Finite Group Databasebeta

← #11740922

Subgroups — #11740922

Order 896.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#544C211yesmaximal, characteristic, complemented
2#90226C211yesmaximal, characteristic
3#89860C211yesmaximal, characteristic, complemented
4#89859C211yesmaximal, characteristic, complemented
5#89856C211yesmaximal, characteristic, complemented
6#89852C211yesmaximal, characteristic, complemented
7#89839C211yesmaximal, characteristic, complemented
8#89838C211yesmaximal, characteristic, complemented
9#89838C211yesmaximal, characteristic, complemented
10#89836C211yesmaximal, characteristic, complemented
11#89775C211yesmaximal, characteristic, complemented
12#89775C211yesmaximal, characteristic, complemented
13#89772C211yesmaximal, characteristic, complemented
14#89740C211yesmaximal, characteristic, complemented
15#89739C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8—21no—
17C7×C4⋊Q8C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C7×#125—21no—
20C7×#125C2211yescharacteristic
21C7×#125C2211yescharacteristic
22C7×C22⋊Q8—21no—
23C7×C22⋊Q8—21no—
24C7×#117C2211yescharacteristic
25C7×#117C2211yescharacteristic
26C2×C7×C4⋊C4C2211yescharacteristic
27#6493C2211yescharacteristic
28#6493C2211yescharacteristic
29#6457—21no—
30#6457—21no—
31#6447—21no—
32#6447—21no—
33#6437C2211yescharacteristic
34C4×D28C2211yescharacteristic
35C4×D28C2211yescharacteristic
36#6433C2211yescharacteristic
37C22⋊(C7⋊C8)C2211yescharacteristic
38C22⋊(C7⋊C8)C2211yescharacteristic
39#6383C2211yescharacteristic
40#6383C2211yescharacteristic
41#6383C2211yescharacteristic
42#6383C2211yescharacteristic
43#6383C2211yescharacteristic
44#6383C2211yescharacteristic
45#6383C2211yescharacteristic
46#6383C2211yescharacteristic
47#6382C2211yescharacteristic
48#6382C2211yescharacteristic
49#6382C2211yescharacteristic
50#6382C2211yescharacteristic
51#6381C2211yescharacteristic
52#6381C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56C4⋊(C7⋊C8)C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3241—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—21no—
62C22×C4×C7—21no—
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×D28—21no—
67C2×D28—21no—
68C2×D28C2311yescharacteristic
69C2×D28C2311yescharacteristic
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4D411yescharacteristic
80C7×C4⋊C4D411yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4D411yescharacteristic
87C7×C22⋊C4D411yescharacteristic
88C42×C7—21no—
89C42×C7C2311yescharacteristic
90C42×C7C2311yescharacteristic
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99C2×C7⋊C8C2311yescharacteristic
100C2×C7⋊C8C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#544D711yescharacteristic, complemented
104#540—71no—
105#501—71no—
106#499—71no—
107#490—71no—
108#486—71no—
109#482—71no—
110#482—71no—
111#481—71no—
112#480—71no—
113#478—71no—
114#478—71no—
115#476—71no—
116#474—71no—
117#433—71no—
118C23×C7C2×D411yescharacteristic
119C22×D7—21no—
120C22×D7—21no—
121C7×Q8—81no—
122C7×Q8—81no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—81no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—81no—
137C22⋊D7—81no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140D28—21no—
141D28—21no—
142D28—21no—
143D28—21no—
144D28—81no—
145D28—81no—
146C4×D7—81no—
147C4×D7—81no—
148C7⋊C8—21no—
149C7⋊C8—21no—
150C7⋊C8—21no—
151C7⋊C8—21no—
152C4⋊Q8—21no—
153C4⋊Q8C2×D711yescharacteristic
154C4⋊Q8C2×D711yescharacteristic
155#127—71no—
156#125—21no—
157#125—71no—
158#125C2×D711yescharacteristic
