Finite Group Databasebeta

← #11735837

Subgroups — #11735837

Order 896.

448 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#400C211yesmaximal, characteristic, complemented
2#89766C211yesmaximal, characteristic, complemented
3#89763C211yesmaximal, characteristic
4#89759C211yesmaximal, characteristic, complemented
5#89758C211yesmaximal, characteristic, complemented
6#89446C211yesmaximal, characteristic, complemented
7#89442C211yesmaximal, characteristic, complemented
8#89441C211yesmaximal, characteristic, complemented
9#89440C211yesmaximal, characteristic, complemented
10#89440C211yesmaximal, characteristic, complemented
11#89437C211yesmaximal, characteristic
12#89437C211yesmaximal, characteristic
13#89436C211yesmaximal, characteristic, complemented
14#89432C211yesmaximal, characteristic, complemented
15#89427C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×C4⋊Dic7C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37C7×#95C2211yescharacteristic
38C7×#95C2211yescharacteristic
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#2750—71nomaximal, Sylow
52C23×D7C2311yescharacteristic
53C23×D7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C2×D28—22no—
68C2×D28—22no—
69C2×D28—41no—
70C2×C4×D7—22no—
71C2×C4×D7—22no—
72C2×C4×D7—41no—
73C2×C4×D7—41no—
74C2×C4×D7—41no—
75C2×C4×D7—41no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—41no—
78(C2×D7)⋊C4—22no—
79(C2×D7)⋊C4—22no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—22no—
82(C2×D7)⋊C4—41no—
83(C2×D7)⋊C4—41no—
84(C2×D7)⋊C4—41no—
85(C2×D7)⋊C4—41no—
86(C2×D7)⋊C4—41no—
87(C2×D7)⋊C4—41no—
88C4⋊Dic7—41no—
89C4⋊Dic7—41no—
90C4⋊Dic7—41no—
91Dic7⋊C4—22no—
92Dic7⋊C4—41no—
93C4×Dic7—41no—
94#400—71no—
95#400—71no—
96#400D711yescharacteristic, complemented
97#397—71no—
98#396—71no—
99#394—71no—
100#388—71no—
101#388—71no—
102#387—71no—
103#386—71no—
104#386—71no—
105#385—71no—
106#385—71no—
107#381—71no—
108#380—71no—
109C23×C7C2411yescharacteristic
110C22×D7—21no—
111C22×D7—21no—
112C22×D7—21no—
113C22×D7—21no—
114C22×D7—21no—
115C22×D7—21no—
116C22×D7—21no—
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—21no—
120C22×D7—21no—
121C22×D7—21no—
122C22×D7#4112yes—
123C22×D7#4112yes—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
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×C7—21no—
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—
163D28—44no—
164C4×D7—44no—
165C4×D7—44no—
166C22×D4—71no—
167C23×C4—71no—
168C23×C4—71no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4—71no—
172C2×C4⋊C4—71no—
173C2×C4⋊C4—71no—
174C2×C4⋊C4C2×D711yescharacteristic
175C2×C4⋊C4C2×D711yescharacteristic
176C2×C22⋊C4—71no—
177C2×C22⋊C4—71no—
178C2×C22⋊C4—71no—
179C2×C22⋊C4—71no—
180C2×C22⋊C4—71no—
181C2×C22⋊C4—71no—
182C2×C22⋊C4—71no—
183C2×C22⋊C4—71no—
184C2×C22⋊C4—71no—
185C2×C22⋊C4—71no—
186C2×C42—71no—
187#95—71no—
188#95—71no—
189#95—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95—71no—
194#95—71no—
195#95—71no—
196#95C2×D711yescharacteristic
197#95C2×D711yescharacteristic
198#95C2×D711yescharacteristic
199#95C2×D711yescharacteristic
200#95C2×D711yescharacteristic
201C22×C7C2×#4111yescharacteristic
202C22×C7C2×#4111yescharacteristic
203C22×C7C2×#4111yescharacteristic
204C22×C7H(5,𝔽2)11yescharacteristic
205C22×C7H(5,𝔽2)11yescharacteristic
206C22×C7H(5,𝔽2)11yescharacteristic
