Finite Group Databasebeta

← #11741055

Subgroups — #11741055

Order 896.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#553C211yesmaximal, characteristic, complemented
2#90254C211yesmaximal, characteristic, complemented
3#89838C211yesmaximal, characteristic, complemented
4#89837C211yesmaximal, characteristic, complemented
5#89811C211yesmaximal, characteristic, complemented
6#89810C211yesmaximal, characteristic, complemented
7#89807C211yesmaximal, characteristic, complemented
8#89807C211yesmaximal, characteristic, complemented
9#89786C211yesmaximal, characteristic, complemented
10#89783C211yesmaximal, characteristic, complemented
11#89779C211yesmaximal, characteristic, complemented
12#89775C211yesmaximal, characteristic, complemented
13#89770C211yesmaximal, characteristic, complemented
14#89739C211yesmaximal, characteristic, complemented
15#89736C211yesmaximal, characteristic, complemented
16C7×#126—21no—
17C7×#125C2211yescharacteristic
18C7×#123—21no—
19C7×#121—21no—
20C7×#121C2211yescharacteristic
21C7×#121C2211yescharacteristic
22C4×C7×D4—21no—
23C4×C7×D4C2211yescharacteristic
24C4×C7×D4C2211yescharacteristic
25C7×#117C2211yescharacteristic
26C2×C7×C4⋊C4C2211yescharacteristic
27#6501—21no—
28C2×#1141C2211yescharacteristic
29#6493C2211yescharacteristic
30#6493C2211yescharacteristic
31#6489C2211yescharacteristic
32#6489C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6455—21no—
36#6448—21no—
37#6448—21no—
38C4×D28C2211yescharacteristic
39#6432C2211yescharacteristic
40#6406C2211yescharacteristic
41#6406C2211yescharacteristic
42#6406C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6383C2211yescharacteristic
46#6383C2211yescharacteristic
47#6383C2211yescharacteristic
48#6382C2211yescharacteristic
49#6382C2211yescharacteristic
50#6382C2211yescharacteristic
51#6382C2211yescharacteristic
52#6381C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3347—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—
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×Dic14—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4D411yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C7×C22⋊C4—21no—
84C7×C22⋊C4D411yescharacteristic
85C7×C22⋊C4D411yescharacteristic
86C42×C7—21no—
87C42×C7C2311yescharacteristic
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90#1141—22no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100#1136—22no—
101C2×C7⋊C8—22no—
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#555—71no—
107#553D711yescharacteristic, complemented
108#501—71no—
109#482—71no—
110#482—71no—
111#482—71no—
112#480—71no—
113#478—71no—
114#474—71no—
115#469—71no—
116#466—71no—
117#442—71no—
118#435—71no—
119#428—71no—
120#427—71no—
121C23×C7—21no—
122C23×C7C2×D411yescharacteristic
123C23×C7C2×D411yescharacteristic
124C22×D7—21no—
125C7×D4—21no—
126C7×D4—21no—
127C7×D4—41no—
128C7×D4—42no—
129C7×D4—81no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—81no—
138C2×C4×C7—81no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2411yescharacteristic
144C22⋊D7—81no—
145C22⋊D7—81no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149D28—21no—
150D28—21no—
151C4×D7—42no—
152Dic14—81no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C7⋊C8—21no—
156C7⋊C8#4112yes—
157C2×D8—71no—
158C2×M16—71no—
159C22×C8—71no—
160C4⋊Q8—71no—
161#126—21no—
162#125C2×D711yescharacteristic
163#124—141no—
164#124—141no—
165#123—21no—
166#121—21no—
167#121—71no—
168#121—71no—
169#121—141no—
170#121C2×D711yescharacteristic
171#121C2×D711yescharacteristic
172C4×D4—21no—
173C4×D4—71no—
174C4×D4C2×D711yescharacteristic
175C4×D4C2×D711yescharacteristic
176#117—71no—
177#117—71no—
178#117—141no—
179#117C2×D711yescharacteristic
180C2×C4⋊C4C2×D711yescharacteristic
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#107—71no—
186#106—71no—
187#106—71no—
188#106—71no—
189#106—71no—
190C4⋊C8—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197C22⋊C8—71no—
198C22⋊C8—71no—
199#97—71no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7C22×D411yescharacteristic
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7C2×#4111yescharacteristic
221C4×C72−1+411yescharacteristic
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×Q8—141no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—71no—
229C2×D4—141no—
230C2×D4—141no—
231C2×D4C22⋊D711yescharacteristic
232C2×D4C22×D711yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—141no—
236C22×C4C22×D711yescharacteristic
237C22×C4C22×D711yescharacteristic
238D8—142no—
239M16—142no—
240C2×C8—71no—
241C2×C8—71no—
242C2×C8—71no—
243C2×C8—71no—
244C2×C8—142no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—141no—
252C4⋊C4C22⋊D711yescharacteristic
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4C22⋊D711yescharacteristic
266C22⋊C4C22⋊D711yescharacteristic
267C42—21no—
268C42—141no—
269C42—141no—
270C42C22×D711yescharacteristic
271C2×C7—41no—
272C2×C7—41no—
273C2×C7—81no—
274C2×C7#54711yescharacteristic
275C2×C7#57511yescharacteristic
276C2×C7#57611yescharacteristic
277D7—81no—
278C23—21no—
279C23—141no—
280C23C2×C22⋊D711yescharacteristic
281C23C2×C22⋊D711yescharacteristic
282Q8—561no—
283D4—21no—
284D4—21no—
285D4—41no—
286D4—42no—
287D4—81no—
288D4—141no—
289D4—141no—
290D4—561no—
291D4—561no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—282no—
305C2×C4C2×C22⋊D711yescharacteristic
306C2×C4C2×C22⋊D711yescharacteristic
307C2×C4C2×C22⋊D711yescharacteristic
308C2×C4C2×C22⋊D711yescharacteristic
309C2×C4C23×D711yescharacteristic
310C8—72no—
311C8—141no—
312C8—141no—
313C8—141no—
314C7#334711yescharacteristic, complemented, Sylow
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—281no—
325C22—281no—
326C22—281no—
327C22C22×C22⋊D711yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—281no—
336C4—281no—
337C4—281no—
338C4D7×#4111yescharacteristic
339C4#655411yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—561no—
344C2#9051311yescharacteristic
345C2#9051511yescharacteristic
346C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#553C2
socleC22×C7C22×D4
hypercenter#553D7
hypocenterC7#3347
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. #90254C2semidirect
3. #89838C2semidirect
4. #89837C2semidirect
5. #89811C2semidirect
6. #89810C2semidirect
7. #89807C2semidirect
8. #89807C2semidirect
9. #89786C2semidirect
10. #89783C2semidirect
11. #89779C2semidirect
12. #89775C2semidirect
13. #89770C2semidirect
14. #89739C2semidirect
15. #89736C2semidirect
107. #553D7semidirect
314. C7#3347semidirect

Maximal subgroups

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

subgroupindex
C7×#5532
#902542
#898382
#898372
#898112
#898102
#898072
#897862
#897832
#897792
#897752
#897702
#897392
#897362
#33477