Finite Group Databasebeta

← #37575161

Subgroups — #37575161

Order 24448.

540 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D191)⋊C4C211yesmaximal, characteristic
2#27132563C211yesmaximal, characteristic
3#27132563C211yesmaximal, characteristic
4#27132560C211yesmaximal, characteristic, complemented
5#27132558C211yesmaximal, characteristic, complemented
6#27132556C211yesmaximal, characteristic, complemented
7#27132554C211yesmaximal, characteristic, complemented
8#27132552C211yesmaximal, characteristic, complemented
9#27132525C211yesmaximal, characteristic, complemented
10#27132524C211yesmaximal, characteristic, complemented
11#27132521C211yesmaximal, characteristic, complemented
12#27132520C211yesmaximal, characteristic, complemented
13C4×Dic191⋊C4C211yesmaximal, characteristic, complemented
14#27132508C211yesmaximal, characteristic, complemented
15#388×C191C211yesmaximal, characteristic, complemented
16C22×C22⋊D191C2211yescharacteristic
17C23×Dic191C2211yescharacteristic
18C22×C4×D191C2211yescharacteristic
19(C22⋊D191)⋊C4C418yescomplemented
20C2×C4⋊Dic191C2211yescharacteristic
21C2×Dic191⋊C4C2211yescharacteristic
22C2×Dic191⋊C4C2211yescharacteristic
23C2×Dic191⋊C4C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×C22⋊Dic191C2211yescharacteristic
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
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×C22⋊C4×C191C2211yescharacteristic
47C2×C22⋊C4×C191C2211yescharacteristic
48C2×C22⋊C4×C191C2211yescharacteristic
49C2×C42×C191C2211yescharacteristic
50#95×C191C2211yescharacteristic
51#95×C191C2211yescharacteristic
52C23×D191C2311yescharacteristic
53C2×C22⋊D191—22no—
54C2×C22⋊D191—22no—
55C2×C22⋊D191C2×C414yes—
56C22×Dic191—41no—
57C22×Dic191—41no—
58C22×Dic191C2×C414yes—
59C22×Dic191C2311yescharacteristic
60C22×Dic191C2311yescharacteristic
61C22×Dic191C2311yescharacteristic
62C22×Dic191C2311yescharacteristic
63C22×Dic191C2311yescharacteristic
64C22×Dic191C2311yescharacteristic
65C22×Dic191C2311yescharacteristic
66C2×C4×D191—41no—
67C2×C4×D191—41no—
68C2×C4×D191C2×C414yes—
69C4⋊Dic191—41no—
70Dic191⋊C4—22no—
71Dic191⋊C4—22no—
72Dic191⋊C4—41no—
73Dic191⋊C4—41no—
74Dic191⋊C4C2×C414yes—
75C22⋊Dic191—41no—
76C22⋊Dic191—41no—
77(C2×D191)⋊C4—41no—
78(C2×D191)⋊C4—41no—
79(C2×D191)⋊C4—41no—
80(C2×D191)⋊C4—41no—
81(C2×D191)⋊C4C2×C414yes—
82C4×Dic191—22no—
83C4×Dic191—22no—
84C4×Dic191—41no—
85C4×Dic191C2×C414yes—
86C24×C191C2311yescharacteristic
87C22×C4×C191C2311yescharacteristic
88C22×C4×C191C2311yescharacteristic
89C22×C4×C191C2311yescharacteristic
90C22×C4×C191C2311yescharacteristic
91C22×C4×C191C2311yescharacteristic
92C22×C4×C191C2311yescharacteristic
93C4⋊C4×C191—41no—
94C22⋊C4×C191—41no—
95C22⋊C4×C191—41no—
96C22⋊C4×C191C2×C414yes—
97C42×C191—41no—
98C22×D191—21no—
99C22×D191—21no—
100C22×D191—21no—
101C22×D191—21no—
102C22×D191—21no—
103C22×D191—21no—
104C22×D191C22×C412yes—
105C22⋊D191—28no—
