Finite Group Databasebeta

← #11736807

Subgroups — #11736807

Order 896.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90226C211yesmaximal, characteristic
2#90210C211yesmaximal, characteristic, complemented
3C7×#433C211yesmaximal, characteristic, complemented
4#89619C211yesmaximal, characteristic, complemented
5#89619C211yesmaximal, characteristic, complemented
6#89618C211yesmaximal, characteristic, complemented
7#89614C211yesmaximal, characteristic, complemented
8#89509C211yesmaximal, characteristic, complemented
9#89509C211yesmaximal, characteristic, complemented
10#89508C211yesmaximal, characteristic, complemented
11#89507C211yesmaximal, characteristic, complemented
12#89485C211yesmaximal, characteristic, complemented
13#89474C211yesmaximal, characteristic, complemented
14#89466C211yesmaximal, characteristic, complemented
15#89455C211yesmaximal, characteristic, complemented
16C7×#117C2211yescharacteristic
17#6493C2211yescharacteristic
18#6493C2211yescharacteristic
19#6489C2211yescharacteristic
20C2×C4⋊Dic7C2211yescharacteristic
21#6457—21no—
22#6457—21no—
23#6453—21no—
24#6451—21no—
25#6447—21no—
26#6447—21no—
27#6441—21no—
28#6441—21no—
29#6437C2211yescharacteristic
30C4×D28C2211yescharacteristic
31C4×D28C2211yescharacteristic
32#6433C2211yescharacteristic
33#6433C2211yescharacteristic
34#6433C2211yescharacteristic
35#6432C2211yescharacteristic
36#6432C2211yescharacteristic
37C7×C4⋊C8C2211yescharacteristic
38C7×C4⋊C8C2211yescharacteristic
39C7×C22⋊C8C2211yescharacteristic
40C7×C22⋊C8C2211yescharacteristic
41C7×#97C2211yescharacteristic
42C4×C7×C8C2211yescharacteristic
43#6395C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6392C2211yescharacteristic
52#6392C2211yescharacteristic
53#6392C2211yescharacteristic
54#6392C2211yescharacteristic
55#6391C2211yescharacteristic
56#6391C2211yescharacteristic
57#6391C2211yescharacteristic
58#6391C2211yescharacteristic
59#3241—71nomaximal, Sylow
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7—21no—
62C2×C22⋊D7—21no—
63C22×Dic7—21no—
64C2×D28—21no—
65C2×D28—21no—
66C2×D28C2311yescharacteristic
67C2×D28C2311yescharacteristic
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×Dic14—21no—
71C2×Dic14—21no—
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4D411yescharacteristic
77C7×C4⋊C4D411yescharacteristic
78C7×C22⋊C4D411yescharacteristic
79C7×C22⋊C4D411yescharacteristic
80C42×C7C2311yescharacteristic
81C42×C7C2311yescharacteristic
82C22⋊Dic7—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103#544—71no—
104#540—71no—
105#501—71no—
106#499—71no—
107#490—71no—
108#486—71no—
109#482—71no—
110#482—71no—
111#481—71no—
112#480—71no—
113#478—71no—
114#478—71no—
115#476—71no—
116#474—71no—
117#433D711yescharacteristic, complemented
118C23×C7C2×D411yescharacteristic
119C22×D7—21no—
120C22×D7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7C2×D411yescharacteristic
124C2×C4×C7C2×D411yescharacteristic
125C2×C4×C7C2×D411yescharacteristic
126C2×C4×C7C2×D411yescharacteristic
127C2×C4×C7C2×D411yescharacteristic
128C2×C4×C7C2411yescharacteristic
129C22⋊D7—81no—
130C22⋊D7—81no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—81no—
138D28—21no—
139D28—21no—
140D28—21no—
141D28—21no—
142D28—81no—
143D28—81no—
144C4×D7—81no—
145C4×D7—81no—
146Dic14—81no—
147Dic14—81no—
148C7×C8—21no—
149C7×C8—21no—
150C7×C8—21no—
151C7×C8—21no—
152C4⋊Q8—71no—
153C4⋊Q8—71no—
154C4⋊Q8—141no—
155#127—71no—
156#125—71no—
157#125—71no—
158#125—71no—
159#125—141no—
160#123—141no—
161#123—141no—
