Finite Group Databasebeta

← #11736416

Subgroups — #11736416

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#90508C211yesmaximal, characteristic
2#90220C211yesmaximal, characteristic, complemented
3C7×#419C211yesmaximal, characteristic, complemented
4#90000C211yesmaximal, characteristic, complemented
5#89906C211yesmaximal, characteristic, complemented
6#89900C211yesmaximal, characteristic, complemented
7#89775C211yesmaximal, characteristic, complemented
8#89588C211yesmaximal, characteristic, complemented
9#89582C211yesmaximal, characteristic, complemented
10#89581C211yesmaximal, characteristic, complemented
11#89575C211yesmaximal, characteristic, complemented
12#89572C211yesmaximal, characteristic, complemented
13#89570C211yesmaximal, characteristic, complemented
14#89568C211yesmaximal, characteristic, complemented
15#89562C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C2×#1161C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C7×#117C2211yescharacteristic
20#6510—21no—
21#6510C2211yescharacteristic
22#6509—21no—
23#6509C2211yescharacteristic
24Q8×Dic7C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26C2×#1144C2211yescharacteristic
27#6493—21no—
28#6493C2211yescharacteristic
29C4×C22⋊D7—21no—
30C4×C22⋊D7C2211yescharacteristic
31#6487C2211yescharacteristic
32C2×#1132C2211yescharacteristic
33C2×#1131C2211yescharacteristic
34#6460C2211yescharacteristic
35D7×C4⋊C4C2211yescharacteristic
36#6453C2211yescharacteristic
37#6450C2211yescharacteristic
38#6446—21no—
39#6442—21no—
40#6435—21no—
41#6433—21no—
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic
52#6390C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54C8×Dic7C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3327—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—22no—
62C2×C7×Q8—41no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×Q8C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7C2311yescharacteristic
67C2×C22⋊D7—21no—
68C2×C22⋊D7D411yescharacteristic
69#1161—22no—
70#1161—41no—
71C2×D28—21no—
72C2×D28C2311yescharacteristic
73C2×C4×D7—21no—
74C2×C4×D7—21no—
75C2×C4×D7C2311yescharacteristic
76C2×Dic14C2311yescharacteristic
77C7×M16—41no—
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C22⋊C4—21no—
83C42×C7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7D411yescharacteristic
86#1144—22no—
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)⋊C4D411yescharacteristic
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4D411yescharacteristic
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1132—41no—
111#1131—41no—
112#552—71no—
113#548—71no—
114#488—71no—
115#484—71no—
116#482—71no—
117#479—71no—
118#470—71no—
119#468—71no—
120#464—71no—
121#452—71no—
122#451—71no—
123C4×Q16—71no—
124#436—71no—
125#419D711yescharacteristic, complemented
126#417—71no—
127C23×C7C2×D411yescharacteristic
128C22×D7—21no—
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—42no—
134C7×Q8#4112yes—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—42no—
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2411yescharacteristic
144C22⋊D7—41no—
145C22⋊D7—42no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152D28—21no—
153D28—21no—
154D28—42no—
155C4×D7—21no—
156C4×D7—21no—
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—
166C22×Q8C2×D711yescharacteristic
167C2×Q16—71no—
168C2×Q16—71no—
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—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182#121—71no—
183#121—141no—
184C4×Q8—71no—
185C4×Q8—71no—
186C4×D4—71no—
187C4×D4—141no—
188#117—141no—
189#117C2×D711yescharacteristic
190C2×C4⋊C4—71no—
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—
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—22no—
219C4×C7—21no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7C2×#4111yescharacteristic
224C4×C7H(5,𝔽2)11yescharacteristic
225Dic7—21no—
226Dic7—21no—
227Dic7—21no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231#41—142no—
232#41—281no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—71no—
237C2×Q8C22×D711yescharacteristic
238C2×Q8C22×D711yescharacteristic
239C2×D4—71no—
240C2×D4—71no—
241C2×D4—141no—
242C2×D4—141no—
243C22×C4—21no—
244C22×C4—71no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4C22×D711yescharacteristic
248Q16—142no—
249Q16—281no—
250SD16—281no—
251M16—41no—
252M16—281no—
253C2×C8—71no—
254C2×C8—71no—
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—141no—
261C4⋊C4—141no—
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—71no—
272C22⋊C4—71no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C42—21no—
280C42—71no—
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—42no—
296Q8—141no—
297Q8—141no—
298Q8#115712yes—
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—42no—
311C2×C4—71no—
312C2×C4—71no—
313C2×C4—71no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
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—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—141no—
344C4—141no—
345C4—141no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4C2×#115711yescharacteristic
350C4#654811yescharacteristic
351C2—41no—
352C2—281no—
353C2—561no—
354C2#9018111yescharacteristic
355C2#9046811yescharacteristic
356C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#419C2
socleC22×C7C22×D4
prime radical#90508C2
hypercenter#419D7
hypocenterC7#3327
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. #90220C2semidirect
3. C7×#419C2semidirect
4. #90000C2semidirect
5. #89906C2semidirect
6. #89900C2semidirect
7. #89775C2semidirect
8. #89588C2semidirect
9. #89582C2semidirect
10. #89581C2semidirect
11. #89575C2semidirect
12. #89572C2semidirect
13. #89570C2semidirect
14. #89568C2semidirect
15. #89562C2semidirect
125. #419D7semidirect
327. C7#3327semidirect

Maximal subgroups

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

subgroupindex
#905082
#902202
C7×#4192
#900002
#899062
#899002
#897752
#895882
#895822
#895812
#895752
#895722
#895702
#895682
#895622
#33277