Finite Group Databasebeta

← #11735175

Subgroups — #11735175

Order 896.

472 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×#397C211yesmaximal, characteristic, complemented
2#89763C211yesmaximal, characteristic, complemented
3#89758C211yesmaximal, characteristic, complemented
4#89733C211yesmaximal, characteristic, complemented
5#89733C211yesmaximal, characteristic, complemented
6#89732C211yesmaximal, characteristic
7#89731C211yesmaximal, characteristic, complemented
8#89728C211yesmaximal, characteristic, complemented
9#89719C211yesmaximal, characteristic, complemented
10#89446C211yesmaximal, characteristic, complemented
11#89445C211yesmaximal, characteristic, complemented
12#89445C211yesmaximal, characteristic, complemented
13#89440C211yesmaximal, characteristic, complemented
14#89437C211yesmaximal, characteristic, complemented
15#89432C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×Dic14C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×C4⋊Dic7C2211yescharacteristic
37C2×C4⋊Dic7C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
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#2824—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C23×D7C2311yescharacteristic
54C23×D7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7—22no—
62C2×C22⋊D7—41no—
63C2×C22⋊D7—41no—
64C2×C22⋊D7—41no—
65C2×C22⋊D7—41no—
66C2×C22⋊D7—41no—
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C2×D28—22no—
74C2×D28—41no—
75C2×D28—41no—
76C2×C4×D7—41no—
77C2×C4×D7—41no—
78C2×C4×D7—41no—
79C2×Dic14—41no—
80C2×Dic14—41no—
81C2×Dic14—41no—
82C7×C4⋊C4—41no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—41no—
86C22⋊Dic7—41no—
87C22⋊Dic7—41no—
88(C2×D7)⋊C4—22no—
89(C2×D7)⋊C4—41no—
90(C2×D7)⋊C4—41no—
91(C2×D7)⋊C4—41no—
92(C2×D7)⋊C4—41no—
93(C2×D7)⋊C4—41no—
94(C2×D7)⋊C4—41no—
95(C2×D7)⋊C4—41no—
96(C2×D7)⋊C4—41no—
97C4⋊Dic7—41no—
98C4⋊Dic7—41no—
99Dic7⋊C4—41no—
100#400—71no—
101#400—71no—
102#397—71no—
103#397D711yescharacteristic, complemented
104#394—71no—
105#394—71no—
106#394—71no—
107#394—71no—
108#393—71no—
109#393—71no—
110#393—71no—
111#392—71no—
112#386—71no—
113#386—71no—
114#385—71no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7C2411yescharacteristic
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7#4112yes—
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×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—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C22⋊D7—44no—
156C22⋊D7—44no—
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—
175D28—44no—
176C4×D7—44no—
177Dic14—44no—
178C22×Q8—71no—
179C22×D4—71no—
180C22×D4—71no—
181C22×D4—71no—
182C23×C4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4C2×D711yescharacteristic
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⋊C4—71no—
198C2×C22⋊C4C2×D711yescharacteristic
199C2×C22⋊C4C2×D711yescharacteristic
200C2×C22⋊C4C2×D711yescharacteristic
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#95C2×D711yescharacteristic
211#95C2×D711yescharacteristic
212#95C2×D711yescharacteristic
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7C2×#4111yescharacteristic
221C22×C7H(5,𝔽2)11yescharacteristic
222C22×C7H(5,𝔽2)11yescharacteristic
223C22×C7H(5,𝔽2)11yescharacteristic
224C22×C7H(5,𝔽2)11yescharacteristic
225C22×C7H(5,𝔽2)11yescharacteristic
226C22×C72−1+411yescharacteristic
227C2×D7—22no—
228C2×D7—22no—
229C2×D7—22no—
230C2×D7—41no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—41no—
237C2×D7—41no—
238C2×D7—41no—
239C2×D7—41no—
240C2×D7—41no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—41no—
247Dic7—41no—
248Dic7—41no—
249Dic7—41no—
250Dic7—41no—
251Dic7—41no—
252Dic7—41no—
253C24—71no—
254C24—71no—
255C24C22×D711yescharacteristic
256C2×Q8—281no—
257C2×Q8—281no—
258C2×Q8—281no—
259C2×D4—142no—
260C2×D4—142no—
261C2×D4—281no—
262C2×D4—281no—
263C2×D4—281no—
264C2×D4—281no—
265C2×D4—281no—
266C2×D4—281no—
267C2×D4—281no—
268C22×C4—71no—
269C22×C4—71no—
270C22×C4—71no—
271C22×C4—71no—
272C22×C4—71no—
273C22×C4—71no—
274C22×C4—281no—
275C22×C4—281no—
276C22×C4—281no—
277C22×C4C22×D711yescharacteristic
278C22×C4C22×D711yescharacteristic
279C22×C4C22×D711yescharacteristic
280C22×C4C22×D711yescharacteristic
281C22×C4C22×D711yescharacteristic
282C22×C4C22×D711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—281no—
285C4⋊C4—281no—
286C4⋊C4—281no—
287C22⋊C4—41no—
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—
300C22⋊C4—281no—
301C2×C7—81no—
302C2×C7#53811yescharacteristic
303C2×C7#53811yescharacteristic
304C2×C7#53911yescharacteristic
305C2×C7#56011yescharacteristic
306C2×C7H(3,𝔽4)11yescharacteristic
307C2×C7#56211yescharacteristic
308C2×C7#56211yescharacteristic
309D7—42no—
310D7—81no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—72no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23—141no—
323C23—141no—
324C23—141no—
325C23—141no—
326C23—141no—
327C23—141no—
328C23—141no—
329C23—141no—
330C23—141no—
331C23—141no—
332C23C23×D711yescharacteristic
333Q8—284no—
334D4—284no—
335D4—284no—
336D4—284no—
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—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
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—141no—
372C2×C4—141no—
373C2×C4—284no—
374C7#282411yescharacteristic, complemented, Sylow
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—142no—
383C22—142no—
384C22—142no—
385C22—281no—
386C22—281no—
387C22—281no—
388C22—281no—
389C22—281no—
390C22—281no—
391C22—281no—
392C22—281no—
393C22—281no—
394C22—281no—
395C22—281no—
396C22#654811yescharacteristic
397C22#654811yescharacteristic
398C22D7×#4111yescharacteristic
399C22#655311yescharacteristic
400C22#655311yescharacteristic
401C22#655311yescharacteristic
402C22#655411yescharacteristic
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—281no—
410C4—281no—
411C4—281no—
412C4—281no—
413C4—281no—
414C4—281no—
415C2—81no—
416C2—282no—
417C2—561no—
418C2#9030411yescharacteristic
419C2#9035711yescharacteristic
420C2#9035811yescharacteristic
421C2#9035811yescharacteristic
422C2#9037411yescharacteristic
423C2#9039511yescharacteristic
424C2#9039811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radical#89732C2
hypercenter#397D7
hypocenterC7#2824
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C7×#397C2semidirect
2. #89763C2semidirect
3. #89758C2semidirect
4. #89733C2semidirect
5. #89733C2semidirect
7. #89731C2semidirect
8. #89728C2semidirect
9. #89719C2semidirect
10. #89446C2semidirect
11. #89445C2semidirect
12. #89445C2semidirect
13. #89440C2semidirect
14. #89437C2semidirect
15. #89432C2semidirect
103. #397D7semidirect
374. C7#2824semidirect

Maximal subgroups

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

subgroupindex
C7×#3972
#897632
#897582
#897332
#897322
#897312
#897282
#897192
#894462
#894452
#894402
#894372
#894322
#28247