Finite Group Databasebeta

← #11737206

Subgroups — #11737206

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
1#90261C211yesmaximal, characteristic, complemented
2#90230C211yesmaximal, characteristic
3C7×#440C211yesmaximal, characteristic
4#89936C211yesmaximal, characteristic
5SD16×Dic7C211yesmaximal, characteristic
6#89794C211yesmaximal, characteristic, complemented
7#89780C211yesmaximal, characteristic
8#89650C211yesmaximal, characteristic, complemented
9#89648C211yesmaximal, characteristic, complemented
10#89568C211yesmaximal, characteristic, complemented
11#89563C211yesmaximal, characteristic, complemented
12#89538C211yesmaximal, characteristic
13#89535C211yesmaximal, characteristic
14#89474C211yesmaximal, characteristic, complemented
15#89473C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21D4×Dic7C2211yescharacteristic
22D4×Dic7C2211yescharacteristic
23#6487—21no—
24C2×Dic7⋊C4—21no—
25#6453—21no—
26#6453—21no—
27#6453C2211yescharacteristic
28#6453C2211yescharacteristic
29#6453C2211yescharacteristic
30#6451C2211yescharacteristic
31#6441—21no—
32#6441—21no—
33#6433C2211yescharacteristic
34#6432—21no—
35#6432C2211yescharacteristic
36C4×Dic14—21no—
37C4×Dic14C2211yescharacteristic
38C7×#107C2211yescharacteristic
39C7×#103C2211yescharacteristic
40C7×#102C2211yescharacteristic
41C7×#97C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6392C2211yescharacteristic
47#6392C2211yescharacteristic
48#6391C2211yescharacteristic
49#6391C2211yescharacteristic
50#6389C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3388—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×Dic7—21no—
64C22×Dic7—21no—
65C2×Dic14—21no—
66C2×Dic14—21no—
67C2×Dic14D411yescharacteristic
68C7×SD16Q814yescomplemented
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C22⋊C4—21no—
75C42×C7—21no—
76C42×C7C2311yescharacteristic
77C22⋊Dic7—21no—
78C22⋊Dic7—21no—
79C4⋊Dic7—21no—
80C4⋊Dic7—21no—
81C4⋊Dic7—21no—
82C4⋊Dic7—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7D411yescharacteristic
86C4⋊Dic7C2311yescharacteristic
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⋊C4D411yescharacteristic
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7C2311yescharacteristic
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104#557—71no—
105#554—71no—
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#440D711yescharacteristic
118C4×SD16—71no—
119C23×C7—21no—
120C7×Q8—21no—
121C7×Q8C2×Q812yes—
122C7×D4—21no—
123C7×D4C2×Q812yes—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—81no—
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2411yescharacteristic
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—42no—
136C2×Dic7—81no—
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141Dic14—41no—
142Dic14—42no—
143Dic14—81no—
144C7×C8—21no—
145C7×C8C2×Q812yes—
146C7⋊C8—21no—
147C7⋊C8—21no—
148C2×SD16C2×D711yescharacteristic
149C4⋊Q8—71no—
150C4⋊Q8—71no—
151C4⋊Q8—141no—
152#125—71no—
153#125—71no—
154#125—71no—
155#125—71no—
156#125—141no—
157#125—141no—
158C22⋊Q8—141no—
159C22⋊Q8—141no—
160C22⋊Q8—141no—
161C4×Q8—71no—
162C4×Q8—71no—
163C4×Q8—71no—
164C4×Q8—141no—
165C4×Q8C2×D711yescharacteristic
166C4×D4—71no—
167C4×D4—71no—
168C4×D4C2×D711yescharacteristic
169C2×C4⋊C4—141no—
170#107—71no—
171#107—71no—
172#107—71no—
173#107C2×D711yescharacteristic
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#103C2×D711yescharacteristic
185#102—71no—
186#102—71no—
187#102C2×D711yescharacteristic
188#97—71no—
189#97C2×D711yescharacteristic
190C4×C8—71no—
191C22×C7—22no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7C22×D411yescharacteristic
195C4×C7—22no—
196C4×C7—21no—
197C4×C7—21no—
198C4×C7—41no—
199C4×C7—41no—
200C4×C7C22×Q811yescharacteristic
201C4×C72−1+411yescharacteristic
202Dic7—21no—
203Dic7—21no—
204Dic7—21no—
205Dic7—21no—
206Dic7—41no—
207Dic7—41no—
208Dic7—41no—
209Dic7—41no—
210C2×Q8—71no—
211C2×Q8—141no—
212C2×Q8—141no—
213C2×Q8C22×D711yescharacteristic
214C2×D4C22×D711yescharacteristic
215C22×C4—21no—
216C22×C4—141no—
217C22×C4—141no—
218SD16Dic1414yescomplemented
219C2×C8—71no—
220C2×C8—71no—
221C2×C8C22×D711yescharacteristic
222C2×C8C22×D711yescharacteristic
223C4⋊C4—21no—
224C4⋊C4—71no—
225C4⋊C4—71no—
226C4⋊C4—71no—
227C4⋊C4—71no—
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⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4C22×D711yescharacteristic
244C4⋊C4C22×D711yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C42—21no—
249C42—71no—
250C42—71no—
251C42—141no—
252C42—141no—
253C42C22×D711yescharacteristic
254C2×C7—42no—
255C2×C7D4×Q811yescharacteristic
256C2×C7#57511yescharacteristic
257C2×C7#57711yescharacteristic
258C23—21no—
259Q8—21no—
260Q8—281no—
261Q8—282no—
262Q8—561no—
263Q8C2×Dic1412yes—
264D4—21no—
265D4C2×Dic1412yes—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—71no—
270C2×C4—71no—
271C2×C4—71no—
272C2×C4—71no—
273C2×C4—81no—
274C2×C4—141no—
275C2×C4—141no—
276C2×C4—141no—
277C2×C4—141no—
278C2×C4—282no—
279C2×C4—561no—
280C2×C4D4×D711yescharacteristic
281C2×C4D4×D711yescharacteristic
282C2×C4C23×D711yescharacteristic
283C8—21no—
284C8—141no—
285C8—141no—
286C8C2×Dic1412yes—
287C7#338811yescharacteristic, complemented, Sylow
288C22—22no—
289C22—41no—
290C22—41no—
291C22C2×D4×D711yescharacteristic
292C4—21no—
293C4—21no—
294C4—22no—
295C4—41no—
296C4—41no—
297C4—141no—
298C4—141no—
299C4—141no—
300C4—141no—
301C4—281no—
302C4—281no—
303C4—281no—
304C4—281no—
305C4C22×Dic1411yescharacteristic
306C4#655411yescharacteristic
307C2—42no—
308C2D4×Dic1411yescharacteristic
309C2#9046611yescharacteristic
310C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
prime radicalC2×C7×D4C23
hypercenter#440D7
hypocenterC7#3388
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. #90261C2semidirect
6. #89794C2semidirect
8. #89650C2semidirect
9. #89648C2semidirect
10. #89568C2semidirect
11. #89563C2semidirect
14. #89474C2semidirect
15. #89473C2semidirect
68. C7×SD16Q8semidirect
218. SD16Dic14semidirect
287. C7#3388semidirect

Maximal subgroups

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

subgroupindex
#902612
#902302
C7×#4402
#899362
SD16×Dic72
#897942
#897802
#896502
#896482
#895682
#895632
#895382
#895352
#894742
#894732
#33887