Finite Group Databasebeta

← #11738194

Subgroups — #11738194

Order 896.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90334C211yesmaximal, characteristic
2#90316C211yesmaximal, characteristic, complemented
3C7×#467C211yesmaximal, characteristic, complemented
4#89957C211yesmaximal, characteristic, complemented
5#89954C211yesmaximal, characteristic, complemented
6#89886C211yesmaximal, characteristic
7#89874C211yesmaximal, characteristic, complemented
8#89824C211yesmaximal, characteristic, complemented
9#89821C211yesmaximal, characteristic
10#89663C211yesmaximal, characteristic, complemented
11#89647C211yesmaximal, characteristic, complemented
12#89590C211yesmaximal, characteristic, complemented
13#89571C211yesmaximal, characteristic, complemented
14#89570C211yesmaximal, characteristic, complemented
15#89563C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C22×C7×C8C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C7×C22⋊Q8C2211yescharacteristic
20#6510C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23#6507C2211yescharacteristic
24C2×#1144C2211yescharacteristic
25C2×#1143C2211yescharacteristic
26#6493C2211yescharacteristic
27#6489C2211yescharacteristic
28#6487C2211yescharacteristic
29C2×#1136C2211yescharacteristic
30C2×#1132C2211yescharacteristic
31#6461—21no—
32#6460—21no—
33#6456C2211yescharacteristic
34#6453—21no—
35#6453C2211yescharacteristic
36#6452—21no—
37#6451C2211yescharacteristic
38#6450C2211yescharacteristic
39#6448—21no—
40#6446—21no—
41#6442—21no—
42#6442—21no—
43C7×#107C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6395C2211yescharacteristic
49#6391C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3258—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×Dic14—21no—
67C2×Dic14C2311yescharacteristic
68C7×Q16—41no—
69C2×C7×C8—41no—
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C22⋊C4—21no—
77C7×C22⋊C4—21no—
78C22⋊Dic7—21no—
79C22⋊Dic7D411yescharacteristic
80C22⋊Dic7D411yescharacteristic
81#1144—41no—
82#1143—41no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7D411yescharacteristic
91C4⋊Dic7D411yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7C2311yescharacteristic
104C4×Dic7C2311yescharacteristic
105#1136—41no—
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108#1132—41no—
109#542—71no—
110#541—71no—
111#479—71no—
112#478—71no—
113#477—71no—
114#475—71no—
115#470—71no—
116#468—71no—
117#467D711yescharacteristic, complemented
118#465—71no—
119#464—71no—
120#464—71no—
121#462—71no—
122#461—71no—
123#424—71no—
124C23×C7C2×D411yescharacteristic
125C22×D7—21no—
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C22⋊D7—81no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146D28—21no—
147D28—21no—
148C4×D7—81no—
149Dic14—21no—
150Dic14—21no—
151Dic14—81no—
152C7×C8—21no—
153C7×C8—21no—
154C7⋊C8—21no—
155C7⋊C8—21no—
156C2×Q16—71no—
157C2×Q16C2×D711yescharacteristic
158C2×SD16—71no—
159C2×SD16—71no—
160C2×M16—71no—
161C22×C8C2×D711yescharacteristic
162C4⋊Q8—71no—
163C4⋊Q8—71no—
164#126—141no—
165#126—141no—
166#126—141no—
167#125—71no—
168#125—141no—
169#125—141no—
170#124—71no—
171#124—141no—
172#123—141no—
173C22⋊Q8—71no—
174C22⋊Q8—141no—
175C22⋊Q8C2×D711yescharacteristic
176C22⋊Q8C2×D711yescharacteristic
177#121—71no—
178C4×Q8—71no—
179C4×Q8—71no—
180C4×Q8—71no—
181C4×D4—71no—
182#117—71no—
183#107—71no—
184#107C2×D711yescharacteristic
185#106—71no—
186#106—71no—
187C4⋊C8—71no—
188C4⋊C8—71no—
189C4⋊C8—71no—
190C4⋊C8—71no—
191#103—71no—
192#103—71no—
193#103—71no—
194#103—71no—
195#103C2×D711yescharacteristic
196#103C2×D711yescharacteristic
197#102—71no—
198#102—71no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7C22×D411yescharacteristic
203C2×D7—41no—
204C2×D7—41no—
205C2×D7—41no—
206C4×C7—21no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7H(5,𝔽2)11yescharacteristic
212C4×C72−1+411yescharacteristic
213Dic7—21no—
214Dic7—21no—
215Dic7—21no—
216Dic7—21no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—71no—
221C2×Q8—141no—
222C2×Q8C22×D711yescharacteristic
223C2×Q8C22×D711yescharacteristic
224C2×D4—71no—
225C2×D4—141no—
226C22×C4—141no—
227C22×C4C22×D711yescharacteristic
228Q16—41no—
229Q16—281no—
230SD16—281no—
231SD16—281no—
232M16—281no—
233C2×C8—41no—
234C2×C8—71no—
235C2×C8—71no—
236C2×C8C22×D711yescharacteristic
237C2×C8C22×D711yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—71no—
258C22⋊C4—71no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C42—71no—
265C42—71no—
266C42—141no—
267C42—141no—
268C42—141no—
269C2×C7—41no—
270C2×C7#53711yescharacteristic
271C2×C7#57711yescharacteristic
272C2×C7#57811yescharacteristic
273D7—81no—
274C23—141no—
275C23D4×D711yescharacteristic
276Q8—21no—
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—141no—
281Q8—141no—
282Q8—561no—
283D4—141no—
284D4—141no—
285D4—561no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—71no—
293C2×C4—71no—
294C2×C4—71no—
295C2×C4—71no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—561no—
300C2×C4D4×D711yescharacteristic
301C2×C4C23×D711yescharacteristic
302C8—21no—
303C8—21no—
304C8—141no—
305C8—141no—
306C7#325811yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22—21no—
310C22—281no—
311C22—281no—
312C22—281no—
313C22C2×D4×D711yescharacteristic
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—141no—
320C4—141no—
321C4—141no—
322C4—141no—
323C4—281no—
324C4—281no—
325C4—281no—
326C4#654811yescharacteristic
327C4#655411yescharacteristic
328C2—41no—
329C2—561no—
330C2#9028811yescharacteristic
331C2#9045711yescharacteristic
332C2#9046111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#467C2
socleC22×C7C22×D4
prime radical#6493C22
hypercenter#467D7
hypocenterC7#3258
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #90316C2semidirect
3. C7×#467C2semidirect
4. #89957C2semidirect
5. #89954C2semidirect
7. #89874C2semidirect
8. #89824C2semidirect
10. #89663C2semidirect
11. #89647C2semidirect
12. #89590C2semidirect
13. #89571C2semidirect
14. #89570C2semidirect
15. #89563C2semidirect
117. #467D7semidirect
306. C7#3258semidirect

Maximal subgroups

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

subgroupindex
#903342
#903162
C7×#4672
#899572
#899542
#898862
#898742
#898242
#898212
#896632
#896472
#895902
#895712
#895702
#895632
#32587