Finite Group Databasebeta

← #11736919

Subgroups — #11736919

Order 896.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90254C211yesmaximal, characteristic
2#90232C211yesmaximal, characteristic, complemented
3C7×#435C211yesmaximal, characteristic, complemented
4#89906C211yesmaximal, characteristic, complemented
5#89893C211yesmaximal, characteristic, complemented
6#89886C211yesmaximal, characteristic, complemented
7#89878C211yesmaximal, characteristic, complemented
8#89623C211yesmaximal, characteristic, complemented
9#89619C211yesmaximal, characteristic, complemented
10#89509C211yesmaximal, characteristic, complemented
11#89507C211yesmaximal, characteristic, complemented
12#89495C211yesmaximal, characteristic, complemented
13#89495C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15#89474C211yesmaximal, characteristic, complemented
16C2×C7×M16C2211yescharacteristic
17C22×C7×C8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19#6501—21no—
20#6498—21no—
21#6493C2211yescharacteristic
22#6493C2211yescharacteristic
23#6489C2211yescharacteristic
24#6489C2211yescharacteristic
25#6489C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27#6487C2211yescharacteristic
28#6487C2211yescharacteristic
29C2×#1132C2211yescharacteristic
30#6455—21no—
31#6452—21no—
32#6448—21no—
33#6448—21no—
34#6442—21no—
35#6440—21no—
36C4×D28C2211yescharacteristic
37#6433C2211yescharacteristic
38#6432C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42C7×C22⋊C8C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47#6392C2211yescharacteristic
48#6392C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6391C2211yescharacteristic
53#6391C2211yescharacteristic
54#6391C2211yescharacteristic
55#6391C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3348—71nomaximal, Sylow
60C2×C7×D4D411yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66C2×D28C2311yescharacteristic
67C2×C4×D7—21no—
68C2×Dic14—21no—
69C2×Dic14C2311yescharacteristic
70C7×M16—22no—
71C2×C7×C8—22no—
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4D411yescharacteristic
77C7×C22⋊C4D411yescharacteristic
78C7×C22⋊C4D411yescharacteristic
79C42×C7C2311yescharacteristic
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—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—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105#1132—22no—
106#555—71no—
107#552—71no—
108#501—71no—
109#485—71no—
110#485—71no—
111#482—71no—
112#481—71no—
113#478—71no—
114#475—71no—
115#468—71no—
116#465—71no—
117#440—71no—
118#435D711yescharacteristic, complemented
119#428—71no—
120#427—71no—
121C23×C7C2×D411yescharacteristic
122C23×C7C2×D411yescharacteristic
123C22×D7—21no—
124C7×D4—41no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C22⋊D7—81no—
135C22⋊D7—81no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—42no—
144D28—21no—
145D28—21no—
146C4×D7—42no—
147Dic14—21no—
148Dic14—21no—
149Dic14—81no—
150C7×C8—21no—
151C7×C8—21no—
152C7×C8—21no—
153C7×C8#4112yes—
154C2×SD16—71no—
155C2×M16C2×D711yescharacteristic
156C22×C8C2×D711yescharacteristic
157C4⋊Q8—71no—
158#126—141no—
159#125—71no—
160#125—141no—
161#124—141no—
162#124—141no—
163#123—141no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166#121—71no—
167#121—71no—
168#121—141no—
169C4×Q8—71no—
170C4×D4—71no—
171C4×D4C2×D711yescharacteristic
172#117—71no—
173#117—71no—
174#117—71no—
175#117—141no—
176#117—141no—
177C2×C4⋊C4—71no—
178#107—71no—
179#107—71no—
180#107—71no—
181#107—71no—
182#106—71no—
183#106—71no—
184#106—71no—
185#106—71no—
186#106—71no—
187C4⋊C8C2×D711yescharacteristic
188#103—71no—
189#103—71no—
190#103—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194C22⋊C8C2×D711yescharacteristic
195C22⋊C8C2×D711yescharacteristic
196#97C2×D711yescharacteristic
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7C22×D411yescharacteristic
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7C2×#4111yescharacteristic
212C4×C72−1+411yescharacteristic
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—71no—
221C2×Q8—141no—
222C2×D4—71no—
223C2×D4—141no—
224C2×D4—141no—
225C2×D4D2811yescharacteristic
226C22×C4—141no—
227C22×C4—141no—
228C22×C4C22×D711yescharacteristic
229C22×C4C22×D711yescharacteristic
230SD16—142no—
231M16—22no—
232C2×C8—22no—
233C2×C8C22×D711yescharacteristic
234C2×C8C22×D711yescharacteristic
235C2×C8C22×D711yescharacteristic
236C2×C8C22×D711yescharacteristic
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4D2811yescharacteristic
251C22⋊C4—141no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4D2811yescharacteristic
260C22⋊C4D2811yescharacteristic
261C42—141no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7—41no—
267C2×C7—41no—
268C2×C7#54711yescharacteristic
269C2×C7#57511yescharacteristic
270C2×C7#57711yescharacteristic
271D7—81no—
272C23—141no—
273C23C2×D2811yescharacteristic
274C23C2×D2811yescharacteristic
275Q8—141no—
276Q8—141no—
277Q8—561no—
278D4—41no—
279D4—141no—
280D4—141no—
281D4—561no—
282D4—561no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—282no—
295C2×C4—282no—
296C2×C4C2×D2811yescharacteristic
297C2×C4C2×D2811yescharacteristic
298C2×C4C2×D2811yescharacteristic
299C2×C4C2×D2811yescharacteristic
300C2×C4C23×D711yescharacteristic
301C8—21no—
302C8—21no—
303C8—21no—
304C8#115912yes—
305C7#334811yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—281no—
313C22—281no—
314C22—281no—
315C22C22×D2811yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—281no—
321C4—281no—
322C4—281no—
323C4—281no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4C2×#115911yescharacteristic
328C4#655411yescharacteristic
329C2—41no—
330C2—41no—
331C2—561no—
332C2#9024611yescharacteristic
333C2#9043911yescharacteristic
334C2#9044211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
prime radical#90254C2
hypercenter#435D7
hypocenterC7#3348
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #90232C2semidirect
3. C7×#435C2semidirect
4. #89906C2semidirect
5. #89893C2semidirect
6. #89886C2semidirect
7. #89878C2semidirect
8. #89623C2semidirect
9. #89619C2semidirect
10. #89509C2semidirect
11. #89507C2semidirect
12. #89495C2semidirect
13. #89495C2semidirect
14. #89482C2semidirect
15. #89474C2semidirect
118. #435D7semidirect
305. C7#3348semidirect

Maximal subgroups

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

subgroupindex
#902542
#902322
C7×#4352
#899062
#898932
#898862
#898782
#896232
#896192
#895092
#895072
#894952
#894822
#894742
#33487