Finite Group Databasebeta

← #11738608

Subgroups — #11738608

Order 896.

348 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90381C211yesmaximal, characteristic, complemented
2#90234C211yesmaximal, characteristic, complemented
3C7×#478C211yesmaximal, characteristic, complemented
4#89837C211yesmaximal, characteristic, complemented
5#89786C211yesmaximal, characteristic
6#89663C211yesmaximal, characteristic, complemented
7#89650C211yesmaximal, characteristic, complemented
8#89643C211yesmaximal, characteristic, complemented
9#89628C211yesmaximal, characteristic, complemented
10#89618C211yesmaximal, characteristic, complemented
11#89600C211yesmaximal, characteristic, complemented
12#89557C211yesmaximal, characteristic
13#89538C211yesmaximal, characteristic, complemented
14#89531C211yesmaximal, characteristic, complemented
15#89528C211yesmaximal, characteristic
16C7×#125C2211yescharacteristic
17C4×C7×D4C2211yescharacteristic
18D4×Dic7C2211yescharacteristic
19D4×Dic7C2211yescharacteristic
20C2×#1141C2211yescharacteristic
21C2×C4⋊Dic7—21no—
22#6487—21no—
23#6460—21no—
24#6459—21no—
25#6459—21no—
26#6456C2211yescharacteristic
27#6456C2211yescharacteristic
28D7×C4⋊C4—21no—
29#6453C2211yescharacteristic
30#6453C2211yescharacteristic
31#6453C2211yescharacteristic
32#6451C2211yescharacteristic
33#6441—21no—
34#6441—21no—
35C4×D28C2211yescharacteristic
36#6432C2211yescharacteristic
37C4×Dic14C2211yescharacteristic
38C7×#107C2211yescharacteristic
39C7×#106C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×#102C2211yescharacteristic
42C7×#102C2211yescharacteristic
43#6406C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46#6392C2211yescharacteristic
47#6392C2211yescharacteristic
48#6389C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6383C2211yescharacteristic
53#6382C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3389—71nomaximal, Sylow
60C2×C7×D4C2311yescharacteristic
61C22×C4×C7—21no—
62C22×Dic7—21no—
63C22×Dic7—21no—
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×Dic14—21no—
69C2×Dic14—21no—
70C2×Dic14D411yescharacteristic
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#1141Q814yescomplemented
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7D411yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7C2311yescharacteristic
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104#554—71no—
105#554—71no—
106#501—71no—
107#501—71no—
108#499—71no—
109#478—71no—
110#478—71no—
111#478—71no—
112#478D711yescharacteristic, complemented
113#474—71no—
114#474—71no—
115#446—71no—
116#442—71no—
117#442—71no—
118C4×D8—71no—
119C23×C7—21no—
120C22×D7—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—81no—
134C2×Dic7—81no—
135C2×Dic7C2×D411yescharacteristic
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139D28—21no—
140D28C2×Q812yes—
141C4×D7—42no—
142C4×D7—81no—
143C4×D7—81no—
144Dic14—41no—
145Dic14—81no—
146Dic14—81no—
147C7×C8—21no—
148C7×C8—21no—
149C7⋊C8—21no—
150C7⋊C8C2×Q812yes—
151C2×D8—71no—
152C4⋊Q8—71no—
153C4⋊Q8—71no—
154#125—71no—
155#125—71no—
156#125—71no—
157#125C2×D711yescharacteristic
158C22⋊Q8—141no—
159C22⋊Q8—141no—
160C22⋊Q8—141no—
161C22⋊Q8—141no—
162C22⋊Q8—141no—
163C22⋊Q8—141no—
164C4×Q8—71no—
165C4×D4—71no—
166C4×D4—71no—
167C4×D4—71no—
168C4×D4—71no—
169C4×D4—71no—
170C4×D4C2×D711yescharacteristic
171C2×C4⋊C4—141no—
172C2×C4⋊C4—141no—
173#107—71no—
174#107—71no—
175#107—71no—
176#107—71no—
177#107C2×D711yescharacteristic
178#106—71no—
179#106—71no—
180#106—71no—
181#106C2×D711yescharacteristic
182C4⋊C8—71no—
183C4⋊C8—71no—
184C4⋊C8C2×D711yescharacteristic
185#102—71no—
186#102—71no—
187#102—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
198C2×D7—22no—
199C2×D7—41no—
200C2×D7—41no—
201C4×C7—21no—
202C4×C7—21no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7C22×Q811yescharacteristic
207C4×C72−1+411yescharacteristic
208Dic7—21no—
209Dic7—21no—
210Dic7—21no—
211Dic7—21no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215C2×Q8—71no—
216C2×Q8—141no—
217C2×Q8—141no—
218C2×D4—71no—
219C2×D4C22×D711yescharacteristic
220C22×C4—21no—
221C22×C4—141no—
222C22×C4—141no—
223C22×C4—141no—
224C22×C4—141no—
225C22×C4—141no—
226D8—74no—
227C2×C8—71no—
228C2×C8—71no—
229C2×C8C22×D711yescharacteristic
230C2×C8C22×D711yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—71no—
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
250C22⋊C4—21no—
251C22⋊C4—141no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C42—71no—
257C42—71no—
258C42C22×D711yescharacteristic
259C2×C7—42no—
260C2×C7D4×Q811yescharacteristic
261C2×C7#57511yescharacteristic
262C2×C7#57611yescharacteristic
263D7—42no—
264C23—21no—
265C23—141no—
266Q8—281no—
267Q8—561no—
268Q8—561no—
269D4—21no—
270D4—72no—
271D4—141no—
272D4Q8×D712yes—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—71no—
278C2×C4—71no—
279C2×C4—71no—
280C2×C4—71no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—282no—
285C2×C4—561no—
286C2×C4—561no—
287C2×C4—561no—
288C2×C4—561no—
289C2×C4D4×D711yescharacteristic
290C2×C4D4×D711yescharacteristic
291C2×C4C23×D711yescharacteristic
292C8—21no—
293C8—21no—
294C8—72no—
295C8—141no—
296C7#338911yescharacteristic, complemented, Sylow
297C22—22no—
298C22—41no—
299C22—41no—
300C22—142no—
301C22—281no—
302C22—281no—
303C22C2×D4×D711yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—141no—
310C4—141no—
311C4—141no—
312C4—141no—
313C4—281no—
314C4—281no—
315C4—281no—
316C4C2×Q8×D711yescharacteristic
317C4#655411yescharacteristic
318C2—42no—
319C2—282no—
320C2#9032311yescharacteristic
321C2#9046011yescharacteristic
322C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#478C2
socleC22×C7C22×D4
prime radicalC2×#1141C22
hypercenter#478D7
hypocenterC7#3389
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. #90381C2semidirect
2. #90234C2semidirect
3. C7×#478C2semidirect
4. #89837C2semidirect
6. #89663C2semidirect
7. #89650C2semidirect
8. #89643C2semidirect
9. #89628C2semidirect
10. #89618C2semidirect
11. #89600C2semidirect
13. #89538C2semidirect
14. #89531C2semidirect
82. #1141Q8semidirect
112. #478D7semidirect
296. C7#3389semidirect

Maximal subgroups

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

subgroupindex
#903812
#902342
C7×#4782
#898372
#897862
#896632
#896502
#896432
#896282
#896182
#896002
#895572
#895382
#895312
#895282
#33897