Finite Group Databasebeta

← #11740973

Subgroups — #11740973

Order 896.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#547C211yesmaximal, characteristic, complemented
2C2×#6514C211yesmaximal, characteristic, complemented
3C2×#6513C211yesmaximal, characteristic, complemented
4#90243C211yesmaximal, characteristic, complemented
5#90011C211yesmaximal, characteristic, complemented
6#90010C211yesmaximal, characteristic, complemented
7#89862C211yesmaximal, characteristic, complemented
8#89858C211yesmaximal, characteristic, complemented
9#89823C211yesmaximal, characteristic, complemented
10#89822C211yesmaximal, characteristic, complemented
11#89821C211yesmaximal, characteristic, complemented
12#89819C211yesmaximal, characteristic, complemented
13#89790C211yesmaximal, characteristic, complemented
14#89788C211yesmaximal, characteristic, complemented
15#89783C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×C7×#41C2211yescharacteristic
18C2×#1157C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C7×C4⋊Q8C2211yescharacteristic
21C7×#123—21no—
22C7×#123—21no—
23C7×C22⋊Q8C2211yescharacteristic
24C7×C22⋊Q8C2211yescharacteristic
25C7×#121—21no—
26C4×C7×D4C2211yescharacteristic
27C4×C7×D4C2211yescharacteristic
28C2×C7×C4⋊C4—21no—
29#6514—24no—
30#6513—24no—
31C2×#1144C2211yescharacteristic
32C2×#1144C2211yescharacteristic
33C2×#1144C2211yescharacteristic
34C2×#1144C2211yescharacteristic
35C2×#1143C2211yescharacteristic
36C2×#1143C2211yescharacteristic
37#6502—21no—
38C2×#1142C2211yescharacteristic
39C2×#1142C2211yescharacteristic
40C2×#1141C2211yescharacteristic
41#6493C2211yescharacteristic
42#6487C2211yescharacteristic
43C2×#1136C2211yescharacteristic
44C22×C7⋊C8C2211yescharacteristic
45#6459—21no—
46#6448—21no—
47(C22⋊D7)⋊C4—21no—
48#6438C2211yescharacteristic
49C4×Dic14C2211yescharacteristic
50#6409C2211yescharacteristic
51#6409C2211yescharacteristic
52#6406C2211yescharacteristic
53C22⋊(C7⋊C8)C2211yescharacteristic
54C22⋊(C7⋊C8)C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6383C2211yescharacteristic
58#6381C2211yescharacteristic
59C4⋊(C7⋊C8)C2211yescharacteristic
60#6377C2211yescharacteristic
61#3269—71nomaximal, Sylow
62C7×#41—22no—
63C7×#41—22no—
64C7×#41—41no—
65C7×#41—41no—
66C2×C7×Q8C2311yescharacteristic
67C2×C7×Q8C2311yescharacteristic
68C2×C7×D4—21no—
69C2×C7×D4—21no—
70C2×C7×D4D411yescharacteristic
71C2×C7×D4C2311yescharacteristic
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7C2311yescharacteristic
76C22×C4×C7C2311yescharacteristic
77C2×C22⋊D7—21no—
78C2×C22⋊D7—21no—
79C22×Dic7—21no—
80#1157—22no—
81#1157—41no—
82C2×D28C2311yescharacteristic
83C2×C4×D7—21no—
84C2×Dic14—22no—
85C2×Dic14—21no—
86C2×Dic14—41no—
87C2×Dic14C2311yescharacteristic
88C2×Dic14C2311yescharacteristic
89C7×C4⋊C4—21no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—21no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4D411yescharacteristic
94C7×C4⋊C4C2311yescharacteristic
95C7×C22⋊C4—21no—
96C7×C22⋊C4—21no—
97C7×C22⋊C4—21no—