162C22⋊Q8—141no—
163C22⋊Q8—141no—
164#121—71no—
165#121—71no—
166#121—141no—
167#121—141no—
168C4×D4—71no—
169C4×D4—71no—
170#117—71no—
171#117C2×D711yescharacteristic
172C2×C4⋊C4—71no—
173#107—71no—
174#107—71no—
175#107—71no—
176#107—71no—
177#106—71no—
178#106—71no—
179#106—71no—
180#106—71no—
181C4⋊C8C2×D711yescharacteristic
182C4⋊C8C2×D711yescharacteristic
183#102—71no—
184#102—71no—
185#102—71no—
186#102—71no—
187#102—71no—
188#102—71no—
189#102—71no—
190#102—71no—
191C22⋊C8C2×D711yescharacteristic
192C22⋊C8C2×D711yescharacteristic
193#97C2×D711yescharacteristic
194C4×C8C2×D711yescharacteristic
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7C22×D411yescharacteristic
199C2×D7—41no—
200C2×D7—41no—
201C2×D7—41no—
202C2×D7—41no—
203C2×D7—41no—
204C2×D7—41no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C72−1+411yescharacteristic
211C4×C72−1+411yescharacteristic
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218C2×Q8—141no—
219C2×Q8—141no—
220C2×D4—71no—
221C2×D4—71no—
222C2×D4—141no—
223C2×D4—141no—
224C2×D4—141no—
225C2×D4—141no—
226C22×C4—141no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4C22×D711yescharacteristic
230C2×C8C22×D711yescharacteristic
231C2×C8C22×D711yescharacteristic
232C2×C8C22×D711yescharacteristic
233C2×C8C22×D711yescharacteristic
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4D2811yescharacteristic
250C4⋊C4D2811yescharacteristic
251C22⋊C4—141no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4D2811yescharacteristic
257C22⋊C4D2811yescharacteristic
258C42—141no—
259C42C22×D711yescharacteristic
260C42C22×D711yescharacteristic
261C2×C7—41no—
262C2×C7#53611yescharacteristic
263C2×C7#57611yescharacteristic
264C2×C7#57711yescharacteristic
265D7—81no—
266D7—81no—
267C23—141no—
268C23—141no—
269C23C2×D2811yescharacteristic
270Q8—561no—
271Q8—561no—
272D4—141no—
273D4—141no—
274D4—141no—
275D4—141no—
276D4—561no—
277D4—561no—
278D4—561no—
279D4—561no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—561no—
289C2×C4—561no—
290C2×C4—561no—
291C2×C4C2×D2811yescharacteristic
292C2×C4C2×D2811yescharacteristic
293C2×C4C2×D2811yescharacteristic
294C2×C4C2×D2811yescharacteristic
295C2×C4C2×D2811yescharacteristic
296C2×C4C23×D711yescharacteristic
297C8—21no—
298C8—21no—
299C8—21no—
300C8—21no—
301C7#324111yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22—21no—
305C22—281no—
306C22—281no—
307C22—281no—
308C22—281no—
309C22—281no—
310C22—281no—
311C22C22×D2811yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—281no—
318C4—281no—
319C4—281no—
320C4—281no—
321C4—281no—
322C4—281no—
323C4#655411yescharacteristic
324C4#655411yescharacteristic
325C2—41no—
326C2—561no—
327C2—561no—
328C2#9021711yescharacteristic
329C2#9044211yescharacteristic
330C2#9044311yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#433C2
socleC22×C7C22×D4
prime radical#90226C2
hypercenter#433D7
hypocenterC7#3241
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #90210C2semidirect
3. C7×#433C2semidirect
4. #89619C2semidirect
5. #89619C2semidirect
6. #89618C2semidirect
7. #89614C2semidirect
8. #89509C2semidirect
9. #89509C2semidirect
10. #89508C2semidirect
11. #89507C2semidirect
12. #89485C2semidirect
13. #89474C2semidirect
14. #89466C2semidirect
15. #89455C2semidirect
117. #433D7semidirect
301. C7#3241semidirect

Maximal subgroups

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

subgroupindex
#902262
#902102
C7×#4332
#896192
#896182
#896142
#895092
#895082
#895072
#894852
#894742
#894662
#894552
#32417