Finite Group Databasebeta

← #11738926

Subgroups — #11738926

Order 896.

374 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#90332C211yesmaximal, characteristic, complemented
2#90198C211yesmaximal, characteristic
3C7×#484C211yesmaximal, characteristic, complemented
4#89821C211yesmaximal, characteristic, complemented
5#89736C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89657C211yesmaximal, characteristic, complemented
8#89653C211yesmaximal, characteristic, complemented
9#89646C211yesmaximal, characteristic, complemented
10#89584C211yesmaximal, characteristic, complemented
11#89581C211yesmaximal, characteristic, complemented
12#89563C211yesmaximal, characteristic, complemented
13#89560C211yesmaximal, characteristic, complemented
14#89509C211yesmaximal, characteristic, complemented
15#89501C211yesmaximal, characteristic, complemented
16C2×#1157—21no—
17C7×C22⋊Q8C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19#6509C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1143C2211yescharacteristic
22#6493C2211yescharacteristic
23#6492—21no—
24C4×C22⋊D7C2211yescharacteristic
25#6489C2211yescharacteristic
26C2×#1136C2211yescharacteristic
27C2×#1131C2211yescharacteristic
28#6461—21no—
29#6459—21no—
30#6458C2211yescharacteristic
31#6457—21no—
32#6456C2211yescharacteristic
33#6455C2211yescharacteristic
34D7×C4⋊C4C2211yescharacteristic
35#6451C2211yescharacteristic
36#6451C2211yescharacteristic
37#6450—21no—
38#6448—21no—
39#6442—21no—
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6392C2211yescharacteristic
50#6391C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3353—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7—21no—
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7D411yescharacteristic
65#1157—44no—
66C2×D28—21no—
67C2×D28C2311yescharacteristic
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7C2311yescharacteristic
71C2×Dic14—21no—
72C2×Dic14—21no—
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7D411yescharacteristic
83#1143—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⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103#1136—22no—
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#1131—22no—
107#556—71no—
108#547—71no—
109#500—71no—
110#484—71no—
111#484D711yescharacteristic, complemented
112#482—71no—
113#482—71no—
114#475—71no—
115#474—71no—
116#468—71no—
117#468—71no—
118C4×SD16—71no—
119#436—71no—
120#428—71no—
121#428—71no—
122C23×C7C2×D411yescharacteristic
123C22×D7—21no—
124C22×D7C2×D411yescharacteristic
125C7×Q8—21no—
126C7×Q8—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—42no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—22no—
137C22⋊D7—42no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144D28—21no—
145D28—21no—
146D28—42no—
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—42no—
150C4×D7—42no—
151Dic14—42no—
152Dic14—42no—
153C7×C8—21no—
154C7×C8—21no—
155C7⋊C8—21no—
156C7⋊C8#4112yescomplemented
157C2×#41—141no—
158C2×SD16—71no—
159C2×M16—71no—
160C2×M16—71no—
161C4⋊Q8—71no—
162C4⋊Q8—71no—
163#126—141no—
164#126—141no—
165#124—141no—
166#123—141no—
167#123—141no—
168C22⋊Q8—71no—
169C22⋊Q8—141no—
170C22⋊Q8C2×D711yescharacteristic
171#121—71no—
172#121—71no—
173C4×Q8—71no—
174C4×Q8—141no—
175C4×D4—71no—
176C4×D4—71no—
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#107C2×D711yescharacteristic
186#107C2×D711yescharacteristic
187#106—71no—
188#106—71no—
189#106—71no—
190C4⋊C8—71no—
191#103—71no—
192#103C2×D711yescharacteristic
193#103C2×D711yescharacteristic
194#102—71no—
195#102—71no—
196#102—71no—
197C22⋊C8—71no—
198C22⋊C8C2×D711yescharacteristic
199C4×C8—71no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7C22×D411yescharacteristic
204C2×D7—21no—
205C2×D7—21no—
206C2×D7—21no—
207C2×D7—41no—
208C2×D7—41no—
209C2×D7—41no—
210C4×C7—21no—
211C4×C7—41no—
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—
222Dic7AGL(1,ℤ/8)12yes—
223#41—284no—
224C2×Q8—141no—
225C2×Q8—141no—
226C2×Q8C22×D711yescharacteristic
227C2×D4—71no—
228C2×D4—71no—
229C2×D4—141no—
230C2×D4—141no—
231C22×C4—21no—
232C22×C4—71no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4C22×D711yescharacteristic
236SD16—142no—
237M16—142no—
238M16—142no—
239C2×C8—71no—
240C2×C8—71no—
241C2×C8C22×D711yescharacteristic
242C2×C8C22×D711yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
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⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—71no—
259C22⋊C4—71no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C42—71no—
267C42—141no—
268C42—141no—
269C42—141no—
270C2×C7—41no—
271C2×C7#54711yescharacteristic
272C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
273C2×C7#57811yescharacteristic
274D7—41no—
275D7—81no—
276C23—71no—
277C23—141no—
278C23D4×D711yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—282no—
282Q8—282no—
283D4—142no—
284D4—141no—
285D4—141no—
286D4—282no—
287D4—282no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—71no—
296C2×C4—71no—
297C2×C4—71no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—282no—
304C2×C4—282no—
305C2×C4D4×D711yescharacteristic
306C2×C4C23×D711yescharacteristic
307C8—21no—
308C8—21no—
309C8—72no—
310C8—141no—
311C7#335311yescharacteristic, complemented, Sylow
312C22—21no—
313C22—21no—
314C22—21no—
315C22—141no—
316C22—141no—
317C22—141no—
318C22—281no—
319C22—281no—
320C22—281no—
321C22C2×D4×D711yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—72no—
328C4—141no—
329C4—141no—
330C4—281no—
331C4—281no—
332C4—281no—
333C4D7×#4111yescharacteristic
334C4#655411yescharacteristic
335C2—41no—
336C2—281no—
337C2—561no—
338C2#9034611yescharacteristic
339C2#9045711yescharacteristic
340C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#484C2
socleC22×C7C22×D4
prime radical#90198C2
hypercenter#484D7
hypocenterC7#3353
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #90332C2semidirect
3. C7×#484C2semidirect
4. #89821C2semidirect
5. #89736C2semidirect
6. #89662C2semidirect
7. #89657C2semidirect
8. #89653C2semidirect
9. #89646C2semidirect
10. #89584C2semidirect
11. #89581C2semidirect
12. #89563C2semidirect
13. #89560C2semidirect
14. #89509C2semidirect
15. #89501C2semidirect
111. #484D7semidirect
156. C7⋊C8#41semidirect
311. C7#3353semidirect

Maximal subgroups

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

subgroupindex
#903322
#901982
C7×#4842
#898212
#897362
#896622
#896572
#896532
#896462
#895842
#895812
#895632
#895602
#895092
#895012
#33537