Finite Group Databasebeta

← #11733745

Subgroups — #11733745

Order 896.

532 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×#6498C211yesmaximal, characteristic
2C7×#390C211yesmaximal, characteristic
3#89992C211yesmaximal, characteristic
4#89750C211yesmaximal, characteristic, complemented
5#89744C211yesmaximal, characteristic, complemented
6#89718C211yesmaximal, characteristic, complemented
7#89717C211yesmaximal, characteristic, complemented
8#89716C211yesmaximal, characteristic, complemented
9#89716C211yesmaximal, characteristic, complemented
10#89716C211yesmaximal, characteristic, complemented
11#89714C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
14#89705C211yesmaximal, characteristic, complemented
15C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C42×C7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31#6498C418yescomplemented
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52#2335—71nomaximal, Sylow
53C24×C7C2311yescharacteristic
54C24×C7C2311yescharacteristic
55C2×C7×D4—41no—
56C2×C7×D4—41no—
57C2×C7×D4C2×C414yes—
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×Dic7—22no—
64C22×Dic7—41no—
65C22×Dic7—41no—
66C22×Dic7—41no—
67C22×Dic7—41no—
68C22×Dic7C2×C414yescomplemented
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C7×C4⋊C4—41no—
78C7×C22⋊C4—22no—
79C7×C22⋊C4—22no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—41no—
82C42×C7—41no—
83C22⋊Dic7—22no—
84C22⋊Dic7—41no—
85C22⋊Dic7—41no—
86C22⋊Dic7C2×C414yes—
87C22⋊Dic7C2×C414yes—
88C22⋊Dic7C2×C414yescomplemented
89Dic7⋊C4—22no—
90Dic7⋊C4—41no—
91Dic7⋊C4—41no—
92Dic7⋊C4—41no—
93Dic7⋊C4C2×C414yescomplemented
94Dic7⋊C4C2×C414yescomplemented
95C4×Dic7—22no—
96C4×Dic7—41no—
97C4×Dic7—41no—
98C4×Dic7—41no—
99C2×#123—71no—
100#390D711yescharacteristic
101#389—71no—
102#388—71no—
103#388—71no—
104#388—71no—
105#388—71no—
106#387—71no—
107#387—71no—
108#386—71no—
109#385—71no—
110#380—71no—
111C4×C4⋊C4—71no—
112C4×C22⋊C4—71no—
113C4×C22⋊C4—71no—
114C23×C7—21no—
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×C7C22×C412yes—
127C23×C7C22×C412yes—
128C23×C7C2411yescharacteristic
129C7×D4—44no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7C22×C412yes—
144C2×C4×C7#4112yes—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
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—44no—
165C2×Dic7—44no—
166C2×Dic7C22×C412yes—
167C2×Dic7C22×C412yes—
168C2×Dic7C22×C412yes—
169C2×Dic7C22×C412yes—
170C2×Dic7#4112yes—
171C22×D4C2×D711yescharacteristic
172C23×C4—71no—
173C23×C4—71no—
174#123—78no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4C2×D711yescharacteristic
182C2×C22⋊C4—71no—
183C2×C22⋊C4—71no—
184C2×C22⋊C4—71no—
185C2×C22⋊C4—71no—
186C2×C22⋊C4—71no—
187C2×C22⋊C4—71no—
188C2×C22⋊C4C2×D711yescharacteristic
189C2×C22⋊C4C2×D711yescharacteristic
190C2×C22⋊C4C2×D711yescharacteristic
191C2×C22⋊C4C2×D711yescharacteristic
192C2×C42—71no—
193C2×C42—71no—
194C2×C42—71no—
195C2×C42—71no—
196C2×C42C2×D711yescharacteristic
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207C22×C7—22no—
208C22×C7—22no—
209C22×C7—22no—
210C22×C7—22no—
211C22×C7—22no—
212C22×C7—22no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7C23×C411yescharacteristic
222C22×C7C2×#4111yescharacteristic
223C22×C7C2×#4111yescharacteristic
224C22×C7H(5,𝔽2)11yescharacteristic
