Finite Group Databasebeta

← #11737203

Subgroups — #11737203

Order 896.

332 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#90260C211yesmaximal, characteristic, complemented
2#90233C211yesmaximal, characteristic
3C7×#440C211yesmaximal, characteristic
4#89938C211yesmaximal, characteristic
5#89938C211yesmaximal, characteristic
6#89803C211yesmaximal, characteristic, complemented
7#89781C211yesmaximal, characteristic
8#89649C211yesmaximal, characteristic, complemented
9#89647C211yesmaximal, characteristic, complemented
10#89562C211yesmaximal, characteristic, complemented
11#89561C211yesmaximal, characteristic, complemented
12#89540C211yesmaximal, characteristic
13#89537C211yesmaximal, characteristic
14#89475C211yesmaximal, characteristic, complemented
15#89473C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1144C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23#6499C2211yescharacteristic
24#6499C2211yescharacteristic
25#6489—21no—
26#6487—21no—
27#6453—21no—
28#6453C2211yescharacteristic
29#6451—21no—
30#6451C2211yescharacteristic
31#6450C2211yescharacteristic
32#6450C2211yescharacteristic
33#6442—21no—
34#6442—21no—
35#6433—21no—
36#6433C2211yescharacteristic
37#6432C2211yescharacteristic
38C4×Dic14—21no—
39C4×Dic14C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6390C2211yescharacteristic
47#6390C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50#6389C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3377—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×Dic14—21no—
64C2×Dic14D411yescharacteristic
65C2×Dic14C2311yescharacteristic
66C2×Dic14C2311yescharacteristic
67C7×SD16—22no—
68C2×C7×C8C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C7×C4⋊C4—21no—
71C7×C4⋊C4C2311yescharacteristic
72C7×C4⋊C4C2311yescharacteristic
73C7×C22⋊C4—21no—
74C42×C7—21no—
75C42×C7C2311yescharacteristic
76C22⋊Dic7—21no—
77C22⋊Dic7—21no—
78C22⋊Dic7—21no—
79C22⋊Dic7—21no—
80#1144—22no—
81#1144—22no—
82C4⋊Dic7—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7D411yescharacteristic
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×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#557—71no—
107#556—71no—
108#491—71no—
109#489—71no—
110#488—71no—
111#479—71no—
112#478—71no—
113#477—71no—
114#464—71no—
115#464—71no—
116#462—71no—
117#446—71no—
118#441—71no—
119#440D711yescharacteristic
120C4×Q16—71no—
121C23×C7—21no—
122C7×Q8—21no—
123C7×Q8#4112yes—
124C7×D4—21no—
125C7×D4—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—42no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141Dic14—21no—
142Dic14—21no—
143Dic14—21no—
144Dic14—21no—
145Dic14—41no—
146Dic14—42no—
147C7×C8—21no—
148C7×C8—21no—
149C7⋊C8—21no—
150C7⋊C8—21no—
151C2×Q16—71no—
152C2×Q16—71no—
153C2×SD16C2×D711yescharacteristic
154C4⋊Q8—71no—
155C4⋊Q8—71no—
156C4⋊Q8—141no—
157#126—141no—
158#126—141no—
159#125—71no—
160#125—71no—
161#125—141no—
162#125—141no—
163#124—71no—
164#124—71no—
165C22⋊Q8—141no—
166C4×Q8—71no—
167C4×Q8—71no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—141no—
172C4×Q8C2×D711yescharacteristic
173C4×D4C2×D711yescharacteristic
174#117—141no—
175#107—71no—
176#107C2×D711yescharacteristic
177#106—71no—
178C4⋊C8—71no—
179C4⋊C8—71no—
180C4⋊C8—71no—
181#103—71no—
182#103—71no—
183#103—71no—
184#103—71no—
185#103—71no—
186#103—71no—
187#103C2×D711yescharacteristic
188#102—71no—
189#102—71no—
190#102C2×D711yescharacteristic
191#97—71no—
192#97C2×D711yescharacteristic
193C4×C8—71no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7C22×D411yescharacteristic
198C4×C7—22no—
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7C2×#4111yescharacteristic
204C4×C72−1+411yescharacteristic
205Dic7—21no—
206Dic7—21no—
207Dic7—21no—
208Dic7—21no—
209Dic7—41no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213C2×Q8—71no—
214C2×Q8—71no—
215C2×Q8—71no—
216C2×Q8—141no—
217C2×Q8C22×D711yescharacteristic
218C2×D4C22×D711yescharacteristic
219C22×C4—21no—
220Q16—142no—
221Q16—142no—
222SD16—22no—
223C2×C8—71no—
224C2×C8—71no—
225C2×C8C22×D711yescharacteristic
226C2×C8C22×D711yescharacteristic
227C4⋊C4—21no—
228C4⋊C4—71no—
229C4⋊C4—71no—
230C4⋊C4—71no—
231C4⋊C4—71no—
232C4⋊C4—71no—
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⋊C4—141no—
244C4⋊C4C22×D711yescharacteristic
245C4⋊C4C22×D711yescharacteristic
246C22⋊C4—21no—
247C22⋊C4—141no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C42—21no—
252C42—71no—
253C42—71no—
254C42—141no—
255C42—141no—
256C42—141no—
257C42—141no—
258C42C22×D711yescharacteristic
259C2×C7—81no—
260C2×C7#54811yescharacteristic
261C2×C7#57511yescharacteristic
262C2×C7#57811yescharacteristic
263C23—21no—
264Q8—21no—
265Q8—141no—
266Q8—141no—
267Q8—141no—
268Q8—141no—
269Q8—281no—
270Q8—282no—
271Q8#115712yes—
272D4—21no—
273D4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—71no—
279C2×C4—71no—
280C2×C4—71no—
281C2×C4—71no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4D4×D711yescharacteristic
287C2×C4D4×D711yescharacteristic
288C2×C4C23×D711yescharacteristic
289C8—21no—
290C8—21no—
291C8—141no—
292C8—141no—
293C7#337711yescharacteristic, complemented, Sylow
294C22—41no—
295C22—41no—
296C22—41no—
297C22C2×D4×D711yescharacteristic
298C4—21no—
299C4—21no—
300C4—22no—
301C4—41no—
302C4—41no—
303C4—141no—
304C4—141no—
305C4—141no—
306C4—141no—
307C4—281no—
308C4—281no—
309C4—281no—
310C4—281no—
311C4C2×#115711yescharacteristic
312C4#655411yescharacteristic
313C2—81no—
314C2#9024311yescharacteristic
315C2#9046411yescharacteristic
316C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
prime radicalC2×C7×D4C23
hypercenter#440D7
hypocenterC7#3377
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. #90260C2semidirect
6. #89803C2semidirect
8. #89649C2semidirect
9. #89647C2semidirect
10. #89562C2semidirect
11. #89561C2semidirect
14. #89475C2semidirect
15. #89473C2semidirect
293. C7#3377semidirect

Maximal subgroups

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

subgroupindex
#902602
#902332
C7×#4402
#899382
#898032
#897812
#896492
#896472
#895622
#895612
#895402
#895372
#894752
#894732
#33777