Finite Group Databasebeta

← #11738097

Subgroups — #11738097

Order 896.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90326C211yesmaximal, characteristic, complemented
2#90309C211yesmaximal, characteristic
3C7×#465C211yesmaximal, characteristic, complemented
4#89946C211yesmaximal, characteristic, complemented
5#89936C211yesmaximal, characteristic, complemented
6#89883C211yesmaximal, characteristic, complemented
7C2×#6392C211yesmaximal, characteristic, complemented
8#89816C211yesmaximal, characteristic, complemented
9#89806C211yesmaximal, characteristic, complemented
10#89633C211yesmaximal, characteristic, complemented
11#89628C211yesmaximal, characteristic, complemented
12#89586C211yesmaximal, characteristic, complemented
13#89568C211yesmaximal, characteristic, complemented
14#89546C211yesmaximal, characteristic, complemented
15#89531C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C22×C7×C8C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C7×#121C2211yescharacteristic
20#6509C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22#6502—21no—
23#6501C2211yescharacteristic
24D4×Dic7—21no—
25D4×Dic7C2211yescharacteristic
26C4×C22⋊D7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×#1131C2211yescharacteristic
30#6460—21no—
31#6455C2211yescharacteristic
32D7×C4⋊C4C2211yescharacteristic
33#6453C2211yescharacteristic
34#6452—21no—
35#6451C2211yescharacteristic
36#6449—21no—
37(C22⋊D7)⋊C4—21no—
38#6442—21no—
39#6441—21no—
40C7×#106C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47C22⋊(C7⋊C8)C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6392C2211yescharacteristic
52#6392C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3346—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7D411yescharacteristic
66C22×Dic7—21no—
67C22×Dic7—21no—
68C22×Dic7—21no—
69C2×C4×D7—21no—
70C2×C4×D7C2311yescharacteristic
71C2×Dic14—21no—
72C7×SD16—22no—
73C2×C7×C8—22no—
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7D411yescharacteristic
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4D411yescharacteristic
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#1131—22no—
106#554—71no—
107#553—71no—
108#501—71no—
109#485—71no—
110#484—71no—
111#481—71no—
112#480—71no—
113#479—71no—
114#474—71no—
115#468—71no—
116#465D711yescharacteristic, complemented
117#440—71no—
118#435—71no—
119#428—71no—
120C2×#107—71no—
121C23×C7—21no—
122C23×C7C2×D411yescharacteristic
123C22×D7C2×D411yescharacteristic
124C7×Q8—21no—
125C7×Q8—21no—
126C7×D4—21no—
127C7×D4—21no—
128C7×D4—42no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—22no—
137C22⋊D7—81no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—81no—
143C2×Dic7—81no—
144C2×Dic7—81no—
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148C4×D7—21no—
149C4×D7—21no—
150C4×D7—81no—
151Dic14—81no—
152C7×C8—21no—
153C7×C8#4112yes—
154C7⋊C8—21no—
155C7⋊C8—21no—
156C2×SD16C2×D711yescharacteristic
157C2×M16—71no—
158C22×C8C2×D711yescharacteristic
159C4⋊Q8—71no—
160#126—141no—
161#125—71no—
162#125—141no—
163#123—141no—
164C22⋊Q8—71no—
165C22⋊Q8—141no—
166C22⋊Q8—141no—
167C22⋊Q8C2×D711yescharacteristic
168#121—71no—
169#121—141no—
170#121C2×D711yescharacteristic
171C4×Q8—71no—
172C4×D4—71no—
173C4×D4—71no—
174C4×D4—141no—
175C4×D4—141no—
176#117—71no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—71no—
180#107—71no—
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#107—71no—
186#106—71no—
187#106—71no—
188#106C2×D711yescharacteristic
189C4⋊C8—71no—
190#103—71no—
191#103—71no—
192#103C2×D711yescharacteristic
193#102—71no—
194#102—71no—
195#102C2×D711yescharacteristic
196C22⋊C8—71no—
197C22⋊C8—71no—
198#97—71no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7#13712yes—
205C22×C7C22×D411yescharacteristic
206C2×D7—21no—
207C2×D7—21no—
208C2×D7—21no—
209C4×C7—21no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7C2×#4111yescharacteristic
214C4×C72−1+411yescharacteristic
215Dic7—21no—
216Dic7—21no—
217Dic7—21no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222C2×Q8—141no—
223C2×Q8C22×D711yescharacteristic
224C2×D4—21no—
225C2×D4—71no—
226C2×D4—141no—
227C2×D4C22×D711yescharacteristic
228C22×C4—71no—
229C22×C4—141no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4C22×D711yescharacteristic
234SD16—22no—
235M16—142no—
236C2×C8—22no—
237C2×C8—71no—
238C2×C8—71no—
239C2×C8C22×D711yescharacteristic
240C2×C8C22×D711yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—71no—
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4C22×D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—71no—
260C22⋊C4—71no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C42—71no—
266C42—141no—
267C42—141no—
268C2×C7—22no—
269C2×C7—81no—
270C2×C7#54711yescharacteristic
271C2×C7C2×#13711yescharacteristic
272C2×C7#57611yescharacteristic
273D7—41no—
274C23—21no—
275C23—71no—
276C23D4×D711yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—561no—
280D4—21no—
281D4—21no—
282D4—42no—
283D4—142no—
284D4—561no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—71no—
291C2×C4—71no—
292C2×C4—71no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—561no—
300C2×C4—561no—
301C2×C4—561no—
302C2×C4—561no—
303C2×C4D4×D711yescharacteristic
304C2×C4C23×D711yescharacteristic
305C8—21no—
306C8—141no—
307C8—141no—
308C8#115912yes—
309C7#334611yescharacteristic, complemented, Sylow
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—141no—
316C22—141no—
317C22—141no—
318C22#647812yes—
319C22C2×D4×D711yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—141no—
325C4—141no—
326C4—141no—
327C4—281no—
328C4—281no—
329C4—281no—
330C4—281no—
331C4C2×#115911yescharacteristic
332C4#655411yescharacteristic
333C2—22no—
334C2—81no—
335C2—281no—
336C2#9030211yescharacteristic
337C2C2×#647811yescharacteristic
338C2#9046011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#465C2
socleC22×C7C22×D4
prime radical#90309C2
hypercenter#465D7
hypocenterC7#3346
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #90326C2semidirect
3. C7×#465C2semidirect
4. #89946C2semidirect
5. #89936C2semidirect
6. #89883C2semidirect
7. C2×#6392C2semidirect
8. #89816C2semidirect
9. #89806C2semidirect
10. #89633C2semidirect
11. #89628C2semidirect
12. #89586C2semidirect
13. #89568C2semidirect
14. #89546C2semidirect
15. #89531C2semidirect
116. #465D7semidirect
309. C7#3346semidirect

Maximal subgroups

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

subgroupindex
#903262
#903092
C7×#4652
#899462
#899362
#898832
C2×#63922
#898162
#898062
#896332
#896282
#895862
#895682
#895462
#895312
#33467