Finite Group Databasebeta

← #11741117

Subgroups — #11741117

Order 896.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#556C211yesmaximal, characteristic, complemented
2#90276C211yesmaximal, characteristic, complemented
3#89858C211yesmaximal, characteristic, complemented
4#89855C211yesmaximal, characteristic, complemented
5#89831C211yesmaximal, characteristic, complemented
6#89830C211yesmaximal, characteristic
7#89829C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89802C211yesmaximal, characteristic, complemented
10#89800C211yesmaximal, characteristic, complemented
11#89799C211yesmaximal, characteristic, complemented
12C4×#1143C211yesmaximal, characteristic, complemented
13#89796C211yesmaximal, characteristic, complemented
14#89788C211yesmaximal, characteristic, complemented
15#89786C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#126—21no—
18C7×#126—21no—
19C7×#124C2211yescharacteristic
20C7×#124C2211yescharacteristic
21C7×C22⋊Q8—21no—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×Q8C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4C2211yescharacteristic
26C7×#117—21no—
27#6510—21no—
28C2×#1144C2211yescharacteristic
29C2×#1144C2211yescharacteristic
30C2×#1143C2211yescharacteristic
31C2×#1143C2211yescharacteristic
32C2×#1143C2211yescharacteristic
33C2×#1143C2211yescharacteristic
34C2×#1142C2211yescharacteristic
35C2×#1141C2211yescharacteristic
36C2×#1141C2211yescharacteristic
37#6457—21no—
38#6457—21no—
39#6456—21no—
40#6438C2211yescharacteristic
41#6438C2211yescharacteristic
42#6437C2211yescharacteristic
43C4×D28C2211yescharacteristic
44C4×D28C2211yescharacteristic
45C4×Dic14C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6384C2211yescharacteristic
50#6383C2211yescharacteristic
51#6383C2211yescharacteristic
52#6382C2211yescharacteristic
53C4⋊(C7⋊C8)C2211yescharacteristic
54C4⋊(C7⋊C8)C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56#6377C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3401—71nomaximal, Sylow
60C2×C7×Q8D411yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C2×D28—21no—
66C2×D28—21no—
67C2×D28C2311yescharacteristic
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×Dic14C2311yescharacteristic
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4D411yescharacteristic
77C7×C4⋊C4D411yescharacteristic
78C7×C4⋊C4D411yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C7×C22⋊C4—21no—
84C7×C22⋊C4—21no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7C2311yescharacteristic
88C42×C7C2311yescharacteristic
89C42×C7C2311yescharacteristic
90#1144—22no—
91#1144—41no—
92#1143—22no—
93#1143—41no—
94#1143—41no—
95#1143—41no—
96#1142—41no—
97#1141—22no—
98#1141—41no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4—21no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4—21no—
103(C2×D7)⋊C4—21no—
104(C2×D7)⋊C4—21no—
105C4⋊Dic7C2311yescharacteristic
106Dic7⋊C4—21no—
107C4×Dic7—21no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#559—71no—
113#556D711yescharacteristic, complemented
114#497—71no—
115#496—71no—
116#495—71no—
117#464—71no—
118#464—71no—
119#463—71no—
120#463—71no—
121#461—71no—
122#460—71no—
123#446—71no—
124#442—71no—
125#441—71no—
126C4×SD16—71no—
127C23×C7—21no—
128C22×D7—21no—
129C22×D7—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×Q8—41no—
135C7×D4—21no—
136C7×D4—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—42no—
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2411yescharacteristic
148C2×Dic7—21no—
149C2×Dic7—21no—
150D28—21no—
151D28—21no—
152D28—21no—
153D28—21no—
154D28—42no—
155D28—81no—
156C4×D7—81no—
157C4×D7—81no—
158Dic14—21no—
159Dic14—21no—
160C7⋊C8—21no—
161C7⋊C8—21no—
162C7⋊C8—21no—
163C7⋊C8#4112yes—
164C2×Q16—71no—
165C2×Q16—71no—
