Finite Group Databasebeta

← #37576005

Subgroups — #37576005

Order 24448.

596 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259294C211yesmaximal, characteristic
2C2×(C22⋊D191)⋊C4C211yesmaximal, characteristic, complemented
3#27132591C211yesmaximal, characteristic, complemented
4#27132589C211yesmaximal, characteristic, complemented
5#27132578C211yesmaximal, characteristic, complemented
6#27132563C211yesmaximal, characteristic, complemented
7#27132558C211yesmaximal, characteristic, complemented
8#27132543C211yesmaximal, characteristic, complemented
9#27132539C211yesmaximal, characteristic, complemented
10#27132527C211yesmaximal, characteristic, complemented
11#27132525C211yesmaximal, characteristic, complemented
12#27132524C211yesmaximal, characteristic, complemented
13#27132521C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic191C211yesmaximal, characteristic, complemented
15#394×C191C211yesmaximal, characteristic, complemented
16C22×C22⋊D191C2211yescharacteristic
17C22×C22⋊D191C2211yescharacteristic
18C23×Dic191C2211yescharacteristic
19C23×Dic191C2211yescharacteristic
20C22×C4×D191C2211yescharacteristic
21#20259294—24no—
22(C22⋊D191)⋊C4—24no—
23C2×Dic191⋊C4C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×Dic191⋊C4C2211yescharacteristic
27C2×C22⋊Dic191C2211yescharacteristic
28C2×C22⋊Dic191C2211yescharacteristic
29C2×C22⋊Dic191C2211yescharacteristic
30C2×C22⋊Dic191C2211yescharacteristic
31C2×(C2×D191)⋊C4C2211yescharacteristic
32C2×(C2×D191)⋊C4C2211yescharacteristic
33C2×(C2×D191)⋊C4C2211yescharacteristic
34C2×(C2×D191)⋊C4C2211yescharacteristic
35C2×C4×Dic191C2211yescharacteristic
36C2×C4×Dic191C2211yescharacteristic
37C191⋊#95C2211yescharacteristic
38C191⋊#95C2211yescharacteristic
39C191⋊#95C2211yescharacteristic
40C191⋊#95C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C191⋊#95C2211yescharacteristic
46C22×D4×C191C2211yescharacteristic
47C2×C4⋊C4×C191C2211yescharacteristic
48C2×C22⋊C4×C191C2211yescharacteristic
49C2×C22⋊C4×C191C2211yescharacteristic
50C2×C22⋊C4×C191C2211yescharacteristic
51C2×C22⋊C4×C191C2211yescharacteristic
52#95×C191C2211yescharacteristic
53C23×D191C2311yescharacteristic
54C2×C22⋊D191—22no—
55C2×C22⋊D191—22no—
56C2×C22⋊D191—22no—
57C2×C22⋊D191—22no—
58C2×C22⋊D191—41no—
59C2×C22⋊D191D414yes—
60C22×Dic191—22no—
61C22×Dic191—22no—
62C22×Dic191—22no—
63C22×Dic191—22no—
64C22×Dic191—22no—
65C22×Dic191—41no—
66C22×Dic191C2311yescharacteristic
67C22×Dic191C2311yescharacteristic
68C22×Dic191C2311yescharacteristic
69C22×Dic191C2311yescharacteristic
70C22×Dic191C2311yescharacteristic
71C22×Dic191C2311yescharacteristic
72C22×Dic191C2311yescharacteristic
73C2×C4×D191—22no—
74C2×C4×D191—22no—
75C2×C4×D191—41no—
76Dic191⋊C4—22no—
77Dic191⋊C4—22no—
78Dic191⋊C4—22no—
79Dic191⋊C4—41no—
80C22⋊Dic191—22no—
81C22⋊Dic191—22no—
82C22⋊Dic191—41no—
83C22⋊Dic191—41no—
84(C2×D191)⋊C4—22no—
85(C2×D191)⋊C4—22no—
86(C2×D191)⋊C4—41no—
87(C2×D191)⋊C4—41no—
88C4×Dic191—22no—
89C4×Dic191—22no—
90C24×C191C2311yescharacteristic
91C24×C191C2311yescharacteristic
92C2×D4×C191—22no—
93C2×D4×C191—22no—
94C2×D4×C191—41no—
95C22×C4×C191C2311yescharacteristic
96C22×C4×C191C2311yescharacteristic
97C22×C4×C191C2311yescharacteristic
98C22×C4×C191C2311yescharacteristic
99C22×C4×C191C2311yescharacteristic
