Finite Group Databasebeta

← #11740984

Subgroups — #11740984

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×#548C211yesmaximal, characteristic, complemented
2#90254C211yesmaximal, characteristic, complemented
3#90010C211yesmaximal, characteristic, complemented
4#90005C211yesmaximal, characteristic, complemented
5#89999C211yesmaximal, characteristic, complemented
6#89997C211yesmaximal, characteristic, complemented
7#89829C211yesmaximal, characteristic, complemented
8#89827C211yesmaximal, characteristic, complemented
9#89820C211yesmaximal, characteristic, complemented
10#89817C211yesmaximal, characteristic, complemented
11#89811C211yesmaximal, characteristic, complemented
12#89807C211yesmaximal, characteristic, complemented
13#89798C211yesmaximal, characteristic, complemented
14#89792C211yesmaximal, characteristic, complemented
15#89783C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic
18C7×#124—21no—
19C7×#124C2211yescharacteristic
20C7×C22⋊Q8—21no—
21C7×C22⋊Q8C2211yescharacteristic
22C7×#121—21no—
23C7×#121C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4—21no—
26C4×C7×D4C2211yescharacteristic
27C2×#1143C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29#6501—21no—
30C2×#1141C2211yescharacteristic
31#6493C2211yescharacteristic
32#6493C2211yescharacteristic
33#6489C2211yescharacteristic
34#6489C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6455—21no—
38#6448—21no—
39#6448—21no—
40C4×D28C2211yescharacteristic
41#6432C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50#6406C2211yescharacteristic
51C22⋊(C7⋊C8)C2211yescharacteristic
52C22⋊(C7⋊C8)C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3321—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—41no—
62C2×C7×Q8—21no—
63C2×C7×Q8—22no—
64C2×C7×Q8—41no—
65C2×C7×Q8C2311yescharacteristic
66C2×C7×Q8C2311yescharacteristic
67C2×C7×D4—21no—
68C2×C7×D4—21no—
69C2×C7×D4D411yescharacteristic
70C2×C7×D4C2311yescharacteristic
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C2×C22⋊D7—21no—
76C2×C22⋊D7—21no—
77C2×D28C2311yescharacteristic
78C2×C4×D7—21no—
79C2×Dic14—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4D411yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C7×C22⋊C4—21no—
88C7×C22⋊C4D411yescharacteristic
89C7×C22⋊C4D411yescharacteristic
90C42×C7—21no—
91C42×C7C2311yescharacteristic
92C22⋊Dic7—21no—
93C22⋊Dic7—21no—
94#1143—22no—
95#1143—41no—
96#1141—41no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
103C4⋊Dic7C2311yescharacteristic
104C4×Dic7—21no—
105C4×Dic7—21no—
106#1136—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#555—71no—
113#548D711yescharacteristic, complemented
114#489—71no—
115#485—71no—
116#482—71no—
117#475—71no—
118#469—71no—
119#465—71no—
120#463—71no—
121#449—71no—
122#448—71no—
123#440—71no—
124#435—71no—
125#419—71no—
126#416—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×Q8—42no—
135C7×Q8#4112yes—
136C7×D4—21no—
137C7×D4—21no—
138C7×D4—41no—
139C7×D4—42no—
140C7×D4—42no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7C2×D411yescharacteristic
151C2×C4×C7C2×D411yescharacteristic
152C2×C4×C7C2×D411yescharacteristic
153C2×C4×C7C2×D411yescharacteristic
154C2×C4×C7C2411yescharacteristic
155C22⋊D7—81no—
156C22⋊D7—81no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160D28—21no—
161D28—21no—
162C4×D7—42no—
163Dic14—81no—
164C7⋊C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41C2×D711yescharacteristic
169C22×Q8C2×D711yescharacteristic
170C2×SD16—71no—
171C2×SD16—71no—
172C2×D8—71no—
173C2×M16—71no—
174C22×C8—71no—
175C4⋊Q8—71no—
176#124—21no—
177#124—141no—
178#124—141no—
179#124C2×D711yescharacteristic
180C22⋊Q8—21no—
181C22⋊Q8C2×D711yescharacteristic
182#121—21no—
183#121—71no—
184#121—71no—
185#121—141no—
186#121C2×D711yescharacteristic
187C4×Q8C2×D711yescharacteristic
188C4×D4—21no—
189C4×D4—71no—
190C4×D4C2×D711yescharacteristic
191#117—71no—
192#117—71no—
193#117—141no—
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—
210#97—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—22no—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7C2×#4111yescharacteristic
232C4×C7H(5,𝔽2)11yescharacteristic
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236#41—22no—
237#41—41no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—141no—
242C2×Q8C22×D711yescharacteristic
243C2×Q8C22×D711yescharacteristic
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—71no—
247C2×D4—141no—
248C2×D4—141no—
249C2×D4C22⋊D711yescharacteristic
250C2×D4C22×D711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—141no—
254C22×C4C22×D711yescharacteristic
255C22×C4C22×D711yescharacteristic
256SD16—142no—
257SD16—281no—
258D8—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—71no—
268C4⋊C4—71no—
269C4⋊C4—71no—
270C4⋊C4—141no—
271C4⋊C4C22⋊D711yescharacteristic
272C4⋊C4C22×D711yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4C22⋊D711yescharacteristic
283C22⋊C4C22⋊D711yescharacteristic
284C42—21no—
285C42—141no—
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—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—561no—
304Q8#115912yes—
305D4—21no—
306D4—21no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—141no—
311D4—141no—
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—42no—
322C2×C4—42no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
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#332111yescharacteristic, 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—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—281no—
358C4—281no—
359C4—281no—
360C4C2×#115911yescharacteristic
361C4#654811yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—561no—
366C2#9049711yescharacteristic
367C2#9051311yescharacteristic
368C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#548C2
socleC22×C7C22×D4
hypercenter#548D7
hypocenterC7#3321
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C7×#548C2semidirect
2. #90254C2semidirect
3. #90010C2semidirect
4. #90005C2semidirect
5. #89999C2semidirect
6. #89997C2semidirect
7. #89829C2semidirect
8. #89827C2semidirect
9. #89820C2semidirect
10. #89817C2semidirect
11. #89811C2semidirect
12. #89807C2semidirect
13. #89798C2semidirect
14. #89792C2semidirect
15. #89783C2semidirect
113. #548D7semidirect
336. C7#3321semidirect

Maximal subgroups

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

subgroupindex
C7×#5482
#902542
#900102
#900052
#899992
#899972
#898292
#898272
#898202
#898172
#898112
#898072
#897982
#897922
#897832
#33217