98C7×C22⋊C4D411yescharacteristic
99C7×C22⋊C4D411yescharacteristic
100C42×C7C2311yescharacteristic
101C22⋊Dic7—21no—
102#1144—22no—
103#1144—22no—
104#1144—22no—
105#1144—41no—
106#1143—22no—
107#1143—41no—
108#1142—22no—
109#1142D414yescomplemented
110#1141—22no—
111(C2×D7)⋊C4—21no—
112(C2×D7)⋊C4—21no—
113(C2×D7)⋊C4—21no—
114C4⋊Dic7C2311yescharacteristic
115Dic7⋊C4—21no—
116Dic7⋊C4—21no—
117C4×Dic7—21no—
118#1136—22no—
119C2×C7⋊C8—22no—
120C2×C7⋊C8C2311yescharacteristic
121C2×C7⋊C8C2311yescharacteristic
122C2×C7⋊C8C2311yescharacteristic
123C2×C7⋊C8C2311yescharacteristic
124C2×#137—71no—
125C2×#135—71no—
126#548—71no—
127#547D711yescharacteristic, complemented
128#497—71no—
129#468—71no—
130#467—71no—
131#463—71no—
132#462—71no—
133#452—71no—
134#452—71no—
135#451—71no—
136#449—71no—
137#440—71no—
138#435—71no—
139C23×C7—21no—
140C23×C7C2×D411yescharacteristic
141C23×C7C2×D411yescharacteristic
142C22×D7—21no—
143C7×Q8—21no—
144C7×Q8—21no—
145C7×Q8—21no—
146C7×Q8—21no—
147C7×D4—21no—
148C7×D4—41no—
149C7×D4—42no—
150C7×D4—42no—
151C7×D4C2×D412yes—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7C2×D411yescharacteristic
163C2×C4×C7C2×D411yescharacteristic
164C2×C4×C7C2×D411yescharacteristic
165C2×C4×C7C2×D411yescharacteristic
166C2×C4×C7C2411yescharacteristic
167C22⋊D7—42no—
168C22⋊D7—42no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—42no—
173D28—21no—
174D28—21no—
175C4×D7—42no—
176Dic14—21no—
177Dic14—21no—
178Dic14—21no—
179Dic14—42no—
180Dic14C2×D412yes—
181C7⋊C8—21no—
182C7⋊C8—21no—
183C7⋊C8—21no—
184C7⋊C8C2×D412yes—
185C2×#41—71no—
186C2×#41C2×D711yescharacteristic
187C2×#41C2×D711yescharacteristic
188C22×Q8—71no—
189#137—144no—
190#135—144no—
191C2×Q16—71no—
192C2×Q16—71no—
193C2×Q16—71no—
194C2×Q16—71no—
195C2×SD16—71no—
196C2×SD16—71no—
197C2×SD16—71no—
198C2×SD16—71no—
199C2×D8—71no—
200C2×M16—71no—
201C22×C8—71no—
202C4⋊Q8C2×D711yescharacteristic
203#124—71no—
204#124—141no—
205#123—21no—
206#123—21no—
207C22⋊Q8—71no—
208C22⋊Q8—141no—
209C22⋊Q8C2×D711yescharacteristic
210C22⋊Q8C2×D711yescharacteristic
211#121—21no—
212#121—71no—
213#121—141no—
214C4×Q8—71no—
215C4×D4—141no—
216C4×D4C2×D711yescharacteristic
217C4×D4C2×D711yescharacteristic
218C2×C4⋊C4—21no—
219#107—71no—
220C4⋊C8—71no—
221#103—71no—
222#103—71no—
223#103—71no—
224#103—71no—
225#102—71no—
226#102—71no—
227C22⋊C8—71no—
228C22⋊C8—71no—
229#97—71no—
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—22no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7C22×D411yescharacteristic
240C2×D7—41no—
241C2×D7—41no—
242C2×D7—41no—
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—41no—
248C4×C7—41no—
249C4×C7—41no—
250C4×C7C22×D411yescharacteristic
251C4×C7H(5,𝔽2)11yescharacteristic
252Dic7—22no—
253Dic7—41no—
254Dic7—41no—
255#41—22no—
256#41—22no—
257#41—41no—
258#41—41no—
259#41—142no—
260#41—281no—
261C2×Q8—71no—
262C2×Q8—71no—
263C2×Q8—142no—
264C2×Q8—141no—
265C2×Q8—281no—
266C2×Q8C22×D711yescharacteristic
267C2×Q8C22×D711yescharacteristic
