Finite Group Databasebeta

← #11738276

Subgroups — #11738276

Order 896.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90333C211yesmaximal, characteristic, complemented
2#90307C211yesmaximal, characteristic
3C7×#468C211yesmaximal, characteristic, complemented
4#89943C211yesmaximal, characteristic, complemented
5#89936C211yesmaximal, characteristic, complemented
6#89899C211yesmaximal, characteristic, complemented
7#89893C211yesmaximal, characteristic, complemented
8#89817C211yesmaximal, characteristic, complemented
9#89806C211yesmaximal, characteristic, complemented
10#89652C211yesmaximal, characteristic, complemented
11#89648C211yesmaximal, characteristic, complemented
12#89586C211yesmaximal, characteristic, complemented
13#89563C211yesmaximal, characteristic, complemented
14#89551C211yesmaximal, characteristic, complemented
15#89538C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C2×C7×M16C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C7×#121C2211yescharacteristic
20#6509C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22#6502—21no—
23#6501C2211yescharacteristic
24#6499—21no—
25D4×Dic7C2211yescharacteristic
26C4×C22⋊D7C2211yescharacteristic
27#6489C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6461—21no—
31#6455C2211yescharacteristic
32#6455C2211yescharacteristic
33#6453—21no—
34#6453C2211yescharacteristic
35#6451C2211yescharacteristic
36#6449—21no—
37#6448—21no—
38#6440—21no—
39#6440—21no—
40C7×#107C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47C22⋊(C7⋊C8)C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6391C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3348—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7D411yescharacteristic
66C22×Dic7—21no—
67C22×Dic7—21no—
68C2×C4×D7C2311yescharacteristic
69C2×Dic14—21no—
70C7×SD16—22no—
71C7×M16—22no—
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C22⋊C4—21no—
78C7×C22⋊C4—21no—
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7D411yescharacteristic
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4D411yescharacteristic
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C8×D7—22no—
106#555—71no—
107#552—71no—
108#501—71no—
109#485—71no—
110#485—71no—
111#482—71no—
112#481—71no—
113#478—71no—
114#475—71no—
115#468D711yescharacteristic, complemented
116#465—71no—
117#440—71no—
118#435—71no—
119#428—71no—
120#427—71no—
121C23×C7—21no—
122C23×C7C2×D411yescharacteristic
123C22×D7C2×D411yescharacteristic
124C7×Q8—21no—
125C7×Q8—21no—
126C7×D4—21no—
127C7×D4—21no—
128C7×D4—81no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—41no—
137C22⋊D7—81no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C4×D7—21no—
148C4×D7—21no—
149Dic14—81no—
150C7×C8—21no—
151C7×C8#4112yes—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C2×SD16C2×D711yescharacteristic
155C2×M16C2×D711yescharacteristic
156C22×C8—71no—
157C4⋊Q8—71no—
158#126—141no—
159#125—71no—
160#125—141no—
161#124—141no—
162#124—141no—
163#123—141no—
164C22⋊Q8—71no—
165C22⋊Q8C2×D711yescharacteristic
166#121—71no—
167#121—141no—
168#121C2×D711yescharacteristic
169C4×Q8—71no—
170C4×D4—71no—
171C4×D4—71no—
172#117—71no—
173#117—71no—
174#117—71no—
175#117—141no—
176#117—141no—
177C2×C4⋊C4—71no—
178#107—71no—
179#107—71no—
180#107—71no—
181#107C2×D711yescharacteristic
182#106—71no—
183#106—71no—
184#106—71no—
185#106—71no—
186#106—71no—
187C4⋊C8—71no—
188#103—71no—
189#103—71no—
190#103C2×D711yescharacteristic
191#102—71no—
192#102—71no—
193#102C2×D711yescharacteristic
194C22⋊C8—71no—
195C22⋊C8—71no—
196#97—71no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7C22×D411yescharacteristic
204C2×D7—21no—
205C2×D7—21no—
206C2×D7—21no—
207C4×C7—21no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7C2×#4111yescharacteristic
212C4×C72−1+411yescharacteristic
213Dic7—21no—
214Dic7—21no—
215Dic7—21no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—141no—
221C2×Q8C22×D711yescharacteristic
222C2×D4—21no—
223C2×D4—71no—
224C2×D4—141no—
225C2×D4C22×D711yescharacteristic
226C22×C4—71no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4C22×D711yescharacteristic
230SD16—22no—
231M16—22no—
232C2×C8—71no—
233C2×C8—71no—
234C2×C8—142no—
235C2×C8C22×D711yescharacteristic
236C2×C8C22×D711yescharacteristic
237C4⋊C4—21no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
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—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C42—71no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42—141no—
266C2×C7—41no—
267C2×C7—81no—
268C2×C7#54711yescharacteristic
269C2×C7#57511yescharacteristic
270C2×C7#57711yescharacteristic
271D7—41no—
272C23—21no—
273C23—71no—
274C23D4×D711yescharacteristic
275Q8—21no—
276Q8—21no—
277Q8—561no—
278D4—21no—
279D4—21no—
280D4—81no—
281D4—281no—
282D4—561no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—71no—
289C2×C4—71no—
290C2×C4—71no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4D4×D711yescharacteristic
300C2×C4C23×D711yescharacteristic
301C8—21no—
302C8—141no—
303C8—141no—
304C8#115912yes—
305C7#334811yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—141no—
313C22—141no—
314C22—141no—
315C22C2×D4×D711yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—141no—
321C4—141no—
322C4—141no—
323C4—281no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4C2×#115911yescharacteristic
328C4#655411yescharacteristic
329C2—41no—
330C2—81no—
331C2—281no—
332C2#9030211yescharacteristic
333C2#9046611yescharacteristic
334C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#468C2
socleC22×C7C22×D4
prime radical#90307C2
hypercenter#468D7
hypocenterC7#3348
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #90333C2semidirect
3. C7×#468C2semidirect
4. #89943C2semidirect
5. #89936C2semidirect
6. #89899C2semidirect
7. #89893C2semidirect
8. #89817C2semidirect
9. #89806C2semidirect
10. #89652C2semidirect
11. #89648C2semidirect
12. #89586C2semidirect
13. #89563C2semidirect
14. #89551C2semidirect
15. #89538C2semidirect
115. #468D7semidirect
305. C7#3348semidirect

Maximal subgroups

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

subgroupindex
#903332
#903072
C7×#4682
#899432
#899362
#898992
#898932
#898172
#898062
#896522
#896482
#895862
#895632
#895512
#895382
#33487