166C2×SD16—71no—
167C2×SD16—71no—
168C2×SD16—71no—
169C2×SD16—71no—
170C2×SD16—71no—
171C2×D8—71no—
172C2×D8—71no—
173C4⋊Q8C2×D711yescharacteristic
174#127—71no—
175#126—21no—
176#126—21no—
177#124—71no—
178#124—71no—
179#124—141no—
180#124C2×D711yescharacteristic
181#124C2×D711yescharacteristic
182#123—141no—
183#123—141no—
184C22⋊Q8—21no—
185C4×Q8—71no—
186C4×Q8C2×D711yescharacteristic
187C4×Q8C2×D711yescharacteristic
188C4×Q8C2×D711yescharacteristic
189C4×D4—71no—
190C4×D4—71no—
191C4×D4—141no—
192C4×D4C2×D711yescharacteristic
193#117—21no—
194#106—71no—
195C4⋊C8—71no—
196C4⋊C8—71no—
197C4⋊C8—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#102—71no—
202#102—71no—
203#102—71no—
204#97—71no—
205#97—71no—
206C4×C8—71no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7C22×D411yescharacteristic
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7C2×#4111yescharacteristic
227C4×C7H(5,𝔽2)11yescharacteristic
228Dic7—41no—
229Dic7—41no—
230C2×Q8—71no—
231C2×Q8C22⋊D711yescharacteristic
232C2×Q8C22×D711yescharacteristic
233C2×Q8C22×D711yescharacteristic
234C2×D4—71no—
235C2×D4—71no—
236C2×D4—141no—
237C2×D4—141no—
238C2×D4C22×D711yescharacteristic
239C22×C4—21no—
240C22×C4—141no—
241C22×C4—141no—
242Q16—142no—
243Q16—281no—
244SD16—142no—
245SD16—281no—
246SD16—281no—
247SD16—281no—
248SD16—281no—
249D8—142no—
250D8—281no—
251C2×C8—71no—
252C2×C8—71no—
253C2×C8—71no—
254C2×C8—71no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—71no—
260C4⋊C4—141no—
261C4⋊C4C22⋊D711yescharacteristic
262C4⋊C4C22⋊D711yescharacteristic
263C4⋊C4C22⋊D711yescharacteristic
264C4⋊C4C22×D711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C42—21no—
277C42—21no—
278C42—141no—
279C42C22×D711yescharacteristic
280C42C22×D711yescharacteristic
281C42C22×D711yescharacteristic
282C2×C7—81no—
283C2×C7#55011yescharacteristic
284C2×C7#57511yescharacteristic
285C2×C7#57711yescharacteristic
286D7—81no—
287D7—81no—
288C23—21no—
289C23—141no—
290C23—141no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—41no—
296Q8—141no—
297Q8—141no—
298D4—21no—
299D4—21no—
300D4—141no—
301D4—141no—
302D4—141no—
303D4—141no—
304D4—282no—
305D4—561no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—561no—
313C2×C4—561no—
314C2×C4C2×C22⋊D711yescharacteristic
315C2×C4C2×C22⋊D711yescharacteristic
316C2×C4C2×C22⋊D711yescharacteristic
317C2×C4C2×C22⋊D711yescharacteristic
318C2×C4C2×C22⋊D711yescharacteristic
319C2×C4C2×C22⋊D711yescharacteristic
320C2×C4C23×D711yescharacteristic
321C8—72no—
322C8—141no—
323C8—141no—
324C8—141no—
325C7#340111yescharacteristic, complemented, Sylow
326C22—41no—
327C22—41no—
328C22—41no—
329C22—281no—
330C22—281no—
331C22—281no—
332C22—281no—
333C22—281no—
334C22—281no—
335C22C22×C22⋊D711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—281no—
346C4—281no—
347C4D7×#4111yescharacteristic
348C4#655311yescharacteristic
349C2—81no—
350C2—561no—
351C2—561no—
352C2#9051311yescharacteristic
353C2#9052511yescharacteristic
354C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#556C2
socleC22×C7C22×D4
prime radical#89830C2
hypercenter#556D7
hypocenterC7#3401
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C7×#556C2semidirect
2. #90276C2semidirect
3. #89858C2semidirect
4. #89855C2semidirect
5. #89831C2semidirect
7. #89829C2semidirect
8. #89826C2semidirect
9. #89802C2semidirect
10. #89800C2semidirect
11. #89799C2semidirect
12. C4×#1143C2semidirect
13. #89796C2semidirect
14. #89788C2semidirect
15. #89786C2semidirect
113. #556D7semidirect
325. C7#3401semidirect

Maximal subgroups

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

subgroupindex
C7×#5562
#902762
#898582
#898552
#898312
#898302
#898292
#898262
#898022
#898002
#897992
C4×#11432
#897962
#897882
#897862
#34017