Finite Group Databasebeta

← #11738805

Subgroups — #11738805

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
1#90310C211yesmaximal, characteristic
2#90210C211yesmaximal, characteristic, complemented
3C7×#482C211yesmaximal, characteristic, complemented
4#89805C211yesmaximal, characteristic, complemented
5#89775C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89648C211yesmaximal, characteristic, complemented
8#89643C211yesmaximal, characteristic, complemented
9#89630C211yesmaximal, characteristic, complemented
10#89536C211yesmaximal, characteristic, complemented
11#89535C211yesmaximal, characteristic, complemented
12#89533C211yesmaximal, characteristic, complemented
13#89532C211yesmaximal, characteristic, complemented
14#89507C211yesmaximal, characteristic, complemented
15#89492C211yesmaximal, characteristic, complemented
16C7×#121C2211yescharacteristic
17C7×#117C2211yescharacteristic
18#6503C2211yescharacteristic
19#6502—21no—
20#6501—21no—
21D4×Dic7C2211yescharacteristic
22#6493C2211yescharacteristic
23#6489C2211yescharacteristic
24C2×C4⋊Dic7C2211yescharacteristic
25#6456C2211yescharacteristic
26#6453C2211yescharacteristic
27#6453C2211yescharacteristic
28#6453C2211yescharacteristic
29#6451C2211yescharacteristic
30#6451C2211yescharacteristic
31#6449—21no—
32#6446—21no—
33#6441—21no—
34#6441—21no—
35#6433—21no—
36#6432—21no—
37C7×#107C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×#102C2211yescharacteristic
40C7×#102C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42#6406C2211yescharacteristic
43#6406C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47#6391C2211yescharacteristic
48#6391C2211yescharacteristic
49#6389C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3241—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C22×Dic7—21no—
67C22×Dic7—21no—
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×Dic14—21no—
71C2×Dic14—21no—
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C22⋊C4—21no—
78C7×C22⋊C4—21no—
79C42×C7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7D411yescharacteristic
82C22⋊Dic7D411yescharacteristic
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7D411yescharacteristic
90C4⋊Dic7D411yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99C4×Dic7C2311yescharacteristic
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#544—71no—
104#540—71no—
105#501—71no—
106#499—71no—
107#490—71no—
108#486—71no—
109#482—71no—
110#482D711yescharacteristic, complemented
111#481—71no—
112#480—71no—
113#478—71no—
114#478—71no—
115#476—71no—
116#474—71no—
117#433—71no—
118C23×C7—21no—
119C23×C7C2×D411yescharacteristic
120C22×D7—21no—
121C7×D4—21no—
122C7×D4—21no—
123C7×D4—81no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2411yescharacteristic
131C22⋊D7—81no—
132C22⋊D7—81no—
133C22⋊D7—81no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—81no—
138C2×Dic7—81no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143D28—21no—
144D28—21no—
145C4×D7—81no—
146Dic14—81no—
147Dic14—81no—
148C7×C8—21no—
149C7×C8—21no—
150C7⋊C8—21no—
151C7⋊C8—21no—
152C4⋊Q8—71no—
153C4⋊Q8—71no—
154C4⋊Q8—141no—
155#127—71no—
156#125—71no—
157#125—71no—
158#125—71no—
159#125—141no—
160#123—141no—
161#123—141no—
162C22⋊Q8—141no—
163C22⋊Q8—141no—
164#121—71no—
165#121—141no—
166#121—141no—
167#121C2×D711yescharacteristic
168C4×D4—71no—
169C4×D4—71no—
170#117—71no—
171#117C2×D711yescharacteristic
172C2×C4⋊C4—71no—
173#107—71no—
174#107—71no—
175#107—71no—
176#107C2×D711yescharacteristic
177#106—71no—
178#106—71no—
179#106—71no—
180#106C2×D711yescharacteristic
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#102C2×D711yescharacteristic
190#102C2×D711yescharacteristic
191C22⋊C8—71no—
192C22⋊C8C2×D711yescharacteristic
193#97—71no—
194C4×C8—71no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7C22×D411yescharacteristic
202C2×D7—41no—
203C2×D7—41no—
204C2×D7—41no—
205C4×C7—21no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C72−1+411yescharacteristic
210C4×C72−1+411yescharacteristic
211Dic7—21no—
212Dic7—21no—
213Dic7—21no—
214Dic7—21no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218C2×Q8—141no—
219C2×Q8—141no—
220C2×D4—21no—
221C2×D4—71no—
222C2×D4—141no—
223C2×D4—141no—
224C2×D4—141no—
225C2×D4C22×D711yescharacteristic
226C22×C4—141no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4C22×D711yescharacteristic
230C2×C8—71no—
231C2×C8—71no—
232C2×C8C22×D711yescharacteristic
233C2×C8C22×D711yescharacteristic
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4C22×D711yescharacteristic
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—71no—
254C22⋊C4—71no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C42—21no—
259C42—71no—
260C42—71no—
261C2×C7—41no—
262C2×C7—81no—
263C2×C7#53611yescharacteristic
264C2×C7#57611yescharacteristic
265C2×C7#57711yescharacteristic
266D7—81no—
267C23—21no—
268C23—141no—
269C23D4×D711yescharacteristic
270Q8—561no—
271Q8—561no—
272D4—21no—
273D4—21no—
274D4—81no—
275D4—141no—
276D4—141no—
277D4—561no—
278D4—561no—
279D4—561no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—71no—
286C2×C4—71no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—561no—
293C2×C4—561no—
294C2×C4—561no—
295C2×C4D4×D711yescharacteristic
296C2×C4C23×D711yescharacteristic
297C8—21no—
298C8—21no—
299C8—141no—
300C8—141no—
301C7#324111yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—281no—
309C22—281no—
310C22—281no—
311C22C2×D4×D711yescharacteristic
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—141no—
317C4—141no—
318C4—141no—
319C4—141no—
320C4—281no—
321C4—281no—
322C4—281no—
323C4#655411yescharacteristic
324C4#655411yescharacteristic
325C2—41no—
326C2—81no—
327C2—561no—
328C2#9033911yescharacteristic
329C2#9046111yescharacteristic
330C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#482C2
socleC22×C7C22×D4
prime radical#90310C2
hypercenter#482D7
hypocenterC7#3241
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #90210C2semidirect
3. C7×#482C2semidirect
4. #89805C2semidirect
5. #89775C2semidirect
6. #89662C2semidirect
7. #89648C2semidirect
8. #89643C2semidirect
9. #89630C2semidirect
10. #89536C2semidirect
11. #89535C2semidirect
12. #89533C2semidirect
13. #89532C2semidirect
14. #89507C2semidirect
15. #89492C2semidirect
110. #482D7semidirect
301. C7#3241semidirect

Maximal subgroups

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

subgroupindex
#903102
#902102
C7×#4822
#898052
#897752
#896622
#896482
#896432
#896302
#895362
#895352
#895332
#895322
#895072
#894922
#32417