Finite Group Databasebeta

← #11738947

Subgroups — #11738947

Order 896.

322 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90313C211yesmaximal, characteristic
2#90208C211yesmaximal, characteristic
3C7×#485C211yesmaximal, characteristic
4#89817C211yesmaximal, characteristic
5#89768C211yesmaximal, characteristic
6#89648C211yesmaximal, characteristic, complemented
7#89647C211yesmaximal, characteristic, complemented
8#89628C211yesmaximal, characteristic, complemented
9#89626C211yesmaximal, characteristic, complemented
10#89569C211yesmaximal, characteristic, complemented
11#89569C211yesmaximal, characteristic, complemented
12#89566C211yesmaximal, characteristic, complemented
13#89563C211yesmaximal, characteristic, complemented
14#89495C211yesmaximal, characteristic
15#89491C211yesmaximal, characteristic
16C7×C22⋊Q8C2211yescharacteristic
17C7×#117C2211yescharacteristic
18Q8×Dic7C2211yescharacteristic
19#6507C2211yescharacteristic
20#6489C2211yescharacteristic
21#6487C2211yescharacteristic
22C2×C4×Dic7C2211yescharacteristic
23#6453C2211yescharacteristic
24#6453C2211yescharacteristic
25#6452—21no—
26#6451C2211yescharacteristic
27#6451C2211yescharacteristic
28#6451C2211yescharacteristic
29#6450—21no—
30#6450C2211yescharacteristic
31#6441—21no—
32#6441—21no—
33#6441—21no—
34#6440—21no—
35C4×Dic14—21no—
36C4×Dic14—21no—
37C7×#107C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×#103C2211yescharacteristic
40C7×#103C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6392C2211yescharacteristic
46#6391C2211yescharacteristic
47#6390C2211yescharacteristic
48#6390C2211yescharacteristic
49#6389C2211yescharacteristic
50#6389C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3237—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×Dic7D411yescharacteristic
63C22×Dic7D411yescharacteristic
64C2×Dic14—21no—
65C2×Dic14—21no—
66C2×Dic14—21no—
67C2×Dic14C2311yescharacteristic
68C2×C7×C8C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C7×C4⋊C4—21no—
71C7×C4⋊C4C2311yescharacteristic
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C22⋊C4—21no—
75C7×C22⋊C4—21no—
76C42×C7—21no—
77C22⋊Dic7—21no—
78C22⋊Dic7—21no—
79C4⋊Dic7—21no—
80C4⋊Dic7—21no—
81C4⋊Dic7—21no—
82C4⋊Dic7C2311yescharacteristic
83C4⋊Dic7C2311yescharacteristic
84C4⋊Dic7C2311yescharacteristic
85Dic7⋊C4—21no—
86Dic7⋊C4—21no—
87Dic7⋊C4—21no—
88Dic7⋊C4—21no—
89Dic7⋊C4—21no—
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94C4×Dic7—21no—
95C4×Dic7—21no—
96C4×Dic7—21no—
97C4×Dic7D411yescharacteristic
98C4×Dic7D411yescharacteristic
99C4×Dic7C2311yescharacteristic
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#533—71no—
104#533—71no—
105#501—71no—
106#501—71no—
107#491—71no—
108#491—71no—
109#485—71no—
110#485—71no—
111#485—71no—
112#485D711yescharacteristic
113#477—71no—
114#477—71no—
115#475—71no—
116#475—71no—
117#432—71no—
118C23×C7C2×D411yescharacteristic
119C7×Q8—21no—
120C7×Q8—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7C2×D411yescharacteristic
128C2×C4×C7C2411yescharacteristic
129C2×Dic7—22no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—41no—
135C2×Dic7C2×D411yescharacteristic
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139Dic14—21no—
140Dic14—21no—
141Dic14—42no—
142Dic14—42no—
143Dic14—42no—
144C7×C8—21no—
145C7×C8—21no—
146C7⋊C8—21no—
147C7⋊C8—21no—
148C4⋊Q8—71no—
149C4⋊Q8—71no—
150C4⋊Q8—71no—
151C4⋊Q8—71no—
152#125—71no—
