Finite Group Databasebeta

← #11738683

Subgroups — #11738683

Order 896.

324 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
1D7×#125C211yesmaximal, characteristic
2#90269C211yesmaximal, characteristic
3C7×#479C211yesmaximal, characteristic, complemented
4#89835C211yesmaximal, characteristic, complemented
5#89794C211yesmaximal, characteristic, complemented
6#89656C211yesmaximal, characteristic
7#89648C211yesmaximal, characteristic, complemented
8#89634C211yesmaximal, characteristic
9#89628C211yesmaximal, characteristic, complemented
10#89607C211yesmaximal, characteristic
11#89601C211yesmaximal, characteristic, complemented
12#89581C211yesmaximal, characteristic
13#89577C211yesmaximal, characteristic
14#89569C211yesmaximal, characteristic, complemented
15#89569C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18#6509C2211yescharacteristic
19#6507C2211yescharacteristic
20#6460C2211yescharacteristic
21#6459—21no—
22D7×C4⋊C4—21no—
23D7×C4⋊C4—21no—
24D7×C4⋊C4C2211yescharacteristic
25D7×C4⋊C4C2211yescharacteristic
26#6453C2211yescharacteristic
27#6453C2211yescharacteristic
28#6452—21no—
29#6452—21no—
30#6452—21no—
31#6451C2211yescharacteristic
32#6435—21no—
33C42×D7C2211yescharacteristic
34#6433C2211yescharacteristic
35C4×Dic14—21no—
36C4×Dic14C2211yescharacteristic
37C7×#107C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×C4⋊C8C2211yescharacteristic
40C7×#103C2211yescharacteristic
41C7×#103C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44(C2×D7)⋊C8C2211yescharacteristic
45(C2×D7)⋊C8C2211yescharacteristic
46#6392C2211yescharacteristic
47#6391C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50#6389C2211yescharacteristic
51#6389C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3236—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C4×D7—21no—
62C2×C4×D7—21no—
63C2×C4×D7D411yescharacteristic
64C2×C4×D7D411yescharacteristic
65C2×C4×D7C2311yescharacteristic
66C2×Dic14—21no—
67C2×Dic14C2311yescharacteristic
68C2×C7×C8C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C42×C7—21no—
77C42×C7C2311yescharacteristic
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80C4⋊Dic7—21no—
81C4⋊Dic7—21no—
82C4⋊Dic7—21no—
83C4⋊Dic7C2311yescharacteristic
84C4⋊Dic7C2311yescharacteristic
85C4⋊Dic7C2311yescharacteristic
86Dic7⋊C4—21no—
87Dic7⋊C4—21no—
88Dic7⋊C4—21no—
89Dic7⋊C4—21no—
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97C4×Dic7—21no—
98C4×Dic7D411yescharacteristic
99C4×Dic7D411yescharacteristic
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#533—71no—
104C2×#125—71no—
105#501—71no—
106#501—71no—
107#491—71no—
108#491—71no—
109#484—71no—
110#484—71no—
111#483—71no—
112#483—71no—
113#479—71no—
114#479—71no—
115#479—71no—
116#479D711yescharacteristic, complemented
117#432—71no—
118C22×D7C2×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×C7C2×D411yescharacteristic
126C2×C4×C7C2×D411yescharacteristic
127C2×C4×C7C2411yescharacteristic
128C2×Dic7—21no—
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7C2×D411yescharacteristic
133C2×Dic7C2×D411yescharacteristic
134C2×Dic7C2×D411yescharacteristic
135C4×D7—22no—
136C4×D7—21no—
137C4×D7—21no—
138C4×D7—41no—
139C4×D7—42no—
140C4×D7—42no—
141Dic14—21no—
142Dic14—21no—
