Finite Group Databasebeta

← #11738736

Subgroups — #11738736

Order 896.

420 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90302C211yesmaximal, characteristic, complemented
2#90198C211yesmaximal, characteristic, complemented
3C7×#480C211yesmaximal, characteristic, complemented
4#89810C211yesmaximal, characteristic, complemented
5C2×#6381C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89654C211yesmaximal, characteristic, complemented
8D7×#107C211yesmaximal, characteristic, complemented
9#89646C211yesmaximal, characteristic, complemented
10#89547C211yesmaximal, characteristic, complemented
11#89546C211yesmaximal, characteristic, complemented
12#89531C211yesmaximal, characteristic, complemented
13#89528C211yesmaximal, characteristic, complemented
14#89508C211yesmaximal, characteristic, complemented
15(C22⋊D7)⋊C8C211yesmaximal, characteristic, complemented
16C2×#1159—21no—
17C2×#1157—21no—
18C7×#121C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20#6501C2211yescharacteristic
21#6498—21no—
22D4×Dic7C2211yescharacteristic
23C2×#1141C2211yescharacteristic
24#6493C2211yescharacteristic
25#6492—21no—
26C4×C22⋊D7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C22×C7⋊C8C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6459—21no—
31#6458C2211yescharacteristic
32#6457—21no—
33#6456C2211yescharacteristic
34D7×C4⋊C4C2211yescharacteristic
35D7×C4⋊C4C2211yescharacteristic
36#6451C2211yescharacteristic
37#6451C2211yescharacteristic
38#6446—21no—
39#6441—21no—
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6406C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6383C2211yescharacteristic
54#6381C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3334—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7D411yescharacteristic
67C22×Dic7—21no—
68C22×Dic7—21no—
69#1159—44no—
70#1157—44no—
71C2×D28—21no—
72C2×D28C2311yescharacteristic
73C2×C4×D7—21no—
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7C2311yescharacteristic
77C2×Dic14—21no—
78C2×Dic14—21no—
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7D411yescharacteristic
89#1141—22no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4D411yescharacteristic
94C4⋊Dic7—21no—
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4D411yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8—22no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C8×D7—22no—
108#547—71no—
109#547—71no—
110#500—71no—
111#480—71no—
112#480—71no—
113#480—71no—
114#480D711yescharacteristic, complemented
115#474—71no—
116#474—71no—
117#466—71no—
118#466—71no—
119C4×D8—71no—
120C8×D4—71no—
121C2×#107—71no—
122C2×#107—71no—
123C23×C7—21no—
124C23×C7C2×D411yescharacteristic
125C22×D7—21no—
126C22×D7C2×D411yescharacteristic
127C7×D4—21no—
128C7×D4—21no—
129C7×D4—42no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—42no—
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C22⋊D7—42no—
139C22⋊D7—42no—
140C22⋊D7D814yescomplemented
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—42no—
145C2×Dic7—42no—
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149D28—21no—
150D28—21no—
151D28—42no—
152C4×D7—21no—
153C4×D7—21no—
154C4×D7—42no—
155C4×D7—42no—
156C4×D7—42no—
157Dic14—42no—
158Dic14—42no—
159C7×C8—21no—
160C7×C8—21no—
161C7⋊C8—21no—
162C7⋊C8#4112yescomplemented
163C2×#41—141no—
164C2×#41—141no—
165C2×D8—71no—
166C22×C8—71no—
167C22×C8—71no—
168C4⋊Q8—71no—
169C4⋊Q8—71no—
170#123—141no—
171#123—141no—
172#123—141no—
173#123—141no—
174C22⋊Q8—141no—
175C22⋊Q8—141no—
176#121—71no—
177#121—71no—
178#121—71no—
179#121C2×D711yescharacteristic
180C4×D4—71no—
181C4×D4—71no—