225C22×C7H(5,𝔽2)11yescharacteristic
226C22×C7H(5,𝔽2)11yescharacteristic
227C22×C72−1+411yescharacteristic
228C4×C7—22no—
229C4×C7—22no—
230C4×C7—41no—
231C4×C7—41no—
232C4×C7—41no—
233Dic7—22no—
234Dic7—22no—
235Dic7—22no—
236Dic7—22no—
237Dic7—22no—
238Dic7—41no—
239Dic7—41no—
240Dic7—41no—
241C24C22×D711yescharacteristic
242C24C22×D711yescharacteristic
243C2×D4—41no—
244C2×D4—41no—
245C2×D4C4×D714yes—
246C22×C4—71no—
247C22×C4—71no—
248C22×C4—71no—
249C22×C4—71no—
250C22×C4—71no—
251C22×C4—74no—
252C22×C4—71no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—142no—
256C22×C4—281no—
257C22×C4—281no—
258C22×C4—281no—
259C22×C4—281no—
260C22×C4C22×D711yescharacteristic
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263C22×C4C22×D711yescharacteristic
264C22×C4C22×D711yescharacteristic
265C4⋊C4—41no—
266C4⋊C4—74no—
267C4⋊C4—74no—
268C4⋊C4—142no—
269C4⋊C4—281no—
270C4⋊C4—281no—
271C4⋊C4—281no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—74no—
277C22⋊C4—74no—
278C22⋊C4—74no—
279C22⋊C4—142no—
280C22⋊C4—281no—
281C22⋊C4—281no—
282C42—41no—
283C42—142no—
284C42—281no—
285C42—281no—
286C42—281no—
287C2×C7—42no—
288C2×C7—42no—
289C2×C7C4×#4111yescharacteristic
290C2×C7#51811yescharacteristic
291C2×C7#52011yescharacteristic
292C2×C7#53811yescharacteristic
293C2×C7#53911yescharacteristic
294C2×C7#54011yescharacteristic
295C2×C7#54211yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C2×C4×D712yes—
309C23C2×C4×D712yes—
310C23C23×D711yescharacteristic
311D4—44no—
312C2×C4—21no—
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—72no—
326C2×C4—72no—
327C2×C4—72no—
328C2×C4—72no—
329C2×C4—72no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—284no—
350C2×C4—284no—
351C2×C4C2×C4×D712yes—
352C2×C4#115912yes—
353C7#233511yescharacteristic, complemented, Sylow
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C22×C4×D711yescharacteristic
369C22C2×#115911yescharacteristic
370C22#654811yescharacteristic
371C22#654811yescharacteristic
372C22#654811yescharacteristic
373C22D7×#4111yescharacteristic
374C22#655411yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—142no—
381C4—142no—
382C4—142no—
383C4—142no—
384C4—142no—
385C4—281no—
386C4—281no—
387C4—281no—
388C2—42no—
389C2—42no—
390C2#9017711yescharacteristic
391C2C4×#115911yescharacteristic
392C2#9023711yescharacteristic
393C2#9029311yescharacteristic
394C2#9030411yescharacteristic
395C2#9036011yescharacteristic
396C2#9041011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#390C2
socle, cosocleC23×C7C24
prime radicalC22×C7×D4C22
hypercenter#390D7
hypocenterC7#2335
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
4. #89750C2semidirect
5. #89744C2semidirect
6. #89718C2semidirect
7. #89717C2semidirect
8. #89716C2semidirect
9. #89716C2semidirect
10. #89716C2semidirect
11. #89714C2semidirect
12. C22⋊C4×Dic7C2semidirect
13. C22⋊C4×Dic7C2semidirect
14. #89705C2semidirect
15. C4×Dic7⋊C4C2semidirect
31. #6498C4semidirect
68. C22×Dic7C2×C4semidirect
88. C22⋊Dic7C2×C4semidirect
93. Dic7⋊C4C2×C4semidirect
94. Dic7⋊C4C2×C4semidirect
353. C7#2335semidirect

Maximal subgroups

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

subgroupindex
C2×#64982
C7×#3902
#899922
#897502
#897442
#897182
#897172
#897162
#897142
C22⋊C4×Dic72
#897052
C4×Dic7⋊C42
#23357