Finite Group Databasebeta

← #11740909

Subgroups — #11740909

Order 896.

376 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×#542C211yesmaximal, characteristic, complemented
2#90217C211yesmaximal, characteristic, complemented
3#89865C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89832C211yesmaximal, characteristic, complemented
6#89831C211yesmaximal, characteristic, complemented
7#89822C211yesmaximal, characteristic, complemented
8#89819C211yesmaximal, characteristic, complemented
9#89812C211yesmaximal, characteristic, complemented
10#89809C211yesmaximal, characteristic
11#89803C211yesmaximal, characteristic, complemented
12#89800C211yesmaximal, characteristic, complemented
13#89791C211yesmaximal, characteristic, complemented
14#89788C211yesmaximal, characteristic, complemented
15#89772C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C7×#126—21no—
20C7×#124—21no—
21C7×#124C2211yescharacteristic
22C7×#123—21no—
23C7×C22⋊Q8—21no—
24C7×C22⋊Q8C2211yescharacteristic
25C7×#121C2211yescharacteristic
26C4×C7×Q8C2211yescharacteristic
27C4×C7×D4C2211yescharacteristic
28C7×#117C2211yescharacteristic
29C2×#1144C2211yescharacteristic
30C2×#1144C2211yescharacteristic
31C2×#1144C2211yescharacteristic
32C2×#1143C2211yescharacteristic
33C2×#1142C2211yescharacteristic
34C2×#1142C2211yescharacteristic
35C2×#1142C2211yescharacteristic
36C2×#1141C2211yescharacteristic
37#6460—21no—
38#6460—21no—
39#6449—21no—
40#6449—21no—
41#6438C2211yescharacteristic
42#6438C2211yescharacteristic
43#6432C2211yescharacteristic
44#6432C2211yescharacteristic
45C22⋊(C7⋊C8)C2211yescharacteristic
46C22⋊(C7⋊C8)C2211yescharacteristic
47#6384C2211yescharacteristic
48#6384C2211yescharacteristic
49#6384C2211yescharacteristic
50#6384C2211yescharacteristic
51#6384C2211yescharacteristic
52#6384C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56C4⋊(C7⋊C8)C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3225—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—21no—
67C22×Dic7—21no—
68#1157—41no—
69#1157—41no—
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×Dic14—21no—
73C2×Dic14—41no—
74C2×Dic14—41no—
75C2×Dic14C2311yescharacteristic
76C2×Dic14C2311yescharacteristic
77C2×Dic14C2311yescharacteristic
78C7×C4⋊C4—21no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4D411yescharacteristic
82C7×C4⋊C4D411yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C7×C22⋊C4—21no—
88C7×C22⋊C4—21no—
89C7×C22⋊C4—21no—
90C7×C22⋊C4D411yescharacteristic
91C7×C22⋊C4D411yescharacteristic
92C42×C7—21no—
93C42×C7C2311yescharacteristic
94C42×C7C2311yescharacteristic
95#1144—41no—
96#1144—41no—
97#1144—41no—
98#1143—41no—
99#1142—41no—
100#1142—41no—
101#1142—41no—
102#1141—41no—
103(C2×D7)⋊C4—21no—
104(C2×D7)⋊C4—21no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107C4⋊Dic7—21no—
108C4⋊Dic7—21no—
109C4⋊Dic7—21no—
110C4⋊Dic7—21no—
111C2×C7⋊C8C2311yescharacteristic
112C2×C7⋊C8C2311yescharacteristic
113C2×C7⋊C8C2311yescharacteristic
114C2×C7⋊C8C2311yescharacteristic
115#542D711yescharacteristic, complemented
116#536—71no—
117#497—71no—
118#495—71no—
119#489—71no—
120#487—71no—
121#464—71no—
122#463—71no—
123#462—71no—
124#461—71no—
125#452—71no—
126#451—71no—
127#449—71no—
128#448—71no—
129#433—71no—
130C23×C7—21no—
131C23×C7C2×D411yescharacteristic
132C22×D7—21no—
133C7×Q8—21no—
134C7×Q8—21no—
135C7×Q8—81no—
136C7×D4—21no—
137C7×D4—21no—
138C7×D4—81no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—81no—
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2411yescharacteristic
151C22⋊D7—42no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—42no—
156D28—21no—
157D28—21no—
158C4×D7—42no—
159Dic14—21no—
160Dic14—21no—
161Dic14—21no—
162Dic14—21no—
163Dic14—21no—
164Dic14—21no—
165Dic14—42no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C7⋊C8—21no—
169C7⋊C8—21no—
170C2×#41—71no—
