Finite Group Databasebeta

← #11741066

Subgroups — #11741066

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×#553C211yesmaximal, characteristic, complemented
2#90243C211yesmaximal, characteristic, complemented
3#89840C211yesmaximal, characteristic, complemented
4#89835C211yesmaximal, characteristic, complemented
5#89814C211yesmaximal, characteristic, complemented
6#89812C211yesmaximal, characteristic, complemented
7#89811C211yesmaximal, characteristic, complemented
8#89809C211yesmaximal, characteristic, complemented
9C4×#1142C211yesmaximal, characteristic, complemented
10#89788C211yesmaximal, characteristic, complemented
11#89784C211yesmaximal, characteristic, complemented
12#89776C211yesmaximal, characteristic, complemented
13#89775C211yesmaximal, characteristic, complemented
14#89743C211yesmaximal, characteristic, complemented
15#89738C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C7×#126—21no—
19C7×#125C2211yescharacteristic
20C7×#123—21no—
21C7×#121—21no—
22C7×#121C2211yescharacteristic
23C7×#121C2211yescharacteristic
24C4×C7×D4—21no—
25C4×C7×D4C2211yescharacteristic
26C4×C7×D4C2211yescharacteristic
27C7×#117C2211yescharacteristic
28C2×C7×C4⋊C4C2211yescharacteristic
29#6502—21no—
30C2×#1142C2211yescharacteristic
31C2×#1142C2211yescharacteristic
32C2×#1141C2211yescharacteristic
33#6493C2211yescharacteristic
34#6487C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C2×#1136C2211yescharacteristic
37#6459—21no—
38#6448—21no—
39(C22⋊D7)⋊C4—21no—
40#6438C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42#6406C2211yescharacteristic
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#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3326—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4—21no—
62C2×C7×D4D411yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C22×Dic7—21no—
71#1157—22no—
72#1157—41no—
73C2×D28C2311yescharacteristic
74C2×C4×D7—21no—
75C2×Dic14—22no—
76C2×Dic14—21no—
77C2×Dic14—41no—
78C2×Dic14C2311yescharacteristic
79C2×Dic14C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4D411yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C22⋊C4—21no—
88C7×C22⋊C4—21no—
89C7×C22⋊C4—21no—
90C7×C22⋊C4—21no—
91C7×C22⋊C4D411yescharacteristic
92C7×C22⋊C4D411yescharacteristic
93C42×C7—21no—
94C42×C7C2311yescharacteristic
95C22⋊Dic7—21no—
96#1142—22no—
97#1142—41no—
98#1141—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—
107#1136—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#553D711yescharacteristic, complemented
113#548—71no—
114#488—71no—
115#483—71no—
116#482—71no—
117#479—71no—
118#469—71no—
119#468—71no—
120#463—71no—
121#449—71no—
122#448—71no—
123C4×SD16—71no—
124#436—71no—
125#419—71no—
126#417—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—41no—
134C7×D4—42no—
135C7×D4—81no—
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—21no—
143C2×C4×C7—81no—
144C2×C4×C7—81no—
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2411yescharacteristic
150C22⋊D7—42no—
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—42no—
163Dic14#4112yes—
164C7⋊C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C22×Q8—71no—
170C2×SD16—71no—
171C2×SD16—71no—
172C2×D8—71no—
173C2×M16—71no—
174C2×M16—71no—
175#126—21no—
176#125C2×D711yescharacteristic
177#124—71no—
178#124—141no—
179#123—21no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182#121—21no—
183#121—71no—
184#121—141no—
185#121C2×D711yescharacteristic
186#121C2×D711yescharacteristic
187C4×Q8—71no—
188C4×D4—21no—
189C4×D4—141no—
190C4×D4C2×D711yescharacteristic
191C4×D4C2×D711yescharacteristic
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#103—71no—
203#103—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208C22⋊C8—71no—
209C22⋊C8—71no—
210C4×C8—71no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7C22×D411yescharacteristic
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×C7C2×#4111yescharacteristic
232C4×C7H(5,𝔽2)11yescharacteristic
233Dic7—22no—
234Dic7—41no—
235Dic7—41no—
236#41—142no—
237#41—281no—
238C2×Q8—71no—
239C2×Q8—71no—
240C2×Q8—142no—
241C2×Q8—141no—
242C2×Q8—281no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—71no—
246C2×D4—141no—
247C2×D4—141no—
248C2×D4C22⋊D711yescharacteristic
249C2×D4C22×D711yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4C22×D711yescharacteristic
255C22×C4C22×D711yescharacteristic
256SD16—142no—
257SD16—281no—
258D8—281no—
259M16—281no—
260M16—281no—
261C2×C8—71no—
262C2×C8—71no—
263C2×C8—71no—
264C2×C8—71no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—71no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4C22⋊D711yescharacteristic
272C4⋊C4C22×D711yescharacteristic
273C4⋊C4C22×D711yescharacteristic
274C4⋊C4C22×D711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4C22⋊D711yescharacteristic
284C22⋊C4C22⋊D711yescharacteristic
285C42—21no—
286C42—141no—
287C42C22×D711yescharacteristic
288C2×C7—41no—
289C2×C7—41no—
290C2×C7—81no—
291C2×C7#54611yescharacteristic
292C2×C7#57511yescharacteristic
293C2×C7#57711yescharacteristic
294D7—81no—
295C23—21no—
296C23—141no—
297C23C2×C22⋊D711yescharacteristic
298C23C2×C22⋊D711yescharacteristic
299Q8—72no—
300Q8—141no—
301Q8—141no—
302Q8—141no—
303Q8—282no—
304D4—21no—
305D4—21no—
306D4—41no—
307D4—42no—
308D4—81no—
309D4—141no—
310D4—141no—
311D4—282no—
312D4—282no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
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×C4C23×D711yescharacteristic
332C8—141no—
333C8—141no—
334C8—141no—
335C8—141no—
336C7#332611yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—281no—
347C22—281no—
348C22—281no—
349C22C22×C22⋊D711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—142no—
358C4—281no—
359C4—281no—
360C4D7×#4111yescharacteristic
361C4#655311yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—561no—
366C2#9051311yescharacteristic
367C2#9052011yescharacteristic
368C2#9052711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#553C2
socleC22×C7C22×D4
hypercenter#553D7
hypocenterC7#3326
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. C7×#553C2semidirect
2. #90243C2semidirect
3. #89840C2semidirect
4. #89835C2semidirect
5. #89814C2semidirect
6. #89812C2semidirect
7. #89811C2semidirect
8. #89809C2semidirect
9. C4×#1142C2semidirect
10. #89788C2semidirect
11. #89784C2semidirect
12. #89776C2semidirect
13. #89775C2semidirect
14. #89743C2semidirect
15. #89738C2semidirect
112. #553D7semidirect
336. C7#3326semidirect

Maximal subgroups

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

subgroupindex
C7×#5532
#902432
#898402
#898352
#898142
#898122
#898112
#898092
C4×#11422
#897882
#897842
#897762
#897752
#897432
#897382
#33267