Finite Group Databasebeta

← #11738604

Subgroups — #11738604

Order 896.

360 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90376C211yesmaximal, characteristic, complemented
2D4×Dic14C211yesmaximal, characteristic
3C7×#478C211yesmaximal, characteristic
4#89837C211yesmaximal, characteristic, complemented
5#89790C211yesmaximal, characteristic
6#89647C211yesmaximal, characteristic, complemented
7#89646C211yesmaximal, characteristic, complemented
8#89628C211yesmaximal, characteristic, complemented
9#89626C211yesmaximal, characteristic, complemented
10#89622C211yesmaximal, characteristic, complemented
11#89600C211yesmaximal, characteristic, complemented
12#89538C211yesmaximal, characteristic
13#89531C211yesmaximal, characteristic
14#89530C211yesmaximal, characteristic
15#89529C211yesmaximal, characteristic
16C22×Dic14—21no—
17C7×#125C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19D4×Dic7C2211yescharacteristic
20D4×Dic7C2211yescharacteristic
21C2×#1142C2211yescharacteristic
22#6487—21no—
23#6453C2211yescharacteristic
24#6452—21no—
25#6452—21no—
26#6451—21no—
27#6451C2211yescharacteristic
28#6451C2211yescharacteristic
29#6451C2211yescharacteristic
30#6450—21no—
31#6450C2211yescharacteristic
32#6450C2211yescharacteristic
33#6441—21no—
34#6441—21no—
35#6432C2211yescharacteristic
36C4×Dic14C2211yescharacteristic
37C4×Dic14C2211yescharacteristic
38C7×#107C2211yescharacteristic
39C7×#106C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×#102C2211yescharacteristic
42C7×#102C2211yescharacteristic
43#6406C2211yescharacteristic
44#6392C2211yescharacteristic
45#6392C2211yescharacteristic
46#6390C2211yescharacteristic
47#6390C2211yescharacteristic
48#6389C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6384C2211yescharacteristic
53#6382C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3386—71nomaximal, Sylow
60C2×C7×D4C2311yescharacteristic
61C22×C4×C7—21no—
62C22×Dic7—21no—
63C22×Dic7—21no—
64C2×Dic14—21no—
65C2×Dic14—21no—
66C2×Dic14—21no—
67C2×Dic14—21no—
68C2×Dic14—44no—
69C2×Dic14D411yescharacteristic
70C2×Dic14C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C22⋊C4—21no—
79C42×C7C2311yescharacteristic
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82#1142Q814yescomplemented
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7D411yescharacteristic
86C4⋊Dic7C2311yescharacteristic
87C4⋊Dic7C2311yescharacteristic
88C4⋊Dic7C2311yescharacteristic
89Dic7⋊C4—21no—
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×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#557—71no—
106D4×Q8—71no—
107#501—71no—
108#501—71no—
109#500—71no—
110#478—71no—
111#478D711yescharacteristic
112#477—71no—
113#477—71no—
114#475—71no—
115#474—71no—
116#446—71no—
117#440—71no—
118#440—71no—
119C4×SD16—71no—
120C23×C7—21no—
121C7×D4—21no—
122C7×D4C2×Q812yes—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—42no—
127C2×C4×C7C2×D411yescharacteristic
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7C2411yescharacteristic
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—42no—
135C2×Dic7—42no—
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140Dic14—21no—
141Dic14—22no—
142Dic14—42no—
143Dic14—42no—
144Dic14—42no—
145Dic14—42no—
146Dic14C2×Q812yescomplemented
147C7×C8—21no—
148C7×C8—21no—
149C7⋊C8—21no—
150C7⋊C8C2×Q812yescomplemented
151C22×Q8—141no—
152C2×SD16—71no—
153C4⋊Q8—71no—
154C4⋊Q8—71no—
155C4⋊Q8—71no—
156C4⋊Q8—71no—