143Dic14—42no—
144C7×C8—21no—
145C7×C8—21no—
146C7⋊C8—21no—
147C7⋊C8—21no—
148C4⋊Q8—71no—
149C4⋊Q8—71no—
150#125—71no—
151#125—71no—
152#125—71no—
153#125—141no—
154#125—141no—
155#125—141no—
156#125C2×D711yescharacteristic
157C22⋊Q8—71no—
158C22⋊Q8—71no—
159C22⋊Q8—141no—
160C4×Q8—71no—
161C4×Q8—141no—
162C4×Q8C2×D711yescharacteristic
163#117—141no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—141no—
167C2×C4⋊C4—141no—
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⋊C8C2×D711yescharacteristic
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⋊C8—71no—
189#97—71no—
190#97—71no—
191C22×C7C22×D411yescharacteristic
192C2×D7—21no—
193C2×D7—21no—
194C2×D7#13712yes—
195C4×C7—21no—
196C4×C7—21no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C7—41no—
200C4×C7—41no—
201C4×C72−1+411yescharacteristic
202C4×C72−1+411yescharacteristic
203Dic7—21no—
204Dic7—21no—
205Dic7—21no—
206Dic7—41no—
207Dic7—41no—
208Dic7—41no—
209Dic7—41no—
210C2×Q8—71no—
211C2×Q8—141no—
212C2×Q8C22×D711yescharacteristic
213C22×C4—71no—
214C22×C4—71no—
215C22×C4—71no—
216C22×C4—141no—
217C22×C4—141no—
218C2×C8—71no—
219C2×C8—71no—
220C2×C8C22×D711yescharacteristic
221C2×C8C22×D711yescharacteristic
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—71no—
226C4⋊C4—71no—
227C4⋊C4—71no—
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⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4C22×D711yescharacteristic
243C4⋊C4C22×D711yescharacteristic
244C4⋊C4C22×D711yescharacteristic
245C22⋊C4—141no—
246C22⋊C4—141no—
247C42—21no—
248C42—71no—
249C42—71no—
250C42—71no—
251C42—141no—
252C42C22×D711yescharacteristic
253C2×C7#53611yescharacteristic
254C2×C7C2×#13711yescharacteristic
255C2×C7#57511yescharacteristic
256D7—22no—
257C23—71no—
258Q8—21no—
259Q8—21no—
260Q8—141no—
261Q8—141no—
262Q8—282no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—71no—
268C2×C4—71no—
269C2×C4—71no—
270C2×C4—142no—
271C2×C4—141no—
272C2×C4—141no—
273C2×C4—141no—
274C2×C4—141no—
275C2×C4—141no—
276C2×C4—141no—
277C2×C4—281no—
278C2×C4—282no—
279C2×C4—282no—
280C2×C4D4×D711yescharacteristic
281C2×C4D4×D711yescharacteristic
282C2×C4C23×D711yescharacteristic
283C8—21no—
284C8—21no—
285C8—141no—
286C8—141no—
287C7#323611yescharacteristic, complemented, Sylow
288C22—72no—
289C22—141no—
290C22—141no—
291C22C2×D4×D711yescharacteristic
292C4—21no—
293C4—21no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—141no—
299C4—141no—
300C4—141no—
301C4—281no—
302C4—281no—
303C4—281no—
304C4—281no—
305C4#655111yescharacteristic
306C4#655411yescharacteristic
307C2—142no—
308C2#9031911yescharacteristic
309C2#9045911yescharacteristic
310C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#479D7
hypocenterC7#3236
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
3. C7×#479C2semidirect
4. #89835C2semidirect
5. #89794C2semidirect
7. #89648C2semidirect
9. #89628C2semidirect
11. #89601C2semidirect
14. #89569C2semidirect
15. #89569C2semidirect
116. #479D7semidirect
287. C7#3236semidirect

Maximal subgroups

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

subgroupindex
D7×#1252
#902692
C7×#4792
#898352
#897942
#896562
#896482
#896342
#896282
#896072
#896012
#895812
#895772
#895692
#32367