153#125—71no—
154#125—141no—
155C22⋊Q8—71no—
156C22⋊Q8—141no—
157C22⋊Q8—141no—
158C22⋊Q8—141no—
159C22⋊Q8C2×D711yescharacteristic
160C4×Q8—71no—
161C4×Q8—71no—
162C4×Q8—141no—
163C4×Q8—141no—
164C4×Q8—141no—
165#117—71no—
166#117—141no—
167#117C2×D711yescharacteristic
168C2×C42—71no—
169#107—71no—
170#107—71no—
171#107—71no—
172#107C2×D711yescharacteristic
173#106—71no—
174#106—71no—
175#106—71no—
176#106C2×D711yescharacteristic
177C4⋊C8—71no—
178C4⋊C8—71no—
179#103—71no—
180#103—71no—
181#103—71no—
182#103—71no—
183#103—71no—
184#103—71no—
185#103C2×D711yescharacteristic
186#103C2×D711yescharacteristic
187C22⋊C8—71no—
188C22⋊C8C2×D711yescharacteristic
189#97—71no—
190#97—71no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7C22×D411yescharacteristic
195C4×C7—21no—
196C4×C7—41no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C7—41no—
200C4×C72−1+411yescharacteristic
201C4×C72−1+411yescharacteristic
202Dic7—21no—
203Dic7—21no—
204Dic7—21no—
205Dic7—41no—
206Dic7—41no—
207Dic7—41no—
208Dic7—41no—
209Dic7#13712yes—
210C2×Q8—71no—
211C2×Q8—141no—
212C2×Q8—141no—
213C2×Q8—141no—
214C2×Q8C22×D711yescharacteristic
215C22×C4—71no—
216C22×C4—71no—
217C22×C4C22×D711yescharacteristic
218C2×C8—71no—
219C2×C8—71no—
220C2×C8C22×D711yescharacteristic
221C2×C8C22×D711yescharacteristic
222C4⋊C4—21no—
223C4⋊C4—71no—
224C4⋊C4—71no—
225C4⋊C4—71no—
226C4⋊C4—141no—
227C4⋊C4—141no—
228C4⋊C4—141no—
229C4⋊C4—141no—
230C4⋊C4—141no—
231C4⋊C4—141no—
232C4⋊C4—141no—
233C4⋊C4—141no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4C22×D711yescharacteristic
239C4⋊C4C22×D711yescharacteristic
240C4⋊C4C22×D711yescharacteristic
241C22⋊C4—21no—
242C22⋊C4—21no—
243C22⋊C4—141no—
244C22⋊C4—141no—
245C42—21no—
246C42—71no—
247C42—71no—
248C42—71no—
249C42—71no—
250C42—141no—
251C42—141no—
252C42—141no—
253C2×C7—41no—
254C2×C7#53611yescharacteristic
255C2×C7C2×#13711yescharacteristic
256C2×C7#57511yescharacteristic
257C23D4×D711yescharacteristic
258Q8—21no—
259Q8—21no—
260Q8—141no—
261Q8—141no—
262Q8—282no—
263Q8—282no—
264Q8—282no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—71no—
272C2×C4—71no—
273C2×C4—71no—
274C2×C4—71no—
275C2×C4—142no—
276C2×C4—141no—
277C2×C4—141no—
278C2×C4—141no—
279C2×C4—141no—
280C2×C4—281no—
281C2×C4D4×D711yescharacteristic
282C2×C4C23×D711yescharacteristic
283C8—21no—
284C8—21no—
285C8—141no—
286C8—141no—
287C7#323711yescharacteristic, complemented, Sylow
288C22—21no—
289C22—21no—
290C22—21no—
291C22C2×D4×D711yescharacteristic
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—72no—
298C4—141no—
299C4—141no—
300C4—141no—
301C4—281no—
302C4—281no—
303C4—281no—
304C4—281no—
305C4#655411yescharacteristic
306C4#655411yescharacteristic
307C2—41no—
308C2#9033911yescharacteristic
309C2#9045911yescharacteristic
310C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#485C2
socleC22×C7C22×D4
prime radicalC23×C7C2×D4
hypercenter#485D7
hypocenterC7#3237
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
6. #89648C2semidirect
7. #89647C2semidirect
8. #89628C2semidirect
9. #89626C2semidirect
10. #89569C2semidirect
11. #89569C2semidirect
12. #89566C2semidirect
13. #89563C2semidirect
287. C7#3237semidirect

Maximal subgroups

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

subgroupindex
#903132
#902082
C7×#4852
#898172
#897682
#896482
#896472
#896282
#896262
#895692
#895662
#895632
#894952
#894912
#32377