Finite Group Databasebeta

← #11734178

Subgroups — #11734178

Order 896.

498 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#392C211yesmaximal, characteristic
2#89994C211yesmaximal, characteristic, complemented
3#89993C211yesmaximal, characteristic, complemented
4#89992C211yesmaximal, characteristic, complemented
5#89992C211yesmaximal, characteristic, complemented
6#89992C211yesmaximal, characteristic, complemented
7#89992C211yesmaximal, characteristic, complemented
8#89725C211yesmaximal, characteristic, complemented
9#89724C211yesmaximal, characteristic, complemented
10#89722C211yesmaximal, characteristic, complemented
11#89720C211yesmaximal, characteristic, complemented
12#89716C211yesmaximal, characteristic, complemented
13#89715C211yesmaximal, characteristic, complemented
14#89432C211yesmaximal, characteristic, complemented
15#89423C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39C7×#95C2211yescharacteristic
40C7⋊#95C2211yescharacteristic
41C7⋊#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2749—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C24×C7C2311yescharacteristic
54C24×C7C2311yescharacteristic
55C2×C7×D4—22no—
56C2×C7×D4—22no—
57C2×C7×D4—22no—
58C2×C7×D4—22no—
59C2×C7×D4—22no—
60C2×C7×D4—41no—
61C2×C7×D4—41no—
62C2×C7×D4—41no—
63C2×C7×D4—41no—
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×Dic7—22no—
69C22×Dic7—41no—
70C22×Dic7—41no—
71C22×Dic7—41no—
72C22×Dic7—41no—
73C22×Dic7—41no—
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C7×C22⋊C4—22no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4—41no—
85C22⋊Dic7—22no—
86C22⋊Dic7—41no—
87C22⋊Dic7—41no—
88C22⋊Dic7—41no—
89C22⋊Dic7—41no—
90C22⋊Dic7—41no—
91C22⋊Dic7—41no—
92C22⋊Dic7—41no—
93C22⋊Dic7—41no—
94C22⋊Dic7—41no—
95C4⋊Dic7—41no—
96C4⋊Dic7—41no—
97Dic7⋊C4—41no—
98Dic7⋊C4—41no—
99C4×Dic7—22no—
100#400—71no—
101#397—71no—
102#397—71no—
103#396—71no—
104#394—71no—
105#392D711yescharacteristic
106#390—71no—
107#388—71no—
108#386—71no—
109#386—71no—
110#386—71no—
111#386—71no—
112#385—71no—
113#383—71no—
114#381—71no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7#4112yes—
135C23×C7#4112yes—
136C23×C7C2411yescharacteristic
137C7×D4—44no—
138C7×D4—44no—
139C7×D4—44no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7#4112yes—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—44no—
177C2×Dic7—44no—
178C22×D4C2×D711yescharacteristic
179C22×D4C2×D711yescharacteristic
180C22×D4C2×D711yescharacteristic
181C23×C4—71no—
182C23×C4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4—71no—
187C2×C22⋊C4—71no—
188C2×C22⋊C4—71no—
189C2×C22⋊C4—71no—
190C2×C22⋊C4—71no—
191C2×C22⋊C4—71no—
192C2×C22⋊C4—71no—
193C2×C22⋊C4—71no—
194C2×C22⋊C4—71no—
195C2×C22⋊C4—71no—
196C2×C22⋊C4—71no—
197C2×C22⋊C4C2×D711yescharacteristic
198C2×C22⋊C4C2×D711yescharacteristic
199C2×C22⋊C4C2×D711yescharacteristic
200C2×C42—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207#95—71no—
208#95—71no—
209#95—71no—
210#95—71no—
211#95—71no—
212#95C2×D711yescharacteristic
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—22no—
216C22×C7—22no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—41no—
234C22×C7C2×#4111yescharacteristic
235C22×C7C2×#4111yescharacteristic
236C22×C7C2×#4111yescharacteristic
237C22×C7H(5,𝔽2)11yescharacteristic
238C22×C7H(5,𝔽2)11yescharacteristic
239C22×C7H(5,𝔽2)11yescharacteristic
240C22×C7H(5,𝔽2)11yescharacteristic
241C4×C7—22no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7—41no—
245Dic7—41no—
246Dic7—41no—
247Dic7—41no—
248Dic7—41no—
249Dic7—41no—
250Dic7—41no—
251Dic7—41no—
252Dic7—41no—
253C24C22×D711yescharacteristic
254C24C22×D711yescharacteristic
255C24C22×D711yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—71no—
268C22×C4—71no—
269C22×C4—71no—
270C22×C4—71no—
271C22×C4—71no—
272C22×C4—71no—
273C22×C4—142no—
274C22×C4—281no—
275C22×C4—281no—
276C22×C4—281no—
277C22×C4—281no—
278C22×C4—281no—
279C22×C4C22×D711yescharacteristic
280C22×C4C22×D711yescharacteristic
281C22×C4C22×D711yescharacteristic
282C22×C4C22×D711yescharacteristic
283C4⋊C4—281no—
284C4⋊C4—281no—
285C4⋊C4—281no—
286C4⋊C4—281no—
287C22⋊C4—22no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—142no—
291C22⋊C4—281no—
292C22⋊C4—281no—
293C22⋊C4—281no—
294C22⋊C4—281no—
295C22⋊C4—281no—
296C22⋊C4—281no—
297C22⋊C4—281no—
298C22⋊C4—281no—
299C22⋊C4—281no—
300C42—142no—
301C2×C7—42no—
302C2×C7—42no—
303C2×C7—81no—
304C2×C7#53811yescharacteristic
305C2×C7#53811yescharacteristic
306C2×C7#54011yescharacteristic
307C2×C7#55111yescharacteristic
308C2×C7#55111yescharacteristic
309C2×C7#55311yescharacteristic
310C2×C7#56011yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23#115912yes—
331C23#115912yes—
332C23C23×D711yescharacteristic
333D4—44no—
334D4—44no—
335D4—44no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—141no—
357C2×C4—141no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—141no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—141no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—284no—
372C2×C4—284no—
373C2×C4#115912yes—
374C7#274911yescharacteristic, complemented, Sylow
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22C2×#115911yescharacteristic
397C22C2×#115911yescharacteristic
398C22C2×#115911yescharacteristic
399C22#654811yescharacteristic
400C22#654811yescharacteristic
401C22#654811yescharacteristic
402C22#655311yescharacteristic
403C4—22no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—281no—
408C4—281no—
409C4—281no—
410C4—281no—
411C4—281no—
412C4—281no—
413C4—281no—
414C4—281no—
415C2—42no—
416C2—42no—
417C2—81no—
418C2#9027811yescharacteristic
419C2#9027811yescharacteristic
420C2#9028411yescharacteristic
421C2#9030811yescharacteristic
422C2#9030911yescharacteristic
423C2#9031111yescharacteristic
424C2#9041011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC7×#392C2
hypercenter#392D7
hypocenterC7#2749
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
2. #89994C2semidirect
3. #89993C2semidirect
4. #89992C2semidirect
5. #89992C2semidirect
6. #89992C2semidirect
7. #89992C2semidirect
8. #89725C2semidirect
9. #89724C2semidirect
10. #89722C2semidirect
11. #89720C2semidirect
12. #89716C2semidirect
13. #89715C2semidirect
14. #89432C2semidirect
15. #89423C2semidirect
374. C7#2749semidirect

Maximal subgroups

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

subgroupindex
C7×#3922
#899942
#899932
#899922
#897252
#897242
#897222
#897202
#897162
#897152
#894322
#894232
#27497