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—44no—
123C2×Dic191C22×C412yes—
124C2×Dic191C22×C412yes—
125C2×Dic191C22×C412yes—
126C2×Dic191#4112yes—
127C2×Dic191#4112yes—
128C4×D191—44no—
129C23×C191—21no—
130C23×C191—21no—
131C23×C191—21no—
132C23×C191—21no—
133C23×C191—21no—
134C23×C191—21no—
135C23×C191C22×C412yes—
136C23×C191C2411yescharacteristic
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×C191—21no—
149C2×C4×C191—21no—
150C2×C4×C191—21no—
151C2×C4×C191—21no—
152C2×C4×C191—21no—
153C2×C4×C191C22×C412yes—
154C2×C4×C191C22×C412yes—
155C22×C191—22no—
156C22×C191—22no—
157C22×C191—22no—
158C22×C191—41no—
159C22×C191—41no—
160C22×C191—41no—
161C22×C191—41no—
162C22×C191C23×C411yescharacteristic
163C22×C191C2×#4111yescharacteristic
164C22×C191C2×#4111yescharacteristic
165C22×C191H(5,𝔽2)11yescharacteristic
166C22×C191H(5,𝔽2)11yescharacteristic
167C22×C191H(5,𝔽2)11yescharacteristic
168C22×C1912−1+411yescharacteristic
169C2×D191—22no—
170C2×D191—22no—
171C2×D191—22no—
172C2×D191—41no—
173C2×D191—41no—
174C2×D191—41no—
175C2×D191—41no—
176C4×C191—22no—
177C4×C191—22no—
178C4×C191—41no—
179C4×C191—41no—
180C4×C191—41no—
181C4×C191—41no—
182Dic191—22no—
183Dic191—22no—
184Dic191—41no—
185Dic191—41no—
186Dic191—41no—
187Dic191—41no—
188Dic191#11714yes—
189C2×C191—42no—
190C2×C191C2×#11711yescharacteristic
191C2×C191#51811yescharacteristic
192C2×C191#52011yescharacteristic
193C2×C191#54011yescharacteristic
194C2×C191#54011yescharacteristic
195C2×C191#54211yescharacteristic
196C2×C191#54211yescharacteristic
197D191—42no—
198C191#231911yescharacteristic, complemented, Sylow
199#2319—1911nomaximal, Sylow
200C2×C4×D4—1911no—
201#390—1911no—
202#390—1911no—
203#389—1911no—
204#389—1911no—
205#388—1911no—
206#388—1911no—
207#388—1911no—
208#388D19111yescharacteristic, complemented
209#381—1911no—
210#381—1911no—
211#380—1911no—
212#380—1911no—
213C4×C4⋊C4—1911no—
214#376—1911no—
215C22×D4—1911no—
216C23×C4—1911no—
217C23×C4—1911no—
218C4×D4—1918no—
219C2×C4⋊C4—1911no—
220C2×C4⋊C4—1911no—
221C2×C4⋊C4—1911no—
222C2×C4⋊C4—1911no—
223C2×C4⋊C4—1911no—
224C2×C4⋊C4—1911no—
225C2×C4⋊C4C2×D19111yescharacteristic
226C2×C22⋊C4—1911no—
227C2×C22⋊C4—1911no—
228C2×C22⋊C4—1911no—
229C2×C22⋊C4—1911no—
230C2×C22⋊C4—1911no—
231C2×C22⋊C4—1911no—
232C2×C22⋊C4—1911no—
233C2×C22⋊C4C2×D19111yescharacteristic
234C2×C22⋊C4C2×D19111yescharacteristic
235C2×C22⋊C4C2×D19111yescharacteristic
236C2×C42—1911no—
237C2×C42—1911no—
238C2×C42—1911no—
239C2×C42—1911no—
240C2×C42C2×D19111yescharacteristic
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#95C2×D19111yescharacteristic
250#95C2×D19111yescharacteristic
251C24—1911no—
252C24C22×D19111yescharacteristic
253C2×D4—3822no—
254C2×D4—3822no—
255C2×D4—1914no—
256C22×C4—1911no—
257C22×C4—1911no—
258C22×C4—1911no—
259C22×C4—1911no—
260C22×C4—1911no—
261C22×C4—1911no—
262C22×C4—1911no—
263C22×C4—1914no—
264C22×C4—1914no—
