Finite Group Databasebeta

← #11736679

Subgroups — #11736679

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#90224C211yesmaximal, characteristic, complemented
2#90198C211yesmaximal, characteristic
3C7×#428C211yesmaximal, characteristic, complemented
4#89906C211yesmaximal, characteristic, complemented
5#89900C211yesmaximal, characteristic, complemented
6#89775C211yesmaximal, characteristic, complemented
7#89742C211yesmaximal, characteristic, complemented
8#89662C211yesmaximal, characteristic, complemented
9#89658C211yesmaximal, characteristic, complemented
10#89656C211yesmaximal, characteristic, complemented
11#89646C211yesmaximal, characteristic, complemented
12#89638C211yesmaximal, characteristic, complemented
13#89633C211yesmaximal, characteristic, complemented
14#89626C211yesmaximal, characteristic, complemented
15#89624C211yesmaximal, characteristic, complemented
16C2×#1157—21no—
17C2×C7×M16C2211yescharacteristic
18C7×#117C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20#6493C2211yescharacteristic
21#6492—21no—
22C4×C22⋊D7C2211yescharacteristic
23#6487C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25C2×#1131C2211yescharacteristic
26#6460C2211yescharacteristic
27#6459—21no—
28#6458C2211yescharacteristic
29#6457—21no—
30#6456C2211yescharacteristic
31#6455C2211yescharacteristic
32D7×C4⋊C4C2211yescharacteristic
33#6451C2211yescharacteristic
34#6451C2211yescharacteristic
35#6450C2211yescharacteristic
36#6448—21no—
37#6442—21no—
38#6439—21no—
39C4×Dic14—21no—
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6395C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47#6392C2211yescharacteristic
48#6390C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50C8×Dic7C2211yescharacteristic
51#6384C2211yescharacteristic
52#6384C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3353—71nomaximal, Sylow
60C22×C4×C7—21no—
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—21no—
63C2×C22⋊D7D411yescharacteristic
64#1157—44no—
65C2×D28—21no—
66C2×D28C2311yescharacteristic
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7C2311yescharacteristic
70C2×Dic14—21no—
71C2×Dic14—21no—
72C2×Dic14C2311yescharacteristic
73C7×M16—22no—
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C42×C7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7D411yescharacteristic
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4D411yescharacteristic
91C4⋊Dic7—21no—
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
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#1132—22no—
106#1131—22no—
107#556—71no—
108#547—71no—
109#500—71no—
110#484—71no—
111#484—71no—
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#428D711yescharacteristic, complemented
122C23×C7C2×D411yescharacteristic
123C22×D7—21no—
124C22×D7C2×D411yescharacteristic
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—42no—
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C22⋊D7—22no—
135C22⋊D7—42no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142D28—21no—
143D28—21no—
144D28—42no—
145C4×D7—21no—
146C4×D7—21no—
147C4×D7—42no—
148C4×D7—42no—
149Dic14—21no—
150Dic14—21no—
151Dic14—42no—
152Dic14—42no—
153C7×C8—21no—
154C7×C8#4112yescomplemented
155C7⋊C8—21no—
156C7⋊C8—21no—
157C2×#41—141no—
158C2×SD16—71no—
159C2×M16—71no—
160C2×M16C2×D711yescharacteristic
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—71no—
170C22⋊Q8—141no—
171#121—71no—
172#121—71no—
173C4×Q8—71no—
174C4×Q8—141no—
175C4×D4—71no—
176C4×D4—71no—
177#117—71no—
178#117C2×D711yescharacteristic
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#106C2×D711yescharacteristic
189#106C2×D711yescharacteristic
190C4⋊C8—71no—
191#103—71no—
192#103—71no—
193#103—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197C22⋊C8—71no—
198C22⋊C8—71no—
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—71no—
225C2×Q8—141no—
226C2×Q8—141no—
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—22no—
238M16—142no—
239C2×C8—71no—
240C2×C8—71no—
241C2×C8C22×D711yescharacteristic
242C2×C8C22×D711yescharacteristic
243C4⋊C4—21no—
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⋊C4C22×D711yescharacteristic
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—21no—
267C42—71no—
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—141no—
280Q8—141no—
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—141no—
309C8—141no—
310C8#115712yescomplemented
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—
333C4C2×#115711yescharacteristic
334C4#655111yescharacteristic
335C2—41no—
336C2—281no—
337C2—561no—
338C2#9019811yescharacteristic
339C2D7×AGL(1,ℤ/8)11yescharacteristic
340C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radical#90198C2
hypercenter#428D7
hypocenterC7#3353
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #90224C2semidirect
3. C7×#428C2semidirect
4. #89906C2semidirect
5. #89900C2semidirect
6. #89775C2semidirect
7. #89742C2semidirect
8. #89662C2semidirect
9. #89658C2semidirect
10. #89656C2semidirect
11. #89646C2semidirect
12. #89638C2semidirect
13. #89633C2semidirect
14. #89626C2semidirect
15. #89624C2semidirect
121. #428D7semidirect
154. C7×C8#41semidirect
310. C8#1157semidirect
311. C7#3353semidirect

Maximal subgroups

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

subgroupindex
#902242
#901982
C7×#4282
#899062
#899002
#897752
#897422
#896622
#896582
#896562
#896462
#896382
#896332
#896262
#896242
#33537