Finite Group Databasebeta

← #11733193

Subgroups — #11733193

Order 896.

424 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#388C211yesmaximal, characteristic
2#89758C211yesmaximal, characteristic, complemented
3#89749C211yesmaximal, characteristic, complemented
4#89723C211yesmaximal, characteristic
5#89722C211yesmaximal, characteristic
6#89722C211yesmaximal, characteristic
7#89719C211yesmaximal, characteristic
8#89719C211yesmaximal, characteristic
9#89718C211yesmaximal, characteristic
10#89708C211yesmaximal, characteristic, complemented
11#89701C211yesmaximal, characteristic, complemented
12#89432C211yesmaximal, characteristic, complemented
13#89428C211yesmaximal, characteristic, complemented
14#89428C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C23×Dic7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C22×Dic14C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C42×C7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×C4⋊Dic7C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
35C7×#95C2211yescharacteristic
36C7×#95C2211yescharacteristic
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#2763—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C22×C4×C7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×Dic7—41no—
60C22×Dic7—41no—
61C22×Dic7—41no—
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C2×Dic14—22no—
71C2×Dic14—22no—
72C2×Dic14—41no—
73C2×Dic14—41no—
74C2×Dic14—41no—
75C2×Dic14—41no—
76C7×C4⋊C4—41no—
77C7×C22⋊C4—22no—
78C7×C22⋊C4—41no—
79C7×C22⋊C4Q814yescomplemented
80C42×C7—22no—
81C22⋊Dic7—22no—
82C22⋊Dic7—41no—
83C22⋊Dic7—41no—
84C4⋊Dic7—41no—
85C4⋊Dic7—41no—
86C4⋊Dic7—41no—
87C4⋊Dic7—41no—
88C4⋊Dic7—41no—
89Dic7⋊C4—41no—
90Dic7⋊C4—41no—
91C4×Dic7—22no—
92#400—71no—
93#400—71no—
94#395—71no—
95#395—71no—
96#393—71no—
97#393—71no—
98#391—71no—
99#391—71no—
100#388—71no—
101#388D711yescharacteristic
102#385—71no—
103#385—71no—
104#385—71no—
105#382—71no—
106#382—71no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7C2×Q812yes—
114C23×C7C2411yescharacteristic
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
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×C7C2×Q812yes—
131C2×C4×C7C2×Q812yes—
132C2×C4×C7#4112yes—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
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—44no—
157C2×Dic7#4112yes—
158Dic14—44no—
159Dic14—44no—
160C22×Q8—71no—
161C22×Q8—71no—
162C23×C4—71no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4C2×D711yescharacteristic
171C2×C22⋊C4—71no—
172C2×C22⋊C4—71no—
173C2×C22⋊C4—71no—
174C2×C22⋊C4C2×D711yescharacteristic
175C2×C22⋊C4C2×D711yescharacteristic
176C2×C22⋊C4C2×D711yescharacteristic
177C2×C42—71no—
178C2×C42C2×D711yescharacteristic
179#95—71no—
180#95—71no—
181#95—71no—
182#95—71no—
183#95—71no—
184#95—71no—
185#95—71no—
186#95—71no—
187#95—71no—
188#95—71no—
189#95—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95C2×D711yescharacteristic
194#95C2×D711yescharacteristic
195C22×C7—22no—
196C22×C7—22no—
197C22×C7—22no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7C22×Q811yescharacteristic
203C22×C7C2×#4111yescharacteristic
204C22×C7C2×#4111yescharacteristic
205C22×C7H(5,𝔽2)11yescharacteristic
206C22×C7H(5,𝔽2)11yescharacteristic
207C22×C7H(5,𝔽2)11yescharacteristic
208C22×C72−1+411yescharacteristic
209C4×C7—22no—
210C4×C7—22no—
211C4×C7—22no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215Dic7—22no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223C24C22×D711yescharacteristic
224C2×Q8—142no—
225C2×Q8—142no—
226C2×Q8—281no—
227C2×Q8—281no—
228C2×Q8—281no—
229C2×Q8—281no—
230C22×C4—71no—
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—71no—
236C22×C4—71no—
237C22×C4—71no—
238C22×C4—281no—
239C22×C4—281no—
240C22×C4—281no—
241C22×C4C22×D711yescharacteristic
242C22×C4C22×D711yescharacteristic
243C22×C4C22×D711yescharacteristic
244C22×C4C22×D711yescharacteristic
245C22×C4C22×D711yescharacteristic
246C22×C4C22×D711yescharacteristic
247C4⋊C4—41no—
248C4⋊C4—281no—
249C4⋊C4—281no—
250C4⋊C4—281no—
251C4⋊C4—281no—
252C4⋊C4—281no—
253C4⋊C4—281no—
254C4⋊C4—281no—
255C22⋊C4—22no—
256C22⋊C4—41no—
257C22⋊C4—142no—
258C22⋊C4—281no—
259C22⋊C4—281no—
260C22⋊C4Dic1414yescomplemented
261C42—22no—
262C42—142no—
263C2×C7—42no—
264C2×C7#54211yescharacteristic
265C2×C7#54211yescharacteristic
266C2×C7#54411yescharacteristic
267C2×C7#55111yescharacteristic
268C2×C7#55411yescharacteristic
269C2×C7#55411yescharacteristic
270C2×C7H(3,𝔽4)11yescharacteristic
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23C2×Dic1412yes—
278C23C23×D711yescharacteristic
279Q8—284no—
280Q8—284no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—72no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—284no—
321C2×C4C2×Dic1412yes—
322C2×C4C2×Dic1412yes—
323C2×C4#115912yes—
324C7#276311yescharacteristic, complemented, Sylow
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×Dic1411yescharacteristic
333C22C2×#115911yescharacteristic
334C22#654811yescharacteristic
335C22D7×#4111yescharacteristic
336C22#655311yescharacteristic
337C22#655311yescharacteristic
338C22#655411yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—142no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4—281no—
350C4—281no—
351C4—281no—
352C4—281no—
353C2—42no—
354C2#9021011yescharacteristic
355C2#9023011yescharacteristic
356C2#9023411yescharacteristic
357C2#9029411yescharacteristic
358C2#9034911yescharacteristic
359C2#9037411yescharacteristic
360C2#9040111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC24×C7C23
hypercenter#388D7
hypocenterC7#2763
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #89758C2semidirect
3. #89749C2semidirect
10. #89708C2semidirect
11. #89701C2semidirect
12. #89432C2semidirect
13. #89428C2semidirect
14. #89428C2semidirect
15. #89418C2semidirect
79. C7×C22⋊C4Q8semidirect
260. C22⋊C4Dic14semidirect
324. C7#2763semidirect

Maximal subgroups

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

subgroupindex
C7×#3882
#897582
#897492
#897232
#897222
#897192
#897182
#897082
#897012
#894322
#894282
#894182
#27637