100C4⋊C4×C191—41no—
101C22⋊C4×C191—22no—
102C22⋊C4×C191—22no—
103C22⋊C4×C191—41no—
104C22⋊C4×C191—41no—
105C22×D191—21no—
106C22×D191—21no—
107C22×D191—21no—
108C22×D191—21no—
109C22×D191—21no—
110C22×D191—21no—
111C22×D191C2×D412yes—
112C22⋊D191—28no—
113C22⋊D191—44no—
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—21no—
127C2×Dic191—21no—
128C2×Dic191—21no—
129C2×Dic191—21no—
130C2×Dic191—21no—
131C2×Dic191—44no—
132C2×Dic191—44no—
133C2×Dic191C2×D412yes—
134C2×Dic191#4112yes—
135C2×Dic191#4112yes—
136C2×Dic191#4112yes—
137C4×D191—44no—
138C23×C191—21no—
139C23×C191—21no—
140C23×C191—21no—
141C23×C191—21no—
142C23×C191—21no—
143C23×C191—21no—
144C23×C191—21no—
145C23×C191—21no—
146C23×C191—21no—
147C23×C191—21no—
148C23×C191—21no—
149C23×C191—21no—
150C23×C191C2×D412yes—
151C23×C191#4112yes—
152C23×C191C2411yescharacteristic
153D4×C191—44no—
154C2×C4×C191—21no—
155C2×C4×C191—21no—
156C2×C4×C191—21no—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C22×C191—22no—
170C22×C191—22no—
171C22×C191—22no—
172C22×C191—22no—
173C22×C191—22no—
174C22×C191—22no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191—41no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191C22×D411yescharacteristic
184C22×C191C2×#4111yescharacteristic
185C22×C191C2×#4111yescharacteristic
186C22×C191C2×#4111yescharacteristic
187C22×C191C2×#4111yescharacteristic
188C22×C191H(5,𝔽2)11yescharacteristic
189C22×C191H(5,𝔽2)11yescharacteristic
190C2×D191—22no—
191C2×D191—22no—
192C2×D191—22no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—41no—
197C4×C191—41no—
198C4×C191—41no—
199C4×C191—41no—
200C4×C191—41no—
201C4×C191—41no—
202Dic191—22no—
203Dic191—41no—
204Dic191—41no—
205Dic191—41no—
206Dic191—41no—
207Dic191—41no—
208Dic191#12314yes—
209C2×C191—42no—
210C2×C191—42no—
211C2×C191C2×#12311yescharacteristic
212C2×C191#52911yescharacteristic
213C2×C191#54011yescharacteristic
214C2×C191#54611yescharacteristic
215C2×C191#54611yescharacteristic
216C2×C191#55311yescharacteristic
217C2×C191#55511yescharacteristic
218D191—42no—
219C191#242011yescharacteristic, complemented, Sylow
220#2420—1911nomaximal, Sylow
221C2×#121—1911no—
222C2×C4×D4—1911no—
223#397—1911no—
224#397—1911no—
225#394—1911no—
226#394D19111yescharacteristic, complemented
227#390—1911no—
228#389—1911no—
229#386—1911no—
230#386—1911no—
231#382—1911no—
232#381—1911no—
233#381—1911no—
234#380—1911no—
235C4×C22⋊C4—1911no—
236C22×D4—1911no—
237C22×D4—1911no—
238C22×D4C2×D19111yescharacteristic
239C23×C4—1911no—
240C23×C4—1911no—
241C23×C4—1911no—
242#121—3824no—
243C4×D4—3824no—
244C2×C4⋊C4—1911no—
245C2×C4⋊C4—1911no—
246C2×C4⋊C4—1911no—
247C2×C4⋊C4—1911no—
248C2×C4⋊C4C2×D19111yescharacteristic
249C2×C22⋊C4—1911no—
250C2×C22⋊C4—1911no—
251C2×C22⋊C4—1911no—
252C2×C22⋊C4—1911no—
253C2×C22⋊C4—1911no—
254C2×C22⋊C4—1911no—
255C2×C22⋊C4—1911no—
256C2×C22⋊C4—1911no—
257C2×C22⋊C4C2×D19111yescharacteristic
258C2×C22⋊C4C2×D19111yescharacteristic
259C2×C22⋊C4C2×D19111yescharacteristic
260C2×C22⋊C4C2×D19111yescharacteristic
261C2×C42—1911no—
262C2×C42—1911no—
263#95—1911no—
264#95—1911no—
265#95—1911no—
266#95—1911no—
267#95—1911no—
268#95—1911no—
269#95—1911no—
270#95—1911no—
271#95—1911no—
272#95C2×D19111yescharacteristic
273C24—1911no—
274C24C22×D19111yescharacteristic