157C4⋊Q8—141no—
158#125—71no—
159#125—141no—
160#125—141no—
161#125C2×D711yescharacteristic
162C22⋊Q8—141no—
163C22⋊Q8—141no—
164C22⋊Q8—141no—
165C4×Q8—71no—
166C4×Q8—71no—
167C4×Q8—71no—
168C4×Q8—71no—
169C4×Q8—141no—
170C4×D4—71no—
171C4×D4—71no—
172C4×D4C2×D711yescharacteristic
173#107—71no—
174#107—71no—
175#107—71no—
176#107—71no—
177#107—71no—
178#107C2×D711yescharacteristic
179#106—71no—
180#106—71no—
181#106C2×D711yescharacteristic
182C4⋊C8—71no—
183C4⋊C8—71no—
184C4⋊C8C2×D711yescharacteristic
185#103—71no—
186#103—71no—
187#103—71no—
188#102—71no—
189#102C2×D711yescharacteristic
190#102C2×D711yescharacteristic
191#97—71no—
192#97—71no—
193C4×C8—71no—
194C22×C7—22no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7C22×D411yescharacteristic
198C4×C7—21no—
199C4×C7—21no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7C22×Q811yescharacteristic
204C4×C72−1+411yescharacteristic
205Dic7—22no—
206Dic7—21no—
207Dic7—21no—
208Dic7—21no—
209Dic7—41no—
210Dic7—41no—
211Dic7—41no—
212Dic7AGL(1,ℤ/8)12yes—
213C2×Q8—71no—
214C2×Q8—71no—
215C2×Q8—141no—
216C2×Q8—141no—
217C2×Q8—141no—
218C2×Q8—141no—
219C2×Q8—284no—
220C2×D4C22×D711yescharacteristic
221C22×C4—21no—
222C22×C4—141no—
223C22×C4—141no—
224SD16—74no—
225C2×C8—71no—
226C2×C8—71no—
227C2×C8C22×D711yescharacteristic
228C2×C8C22×D711yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—71no—
232C4⋊C4—71no—
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4C22×D711yescharacteristic
247C4⋊C4C22×D711yescharacteristic
248C4⋊C4C22×D711yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—141no—
251C22⋊C4—141no—
252C42—71no—
253C42—71no—
254C42—141no—
255C42—141no—
256C42—141no—
257C42C22×D711yescharacteristic
258C2×C7—42no—
259C2×C7D4×Q811yescharacteristic
260C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
261C2×C7#57811yescharacteristic
262C23—21no—
263Q8—72no—
264Q8—142no—
265Q8—141no—
266Q8—282no—
267Q8—282no—
268Q8—282no—
269Q8—282no—
270D4—21no—
271D4Q8×D712yes—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—71no—
277C2×C4—71no—
278C2×C4—71no—
279C2×C4—71no—
280C2×C4—141no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—282no—
285C2×C4—282no—
286C2×C4D4×D711yescharacteristic
287C2×C4D4×D711yescharacteristic
288C2×C4C23×D711yescharacteristic
289C8—21no—
290C8—21no—
291C8—72no—
292C8—141no—
293C7#338611yescharacteristic, complemented, Sylow
294C22—22no—
295C22—41no—
296C22—41no—
297C22C2×D4×D711yescharacteristic
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—72no—
304C4—141no—
305C4—141no—
306C4—141no—
307C4—142no—
308C4—281no—
309C4—281no—
310C4—281no—
311C4C2×Q8×D711yescharacteristic
312C4#655411yescharacteristic
313C2—42no—
314C2#9032311yescharacteristic
315C2#9046211yescharacteristic
316C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#478C2
socleC22×C7C22×D4
prime radicalC2×C7×D4C23
hypercenter#478D7
hypocenterC7#3386
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #90376C2semidirect
4. #89837C2semidirect
6. #89647C2semidirect
7. #89646C2semidirect
8. #89628C2semidirect
9. #89626C2semidirect
10. #89622C2semidirect
11. #89600C2semidirect
82. #1142Q8semidirect
146. Dic14C2×Q8semidirect
150. C7⋊C8C2×Q8semidirect
293. C7#3386semidirect

Maximal subgroups

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

subgroupindex
#903762
D4×Dic142
C7×#4782
#898372
#897902
#896472
#896462
#896282
#896262
#896222
#896002
#895382
#895312
#895302
#895292
#33867