Finite Group Databasebeta

← #11739195

Subgroups — #11739195

Order 896.

382 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90417C211yesmaximal, characteristic
2#90385C211yesmaximal, characteristic, complemented
3C7×#491C211yesmaximal, characteristic, complemented
4#89862C211yesmaximal, characteristic, complemented
5#89838C211yesmaximal, characteristic, complemented
6#89588C211yesmaximal, characteristic, complemented
7#89586C211yesmaximal, characteristic, complemented
8#89585C211yesmaximal, characteristic, complemented
9#89582C211yesmaximal, characteristic, complemented
10#89582C211yesmaximal, characteristic, complemented
11#89581C211yesmaximal, characteristic, complemented
12#89576C211yesmaximal, characteristic, complemented
13#89575C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15#89478C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×#1161C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C7×#125C2211yescharacteristic
20#6509C2211yescharacteristic
21#6509C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23C2×#1144C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25C2×#1131C2211yescharacteristic
26C2×C8×D7C2211yescharacteristic
27#6461—21no—
28#6460C2211yescharacteristic
29#6460C2211yescharacteristic
30#6458—21no—
31#6457—21no—
32#6457—21no—
33#6457—21no—
34#6455—21no—
35#6455C2211yescharacteristic
36D7×C4⋊C4—21no—
37D7×C4⋊C4C2211yescharacteristic
38#6453—21no—
39C4×D28C2211yescharacteristic
40C4×D28C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46C7×#97C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52#6392C2211yescharacteristic
53#6390C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3315—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62#1161—22no—
63#1161—22no—
64#1161—41no—
65#1161—41no—
66C2×D28—21no—
67C2×D28—21no—
68C2×D28D411yescharacteristic
69C2×D28C2311yescharacteristic
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×C4×D7C2311yescharacteristic
75C2×C4×D7C2311yescharacteristic
76C2×Dic14C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C42×C7C2311yescharacteristic
86#1144—41no—
87#1144—41no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4D411yescharacteristic
95(C2×D7)⋊C4D411yescharacteristic
96C4⋊Dic7—21no—
97C4⋊Dic7—21no—
98C4⋊Dic7D411yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#1132—22no—
110#1131—41no—
111C8×D7—41no—
112#552—71no—
113#547—71no—
114#491D711yescharacteristic, complemented
115#485—71no—
116#484—71no—
117#478—71no—
118#470—71no—
119#467—71no—
120#462—71no—
121#452—71no—
122#452—71no—
123#440—71no—
124#435—71no—
125#417—71no—
126#416—71no—
127C22×D7—21no—
128C22×D7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146D28—21no—
147D28—41no—
148D28—42no—
149D28—42no—
150D28#4112yes—
151C4×D7—21no—
152C4×D7—21no—
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—42no—
156C4×D7—42no—
157C4×D7—42no—
158C4×D7—42no—
159Dic14—21no—
160Dic14—21no—
161C7×C8—21no—
162C7×C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41—71no—
166C2×#41—71no—
167C2×Q16—71no—
168C2×Q16—71no—
169C2×SD16—71no—
170C2×M16—71no—
171C22×C8—71no—
172C4⋊Q8C2×D711yescharacteristic
173#126—141no—
174#125—141no—
175#125C2×D711yescharacteristic
176#123—141no—
177#123—141no—
178#123—141no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183#121—141no—
184C4×Q8—71no—
185C4×D4—71no—
186C4×D4—71no—
187#117—71no—
188#117—141no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—141no—
191#107—71no—
192#107—71no—
193#106—71no—
194C4⋊C8—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103C2×D711yescharacteristic
199#103C2×D711yescharacteristic
200#103C2×D711yescharacteristic
201#103C2×D711yescharacteristic
202#102—71no—
203#102—71no—
204#102—71no—
205C22⋊C8—71no—
206C22⋊C8—71no—
207#97C2×D711yescharacteristic
208C22×C7C22×D411yescharacteristic
209C2×D7—21no—
210C2×D7—21no—
211C2×D7—22no—
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—21no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7C2×#4111yescharacteristic
225C4×C7H(5,𝔽2)11yescharacteristic
226Dic7—21no—
227Dic7—21no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231#41—142no—
232#41—142no—
233#41—281no—
234#41—281no—
235C2×Q8—71no—
236C2×Q8C22×D711yescharacteristic
237C2×Q8C22×D711yescharacteristic
238C2×D4—71no—
239C2×D4—71no—
240C2×D4—141no—
241C2×D4—141no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—141no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—141no—
248Q16—281no—
249Q16—281no—
250SD16—142no—
251M16—281no—
252C2×C8—71no—
253C2×C8—71no—
254C2×C8—281no—
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—71no—
262C4⋊C4—71no—
263C4⋊C4—71no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4C22×D711yescharacteristic
271C4⋊C4C22×D711yescharacteristic
272C22⋊C4—71no—
273C22⋊C4—71no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C42—141no—
281C42—141no—
282C42C22×D711yescharacteristic
283C2×C7#54611yescharacteristic
284C2×C7#57511yescharacteristic
285C2×C7#57811yescharacteristic
286D7—41no—
287D7—41no—
288D7—42no—
289C23—71no—
290C23—71no—
291C23—141no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—141no—
297Q8—141no—
298D4—72no—
299D4—141no—
300D4—281no—
301D4—282no—
302D4—282no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—71no—
308C2×C4—71no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—282no—
317C2×C4—282no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4D4×D711yescharacteristic
321C2×C4D4×D711yescharacteristic
322C2×C4C23×D711yescharacteristic
323C8—21no—
324C8—21no—
325C8—141no—
326C8—141no—
327C7#331511yescharacteristic, complemented, Sylow
328C22—141no—
329C22—141no—
330C22—141no—
331C22—141no—
332C22—141no—
333C22—141no—
334C22—142no—
335C22—281no—
336C22—281no—
337C22C2×D4×D711yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—141no—
345C4—141no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4D7×#4111yescharacteristic
350C4#655311yescharacteristic
351C2—281no—
352C2—281no—
353C2—282no—
354C2#9036711yescharacteristic
355C2#9046811yescharacteristic
356C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#491C2
socleC22×C7C22×D4
prime radical#90417C2
hypercenter#491D7
hypocenterC7#3315
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #90385C2semidirect
3. C7×#491C2semidirect
4. #89862C2semidirect
5. #89838C2semidirect
6. #89588C2semidirect
7. #89586C2semidirect
8. #89585C2semidirect
9. #89582C2semidirect
10. #89582C2semidirect
11. #89581C2semidirect
12. #89576C2semidirect
13. #89575C2semidirect
14. #89482C2semidirect
15. #89478C2semidirect
114. #491D7semidirect
327. C7#3315semidirect

Maximal subgroups

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

subgroupindex
#904172
#903852
C7×#4912
#898622
#898382
#895882
#895862
#895852
#895822
#895812
#895762
#895752
#894822
#894782
#33157