Finite Group Databasebeta

← #11741077

Subgroups — #11741077

Order 896.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#554C211yesmaximal, characteristic
2#90258C211yesmaximal, characteristic, complemented
3#89866C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89837C211yesmaximal, characteristic, complemented
6#89836C211yesmaximal, characteristic, complemented
7#89835C211yesmaximal, characteristic, complemented
8#89835C211yesmaximal, characteristic, complemented
9#89796C211yesmaximal, characteristic, complemented
10#89790C211yesmaximal, characteristic
11#89790C211yesmaximal, characteristic
12C4×#1142C211yesmaximal, characteristic
13#89781C211yesmaximal, characteristic
14#89781C211yesmaximal, characteristic
15#89780C211yesmaximal, characteristic
16C7×C4⋊Q8C2211yescharacteristic
17C7×#125C2211yescharacteristic
18C7×#125C2211yescharacteristic
19C7×C22⋊Q8—21no—
20C7×C22⋊Q8—21no—
21C7×C22⋊Q8—21no—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×D4C2211yescharacteristic
24C4×C7×D4C2211yescharacteristic
25C4×C7×D4C2211yescharacteristic
26C2×C7×C4⋊C4—21no—
27#6507—21no—
28C2×#1142C2211yescharacteristic
29#6452—21no—
30#6452—21no—
31#6450—21no—
32#6433C2211yescharacteristic
33#6433C2211yescharacteristic
34#6432C2211yescharacteristic
35C4×Dic14C2211yescharacteristic
36C4×Dic14C2211yescharacteristic
37C4×Dic14C2211yescharacteristic
38#6406C2211yescharacteristic
39#6406C2211yescharacteristic
40#6406C2211yescharacteristic
41#6384C2211yescharacteristic
42#6384C2211yescharacteristic
43#6384C2211yescharacteristic
44#6382C2211yescharacteristic
45#6382C2211yescharacteristic
46#6382C2211yescharacteristic
47#6382C2211yescharacteristic
48#6382C2211yescharacteristic
49#6381C2211yescharacteristic
50#6381C2211yescharacteristic
51#6381C2211yescharacteristic
52#6381C2211yescharacteristic
53C4⋊(C7⋊C8)C2211yescharacteristic
54C4⋊(C7⋊C8)C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56#6377C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3388—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8D411yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C2×Dic14—21no—
67C2×Dic14C2311yescharacteristic
68C7×C4⋊C4—21no—
69C7×C4⋊C4—21no—
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4D411yescharacteristic
74C7×C4⋊C4D411yescharacteristic
75C7×C4⋊C4D411yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C7×C22⋊C4—21no—
82C42×C7C2311yescharacteristic
83C42×C7C2311yescharacteristic
84C42×C7C2311yescharacteristic
85#1142Q814yescomplemented
86C4⋊Dic7—21no—
87C4⋊Dic7C2311yescharacteristic
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C2×C7⋊C8C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104#557—71no—
105#554D711yescharacteristic
106#501—71no—
107#501—71no—
108#499—71no—
109#479—71no—
110#479—71no—
111#478—71no—
112#478—71no—
113#476—71no—
114#475—71no—
115#446—71no—
116#440—71no—
117#440—71no—
118C4×SD16—71no—
119C23×C7—21no—
120C7×Q8—41no—
121C7×Q8—81no—
122C7×D4—21no—
123C7×D4C2×Q812yes—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—42no—
128C2×C4×C7—81no—
129C2×C4×C7—81no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141Dic14—21no—
142Dic14—42no—
143Dic14C2×Q812yes—
144C7⋊C8—21no—
145C7⋊C8—21no—
146C7⋊C8—21no—
147C7⋊C8C2×Q812yes—
148C2×SD16—71no—
149C4⋊Q8—71no—
150C4⋊Q8—141no—
151C4⋊Q8C2×D711yescharacteristic
152#125—71no—
153#125—71no—
154#125—141no—
155#125—141no—
