Finite Group Databasebeta

← #11741049

Subgroups — #11741049

Order 896.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#552C211yesmaximal, characteristic, complemented
2#90244C211yesmaximal, characteristic
3#89839C211yesmaximal, characteristic, complemented
4#89835C211yesmaximal, characteristic, complemented
5#89821C211yesmaximal, characteristic, complemented
6#89820C211yesmaximal, characteristic, complemented
7#89819C211yesmaximal, characteristic, complemented
8#89819C211yesmaximal, characteristic, complemented
9#89802C211yesmaximal, characteristic, complemented
10#89799C211yesmaximal, characteristic, complemented
11#89783C211yesmaximal, characteristic, complemented
12#89775C211yesmaximal, characteristic, complemented
13#89774C211yesmaximal, characteristic, complemented
14#89740C211yesmaximal, characteristic, complemented
15#89738C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C7×#126—21no—
19C7×#125—21no—
20C7×#125C2211yescharacteristic
21C7×#123—21no—
22C7×C22⋊Q8C2211yescharacteristic
23C7×C22⋊Q8C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4C2211yescharacteristic
26C7×#117—21no—
27C7×#117C2211yescharacteristic
28C2×C7×C4⋊C4C2211yescharacteristic
29C2×#1144C2211yescharacteristic
30C2×#1143C2211yescharacteristic
31C2×#1143C2211yescharacteristic
32#6503—21no—
33#6493C2211yescharacteristic
34#6493C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6456—21no—
38#6447—21no—
39#6447—21no—
40#6437C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42#6409C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6384C2211yescharacteristic
46#6384C2211yescharacteristic
47#6384C2211yescharacteristic
48#6383C2211yescharacteristic
49#6383C2211yescharacteristic
50#6383C2211yescharacteristic
51#6383C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3317—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4D411yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69#1157—22no—
70#1157—22no—
71#1157—41no—
72#1157—41no—
73C2×D28—21no—
74C2×D28—21no—
75C2×D28C2311yescharacteristic
76C2×D28C2311yescharacteristic
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×Dic14C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4—21no—
86C7×C4⋊C4D411yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C22⋊C4—21no—
91C7×C22⋊C4—21no—
92C7×C22⋊C4—21no—
93C7×C22⋊C4D411yescharacteristic
94C7×C22⋊C4D411yescharacteristic
95C42×C7—21no—
96C42×C7—21no—
97C42×C7C2311yescharacteristic
98#1144—22no—
99#1143—41no—
100#1143—41no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4—21no—
103C4⋊Dic7C2311yescharacteristic
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#552D711yescharacteristic, complemented
113#550—71no—
114#490—71no—
115#482—71no—
116#481—71no—
117#479—71no—
118#468—71no—
119#465—71no—
120#460—71no—
121#452—71no—
122#452—71no—
123#441—71no—
124#435—71no—
125#417—71no—
126#416—71no—
127C23×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C7×Q8—21no—
132C7×Q8—21no—
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×C7—42no—
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2411yescharacteristic
148C22⋊D7—42no—
149C22⋊D7—42no—
150C22⋊D7—81no—
151C22⋊D7—81no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154D28—21no—
155D28—21no—
156D28—21no—
157D28—21no—
158D28—42no—
159D28—81no—
160C4×D7—42no—
161C4×D7—42no—
162Dic14—21no—
163Dic14#4112yes—
164C7⋊C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C2×#41—71no—
170C2×Q16—71no—
171C2×SD16—71no—
172C2×SD16—71no—
173C2×M16—71no—
174C22×C8—71no—
175#127—71no—
176#127—141no—
177#126—21no—
178#125—21no—
179#125C2×D711yescharacteristic
180#123—21no—
181C22⋊Q8C2×D711yescharacteristic
182C22⋊Q8C2×D711yescharacteristic
183#121—71no—
184#121—71no—
185#121—141no—
186#121—141no—
187C4×Q8—71no—
188C4×Q8C2×D711yescharacteristic
189C4×D4—141no—
190C4×D4C2×D711yescharacteristic
191#117—21no—
192#117C2×D711yescharacteristic
193C2×C4⋊C4C2×D711yescharacteristic
194#107—71no—
195#106—71no—
196#106—71no—
197C4⋊C8—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208C22⋊C8—71no—
209C22⋊C8—71no—
210#97—71no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7C22×D411yescharacteristic
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C2×D7—41no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—41no—
232C4×C7C2×#4111yescharacteristic
233C4×C7H(5,𝔽2)11yescharacteristic
234Dic7—22no—
235Dic7—41no—
236#41—142no—
237#41—142no—
238#41—281no—
239#41—281no—
240C2×Q8—71no—
241C2×Q8C22×D711yescharacteristic
242C2×D4—71no—
243C2×D4—71no—
244C2×D4—141no—
245C2×D4—141no—
246C2×D4—141no—
247C2×D4—141no—
248C2×D4—141no—
249C2×D4—141no—
250C2×D4C22⋊D711yescharacteristic
251C22×C4—21no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4C22×D711yescharacteristic
255C22×C4C22×D711yescharacteristic
256Q16—142no—
257SD16—281no—
258SD16—281no—
259M16—281no—
260C2×C8—71no—
261C2×C8—71no—
262C2×C8—71no—
263C2×C8—71no—
264C2×C8—281no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—71no—
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—141no—
281C22⋊C4—141no—
282C22⋊C4C22⋊D711yescharacteristic
283C22⋊C4C22⋊D711yescharacteristic
284C42—21no—
285C42—21no—
286C42—141no—
287C42C22×D711yescharacteristic
288C2×C7—41no—
289C2×C7—41no—
290C2×C7#54611yescharacteristic
291C2×C7#57511yescharacteristic
292C2×C7#57711yescharacteristic
293D7—81no—
294D7—81no—
295C23—141no—
296C23—141no—
297C23C2×C22⋊D711yescharacteristic
298C23C2×C22⋊D711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—72no—
302Q8—141no—
303D4—41no—
304D4—141no—
305D4—141no—
306D4—141no—
307D4—141no—
308D4—282no—
309D4—282no—
310D4—282no—
311D4—561no—
312D4—561no—
313D4—561no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
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×C4C23×D711yescharacteristic
332C8—141no—
333C8—141no—
334C8—141no—
335C8—141no—
336C7#331711yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—281no—
344C22—281no—
345C22—281no—
346C22—281no—
347C22—281no—
348C22—281no—
349C22C22×C22⋊D711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—142no—
359C4—281no—
360C4D7×#4111yescharacteristic
361C4#655311yescharacteristic
362C2—41no—
363C2—41no—
364C2—561no—
365C2—561no—
366C2#9051311yescharacteristic
367C2#9052011yescharacteristic
368C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#552C2
socleC22×C7C22×D4
prime radical#90244C2
hypercenter#552D7
hypocenterC7#3317
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
1. C7×#552C2semidirect
3. #89839C2semidirect
4. #89835C2semidirect
5. #89821C2semidirect
6. #89820C2semidirect
7. #89819C2semidirect
8. #89819C2semidirect
9. #89802C2semidirect
10. #89799C2semidirect
11. #89783C2semidirect
12. #89775C2semidirect
13. #89774C2semidirect
14. #89740C2semidirect
15. #89738C2semidirect
112. #552D7semidirect
336. C7#3317semidirect

Maximal subgroups

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

subgroupindex
C7×#5522
#902442
#898392
#898352
#898212
#898202
#898192
#898022
#897992
#897832
#897752
#897742
#897402
#897382
#33177