Finite Group Databasebeta

← #11741163

Subgroups — #11741163

Order 896.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#559C211yesmaximal, characteristic
2#90258C211yesmaximal, characteristic, complemented
3#89847C211yesmaximal, characteristic
4#89843C211yesmaximal, characteristic
5#89828C211yesmaximal, characteristic, complemented
6#89827C211yesmaximal, characteristic, complemented
7#89826C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89802C211yesmaximal, characteristic, complemented
10#89796C211yesmaximal, characteristic, complemented
11#89794C211yesmaximal, characteristic, complemented
12#89790C211yesmaximal, characteristic, complemented
13C4×#1142C211yesmaximal, characteristic, complemented
14#89781C211yesmaximal, characteristic, complemented
15#89780C211yesmaximal, characteristic, complemented
16C7×#127C2211yescharacteristic
17C7×#124—21no—
18C7×#124C2211yescharacteristic
19C7×#124C2211yescharacteristic
20C7×#123—21no—
21C7×#123—21no—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×Q8C2211yescharacteristic
24C4×C7×D4—21no—
25C4×C7×D4C2211yescharacteristic
26C4×C7×D4C2211yescharacteristic
27#6507—21no—
28C2×#1144C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31#6452—21no—
32#6452—21no—
33#6450—21no—
34#6433C2211yescharacteristic
35#6433C2211yescharacteristic
36#6432C2211yescharacteristic
37C4×Dic14C2211yescharacteristic
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40#6409C2211yescharacteristic
41#6409C2211yescharacteristic
42#6409C2211yescharacteristic
43#6406C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6406C2211yescharacteristic
49#6384C2211yescharacteristic
50#6382C2211yescharacteristic
51#6382C2211yescharacteristic
52#6381C2211yescharacteristic
53C4⋊(C7⋊C8)C2211yescharacteristic
54C4⋊(C7⋊C8)C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56#6377C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3371—71nomaximal, Sylow
60C2×C7×Q8D411yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4—21no—
64C2×C7×D4C2311yescharacteristic
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C2×Dic14—21no—
69C2×Dic14C2311yescharacteristic
70C7×C4⋊C4—21no—
71C7×C4⋊C4D411yescharacteristic
72C7×C4⋊C4D411yescharacteristic
73C7×C4⋊C4D411yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C22⋊C4—21no—
76C7×C22⋊C4—21no—
77C7×C22⋊C4—21no—
78C7×C22⋊C4—21no—
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C42×C7—21no—
82C42×C7C2311yescharacteristic
83C42×C7C2311yescharacteristic
84C42×C7C2311yescharacteristic
85#1144—22no—
86#1142—22no—
87#1142—22no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
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
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#559D711yescharacteristic
107#557—71no—
108#490—71no—
109#489—71no—
110#488—71no—
111#479—71no—
112#478—71no—
113#476—71no—
114#464—71no—
115#464—71no—
116#460—71no—
117#446—71no—
118#441—71no—
119#440—71no—
120C4×SD16—71no—
121C23×C7—21no—
122C23×C7—21no—
123C7×Q8—21no—
124C7×Q8—21no—
125C7×Q8—41no—
126C7×D4—21no—
127C7×D4—21no—
128C7×D4—21no—
129C7×D4—21no—
130C7×D4—42no—
131C7×D4—81no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—81no—
135C2×C4×C7—81no—
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2411yescharacteristic
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147Dic14—21no—
148Dic14—42no—
149Dic14#4112yes—
150C7⋊C8—21no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C2×Q16—71no—
155C2×SD16—71no—
156C2×SD16—71no—
157C4⋊Q8—71no—
158C4⋊Q8—141no—
159#127C2×D711yescharacteristic
160#125—71no—
161#125—71no—
162#125—141no—
163#125—141no—
164#124—21no—
165#124C2×D711yescharacteristic
166#124C2×D711yescharacteristic
167#123—21no—
168#123—21no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—71no—
172C4×Q8—141no—
173C4×Q8C2×D711yescharacteristic
174C4×Q8C2×D711yescharacteristic
175C4×D4—21no—
176C4×D4C2×D711yescharacteristic
177C4×D4C2×D711yescharacteristic
178#107—71no—
179#106—71no—
180#106—71no—
181C4⋊C8—71no—
182C4⋊C8—71no—
183C4⋊C8—71no—
184#103—71no—
185#103—71no—
186#103—71no—
187#103—71no—
188#102—71no—
189#102—71no—
190#102—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194#97—71no—
195#97—71no—
196C4×C8—71no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7C22×D411yescharacteristic
204C4×C7—21no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7C2×#4111yescharacteristic
213C4×C72−1+411yescharacteristic
214Dic7—22no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218C2×Q8—71no—
219C2×Q8—141no—
220C2×Q8C22⋊D711yescharacteristic
221C2×Q8C22×D711yescharacteristic
222C2×D4—21no—
223C2×D4—21no—
224C2×D4C22×D711yescharacteristic
225C2×D4C22×D711yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228Q16—142no—
229SD16—142no—
230SD16—142no—
231C2×C8—71no—
232C2×C8—71no—
233C2×C8—71no—
234C2×C8—71no—
235C4⋊C4—21no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4C22⋊D711yescharacteristic
248C4⋊C4C22⋊D711yescharacteristic
249C4⋊C4C22⋊D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C42—21no—
258C42—141no—
259C42—141no—
260C42—141no—
261C42C22×D711yescharacteristic
262C42C22×D711yescharacteristic
263C42C22×D711yescharacteristic
264C2×C7—81no—
265C2×C7—81no—
266C2×C7#54811yescharacteristic
267C2×C7#57511yescharacteristic
268C2×C7#57711yescharacteristic
269C23—21no—
270C23—21no—
271Q8—21no—
272Q8—21no—
273Q8—41no—
274Q8—72no—
275Q8—141no—
276Q8—282no—
277D4—21no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—81no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4C2×C22⋊D711yescharacteristic
292C2×C4C2×C22⋊D711yescharacteristic
293C2×C4C2×C22⋊D711yescharacteristic
294C2×C4C2×C22⋊D711yescharacteristic
295C2×C4C2×C22⋊D711yescharacteristic
296C2×C4C2×C22⋊D711yescharacteristic
297C2×C4C23×D711yescharacteristic
298C8—141no—
299C8—141no—
300C8—141no—
301C8—141no—
302C7#337111yescharacteristic, complemented, Sylow
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C22×C22⋊D711yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—142no—
319C4—281no—
320C4—281no—
321C4—281no—
322C4D7×#4111yescharacteristic
323C4#655411yescharacteristic
324C2—81no—
325C2—81no—
326C2#9051311yescharacteristic
327C2#9052211yescharacteristic
328C2#9052711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#559C2
socleC22×C7C22×D4
prime radicalC7×#127C22
hypercenter#559D7
hypocenterC7#3371
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #90258C2semidirect
5. #89828C2semidirect
6. #89827C2semidirect
7. #89826C2semidirect
8. #89826C2semidirect
9. #89802C2semidirect
10. #89796C2semidirect
11. #89794C2semidirect
12. #89790C2semidirect
13. C4×#1142C2semidirect
14. #89781C2semidirect
15. #89780C2semidirect
302. C7#3371semidirect

Maximal subgroups

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

subgroupindex
C7×#5592
#902582
#898472
#898432
#898282
#898272
#898262
#898022
#897962
#897942
#897902
C4×#11422
#897812
#897802
#33717