171C22×Q8—71no—
172C2×Q16—71no—
173C2×Q16—71no—
174C2×Q16—71no—
175C2×SD16—71no—
176C2×SD16—71no—
177C2×SD16—71no—
178C2×SD16—71no—
179C2×D8—71no—
180C4⋊Q8—71no—
181C4⋊Q8—71no—
182C4⋊Q8C2×D711yescharacteristic
183#126—21no—
184#124—21no—
185#124—71no—
186#124—71no—
187#124C2×D711yescharacteristic
188#123—21no—
189#123—141no—
190#123—141no—
191C22⋊Q8—21no—
192C22⋊Q8—141no—
193C22⋊Q8—141no—
194C22⋊Q8C2×D711yescharacteristic
195#121C2×D711yescharacteristic
196C4×Q8C2×D711yescharacteristic
197C4×D4C2×D711yescharacteristic
198#117C2×D711yescharacteristic
199C4⋊C8—71no—
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#103—71no—
205#103—71no—
206#103—71no—
207#102—71no—
208#102—71no—
209C22⋊C8—71no—
210C22⋊C8—71no—
211#97—71no—
212C4×C8—71no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7C22×D411yescharacteristic
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—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7H(5,𝔽2)11yescharacteristic
232C4×C7H(5,𝔽2)11yescharacteristic
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236#41—281no—
237#41—281no—
238C2×Q8—21no—
239C2×Q8—71no—
240C2×Q8—71no—
241C2×Q8—71no—
242C2×Q8—141no—
243C2×Q8—281no—
244C2×Q8—281no—
245C2×Q8C22×D711yescharacteristic
246C2×D4—21no—
247C2×D4—71no—
248C2×D4—141no—
249C2×D4C22×D711yescharacteristic
250C22×C4—21no—
251C22×C4—141no—
252C22×C4—141no—
253C22×C4C22×D711yescharacteristic
254Q16—281no—
255Q16—281no—
256Q16—281no—
257SD16—281no—
258SD16—281no—
259SD16—281no—
260SD16—281no—
261D8—281no—
262C2×C8—71no—
263C2×C8—71no—
264C2×C8—71no—
265C2×C8—71no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4C22⋊D711yescharacteristic
274C4⋊C4C22⋊D711yescharacteristic
275C4⋊C4C22×D711yescharacteristic
276C4⋊C4C22×D711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4C22⋊D711yescharacteristic
287C22⋊C4C22⋊D711yescharacteristic
288C42—21no—
289C42C22×D711yescharacteristic
290C42C22×D711yescharacteristic
291C2×C7—41no—
292C2×C7—81no—
293C2×C7#53511yescharacteristic
294C2×C7#57711yescharacteristic
295C2×C7#57811yescharacteristic
296D7—81no—
297C23—21no—
298C23—141no—
299C23C2×C22⋊D711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—81no—
303Q8—141no—
304Q8—141no—
305Q8—141no—
306Q8—141no—
307Q8—141no—
308Q8—141no—
309Q8—282no—
310D4—21no—
311D4—21no—
312D4—81no—
313D4—141no—
314D4—141no—
315D4—282no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—282no—
326C2×C4—282no—
327C2×C4C2×C22⋊D711yescharacteristic
328C2×C4C2×C22⋊D711yescharacteristic
329C2×C4C2×C22⋊D711yescharacteristic
330C2×C4C2×C22⋊D711yescharacteristic
331C2×C4C2×C22⋊D711yescharacteristic
332C2×C4C23×D711yescharacteristic
333C8—141no—
334C8—141no—
335C8—141no—
336C8—141no—
337C7#322511yescharacteristic, complemented, Sylow
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—281no—
345C22—281no—
346C22—281no—
347C22C22×C22⋊D711yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—281no—
357C4—281no—
358C4—281no—
359C4#655311yescharacteristic
360C4#655311yescharacteristic
361C2—41no—
362C2—81no—
363C2—561no—
364C2#9052111yescharacteristic
365C2#9052711yescharacteristic
366C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#542C2
socleC22×C7C22×D4
prime radical#89809C2
hypercenter#542D7
hypocenterC7#3225
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. C7×#542C2semidirect
2. #90217C2semidirect
3. #89865C2semidirect
4. #89861C2semidirect
5. #89832C2semidirect
6. #89831C2semidirect
7. #89822C2semidirect
8. #89819C2semidirect
9. #89812C2semidirect
11. #89803C2semidirect
12. #89800C2semidirect
13. #89791C2semidirect
14. #89788C2semidirect
15. #89772C2semidirect
115. #542D7semidirect
337. C7#3225semidirect

Maximal subgroups

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

subgroupindex
C7×#5422
#902172
#898652
#898612
#898322
#898312
#898222
#898192
#898122
#898092
#898032
#898002
#897912
#897882
#897722
#32257