Finite Group Databasebeta

← #11738619

Subgroups — #11738619

Order 896.

368 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#125C211yesmaximal, characteristic, complemented
2#90239C211yesmaximal, characteristic
3C7×#478C211yesmaximal, characteristic, complemented
4#89838C211yesmaximal, characteristic, complemented
5#89781C211yesmaximal, characteristic, complemented
6#89654C211yesmaximal, characteristic, complemented
7#89648C211yesmaximal, characteristic, complemented
8#89635C211yesmaximal, characteristic, complemented
9#89628C211yesmaximal, characteristic, complemented
10#89619C211yesmaximal, characteristic, complemented
11#89607C211yesmaximal, characteristic, complemented
12#89549C211yesmaximal, characteristic, complemented
13#89546C211yesmaximal, characteristic, complemented
14#89536C211yesmaximal, characteristic, complemented
15#89536C211yesmaximal, characteristic, complemented
16C2×#1157—21no—
17C7×#125C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19#6503C2211yescharacteristic
20#6501C2211yescharacteristic
21C4×C22⋊D7—21no—
22#6458C2211yescharacteristic
23D7×C4⋊C4—21no—
24D7×C4⋊C4—21no—
25D7×C4⋊C4C2211yescharacteristic
26D7×C4⋊C4C2211yescharacteristic
27#6453C2211yescharacteristic
28#6453C2211yescharacteristic
29#6452—21no—
30#6452—21no—
31#6451C2211yescharacteristic
32#6448—21no—
33#6447—21no—
34C4×D28C2211yescharacteristic
35C42×D7C2211yescharacteristic
36#6433C2211yescharacteristic
37C7×#107C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×C4⋊C8C2211yescharacteristic
40C7×#102C2211yescharacteristic
41C7×#102C2211yescharacteristic
42#6406C2211yescharacteristic
43#6406C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48#6392C2211yescharacteristic
49#6391C2211yescharacteristic
50#6389C2211yescharacteristic
51#6389C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3234—71nomaximal, Sylow
60C2×C7×D4C2311yescharacteristic
61C22×C4×C7—21no—
62C2×C22⋊D7—21no—
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65#1157—44no—
66C2×D28—21no—
67C2×D28C2311yescharacteristic
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7D411yescharacteristic
72C2×C4×D7D411yescharacteristic
73C2×C4×D7C2311yescharacteristic
74C2×Dic14—21no—
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C22⋊C4—21no—
83C42×C7C2311yescharacteristic
84C22⋊Dic7—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99C4×Dic7D411yescharacteristic
100C4×Dic7D411yescharacteristic
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104#532—71no—
105C2×#125—71no—
106#501—71no—
107#501—71no—
108#490—71no—
109#490—71no—
110#481—71no—
111#481—71no—
112#480—71no—
113#480—71no—
114#478—71no—
115#478—71no—
116#478—71no—
117#478D711yescharacteristic, complemented
118#432—71no—
119C23×C7—21no—
120C22×D7—21no—
121C22×D7C2×D411yescharacteristic
122C7×D4—21no—
123C7×D4—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—42no—
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2411yescharacteristic
131C22⋊D7—42no—
132C22⋊D7—42no—
133C22⋊D7—42no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140D28—21no—
141D28—21no—
142D28—42no—
143C4×D7—22no—
144C4×D7—21no—
145C4×D7—21no—
146C4×D7—41no—
147C4×D7—42no—
148C4×D7—42no—
149C4×D7—42no—
150Dic14—42no—
151C7×C8—21no—
152C7×C8—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C2×#41—141no—
156C4⋊Q8—71no—
157#127—71no—
158#125—71no—
159#125—71no—
160#125—71no—
161#125—141no—
162#125—141no—
163#125C2×D711yescharacteristic
164#124—141no—
165#121—71no—
166#121—71no—
167#121—141no—
168C4×D4—71no—
169C4×D4—141no—
170C4×D4C2×D711yescharacteristic
171C2×C4⋊C4—71no—
172C2×C4⋊C4—71no—
173C2×C4⋊C4—141no—
174C2×C4⋊C4—141no—
175C2×C42—71no—
176#107—71no—
177#107—71no—
178#107—71no—
179#107C2×D711yescharacteristic
180#106—71no—
181#106—71no—
182#106—71no—
183#106C2×D711yescharacteristic
184C4⋊C8—71no—
185C4⋊C8C2×D711yescharacteristic
186#102—71no—
187#102—71no—
188#102—71no—
189#102—71no—
190#102—71no—
191#102—71no—
192#102C2×D711yescharacteristic
193#102C2×D711yescharacteristic
194C22⋊C8—71no—
195C22⋊C8—71no—
196#97—71no—
197#97—71no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7C22×D411yescharacteristic
202C2×D7—21no—
203C2×D7—21no—
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C2×D7AGL(1,ℤ/8)12yes—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C72−1+411yescharacteristic
214C4×C72−1+411yescharacteristic
215Dic7—21no—
216Dic7—21no—
217Dic7—21no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221#41—284no—
222C2×Q8—141no—
223C2×D4—71no—
224C2×D4—141no—
225C2×D4—141no—
226C2×D4—141no—
227C2×D4—141no—
228C2×D4C22×D711yescharacteristic
229C22×C4—21no—
230C22×C4—71no—
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4—141no—
236C2×C8—71no—
237C2×C8—71no—
238C2×C8C22×D711yescharacteristic
239C2×C8C22×D711yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—71no—
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⋊C4—141no—
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C42—71no—
262C42—71no—
263C42—71no—
264C42C22×D711yescharacteristic
265C2×C7—81no—
266C2×C7#53611yescharacteristic
267C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
268C2×C7#57511yescharacteristic
269D7—22no—
270D7—81no—
271C23—21no—
272C23—71no—
273C23—141no—
274Q8—282no—
275D4—21no—
276D4—21no—
277D4—141no—
278D4—141no—
279D4—282no—
280D4—282no—
281D4—282no—
282D4—282no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—71no—
290C2×C4—142no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—281no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4D4×D711yescharacteristic
301C2×C4D4×D711yescharacteristic
302C2×C4C23×D711yescharacteristic
303C8—21no—
304C8—21no—
305C8—141no—
306C8—141no—
307C7#323411yescharacteristic, complemented, Sylow
308C22—41no—
309C22—41no—
310C22—41no—
311C22—72no—
312C22—141no—
313C22—141no—
314C22—281no—
315C22—281no—
316C22—281no—
317C22C2×D4×D711yescharacteristic
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—141no—
324C4—141no—
325C4—141no—
326C4—281no—
327C4—281no—
328C4—281no—
329C4#655111yescharacteristic
330C4#655411yescharacteristic
331C2—81no—
332C2—142no—
333C2—561no—
334C2#9031911yescharacteristic
335C2#9045911yescharacteristic
336C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#478C2
socleC22×C7C22×D4
prime radical#90239C2
hypercenter#478D7
hypocenterC7#3234
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. D7×#125C2semidirect
3. C7×#478C2semidirect
4. #89838C2semidirect
5. #89781C2semidirect
6. #89654C2semidirect
7. #89648C2semidirect
8. #89635C2semidirect
9. #89628C2semidirect
10. #89619C2semidirect
11. #89607C2semidirect
12. #89549C2semidirect
13. #89546C2semidirect
14. #89536C2semidirect
15. #89536C2semidirect
117. #478D7semidirect
307. C7#3234semidirect

Maximal subgroups

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

subgroupindex
D7×#1252
#902392
C7×#4782
#898382
#897812
#896542
#896482
#896352
#896282
#896192
#896072
#895492
#895462
#895362
#32347