—
164C22×C191#12314yes—
165C22×C191C22×D411yescharacteristic
166C22×C191C2×#4111yescharacteristic
167C22×C191C2×#4111yescharacteristic
168C22×C191C2×#4111yescharacteristic
169C22×C191C2×#4111yescharacteristic
170C22×C191H(5,𝔽2)11yescharacteristic
171C22×C1912−1+411yescharacteristic
172C2×D191—22no—
173C2×D191—22no—
174C2×D191—22no—
175C2×D191—41no—
176C2×D191—41no—
177C2×D191—41no—
178C2×D191—41no—
179C4×C191—22no—
180C4×C191—41no—
181C4×C191—41no—
182C4×C191—41no—
183C4×C191—41no—
184C4×C191—41no—
185Dic191—22no—
186Dic191—22no—
187Dic191—41no—
188Dic191—41no—
189Dic191—41no—
190Dic191—41no—
191Dic191—41no—
192C2×C191—24no—
193C2×C191C2×#12311yescharacteristic
194C2×C191#52911yescharacteristic
195C2×C191#53911yescharacteristic
196C2×C191#54611yescharacteristic
197C2×C191#54711yescharacteristic
198C2×C191#55111yescharacteristic
199C2×C191#55211yescharacteristic
200D191—42no—
201C191#241911yescharacteristic, complemented, Sylow
202#2419—1911nomaximal, Sylow
203C2×#123—1911no—
204C2×C4×D4—1911no—
205#400—1911no—
206#398—1911no—
207#397—1911no—
208#397—1911no—
209#387—1911no—
210#387—1911no—
211#386—1911no—
212#385—1911no—
213#385D19111yescharacteristic, complemented
214#381—1911no—
215#381—1911no—
216#380—1911no—
217C2×#95—1911no—
218C22×D4—1911no—
219C23×C4—1911no—
220C23×C4—1911no—
221C23×C4—1911no—
222C23×C4C2×D19111yescharacteristic
223#123—3824no—
224C4×D4—3824no—
225C2×C4⋊C4—1911no—
226C2×C4⋊C4—1911no—
227C2×C4⋊C4—1911no—
228C2×C4⋊C4—1911no—
229C2×C4⋊C4—1911no—
230C2×C4⋊C4C2×D19111yescharacteristic
231C2×C4⋊C4C2×D19111yescharacteristic
232C2×C22⋊C4—1911no—
233C2×C22⋊C4—1911no—
234C2×C22⋊C4—1911no—
235C2×C22⋊C4—1911no—
236C2×C22⋊C4—1911no—
237C2×C22⋊C4—1911no—
238C2×C22⋊C4C2×D19111yescharacteristic
239C2×C22⋊C4C2×D19111yescharacteristic
240C2×C42—1911no—
241#95—1911no—
242#95—1911no—
243#95—1911no—
244#95—1911no—
245#95—1911no—
246#95—1911no—
247#95—1911no—
248#95—1911no—
249#95—1911no—
250#95—1911no—
251#95—1911no—
252#95—1911no—
253#95C2×D19111yescharacteristic
254#95C2×D19111yescharacteristic
255C24—1911no—
256C24C22×D19111yescharacteristic
257C2×D4—3822no—
258C2×D4—3822no—
259C2×D4—1914no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—3822no—
264C22×C4—3822no—
265C22×C4—3822no—
266C22×C4—3822no—
267C22×C4—3822no—
268C22×C4—3822no—
269C22×C4—3822no—
270C22×C4—1911no—
271C22×C4—1911no—
272C22×C4—1911no—
273C22×C4—1911no—
274C22×C4—1911no—
275C22×C4—1911no—
276C22×C4—1911no—
277C22×C4—7641no—
278C22×C4—7641no—
279C22×C4C22×D19111yescharacteristic
280C22×C4C22×D19111yescharacteristic
281C22×C4C22×D19111yescharacteristic
282C22×C4C22×D19111yescharacteristic
283C22×C4C22×D19111yescharacteristic
284C22×C4C22×D19111yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—41no—
287C4⋊C4—3822no—
288C4⋊C4—3822no—
289C4⋊C4—3822no—
290C4⋊C4—3822no—
291C4⋊C4—7641no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—3822no—
295C22⋊C4—3822no—
296C22⋊C4—3822no—
297C22⋊C4—3822no—
298C22⋊C4—3822no—
299C22⋊C4—7641no—
300C42—3822no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—3821no—
306C23—3821no—
307C23—3821no—
308C23—3821no—
309C23—3821no—
310C23—3821no—
311C23—1912no—
312C23D4×D19112yes—
313C23#1569028712yes—
314C23#1569028812yes—
315C23C23×D19111yescharacteristic
316D4—3828no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—44no—
335C2×C4—3821no—
336C2×C4—3821no—
337C2×C4—3821no—
338C2×C4—3821no—
339C2×C4—3821no—
340C2×C4—3821no—
341C2×C4—3821no—
342C2×C4—3821no—
343C2×C4—3821no—
344C2×C4—3821no—
345C2×C4—3821no—
346C2×C4—3821no—
347C2×C4—3821no—
348C2×C4—3821no—
349C2×C4—3821no—
350C2×C4—3821no—
351C2×C4—3821no—
352C2×C4—3821no—
353C2×C4—3821no—
354C2×C4—1912no—
355C2×C4—1912no—
356C2×C4—7644no—
357C2×C4—7644no—
358C2×C4—7644no—
359C2×C4#1569028612yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—3822no—
367C22—3822no—
368C22—3822no—
369C22—7641no—
370C22—7641no—
371C22—7641no—
372C22—7641no—
373C22#2025930714yes—
374C22C2×D4×D19111yescharacteristic
375C22#41×D19111yescharacteristic
376C22C2×#1569028611yescharacteristic
377C22C2×#1569028711yescharacteristic
378C22C2×#1569028811yescharacteristic
379C22#2025936711yescharacteristic
380C22#2025936911yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—3822no—
388C4—3822no—
389C4—7641no—
390C4—7641no—
391C4—7641no—
392C4—7641no—
393C4—7641no—
394C2—24no—
395C2—7642no—
396C2C2×#2025930711yescharacteristic
397C2#2713319311yescharacteristic
398C2#2713325511yescharacteristic
399C2#2713330011yescharacteristic
400C2#2713331111yescharacteristic
401C2#2713334011yescharacteristic
402C2#2713335511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#385×C191C2
prime radicalC22×C22⋊D191C22
hypercenter#385D191
hypocenterC191#2419
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #27132605C2semidirect
4. #27132596C2semidirect
5. #27132592C2semidirect
6. #27132589C2semidirect
7. #27132550C2semidirect
8. #27132549C2semidirect
10. #27132534C2semidirect
11. #27132525C2semidirect
12. #27132524C2semidirect
13. #27132521C2semidirect
14. C2×C191⋊#95C2semidirect
15. #385×C191C2semidirect
201. C191#2419semidirect
213. #385D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593062
C2×C4×C22⋊D1912
#271326052
#271325962
#271325922
#271325892
#271325502
#271325492
#271325402
#271325342
#271325252
#271325242
#271325212
C2×C191⋊#952
#385×C1912
#2419191