Finite Group Databasebeta

← #11741161

Subgroups — #11741161

Order 896.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#559C211yesmaximal, characteristic, complemented
2#90276C211yesmaximal, characteristic, complemented
3#89846C211yesmaximal, characteristic, complemented
4#89844C211yesmaximal, characteristic, complemented
5#89831C211yesmaximal, characteristic, complemented
6#89830C211yesmaximal, characteristic, complemented
7#89829C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89800C211yesmaximal, characteristic, complemented
10#89798C211yesmaximal, characteristic, complemented
11#89796C211yesmaximal, characteristic, complemented
12#89790C211yesmaximal, characteristic, complemented
13#89788C211yesmaximal, characteristic, complemented
14#89787C211yesmaximal, characteristic, complemented
15C4×#1141C211yesmaximal, characteristic, complemented
16C7×#127C2211yescharacteristic
17C7×#124—21no—
18C7×#124C2211yescharacteristic
19C7×#124C2211yescharacteristic
20C7×#123—21no—
21C7×#123—21no—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×Q8C2211yescharacteristic
24C4×C7×D4—21no—
25C4×C7×D4C2211yescharacteristic
26C4×C7×D4C2211yescharacteristic
27#6510—21no—
28C2×#1144C2211yescharacteristic
29C2×#1143C2211yescharacteristic
30C2×#1143C2211yescharacteristic
31C2×#1142C2211yescharacteristic
32C2×#1142C2211yescharacteristic
33C2×#1141C2211yescharacteristic
34C2×#1141C2211yescharacteristic
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#6406C2211yescharacteristic
48#6406C2211yescharacteristic
49#6384C2211yescharacteristic
50#6383C2211yescharacteristic
51#6383C2211yescharacteristic
52#6381C2211yescharacteristic
53C4⋊(C7⋊C8)C2211yescharacteristic
54C4⋊(C7⋊C8)C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56#6377C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3403—71nomaximal, Sylow
60C2×C7×Q8D411yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4—21no—
64C2×C7×D4C2311yescharacteristic
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C2×D28—21no—
69C2×D28—21no—
70C2×D28C2311yescharacteristic
71C2×D28C2311yescharacteristic
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×Dic14C2311yescharacteristic
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—
85C7×C22⋊C4—21no—
86C42×C7—21no—
87C42×C7C2311yescharacteristic
88C42×C7C2311yescharacteristic
89C42×C7C2311yescharacteristic
90#1144—41no—
91#1143—22no—
92#1143—41no—
93#1142—22no—
94#1142—41no—
95#1141—22no—
96#1141—41no—
97#1141—41no—
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#559D711yescharacteristic, complemented
114#496—71no—
115#496—71no—
116#495—71no—
117#464—71no—
118#464—71no—
119#463—71no—
120#463—71no—
121#459—71no—
122#459—71no—
123#446—71no—
124#440—71no—
125#440—71no—
126C4×D8—71no—
127C23×C7—21no—
128C23×C7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—41no—
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—21no—
137C7×D4—21no—
138C7×D4—42no—
139C7×D4—81no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—81no—
143C2×C4×C7—81no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2411yescharacteristic
151C2×Dic7—21no—
152C2×Dic7—21no—
153D28—21no—
154D28—21no—
155D28—21no—
156D28—21no—
157D28—42no—
158D28—81no—
159C4×D7—81no—
160C4×D7—81no—
161Dic14—21no—
162Dic14—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C7⋊C8—21no—
166C7⋊C8#4112yes—
167C2×Q16—71no—
168C2×SD16—71no—
169C2×SD16—71no—
170C2×SD16—71no—
171C2×SD16—71no—
172C2×D8—71no—
173C2×D8—71no—
174C2×D8—71no—
175C2×D8—71no—
176#127—71no—
177#127C2×D711yescharacteristic
178#124—21no—
179#124—71no—