275C24C22×D19111yescharacteristic
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—3822no—
280C2×D4—3822no—
281C2×D4—3822no—
282C2×D4—3822no—
283C2×D4—1914no—
284C2×D4—7641no—
285C22×C4—3822no—
286C22×C4—3822no—
287C22×C4—3822no—
288C22×C4—3822no—
289C22×C4—3822no—
290C22×C4—3822no—
291C22×C4—3822no—
292C22×C4—1911no—
293C22×C4—1911no—
294C22×C4—1911no—
295C22×C4—1911no—
296C22×C4—1911no—
297C22×C4—1911no—
298C22×C4—1911no—
299C22×C4—7641no—
300C22×C4—7641no—
301C22×C4C22×D19111yescharacteristic
302C22×C4C22×D19111yescharacteristic
303C22×C4C22×D19111yescharacteristic
304C22×C4C22×D19111yescharacteristic
305C22×C4C22×D19111yescharacteristic
306C4⋊C4—41no—
307C4⋊C4—3822no—
308C4⋊C4—3822no—
309C4⋊C4—3822no—
310C4⋊C4—7641no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—3822no—
316C22⋊C4—3822no—
317C22⋊C4—3822no—
318C22⋊C4—3822no—
319C22⋊C4—7641no—
320C22⋊C4—7641no—
321C22⋊C4—7641no—
322C22⋊C4—7641no—
323C42—3822no—
324C42—3822no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—3821no—
338C23—3821no—
339C23—3821no—
340C23—3821no—
341C23—3821no—
342C23—3821no—
343C23—1912no—
344C23D4×D19112yes—
345C23#1569028612yes—
346C23C23×D19111yescharacteristic
347D4—44no—
348D4—3828no—
349D4—7644no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—3821no—
368C2×C4—3821no—
369C2×C4—3821no—
370C2×C4—3821no—
371C2×C4—3821no—
372C2×C4—3821no—
373C2×C4—3821no—
374C2×C4—3821no—
375C2×C4—3821no—
376C2×C4—3821no—
377C2×C4—3821no—
378C2×C4—3821no—
379C2×C4—3821no—
380C2×C4—3821no—
381C2×C4—3821no—
382C2×C4—1912no—
383C2×C4—1912no—
384C2×C4—1912no—
385C2×C4—1912no—
386C2×C4—7644no—
387C2×C4—7644no—
388C2×C4—7644no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—3822no—
404C22—3822no—
405C22—3822no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22—7641no—
410C22C2×D4×D19111yescharacteristic
411C22#41×D19111yescharacteristic
412C22#41×D19111yescharacteristic
413C22#41×D19111yescharacteristic
414C22C2×#1569028611yescharacteristic
415C22#2025936711yescharacteristic
416C22#2025936711yescharacteristic
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—3822no—
423C4—1914no—
424C4—7641no—
425C4—7641no—
426C4—7641no—
427C4—7641no—
428C4—7641no—
429C2—42no—
430C2—42no—
431C2—7642no—
432C2#123×D19111yescharacteristic
433C2#2713319011yescharacteristic
434C2#2713326311yescharacteristic
435C2#2713330011yescharacteristic
436C2#2713330211yescharacteristic
437C2#2713335611yescharacteristic
438C2#2713337511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#394×C191C2
prime radicalC2×#20259294C2
hypercenter#394D191
hypocenterC191#2420
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D191)⋊C4C2semidirect
3. #27132591C2semidirect
4. #27132589C2semidirect
5. #27132578C2semidirect
6. #27132563C2semidirect
7. #27132558C2semidirect
8. #27132543C2semidirect
9. #27132539C2semidirect
10. #27132527C2semidirect
11. #27132525C2semidirect
12. #27132524C2semidirect
13. #27132521C2semidirect
14. C22⋊C4×Dic191C2semidirect
15. #394×C191C2semidirect
219. C191#2420semidirect
226. #394D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592942
C2×(C22⋊D191)⋊C42
#271325912
#271325892
#271325782
#271325632
#271325582
#271325432
#271325392
#271325272
#271325252
#271325242
#271325212
C22⋊C4×Dic1912
#394×C1912
#2420191