Finite Group Databasebeta

← #11737253

Subgroups — #11737253

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#90272C211yesmaximal, characteristic, complemented
2#90254C211yesmaximal, characteristic
3C7×#440C211yesmaximal, characteristic, complemented
4#89946C211yesmaximal, characteristic, complemented
5#89943C211yesmaximal, characteristic, complemented
6#89792C211yesmaximal, characteristic, complemented
7#89781C211yesmaximal, characteristic, complemented
8#89653C211yesmaximal, characteristic, complemented
9#89652C211yesmaximal, characteristic, complemented
10#89587C211yesmaximal, characteristic, complemented
11#89586C211yesmaximal, characteristic, complemented
12#89556C211yesmaximal, characteristic, complemented
13#89549C211yesmaximal, characteristic, complemented
14#89478C211yesmaximal, characteristic, complemented
15#89474C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19#6509C2211yescharacteristic
20#6509C2211yescharacteristic
21#6501C2211yescharacteristic
22#6501C2211yescharacteristic
23#6493—21no—
24#6489—21no—
25C2×#1131C2211yescharacteristic
26C2×C8×D7C2211yescharacteristic
27#6457—21no—
28#6455C2211yescharacteristic
29#6455C2211yescharacteristic
30#6455C2211yescharacteristic
31D7×C4⋊C4C2211yescharacteristic
32#6452—21no—
33#6448—21no—
34#6448—21no—
35#6439—21no—
36C4×D28C2211yescharacteristic
37#6435—21no—
38#6433C2211yescharacteristic
39#6432C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6392C2211yescharacteristic
51#6392C2211yescharacteristic
52#6391C2211yescharacteristic
53#6391C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3348—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×D28D411yescharacteristic
66C2×C4×D7—21no—
67C2×C4×D7C2311yescharacteristic
68C2×C4×D7C2311yescharacteristic
69C2×Dic14—21no—
70C7×SD16—22no—
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C22⋊C4—21no—
77C42×C7—21no—
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4D411yescharacteristic
87(C2×D7)⋊C4D411yescharacteristic
88C4⋊Dic7—21no—
89C4⋊Dic7D411yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104#1131—22no—
105C8×D7—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#440D711yescharacteristic, complemented
118#435—71no—
119#428—71no—
120#427—71no—
121C23×C7—21no—
122C22×D7C2×D411yescharacteristic
123C22×D7C2×D411yescharacteristic
124C7×Q8—21no—
125C7×Q8—21no—
126C7×D4—21no—
127C7×D4—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—42no—
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2411yescharacteristic
135C22⋊D7—81no—
136C22⋊D7—81no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143D28—41no—
144C4×D7—21no—
145C4×D7—21no—
146C4×D7—21no—
147C4×D7—21no—
148C4×D7—42no—
149Dic14—81no—
150C7×C8—21no—
151C7×C8#4112yes—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C2×SD16C2×D711yescharacteristic
155C2×M16—71no—
156C22×C8—71no—
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×Q8C2×D711yescharacteristic
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#107C2×D711yescharacteristic
182#106—71no—
183#106—71no—
184#106—71no—
185#106—71no—
186#106—71no—
187C4⋊C8—71no—
188#103—71no—
189#103—71no—
190#103C2×D711yescharacteristic
191#102—71no—
192#102—71no—
193#102C2×D711yescharacteristic
194C22⋊C8—71no—
195C22⋊C8—71no—
196#97C2×D711yescharacteristic
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7C22×D411yescharacteristic
201C2×D7—21no—
202C2×D7—21no—
203C2×D7—21no—
204C2×D7—21no—
205C2×D7—21no—
206C2×D7—21no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7C2×#4111yescharacteristic
213C4×C72−1+411yescharacteristic
214Dic7—21no—
215Dic7—21no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—141no—
221C2×Q8C22×D711yescharacteristic
222C2×D4—71no—
223C2×D4—141no—
224C2×D4—141no—
225C2×D4C22×D711yescharacteristic
226C22×C4—21no—
227C22×C4—71no—
228C22×C4—71no—
229C22×C4—141no—
230SD16—22no—
231M16—142no—
232C2×C8—71no—
233C2×C8—71no—
234C2×C8—142no—
235C2×C8C22×D711yescharacteristic
236C2×C8C22×D711yescharacteristic
237C4⋊C4—21no—
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⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—71no—
253C22⋊C4—71no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C42—21no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7—81no—
267C2×C7#54711yescharacteristic
268C2×C7#57511yescharacteristic
269C2×C7#57711yescharacteristic
270D7—41no—
271D7—41no—
272C23—21no—
273C23—71no—
274C23—71no—
275Q8—21no—
276Q8—21no—
277Q8—561no—
278D4—21no—
279D4—21no—
280D4—281no—
281D4—561no—
282D4—561no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—282no—
298C2×C4D4×D711yescharacteristic
299C2×C4D4×D711yescharacteristic
300C2×C4C23×D711yescharacteristic
301C8—21no—
302C8—141no—
303C8—141no—
304C8#115712yes—
305C7#334811yescharacteristic, complemented, Sylow
306C22—41no—
307C22—41no—
308C22—41no—
309C22—141no—
310C22—141no—
311C22—141no—
312C22—141no—
313C22—141no—
314C22—141no—
315C22C2×D4×D711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—141no—
322C4—141no—
323C4—281no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4C2×#115711yescharacteristic
328C4#655411yescharacteristic
329C2—81no—
330C2—281no—
331C2—281no—
332C2#9024211yescharacteristic
333C2#9046411yescharacteristic
334C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
prime radical#90254C2
hypercenter#440D7
hypocenterC7#3348
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #90272C2semidirect
3. C7×#440C2semidirect
4. #89946C2semidirect
5. #89943C2semidirect
6. #89792C2semidirect
7. #89781C2semidirect
8. #89653C2semidirect
9. #89652C2semidirect
10. #89587C2semidirect
11. #89586C2semidirect
12. #89556C2semidirect
13. #89549C2semidirect
14. #89478C2semidirect
15. #89474C2semidirect
117. #440D7semidirect
305. C7#3348semidirect

Maximal subgroups

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

subgroupindex
#902722
#902542
C7×#4402
#899462
#899432
#897922
#897812
#896532
#896522
#895872
#895862
#895562
#895492
#894782
#894742
#33487