Finite Group Databasebeta

← #11741119

Subgroups — #11741119

Order 896.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#556C211yesmaximal, characteristic, complemented
2#90243C211yesmaximal, characteristic
3#89863C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89823C211yesmaximal, characteristic, complemented
6#89822C211yesmaximal, characteristic, complemented
7#89821C211yesmaximal, characteristic, complemented
8#89819C211yesmaximal, characteristic, complemented
9#89802C211yesmaximal, characteristic, complemented
10#89800C211yesmaximal, characteristic, complemented
11#89783C211yesmaximal, characteristic, complemented
12#89776C211yesmaximal, characteristic, complemented
13#89775C211yesmaximal, characteristic, complemented
14#89774C211yesmaximal, characteristic, complemented
15#89768C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C7×#126—21no—
20C7×#126—21no—
21C7×#124—21no—
22C7×C22⋊Q8C2211yescharacteristic
23C7×C22⋊Q8C2211yescharacteristic
24C4×C7×Q8—21no—
25C4×C7×Q8C2211yescharacteristic
26C4×C7×D4C2211yescharacteristic
27C7×#117C2211yescharacteristic
28C7×#117C2211yescharacteristic
29C2×#1144C2211yescharacteristic
30C2×#1144C2211yescharacteristic
31C2×#1143C2211yescharacteristic
32#6502—21no—
33#6493C2211yescharacteristic
34#6487C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6459—21no—
38#6448—21no—
39(C22⋊D7)⋊C4—21no—
40#6438C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42#6409C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6384C2211yescharacteristic
46#6384C2211yescharacteristic
47#6384C2211yescharacteristic
48#6384C2211yescharacteristic
49#6384C2211yescharacteristic
50#6384C2211yescharacteristic
51#6383C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3313—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—
68#1157—22no—
69#1157—41no—
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×Dic14—22no—
73C2×Dic14—21no—
74C2×Dic14—41no—
75C2×Dic14C2311yescharacteristic
76C2×Dic14C2311yescharacteristic
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4D411yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C7×C22⋊C4—21no—
88C7×C22⋊C4—21no—
89C7×C22⋊C4D411yescharacteristic
90C7×C22⋊C4D411yescharacteristic
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7C2311yescharacteristic
95C22⋊Dic7—21no—
96#1144—22no—
97#1144—41no—
98#1143—41no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4—21no—
101(C2×D7)⋊C4—21no—
102C4⋊Dic7C2311yescharacteristic
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106#1136—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#556D711yescharacteristic, complemented
113#548—71no—
114#489—71no—
115#485—71no—
116#482—71no—
117#477—71no—
118#468—71no—
119#467—71no—
120#464—71no—
121#452—71no—
122#451—71no—
123#441—71no—
124#435—71no—
125#419—71no—
126#416—71no—
127C23×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—42no—
133C7×D4—41no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2411yescharacteristic
147C22⋊D7—42no—
148C22⋊D7—42no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—42no—
153D28—21no—
154D28—21no—
155C4×D7—42no—
156Dic14—21no—
157Dic14—21no—
158Dic14—21no—
159Dic14—42no—
160Dic14#4112yes—
161C7⋊C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41—71no—
166C22×Q8—71no—
167C2×Q16—71no—
168C2×Q16—71no—
169C2×SD16—71no—
170C2×M16—71no—
171C22×C8—71no—
172C4⋊Q8C2×D711yescharacteristic
173#126—21no—
174#126—21no—
175#124—21no—
176#124—71no—
177#124—141no—
178C22⋊Q8—71no—
179C22⋊Q8—141no—
180C22⋊Q8C2×D711yescharacteristic
181C22⋊Q8C2×D711yescharacteristic
182#121—71no—
183#121—141no—
184C4×Q8—21no—
185C4×Q8—71no—
186C4×Q8C2×D711yescharacteristic
187C4×D4—141no—
188C4×D4C2×D711yescharacteristic
189#117C2×D711yescharacteristic
190#117C2×D711yescharacteristic
191#107—71no—
192#107—71no—
193#106—71no—
194C4⋊C8—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205C22⋊C8—71no—
206C22⋊C8—71no—
207#97—71no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7C22×D411yescharacteristic
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7C2×#4111yescharacteristic
227C4×C7H(5,𝔽2)11yescharacteristic
228Dic7—22no—
229Dic7—41no—
230Dic7—41no—
231#41—142no—
232#41—281no—
233C2×Q8—21no—
234C2×Q8—71no—
235C2×Q8—71no—
236C2×Q8—142no—
237C2×Q8—141no—
238C2×Q8—281no—
239C2×Q8C22×D711yescharacteristic
240C2×D4—71no—
241C2×D4—141no—
242C2×D4—141no—
243C2×D4C22⋊D711yescharacteristic
244C22×C4—141no—
245C22×C4—141no—
246C22×C4C22×D711yescharacteristic
247C22×C4C22×D711yescharacteristic
248Q16—142no—
249Q16—281no—
250SD16—281no—
251M16—281no—
252C2×C8—71no—
253C2×C8—71no—
254C2×C8—71no—
255C2×C8—71no—
256C2×C8—281no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—71no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4C22⋊D711yescharacteristic
265C4⋊C4C22×D711yescharacteristic
266C4⋊C4C22×D711yescharacteristic
267C4⋊C4C22×D711yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4C22⋊D711yescharacteristic
277C22⋊C4C22⋊D711yescharacteristic
278C42—21no—
279C42—21no—
280C42—21no—
281C42—141no—
282C42C22×D711yescharacteristic
283C2×C7—41no—
284C2×C7—41no—
285C2×C7#54611yescharacteristic
286C2×C7#57511yescharacteristic
287C2×C7#57811yescharacteristic
288D7—81no—
289C23—141no—
290C23C2×C22⋊D711yescharacteristic
291C23C2×C22⋊D711yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—42no—
295Q8—72no—
296Q8—141no—
297Q8—141no—
298Q8—141no—
299Q8—282no—
300D4—41no—
301D4—141no—
302D4—141no—
303D4—282no—
304D4—282no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—282no—
317C2×C4—282no—
318C2×C4C2×C22⋊D711yescharacteristic
319C2×C4C2×C22⋊D711yescharacteristic
320C2×C4C2×C22⋊D711yescharacteristic
321C2×C4C2×C22⋊D711yescharacteristic
322C2×C4C23×D711yescharacteristic
323C8—141no—
324C8—141no—
325C8—141no—
326C8—141no—
327C7#331311yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—281no—
335C22—281no—
336C22—281no—
337C22C22×C22⋊D711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—142no—
347C4—281no—
348C4—281no—
349C4D7×#4111yescharacteristic
350C4#655311yescharacteristic
351C2—41no—
352C2—41no—
353C2—561no—
354C2#9051311yescharacteristic
355C2#9052011yescharacteristic
356C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#556C2
socleC22×C7C22×D4
prime radical#90243C2
hypercenter#556D7
hypocenterC7#3313
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. #89863C2semidirect
4. #89861C2semidirect
5. #89823C2semidirect
6. #89822C2semidirect
7. #89821C2semidirect
8. #89819C2semidirect
9. #89802C2semidirect
10. #89800C2semidirect
11. #89783C2semidirect
12. #89776C2semidirect
13. #89775C2semidirect
14. #89774C2semidirect
15. #89768C2semidirect
112. #556D7semidirect
327. C7#3313semidirect

Maximal subgroups

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

subgroupindex
C7×#5562
#902432
#898632
#898612
#898232
#898222
#898212
#898192
#898022
#898002
#897832
#897762
#897752
#897742
#897682
#33137