207C22×C72−1+411yescharacteristic
208C2×D7—22no—
209C2×D7—22no—
210C2×D7—22no—
211C2×D7—22no—
212C2×D7—22no—
213C2×D7—22no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C4×C7—22no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233Dic7—41no—
234Dic7—41no—
235C24—71no—
236C24—71no—
237C2×D4—142no—
238C2×D4—142no—
239C2×D4—281no—
240C22×C4—71no—
241C22×C4—71no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—71no—
245C22×C4—71no—
246C22×C4—142no—
247C22×C4—142no—
248C22×C4—281no—
249C22×C4—281no—
250C22×C4—281no—
251C22×C4—281no—
252C22×C4C22×D711yescharacteristic
253C22×C4C22×D711yescharacteristic
254C22×C4C22×D711yescharacteristic
255C22×C4C22×D711yescharacteristic
256C22×C4C22×D711yescharacteristic
257C22×C4C22×D711yescharacteristic
258C22×C4C22×D711yescharacteristic
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—142no—
262C4⋊C4—281no—
263C4⋊C4—281no—
264C4⋊C4—281no—
265C4⋊C4—281no—
266C22⋊C4—142no—
267C22⋊C4—142no—
268C22⋊C4—142no—
269C22⋊C4—142no—
270C22⋊C4—281no—
271C22⋊C4—281no—
272C22⋊C4—281no—
273C22⋊C4—281no—
274C22⋊C4—281no—
275C22⋊C4—281no—
276C42—281no—
277C2×C7#53811yescharacteristic
278C2×C7#53811yescharacteristic
279C2×C7#53911yescharacteristic
280C2×C7#55111yescharacteristic
281C2×C7#55211yescharacteristic
282C2×C7#55311yescharacteristic
283C2×C7#56211yescharacteristic
284D7—42no—
285D7—42no—
286C23—72no—
287C23—72no—
288C23—141no—
289C23—141no—
290C23—141no—
291C23—141no—
292C23—141no—
293C23—141no—
294C23—141no—
295C23—141no—
296C23—141no—
297C23—141no—
298C23—141no—
299C23—141no—
300C23C23×D711yescharacteristic
301D4—284no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
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—141no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
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—284no—
341C2×C4—284no—
342C2×C4#116112yes—
343C7#275011yescharacteristic, complemented, Sylow
344C22—142no—
345C22—142no—
346C22—142no—
347C22—142no—
348C22—142no—
349C22—142no—
350C22—281no—
351C22—281no—
352C22—281no—
353C22—281no—
354C22—281no—
355C22—281no—
356C22—281no—
357C22—281no—
358C22#654811yescharacteristic
359C22C2×#116111yescharacteristic
360C22D7×#4111yescharacteristic
361C22D7×#4111yescharacteristic
362C22#655311yescharacteristic
363C22#655311yescharacteristic
364C22#655411yescharacteristic
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—281no—
373C4—281no—
374C4—281no—
375C4—281no—
376C4—281no—
377C4—281no—
378C2—282no—
379C2—282no—
380C2#9033511yescharacteristic
381C2#9035811yescharacteristic
382C2#9037011yescharacteristic
383C2#9038511yescharacteristic
384C2#9038611yescharacteristic
385C2#9039511yescharacteristic
386C2#9039811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC22×D28C22
hypercenter#400D7
hypocenterC7#2750
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C7×#400C2semidirect
2. #89766C2semidirect
4. #89759C2semidirect
5. #89758C2semidirect
6. #89446C2semidirect
7. #89442C2semidirect
8. #89441C2semidirect
9. #89440C2semidirect
10. #89440C2semidirect
13. #89436C2semidirect
14. #89432C2semidirect
15. #89427C2semidirect
96. #400D7semidirect
343. C7#2750semidirect

Maximal subgroups

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

subgroupindex
C7×#4002
#897662
#897632
#897592
#897582
#894462
#894422
#894412
#894402
#894372
#894362
#894322
#894272
#27507