Finite Group Databasebeta

← #11738779

Subgroups — #11738779

Order 896.

366 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90309C211yesmaximal, characteristic, complemented
2#90205C211yesmaximal, characteristic, complemented
3C7×#481C211yesmaximal, characteristic, complemented
4#89810C211yesmaximal, characteristic, complemented
5C2×#6381C211yesmaximal, characteristic, complemented
6#89643C211yesmaximal, characteristic, complemented
7#89638C211yesmaximal, characteristic, complemented
8#89632C211yesmaximal, characteristic, complemented
9#89630C211yesmaximal, characteristic, complemented
10#89551C211yesmaximal, characteristic, complemented
11#89547C211yesmaximal, characteristic, complemented
12#89535C211yesmaximal, characteristic, complemented
13#89528C211yesmaximal, characteristic, complemented
14#89507C211yesmaximal, characteristic, complemented
15(C22⋊D7)⋊C8C211yesmaximal, characteristic, complemented
16C7×#121C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18#6502—21no—
19#6501C2211yescharacteristic
20D4×Dic7—21no—
21D4×Dic7C2211yescharacteristic
22C2×#1141C2211yescharacteristic
23#6493—21no—
24#6493C2211yescharacteristic
25C4×C22⋊D7—21no—
26C4×C22⋊D7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C22×C7⋊C8C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6461—21no—
31#6458C2211yescharacteristic
32#6457—21no—
33#6456C2211yescharacteristic
34#6455C2211yescharacteristic
35#6455C2211yescharacteristic
36#6453C2211yescharacteristic
37#6453C2211yescharacteristic
38#6449—21no—
39#6442—21no—
40C7×#106C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6406C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6391C2211yescharacteristic
50#6391C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6383C2211yescharacteristic
54#6381C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3339—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—
69C2×D28—21no—
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C22⋊C4—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7D411yescharacteristic
83#1141—22no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4D411yescharacteristic
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8—22no—
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C8×D7—22no—
106#553—71no—
107#553—71no—
108#499—71no—
109#482—71no—
110#482—71no—
111#481—71no—
112#481D711yescharacteristic, complemented
113#478—71no—
114#478—71no—
115#466—71no—
116#466—71no—
117C4×D8—71no—
118C8×D4—71no—
119#427—71no—
120C2×#107—71no—
121C23×C7—21no—
122C23×C7C2×D411yescharacteristic
123C22×D7—21no—
124C22×D7C2×D411yescharacteristic
125C7×D4—21no—
126C7×D4—21no—
127C7×D4—42no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—81no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—22no—
137C22⋊D7—42no—
138C22⋊D7—81no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—81no—
143C2×Dic7—81no—
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147D28—21no—
148D28—21no—
149D28—81no—
150C4×D7—21no—
151C4×D7—21no—
152C4×D7—81no—
153C7×C8—21no—
154C7×C8—21no—
155C7⋊C8—21no—
156C7⋊C8#4112yes—
157C2×D8—71no—
158C22×C8—71no—
159C22×C8—71no—
160#126—141no—
161#126—141no—
162#125—71no—
163#125—71no—
164#123—141no—
165#123—141no—
166#121—71no—
167#121—71no—
168#121—71no—
169#121—141no—
170#121—141no—
171#121C2×D711yescharacteristic
172C4×D4—71no—
173C4×D4—71no—
174C4×D4—71no—
175C4×D4—141no—
176C4×D4—141no—
177#117—71no—
178#117—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4C2×D711yescharacteristic
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#107—71no—
186#106—71no—
187#106—71no—
188#106C2×D711yescharacteristic
189#106C2×D711yescharacteristic
190C4⋊C8—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194#102—71no—
195#102C2×D711yescharacteristic
196#102C2×D711yescharacteristic
197C22⋊C8—71no—
198C22⋊C8C2×D711yescharacteristic
199C4×C8—71no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7#13512yes—
206C22×C7C22×D411yescharacteristic
207C2×D7—21no—
208C2×D7—21no—
209C2×D7—21no—
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C4×C7—21no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7C2×#4111yescharacteristic
218C4×C72−1+411yescharacteristic
219Dic7—21no—
220Dic7—21no—
221Dic7—21no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×D4—21no—
226C2×D4—71no—
227C2×D4—71no—
228C2×D4—141no—
229C2×D4—141no—
230C2×D4—141no—
231C2×D4C22×D711yescharacteristic
232C22×C4—21no—
233C22×C4—71no—
234C22×C4—141no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4C22×D711yescharacteristic
238D8—142no—
239C2×C8—71no—
240C2×C8—71no—
241C2×C8—142no—
242C2×C8—142no—
243C2×C8C22×D711yescharacteristic
244C2×C8C22×D711yescharacteristic
245C4⋊C4—21no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4C22×D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C4⋊C4C22×D711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—71no—
260C22⋊C4—71no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C42—71no—
269C42—141no—
270C42—141no—
271C2×C7—22no—
272C2×C7—81no—
273C2×C7#54711yescharacteristic
274C2×C7C2×#13511yescharacteristic
275C2×C7#57611yescharacteristic
276D7—41no—
277D7—81no—
278C23—21no—
279C23—71no—
280C23—141no—
281C23D4×D711yescharacteristic
282D4—21no—
283D4—21no—
284D4—42no—
285D4—142no—
286D4—141no—
287D4—141no—
288D4—282no—
289D4—561no—
290D4—561no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—71no—
297C2×C4—71no—
298C2×C4—71no—
299C2×C4—81no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—561no—
306C2×C4—561no—
307C2×C4—561no—
308C2×C4D4×D711yescharacteristic
309C2×C4C23×D711yescharacteristic
310C8—21no—
311C8—21no—
312C8—72no—
313C8—141no—
314C7#333911yescharacteristic, complemented, Sylow
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—141no—
321C22—141no—
322C22—141no—
323C22—281no—
324C22—281no—
325C22—281no—
326C22#647912yes—
327C22C2×D4×D711yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—141no—
333C4—141no—
334C4—141no—
335C4—281no—
336C4—281no—
337C4—281no—
338C4D7×#4111yescharacteristic
339C4#655411yescharacteristic
340C2—22no—
341C2—81no—
342C2—281no—
343C2—561no—
344C2#9034611yescharacteristic
345C2C2×#647911yescharacteristic
346C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#481C2
socleC22×C7C22×D4
hypercenter#481D7
hypocenterC7#3339
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #90309C2semidirect
2. #90205C2semidirect
3. C7×#481C2semidirect
4. #89810C2semidirect
5. C2×#6381C2semidirect
6. #89643C2semidirect
7. #89638C2semidirect
8. #89632C2semidirect
9. #89630C2semidirect
10. #89551C2semidirect
11. #89547C2semidirect
12. #89535C2semidirect
13. #89528C2semidirect
14. #89507C2semidirect
15. (C22⋊D7)⋊C8C2semidirect
112. #481D7semidirect
314. C7#3339semidirect

Maximal subgroups

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

subgroupindex
#903092
#902052
C7×#4812
#898102
C2×#63812
#896432
#896382
#896322
#896302
#895512
#895472
#895352
#895282
#895072
(C22⋊D7)⋊C82
#33397