180#124—71no—
181#124—141no—
182#124C2×D711yescharacteristic
183#124C2×D711yescharacteristic
184#123—21no—
185#123—21no—
186#123—141no—
187#123—141no—
188C4×Q8—71no—
189C4×Q8C2×D711yescharacteristic
190C4×Q8C2×D711yescharacteristic
191C4×D4—21no—
192C4×D4—71no—
193C4×D4—71no—
194C4×D4—141no—
195C4×D4C2×D711yescharacteristic
196C4×D4C2×D711yescharacteristic
197#107—71no—
198C4⋊C8—71no—
199C4⋊C8—71no—
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#97—71no—
208#97—71no—
209C4×C8—71no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7C22×D411yescharacteristic
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7C2×#4111yescharacteristic
232C4×C7H(5,𝔽2)11yescharacteristic
233Dic7—41no—
234Dic7—41no—
235C2×Q8—71no—
236C2×Q8C22⋊D711yescharacteristic
237C2×Q8C22×D711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—71no—
241C2×D4—71no—
242C2×D4—141no—
243C2×D4—141no—
244C2×D4C22×D711yescharacteristic
245C2×D4C22×D711yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—141no—
249C22×C4—141no—
250Q16—281no—
251SD16—142no—
252SD16—142no—
253SD16—281no—
254SD16—281no—
255D8—142no—
256D8—281no—
257D8—281no—
258D8—281no—
259C2×C8—71no—
260C2×C8—71no—
261C2×C8—71no—
262C2×C8—71no—
263C4⋊C4—21no—
264C4⋊C4—71no—
265C4⋊C4—141no—
266C4⋊C4C22⋊D711yescharacteristic
267C4⋊C4C22⋊D711yescharacteristic
268C4⋊C4C22⋊D711yescharacteristic
269C4⋊C4C22×D711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C42—21no—
283C42—141no—
284C42C22×D711yescharacteristic
285C42C22×D711yescharacteristic
286C42C22×D711yescharacteristic
287C2×C7—81no—
288C2×C7—81no—
289C2×C7#55011yescharacteristic
290C2×C7#57511yescharacteristic
291C2×C7#57611yescharacteristic
292D7—81no—
293D7—81no—
294C23—21no—
295C23—21no—
296C23—141no—
297C23—141no—
298Q8—21no—
299Q8—21no—
300Q8—41no—
301Q8—141no—
302Q8—141no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—81no—
309D4—141no—
310D4—141no—
311D4—141no—
312D4—141no—
313D4—282no—
314D4—561no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—561no—
322C2×C4—561no—
323C2×C4C2×C22⋊D711yescharacteristic
324C2×C4C2×C22⋊D711yescharacteristic
325C2×C4C2×C22⋊D711yescharacteristic
326C2×C4C2×C22⋊D711yescharacteristic
327C2×C4C2×C22⋊D711yescharacteristic
328C2×C4C2×C22⋊D711yescharacteristic
329C2×C4C23×D711yescharacteristic
330C8—72no—
331C8—141no—
332C8—141no—
333C8—141no—
334C7#340311yescharacteristic, complemented, Sylow
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—281no—
342C22—281no—
343C22—281no—
344C22—281no—
345C22—281no—
346C22—281no—
347C22C22×C22⋊D711yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—281no—
357C4—281no—
358C4D7×#4111yescharacteristic
359C4#655311yescharacteristic
360C2—81no—
361C2—81no—
362C2—561no—
363C2—561no—
364C2#9051311yescharacteristic
365C2#9052511yescharacteristic
366C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#559C2
socleC22×C7C22×D4
hypercenter#559D7
hypocenterC7#3403
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. C7×#559C2semidirect
2. #90276C2semidirect
3. #89846C2semidirect
4. #89844C2semidirect
5. #89831C2semidirect
6. #89830C2semidirect
7. #89829C2semidirect
8. #89826C2semidirect
9. #89800C2semidirect
10. #89798C2semidirect
11. #89796C2semidirect
12. #89790C2semidirect
13. #89788C2semidirect
14. #89787C2semidirect
15. C4×#1141C2semidirect
113. #559D7semidirect
334. C7#3403semidirect

Maximal subgroups

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

subgroupindex
C7×#5592
#902762
#898462
#898442
#898312
#898302
#898292
#898262
#898002
#897982
#897962
#897902
#897882
#897872
C4×#11412
#34037