182C4×D4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4C2×D711yescharacteristic
187#107—71no—
188#107—71no—
189#107—71no—
190#107—71no—
191#107—71no—
192#107—71no—
193#107—71no—
194#107C2×D711yescharacteristic
195#107C2×D711yescharacteristic
196C4⋊C8—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102C2×D711yescharacteristic
202#102C2×D711yescharacteristic
203C22⋊C8—71no—
204C22⋊C8C2×D711yescharacteristic
205C4×C8—71no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7C2×D812yes—
212C22×C7C22×D411yescharacteristic
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7C2×D812yes—
219C4×C7—21no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7C2×#4111yescharacteristic
224C4×C72−1+411yescharacteristic
225Dic7—21no—
226Dic7—21no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7C2×D812yes—
231#41—284no—
232#41—284no—
233C2×Q8—141no—
234C2×Q8—141no—
235C2×D4—21no—
236C2×D4—71no—
237C2×D4—71no—
238C2×D4—141no—
239C2×D4—141no—
240C2×D4—141no—
241C2×D4C22×D711yescharacteristic
242C22×C4—21no—
243C22×C4—71no—
244C22×C4—141no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—141no—
249C22×C4C22×D711yescharacteristic
250D8—142no—
251C2×C8—71no—
252C2×C8—71no—
253C2×C8—142no—
254C2×C8—142no—
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—71no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4C22×D711yescharacteristic
268C4⋊C4C22×D711yescharacteristic
269C4⋊C4C22×D711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—71no—
272C22⋊C4—71no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C42—71no—
279C2×C7—22no—
280C2×C7—81no—
281C2×C7#54711yescharacteristic
282C2×C7C22×D811yescharacteristic
283C2×C7#57811yescharacteristic
284D7—22no—
285D7—81no—
286C23—21no—
287C23—71no—
288C23—141no—
289C23D4×D711yescharacteristic
290Q8—282no—
291Q8—282no—
292D4—21no—
293D4—21no—
294D4—42no—
295D4—74no—
296D4—141no—
297D4—141no—
298D4—282no—
299D4—282no—
300D4—282no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—71no—
308C2×C4—71no—
309C2×C4—71no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—282no—
316C2×C4—282no—
317C2×C4—282no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4D4×D711yescharacteristic
321C2×C4C23×D711yescharacteristic
322C8—21no—
323C8—21no—
324C8—72no—
325C8—141no—
326C7#333411yescharacteristic, complemented, Sylow
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—72no—
333C22—141no—
334C22—141no—
335C22—281no—
336C22—281no—
337C22—281no—
338C22D7×D812yes—
339C22C2×D4×D711yescharacteristic
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—72no—
345C4—141no—
346C4—141no—
347C4—281no—
348C4—281no—
349C4—281no—
350C4D7×#4111yescharacteristic
351C4#655411yescharacteristic
352C2—22no—
353C2—81no—
354C2—142no—
355C2—561no—
356C2#9034611yescharacteristic
357C2C2×D7×D811yescharacteristic
358C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#480C2
socleC22×C7C22×D4
hypercenter#480D7
hypocenterC7#3334
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #90302C2semidirect
2. #90198C2semidirect
3. C7×#480C2semidirect
4. #89810C2semidirect
5. C2×#6381C2semidirect
6. #89662C2semidirect
7. #89654C2semidirect
8. D7×#107C2semidirect
9. #89646C2semidirect
10. #89547C2semidirect
11. #89546C2semidirect
12. #89531C2semidirect
13. #89528C2semidirect
14. #89508C2semidirect
15. (C22⋊D7)⋊C8C2semidirect
114. #480D7semidirect
140. C22⋊D7D8semidirect
162. C7⋊C8#41semidirect
326. C7#3334semidirect

Maximal subgroups

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

subgroupindex
#903022
#901982
C7×#4802
#898102
C2×#63812
#896622
#896542
D7×#1072
#896462
#895472
#895462
#895312
#895282
#895082
(C22⋊D7)⋊C82
#33347