159#125C2×D711yescharacteristic
160#123—141no—
161#123—141no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164#121—71no—
165#121—71no—
166#121—141no—
167#121—141no—
168C4×D4—71no—
169C4×D4—71no—
170#117C2×D711yescharacteristic
171#117C2×D711yescharacteristic
172C2×C4⋊C4C2×D711yescharacteristic
173#107—71no—
174#107—71no—
175#107—71no—
176#107—71no—
177#106—71no—
178#106—71no—
179#106—71no—
180#106—71no—
181C4⋊C8—71no—
182C4⋊C8—71no—
183#102—71no—
184#102—71no—
185#102—71no—
186#102—71no—
187#102—71no—
188#102—71no—
189#102—71no—
190#102—71no—
191C22⋊C8—71no—
192C22⋊C8—71no—
193#97—71no—
194C4×C8—71no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7C22×D411yescharacteristic
199C2×D7—41no—
200C2×D7—41no—
201C2×D7—41no—
202C2×D7—41no—
203C2×D7—41no—
204C2×D7—41no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C72−1+411yescharacteristic
215C4×C72−1+411yescharacteristic
216Dic7—41no—
217Dic7—41no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×D4—71no—
221C2×D4—71no—
222C2×D4—141no—
223C2×D4—141no—
224C2×D4—141no—
225C2×D4—141no—
226C22×C4—21no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4C22×D711yescharacteristic
230C2×C8—71no—
231C2×C8—71no—
232C2×C8—71no—
233C2×C8—71no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4C22⋊D711yescharacteristic
246C4⋊C4C22⋊D711yescharacteristic
247C4⋊C4C22×D711yescharacteristic
248C4⋊C4C22×D711yescharacteristic
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4C22⋊D711yescharacteristic
257C22⋊C4C22⋊D711yescharacteristic
258C42—21no—
259C42C22×D711yescharacteristic
260C42C22×D711yescharacteristic
261C2×C7—41no—
262C2×C7#53611yescharacteristic
263C2×C7#57611yescharacteristic
264C2×C7#57711yescharacteristic
265D7—81no—
266D7—81no—
267C23—141no—
268C23—141no—
269C23C2×C22⋊D711yescharacteristic
270Q8—81no—
271Q8—81no—
272D4—141no—
273D4—141no—
274D4—141no—
275D4—141no—
276D4—561no—
277D4—561no—
278D4—561no—
279D4—561no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—81no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—561no—
290C2×C4—561no—
291C2×C4C2×C22⋊D711yescharacteristic
292C2×C4C2×C22⋊D711yescharacteristic
293C2×C4C2×C22⋊D711yescharacteristic
294C2×C4C2×C22⋊D711yescharacteristic
295C2×C4C2×C22⋊D711yescharacteristic
296C2×C4C23×D711yescharacteristic
297C8—141no—
298C8—141no—
299C8—141no—
300C8—141no—
301C7#324111yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22—21no—
305C22—281no—
306C22—281no—
307C22—281no—
308C22—281no—
309C22—281no—
310C22—281no—
311C22C22×C22⋊D711yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—281no—
322C4—281no—
323C4#655111yescharacteristic
324C4#655111yescharacteristic
325C2—41no—
326C2—561no—
327C2—561no—
328C2#9050711yescharacteristic
329C2#9052611yescharacteristic
330C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#544C2
socleC22×C7C22×D4
prime radical#90226C2
hypercenter#544D7
hypocenterC7#3241
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
1. C7×#544C2semidirect
3. #89860C2semidirect
4. #89859C2semidirect
5. #89856C2semidirect
6. #89852C2semidirect
7. #89839C2semidirect
8. #89838C2semidirect
9. #89838C2semidirect
10. #89836C2semidirect
11. #89775C2semidirect
12. #89775C2semidirect
13. #89772C2semidirect
14. #89740C2semidirect
15. #89739C2semidirect
103. #544D7semidirect
301. C7#3241semidirect

Maximal subgroups

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

subgroupindex
C7×#5442
#902262
#898602
#898592
#898562
#898522
#898392
#898382
#898362
#897752
#897722
#897402
#897392
#32417