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—71no—
271C2×D4—141no—
272C2×D4—141no—
273C2×D4C22⋊D711yescharacteristic
274C2×D4C22×D711yescharacteristic
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—141no—
279C22×C4—141no—
280C22×C4C22×D711yescharacteristic
281C22×C4C22×D711yescharacteristic
282Q16—142no—
283Q16—142no—
284Q16—142no—
285Q16—281no—
286SD16—74no—
287SD16—142no—
288SD16—142no—
289SD16—281no—
290D8—142no—
291M16—142no—
292C2×C8—71no—
293C2×C8—71no—
294C2×C8—71no—
295C2×C8—71no—
296C2×C8—142no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—71no—
302C4⋊C4—141no—
303C4⋊C4—141no—
304C4⋊C4C22⋊D711yescharacteristic
305C4⋊C4C22×D711yescharacteristic
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—141no—
310C22⋊C4—141no—
311C22⋊C4—141no—
312C22⋊C4—141no—
313C22⋊C4C22⋊D711yescharacteristic
314C22⋊C4C22⋊D711yescharacteristic
315C42—141no—
316C42C22×D711yescharacteristic
317C2×C7—41no—
318C2×C7—41no—
319C2×C7—42no—
320C2×C7D4211yescharacteristic
321C2×C7#57511yescharacteristic
322C2×C7#57811yescharacteristic
323D7—81no—
324C23—21no—
325C23—141no—
326C23C2×C22⋊D711yescharacteristic
327C23C2×C22⋊D711yescharacteristic
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—72no—
333Q8—141no—
334Q8—141no—
335Q8—141no—
336Q8—282no—
337D4—21no—
338D4—41no—
339D4—42no—
340D4—42no—
341D4—141no—
342D4—141no—
343D4—282no—
344D4—282no—
345D4D4×D712yes—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—141no—
357C2×C4—141no—
358C2×C4—141no—
359C2×C4—282no—
360C2×C4—282no—
361C2×C4C2×C22⋊D711yescharacteristic
362C2×C4C2×C22⋊D711yescharacteristic
363C2×C4C2×C22⋊D711yescharacteristic
364C2×C4C2×C22⋊D711yescharacteristic
365C2×C4C23×D711yescharacteristic
366C8—72no—
367C8—141no—
368C8—141no—
369C8—141no—
370C7#326911yescharacteristic, complemented, Sylow
371C22—21no—
372C22—21no—
373C22—22no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—281no—
381C22—281no—
382C22—281no—
383C22C22×C22⋊D711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—142no—
392C4—281no—
393C4—281no—
394C4C2×D4×D711yescharacteristic
395C4#654811yescharacteristic
396C2—41no—
397C2—41no—
398C2—42no—
399C2—561no—
400C2D4×C22⋊D711yescharacteristic
401C2#9051311yescharacteristic
402C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#547C2
socleC22×C7C22×D4
hypercenter#547D7
hypocenterC7#3269
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C7×#547C2semidirect
2. C2×#6514C2semidirect
3. C2×#6513C2semidirect
4. #90243C2semidirect
5. #90011C2semidirect
6. #90010C2semidirect
7. #89862C2semidirect
8. #89858C2semidirect
9. #89823C2semidirect
10. #89822C2semidirect
11. #89821C2semidirect
12. #89819C2semidirect
13. #89790C2semidirect
14. #89788C2semidirect
15. #89783C2semidirect
109. #1142D4semidirect
127. #547D7semidirect
370. C7#3269semidirect

Maximal subgroups

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

subgroupindex
C7×#5472
C2×#65142
C2×#65132
#902432
#900112
#900102
#898622
#898582
#898232
#898222
#898212
#898192
#897902
#897882
#897832
#32697