Finite Group Databasebeta

← #11737251

Subgroups — #11737251

Order 896.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90272C211yesmaximal, characteristic, complemented
2#90242C211yesmaximal, characteristic
3C7×#440C211yesmaximal, characteristic, complemented
4#89946C211yesmaximal, characteristic, complemented
5#89943C211yesmaximal, characteristic, complemented
6#89793C211yesmaximal, characteristic, complemented
7#89781C211yesmaximal, characteristic, complemented
8#89653C211yesmaximal, characteristic, complemented
9D7×#107C211yesmaximal, characteristic, complemented
10#89581C211yesmaximal, characteristic, complemented
11#89577C211yesmaximal, characteristic, complemented
12#89551C211yesmaximal, characteristic, complemented
13#89546C211yesmaximal, characteristic, complemented
14#89478C211yesmaximal, characteristic, complemented
15#89474C211yesmaximal, characteristic, complemented
16C2×#1157—21no—
17C2×C7×SD16C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20#6509C2211yescharacteristic
21#6509C2211yescharacteristic
22#6501C2211yescharacteristic
23#6501C2211yescharacteristic
24#6487—21no—
25C2×#1131C2211yescharacteristic
26C2×C8×D7C2211yescharacteristic
27#6457—21no—
28#6455C2211yescharacteristic
29D7×C4⋊C4C2211yescharacteristic
30D7×C4⋊C4C2211yescharacteristic
31D7×C4⋊C4C2211yescharacteristic
32#6452—21no—
33#6446—21no—
34#6446—21no—
35#6439—21no—
36C4×D28C2211yescharacteristic
37#6435—21no—
38#6433C2211yescharacteristic
39#6432C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6392C2211yescharacteristic
51#6392C2211yescharacteristic
52#6391C2211yescharacteristic
53#6391C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3343—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65#1157—44no—
66C2×D28D411yescharacteristic
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7C2311yescharacteristic
71C2×C4×D7C2311yescharacteristic
72C2×Dic14—21no—
73C7×SD16—22no—
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C22⋊C4—21no—
80C42×C7—21no—
81C42×C7C2311yescharacteristic
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4D411yescharacteristic
88(C2×D7)⋊C4D411yescharacteristic
89C4⋊Dic7—21no—
90C4⋊Dic7D411yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#1131—22no—
106C8×D7—22no—
107#552—71no—
108#547—71no—
109#501—71no—
110#484—71no—
111#483—71no—
112#482—71no—
113#480—71no—
114#478—71no—
115#477—71no—
116#468—71no—
117#465—71no—
118#440D711yescharacteristic, complemented
119#435—71no—
120#428—71no—
121C2×#107—71no—
122C23×C7—21no—
123C22×D7C2×D411yescharacteristic
124C22×D7C2×D411yescharacteristic
125C7×Q8—21no—
126C7×Q8—21no—
127C7×D4—21no—
128C7×D4—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—42no—
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—42no—
137C22⋊D7—42no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144D28—22no—
145C4×D7—21no—
146C4×D7—21no—
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—42no—
150C4×D7—42no—
151C4×D7—42no—
152Dic14—42no—
153C7×C8—21no—
154C7×C8#4112yescomplemented
155C7⋊C8—21no—
156C7⋊C8—21no—
157C2×#41—141no—
158C2×SD16C2×D711yescharacteristic
159C2×M16—71no—
160C22×C8—71no—
161C4⋊Q8—71no—
162#126—141no—
163#125—71no—
164#125—141no—
165#123—141no—
166#123—141no—
167#123—141no—
168C22⋊Q8—71no—
169C22⋊Q8—71no—
170C22⋊Q8—141no—
171#121—71no—
172#121—71no—
173C4×Q8C2×D711yescharacteristic
174C4×D4—71no—
175C4×D4C2×D711yescharacteristic
176#117—71no—
177#117—141no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#107—71no—
186#107C2×D711yescharacteristic
187#106—71no—
188#106—71no—
189#106—71no—
190C4⋊C8—71no—
191#103—71no—
192#103—71no—
193#103C2×D711yescharacteristic
194#102—71no—
195#102—71no—
196#102C2×D711yescharacteristic
197C22⋊C8—71no—
198C22⋊C8—71no—
199#97C2×D711yescharacteristic
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7C22×D411yescharacteristic
204C2×D7—21no—
205C2×D7—21no—
206C2×D7—21no—
207C2×D7—21no—
208C2×D7—21no—
209C2×D7AGL(1,ℤ/8)12yes—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7C2×#4111yescharacteristic
216C4×C72−1+411yescharacteristic
217Dic7—21no—
218Dic7—21no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223#41—284no—
224C2×Q8—141no—
225C2×Q8C22×D711yescharacteristic
226C2×D4—71no—
227C2×D4—141no—
228C2×D4—141no—
229C2×D4C22×D711yescharacteristic
230C22×C4—21no—
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4—141no—
236SD16—22no—
237M16—142no—
238C2×C8—71no—
239C2×C8—71no—
240C2×C8—142no—
241C2×C8C22×D711yescharacteristic
242C2×C8C22×D711yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—71no—
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4C22×D711yescharacteristic
258C4⋊C4C22×D711yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—71no—
261C22⋊C4—71no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C42—21no—
268C42—141no—
269C42C22×D711yescharacteristic
270C2×C7—81no—
271C2×C7#54711yescharacteristic
272C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
273C2×C7#57811yescharacteristic
274D7—22no—
275D7—41no—
276C23—21no—
277C23—71no—
278C23—71no—
279Q8—21no—
280Q8—21no—
281Q8—282no—
282D4—21no—
283D4—21no—
284D4—142no—
285D4—282no—
286D4—282no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
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—141no—
300C2×C4—141no—
301C2×C4—282no—
302C2×C4—282no—
303C2×C4—282no—
304C2×C4D4×D711yescharacteristic
305C2×C4D4×D711yescharacteristic
306C2×C4C23×D711yescharacteristic
307C8—21no—
308C8—141no—
309C8—141no—
310C8#115712yescomplemented
311C7#334311yescharacteristic, complemented, Sylow
312C22—41no—
313C22—41no—
314C22—41no—
315C22—72no—
316C22—141no—
317C22—141no—
318C22—141no—
319C22—141no—
320C22—141no—
321C22C2×D4×D711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—141no—
328C4—141no—
329C4—281no—
330C4—281no—
331C4—281no—
332C4—281no—
333C4C2×#115711yescharacteristic
334C4#655411yescharacteristic
335C2—81no—
336C2—142no—
337C2—281no—
338C2#9024211yescharacteristic
339C2D7×AGL(1,ℤ/8)11yescharacteristic
340C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
prime radical#90242C2
hypercenter#440D7
hypocenterC7#3343
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #90272C2semidirect
3. C7×#440C2semidirect
4. #89946C2semidirect
5. #89943C2semidirect
6. #89793C2semidirect
7. #89781C2semidirect
8. #89653C2semidirect
9. D7×#107C2semidirect
10. #89581C2semidirect
11. #89577C2semidirect
12. #89551C2semidirect
13. #89546C2semidirect
14. #89478C2semidirect
15. #89474C2semidirect
118. #440D7semidirect
154. C7×C8#41semidirect
310. C8#1157semidirect
311. C7#3343semidirect

Maximal subgroups

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

subgroupindex
#902722
#902422
C7×#4402
#899462
#899432
#897932
#897812
#896532
D7×#1072
#895812
#895772
#895512
#895462
#894782
#894742
#33437