156#125C2×D711yescharacteristic
157#125C2×D711yescharacteristic
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C4×Q8—71no—
162C4×Q8—71no—
163C4×Q8—71no—
164C4×Q8—141no—
165C4×Q8C2×D711yescharacteristic
166C4×D4C2×D711yescharacteristic
167C4×D4C2×D711yescharacteristic
168C4×D4C2×D711yescharacteristic
169C2×C4⋊C4—21no—
170#107—71no—
171#107—71no—
172#107—71no—
173#107—71no—
174#106—71no—
175#106—71no—
176#106—71no—
177#106—71no—
178#106—71no—
179C4⋊C8—71no—
180C4⋊C8—71no—
181C4⋊C8—71no—
182#103—71no—
183#103—71no—
184#103—71no—
185#102—71no—
186#102—71no—
187#102—71no—
188#97—71no—
189#97—71no—
190C4×C8—71no—
191C22×C7—22no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7C22×D411yescharacteristic
195C4×C7—21no—
196C4×C7—21no—
197C4×C7—21no—
198C4×C7—21no—
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7C22×Q811yescharacteristic
205C4×C72−1+411yescharacteristic
206Dic7—22no—
207Dic7—41no—
208Dic7—41no—
209Dic7—41no—
210C2×Q8—21no—
211C2×Q8—71no—
212C2×Q8—141no—
213C2×Q8C22⋊D711yescharacteristic
214C2×D4C22×D711yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218SD16—74no—
219C2×C8—71no—
220C2×C8—71no—
221C2×C8—71no—
222C2×C8—71no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—71no—
229C4⋊C4—71no—
230C4⋊C4—71no—
231C4⋊C4—141no—
232C4⋊C4—141no—
233C4⋊C4—141no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4C22⋊D711yescharacteristic
240C4⋊C4C22⋊D711yescharacteristic
241C4⋊C4C22⋊D711yescharacteristic
242C4⋊C4C22×D711yescharacteristic
243C4⋊C4C22×D711yescharacteristic
244C4⋊C4C22×D711yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—21no—
247C22⋊C4—21no—
248C42—141no—
249C42—141no—
250C42—141no—
251C42C22×D711yescharacteristic
252C42C22×D711yescharacteristic
253C42C22×D711yescharacteristic
254C2×C7—42no—
255C2×C7D4×Q811yescharacteristic
256C2×C7#57511yescharacteristic
257C2×C7#57711yescharacteristic
258C23—21no—
259Q8—41no—
260Q8—72no—
261Q8—81no—
262Q8—141no—
263Q8—282no—
264D4—21no—
265D4Q8×D712yes—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—141no—
273C2×C4—141no—
274C2×C4—141no—
275C2×C4—141no—
276C2×C4C2×C22⋊D711yescharacteristic
277C2×C4C2×C22⋊D711yescharacteristic
278C2×C4C2×C22⋊D711yescharacteristic
279C2×C4C2×C22⋊D711yescharacteristic
280C2×C4C2×C22⋊D711yescharacteristic
281C2×C4C2×C22⋊D711yescharacteristic
282C2×C4C23×D711yescharacteristic
283C8—72no—
284C8—141no—
285C8—141no—
286C8—141no—
287C7#338811yescharacteristic, complemented, Sylow
288C22—22no—
289C22—41no—
290C22—41no—
291C22C22×C22⋊D711yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—142no—
302C4—281no—
303C4—281no—
304C4—281no—
305C4C2×Q8×D711yescharacteristic
306C4#655111yescharacteristic
307C2—42no—
308C2Q8×C22⋊D711yescharacteristic
309C2#9051311yescharacteristic
310C2#9052711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#554C2
socleC22×C7C22×D4
prime radicalC2×C7×D4C23
hypercenter#554D7
hypocenterC7#3388
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #90258C2semidirect
3. #89866C2semidirect
4. #89852C2semidirect
5. #89837C2semidirect
6. #89836C2semidirect
7. #89835C2semidirect
8. #89835C2semidirect
9. #89796C2semidirect
85. #1142Q8semidirect
287. C7#3388semidirect

Maximal subgroups

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

subgroupindex
C7×#5542
#902582
#898662
#898522
#898372
#898362
#898352
#897962
#897902
C4×#11422
#897812
#897802
#33887