Finite Group Databasebeta

← #11738363

Subgroups — #11738363

Order 896.

380 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#90333C211yesmaximal, characteristic, complemented
2#90324C211yesmaximal, characteristic
3C7×#470C211yesmaximal, characteristic, complemented
4#89958C211yesmaximal, characteristic, complemented
5#89957C211yesmaximal, characteristic, complemented
6#89905C211yesmaximal, characteristic, complemented
7#89900C211yesmaximal, characteristic, complemented
8#89819C211yesmaximal, characteristic, complemented
9#89816C211yesmaximal, characteristic, complemented
10#89638C211yesmaximal, characteristic, complemented
11#89629C211yesmaximal, characteristic, complemented
12#89584C211yesmaximal, characteristic, complemented
13#89574C211yesmaximal, characteristic, complemented
14#89572C211yesmaximal, characteristic, complemented
15#89562C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C2×C7×M16C2211yescharacteristic
20C7×C22⋊Q8C2211yescharacteristic
21C7×C22⋊Q8C2211yescharacteristic
22#6510C2211yescharacteristic
23#6509C2211yescharacteristic
24Q8×Dic7C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26C2×#1143C2211yescharacteristic
27C4×C22⋊D7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×#1131C2211yescharacteristic
30#6461—21no—
31#6459—21no—
32#6458C2211yescharacteristic
33#6456—21no—
34#6455C2211yescharacteristic
35#6453—21no—
36#6453C2211yescharacteristic
37#6450C2211yescharacteristic
38#6449—21no—
39#6448—21no—
40#6447—21no—
41#6440—21no—
42C7×#106C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47C22⋊(C7⋊C8)C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54C8×Dic7C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3327—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7D411yescharacteristic
65C22×Dic7—21no—
66Q8×D7—22no—
67Q8×D7—41no—
68#1157—22no—
69#1157—41no—
70C2×D28—21no—
71C2×D28C2311yescharacteristic
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×C4×D7C2311yescharacteristic
75C2×Dic14—21no—
76C2×Dic14C2311yescharacteristic
77C7×Q16—41no—
78C7×M16—41no—
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C22⋊Dic7D411yescharacteristic
88#1144—22no—
89#1143—41no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4D411yescharacteristic
96C4⋊Dic7—21no—
97C4⋊Dic7—21no—
98C4⋊Dic7—21no—
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4D411yescharacteristic
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111#1131—41no—
112#552—71no—
113#548—71no—
114#488—71no—
115#484—71no—
116#482—71no—
117#479—71no—
118#470D711yescharacteristic, complemented
119#468—71no—
120#464—71no—
121#452—71no—
122#451—71no—
123C4×Q16—71no—
124#436—71no—
125#419—71no—
126#417—71no—
127C23×C7C2×D411yescharacteristic
128C22×D7—21no—
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×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C22⋊D7—41no—
143C22⋊D7—42no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—42no—
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151D28—21no—
152D28—21no—
153D28—42no—
154C4×D7—21no—
155C4×D7—21no—
156C4×D7—42no—
157C4×D7—42no—
158Dic14—21no—
159Dic14—42no—
160Dic14#4112yes—
161C7×C8—21no—
162C7×C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41—71no—
166C22×Q8—71no—
167C2×Q16—71no—
168C2×Q16C2×D711yescharacteristic
169C2×SD16—71no—
170C2×M16—71no—
171C2×M16C2×D711yescharacteristic
172#126—141no—
173#125—71no—
174#125—141no—
175#124—71no—
176#124—141no—
177#123—141no—
178C22⋊Q8—71no—
179C22⋊Q8—141no—
180C22⋊Q8C2×D711yescharacteristic
181C22⋊Q8C2×D711yescharacteristic
182#121—71no—
183#121—141no—
184C4×Q8—71no—
185C4×Q8—71no—
186C4×D4—71no—
187C4×D4—141no—
188#117—71no—
189#117—141no—
190C2×C4⋊C4—71no—
191#107—71no—
192#107—71no—
193#106C2×D711yescharacteristic
194C4⋊C8—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103C2×D711yescharacteristic
201#103C2×D711yescharacteristic
202#102—71no—
203#102—71no—
204#102—71no—
205C22⋊C8—71no—
206C22⋊C8—71no—
207C4×C8—71no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7C22×D411yescharacteristic
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—21no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7C2×#4111yescharacteristic
224C4×C7H(5,𝔽2)11yescharacteristic
225Dic7—22no—
226Dic7—21no—
227Dic7—21no—
228Dic7—21no—
229Dic7—41no—
230Dic7—41no—
231#41—142no—
232#41—281no—
233C2×Q8—71no—
234C2×Q8—141no—
235C2×Q8—142no—
236C2×Q8—281no—
237C2×Q8C22×D711yescharacteristic
238C2×Q8C22×D711yescharacteristic
239C2×D4—71no—
240C2×D4—71no—
241C2×D4—141no—
242C2×D4—141no—
243C22×C4—71no—
244C22×C4—141no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4C22×D711yescharacteristic
248Q16—41no—
249Q16—142no—
250SD16—281no—
251M16—41no—
252M16—281no—
253C2×C8—71no—
254C2×C8—71no—
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—71no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4C22×D711yescharacteristic
269C4⋊C4C22×D711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—71no—
273C22⋊C4—71no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C42—71no—
280C42—141no—
281C42—141no—
282C42—141no—
283C2×C7—41no—
284C2×C7#54611yescharacteristic
285C2×C7#57511yescharacteristic
286C2×C7#57811yescharacteristic
287D7—41no—
288D7—81no—
289C23—71no—
290C23—141no—
291C23D4×D711yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—72no—
297Q8—141no—
298Q8—282no—
299D4—141no—
300D4—141no—
301D4—282no—
302D4—281no—
303D4—282no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—71no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4—282no—
321C2×C4D4×D711yescharacteristic
322C2×C4C23×D711yescharacteristic
323C8—21no—
324C8—21no—
325C8—141no—
326C8—141no—
327C7#332711yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—21no—
331C22—141no—
332C22—141no—
333C22—141no—
334C22—281no—
335C22—281no—
336C22—281no—
337C22C2×D4×D711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—141no—
344C4—141no—
345C4—141no—
346C4—142no—
347C4—281no—
348C4—281no—
349C4#654811yescharacteristic
350C4D7×#4111yescharacteristic
351C2—41no—
352C2—281no—
353C2—561no—
354C2#9030111yescharacteristic
355C2#9046811yescharacteristic
356C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#470C2
socleC22×C7C22×D4
prime radical#90324C2
hypercenter#470D7
hypocenterC7#3327
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #90333C2semidirect
3. C7×#470C2semidirect
4. #89958C2semidirect
5. #89957C2semidirect
6. #89905C2semidirect
7. #89900C2semidirect
8. #89819C2semidirect
9. #89816C2semidirect
10. #89638C2semidirect
11. #89629C2semidirect
12. #89584C2semidirect
13. #89574C2semidirect
14. #89572C2semidirect
15. #89562C2semidirect
118. #470D7semidirect
327. C7#3327semidirect

Maximal subgroups

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

subgroupindex
#903332
#903242
C7×#4702
#899582
#899572
#899052
#899002
#898192
#898162
#896382
#896292
#895842
#895742
#895722
#895622
#33277