265C22×C4—7641no—
266C22×C4—7641no—
267C22×C4—7641no—
268C22×C4—7641no—
269C22×C4C22×D19111yescharacteristic
270C22×C4C22×D19111yescharacteristic
271C22×C4C22×D19111yescharacteristic
272C22×C4C22×D19111yescharacteristic
273C22×C4C22×D19111yescharacteristic
274C22×C4C22×D19111yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—3822no—
277C4⋊C4—3822no—
278C4⋊C4—1914no—
279C4⋊C4—7641no—
280C4⋊C4—7641no—
281C4⋊C4—7641no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—1914no—
285C22⋊C4—7641no—
286C22⋊C4—7641no—
287C22⋊C4—7641no—
288C22⋊C4—7641no—
289C22⋊C4—7641no—
290C22⋊C4—7641no—
291C22⋊C4C4×D19114yes—
292C42—41no—
293C42—3822no—
294C42—3822no—
295C42—1914no—
296C42—7641no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—3821no—
304C23—3821no—
305C23—3821no—
306C23—3821no—
307C23—3821no—
308C23—3821no—
309C23—1912no—
310C23C2×C4×D19112yes—
311C23C23×D19111yescharacteristic
312D4—3828no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
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—3821no—
330C2×C4—3821no—
331C2×C4—3821no—
332C2×C4—3821no—
333C2×C4—3821no—
334C2×C4—3821no—
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—1912no—
346C2×C4—1912no—
347C2×C4—1912no—
348C2×C4—1912no—
349C2×C4—1912no—
350C2×C4—7644no—
351C2×C4—7644no—
352C2×C4C2×C4×D19112yes—
353C2×C4C2×C4×D19112yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3822no—
362C22—3822no—
363C22—3822no—
364C22—7641no—
365C22—7641no—
366C22—7641no—
367C22—7641no—
368C22C22×C4×D19111yescharacteristic
369C22#41×D19111yescharacteristic
370C22#41×D19111yescharacteristic
371C22#2025936711yescharacteristic
372C22#2025936711yescharacteristic
373C22#2025936811yescharacteristic
374C22#2025936911yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3822no—
382C4—3822no—
383C4—1914no—
384C4—7641no—
385C4—7641no—
386C4—7641no—
387C4—7641no—
388C2—42no—
389C2—7642no—
390C2#117×D19111yescharacteristic
391C2#2713313511yescharacteristic
392C2#2713314211yescharacteristic
393C2#2713326311yescharacteristic
394C2#2713326511yescharacteristic
395C2#2713327711yescharacteristic
396C2#2713327911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#388×C191C2
socle, cosocleC23×C191C24
prime radicalC22×C22⋊D191C22
hypercenter#388D191
hypocenterC191#2319
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
4. #27132560C2semidirect
5. #27132558C2semidirect
6. #27132556C2semidirect
7. #27132554C2semidirect
8. #27132552C2semidirect
9. #27132525C2semidirect
10. #27132524C2semidirect
11. #27132521C2semidirect
12. #27132520C2semidirect
13. C4×Dic191⋊C4C2semidirect
14. #27132508C2semidirect
15. #388×C191C2semidirect
19. (C22⋊D191)⋊C4C4semidirect
198. C191#2319semidirect
208. #388D191semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D191)⋊C42
#271325632
#271325602
#271325582
#271325562
#271325542
#271325522
#271325252
#271325242
#271325212
#271325202
C4×Dic191⋊C42
#271325082
#388×C1912
#2319191