Finite Group Databasebeta

← #11741106

Subgroups — #11741106

Order 896.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#556C211yesmaximal, characteristic, complemented
2#90256C211yesmaximal, characteristic
3#89856C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89821C211yesmaximal, characteristic, complemented
6#89820C211yesmaximal, characteristic, complemented
7#89817C211yesmaximal, characteristic, complemented
8#89815C211yesmaximal, characteristic, complemented
9#89798C211yesmaximal, characteristic, complemented
10#89794C211yesmaximal, characteristic, complemented
11#89783C211yesmaximal, characteristic, complemented
12#89775C211yesmaximal, characteristic, complemented
13#89775C211yesmaximal, characteristic, complemented
14#89770C211yesmaximal, characteristic, complemented
15#89767C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#126—21no—
18C7×#126—21no—
19C7×#124—21no—
20C7×C22⋊Q8C2211yescharacteristic
21C7×C22⋊Q8C2211yescharacteristic
22C4×C7×Q8—21no—
23C4×C7×Q8C2211yescharacteristic
24C4×C7×D4C2211yescharacteristic
25C7×#117C2211yescharacteristic
26C7×#117C2211yescharacteristic
27C2×#1143C2211yescharacteristic
28#6502—21no—
29#6493C2211yescharacteristic
30#6493C2211yescharacteristic
31#6489C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6461—21no—
36#6446—21no—
37(C22⋊D7)⋊C4—21no—
38C4×D28C2211yescharacteristic
39#6433C2211yescharacteristic
40#6409C2211yescharacteristic
41#6409C2211yescharacteristic
42#6409C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6383C2211yescharacteristic
46#6383C2211yescharacteristic
47#6383C2211yescharacteristic
48#6382C2211yescharacteristic
49#6382C2211yescharacteristic
50#6382C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3350—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4D411yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C22×Dic7—21no—
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4D411yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C22⋊C4—21no—
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C7×C22⋊C4—21no—
82C7×C22⋊C4D411yescharacteristic
83C7×C22⋊C4D411yescharacteristic
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7C2311yescharacteristic
88C22⋊Dic7—21no—
89#1143—22no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99C4×Dic7—21no—
100#1136—22no—
101C2×C7⋊C8—22no—
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#556D711yescharacteristic, complemented
107#553—71no—
108#501—71no—
109#485—71no—
110#483—71no—
111#482—71no—
112#482—71no—
113#479—71no—
114#476—71no—
115#468—71no—
116#465—71no—
117#440—71no—
118#435—71no—
119#428—71no—
120#427—71no—
121C23×C7C2×D411yescharacteristic
122C23×C7C2×D411yescharacteristic
123C22×D7—21no—
124C7×Q8—21no—
125C7×Q8—21no—
126C7×Q8—42no—
127C7×D4—41no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C22⋊D7—42no—
142C22⋊D7—81no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—81no—
147D28—21no—
148D28—21no—
149C4×D7—81no—
150C7⋊C8—21no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C7⋊C8#4112yes—
154C2×SD16—71no—
155C2×M16—71no—
156C22×C8—71no—
157C4⋊Q8C2×D711yescharacteristic
158#126—21no—
159#126—21no—
160#126—141no—
161#125—71no—
162#124—21no—
163#123—141no—
164C22⋊Q8C2×D711yescharacteristic
165C22⋊Q8C2×D711yescharacteristic
166#121—71no—
167#121—71no—
168#121—141no—
169C4×Q8—21no—
170C4×Q8C2×D711yescharacteristic
171C4×D4—71no—
172C4×D4—141no—
173C4×D4C2×D711yescharacteristic
174#117—71no—
175#117C2×D711yescharacteristic
176#117C2×D711yescharacteristic
177C2×C4⋊C4—71no—
178#107—71no—
179#107—71no—
180#107—71no—
181#107—71no—
182#106—71no—
183#106—71no—
184#106—71no—
185#106—71no—
186#106—71no—
187C4⋊C8—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194C22⋊C8—71no—
195C22⋊C8—71no—
196#97—71no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7C22×D411yescharacteristic
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7C2×#4111yescharacteristic
216C4×C72−1+411yescharacteristic
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—21no—
221C2×Q8C22×D711yescharacteristic
222C2×D4—71no—
223C2×D4—141no—
224C2×D4—141no—
225C2×D4C22⋊D711yescharacteristic
226C22×C4—141no—
227C22×C4—141no—
228C22×C4C22×D711yescharacteristic
229C22×C4C22×D711yescharacteristic
230SD16—142no—
231M16—142no—
232C2×C8—71no—
233C2×C8—71no—
234C2×C8—71no—
235C2×C8—71no—
236C2×C8—142no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4C22⋊D711yescharacteristic
248C4⋊C4C22×D711yescharacteristic
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4C22⋊D711yescharacteristic
260C22⋊C4C22⋊D711yescharacteristic
261C42—21no—
262C42—21no—
263C42—21no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7—41no—
267C2×C7—41no—
268C2×C7#54711yescharacteristic
269C2×C7#57511yescharacteristic
270C2×C7#57711yescharacteristic
271D7—81no—
272C23—141no—
273C23C2×C22⋊D711yescharacteristic
274C23C2×C22⋊D711yescharacteristic
275Q8—21no—
276Q8—21no—
277Q8—42no—
278D4—41no—
279D4—141no—
280D4—141no—
281D4—282no—
282D4—561no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—561no—
295C2×C4—561no—
296C2×C4C2×C22⋊D711yescharacteristic
297C2×C4C2×C22⋊D711yescharacteristic
298C2×C4C2×C22⋊D711yescharacteristic
299C2×C4C2×C22⋊D711yescharacteristic
300C2×C4C23×D711yescharacteristic
301C8—72no—
302C8—141no—
303C8—141no—
304C8—141no—
305C7#335011yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—281no—
313C22—281no—
314C22—281no—
315C22C22×C22⋊D711yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4D7×#4111yescharacteristic
328C4#655411yescharacteristic
329C2—41no—
330C2—41no—
331C2—561no—
332C2#9051311yescharacteristic
333C2#9051511yescharacteristic
334C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#556C2
socleC22×C7C22×D4
prime radical#90256C2
hypercenter#556D7
hypocenterC7#3350
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C7×#556C2semidirect
3. #89856C2semidirect
4. #89852C2semidirect
5. #89821C2semidirect
6. #89820C2semidirect
7. #89817C2semidirect
8. #89815C2semidirect
9. #89798C2semidirect
10. #89794C2semidirect
11. #89783C2semidirect
12. #89775C2semidirect
13. #89775C2semidirect
14. #89770C2semidirect
15. #89767C2semidirect
106. #556D7semidirect
305. C7#3350semidirect

Maximal subgroups

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

subgroupindex
C7×#5562
#902562
#898562
#898522
#898212
#898202
#898172
#898152
#897982
#897942
#897832
#897752
#897702
#897672
#33507