Finite Group Databasebeta

← #11737257

Subgroups — #11737257

Order 896.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90265C211yesmaximal, characteristic, complemented
2#90253C211yesmaximal, characteristic, complemented
3C7×#440C211yesmaximal, characteristic, complemented
4#89944C211yesmaximal, characteristic, complemented
5#89935C211yesmaximal, characteristic, complemented
6C4×#1143C211yesmaximal, characteristic, complemented
7C4×#1141C211yesmaximal, characteristic, complemented
8#89663C211yesmaximal, characteristic, complemented
9#89654C211yesmaximal, characteristic, complemented
10#89583C211yesmaximal, characteristic, complemented
11#89571C211yesmaximal, characteristic, complemented
12#89558C211yesmaximal, characteristic, complemented
13#89549C211yesmaximal, characteristic, complemented
14#89483C211yesmaximal, characteristic, complemented
15#89481C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1143C2211yescharacteristic
22#6503C2211yescharacteristic
23#6501C2211yescharacteristic
24C2×#1141C2211yescharacteristic
25#6493—21no—
26#6492—21no—
27C2×D56C2211yescharacteristic
28#6458C2211yescharacteristic
29#6456—21no—
30#6456C2211yescharacteristic
31D7×C4⋊C4—21no—
32D7×C4⋊C4C2211yescharacteristic
33#6453C2211yescharacteristic
34#6447—21no—
35#6446—21no—
36C4×D28—21no—
37C4×D28C2211yescharacteristic
38C4×D28C2211yescharacteristic
39#6435—21no—
40#6435C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×#97C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6391C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3119—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66#1161—22no—
67#1161—22no—
68C2×D28—21no—
69C2×D28—21no—
70C2×D28C2311yescharacteristic
71C2×D28C2311yescharacteristic
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7C2311yescharacteristic
78C7×SD16—22no—
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C42×C7—21no—
86C42×C7C2311yescharacteristic
87C22⋊Dic7—21no—
88#1143—22no—
89#1141—22no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4D411yescharacteristic
95(C2×D7)⋊C4D411yescharacteristic
96C4⋊Dic7—21no—
97C4⋊Dic7—21no—
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4D411yescharacteristic
103Dic7⋊C4D411yescharacteristic
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108D56—22no—
109#540—71no—
110#520—71no—
111#481—71no—
112#480—71no—
113#479—71no—
114#478—71no—
115#460—71no—
116#459—71no—
117#449—71no—
118#449—71no—
119#442—71no—
120#440D711yescharacteristic, complemented
121C4×SD16—71no—
122C4×D8—71no—
123#433—71no—
124C23×C7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7C2×D411yescharacteristic
128C7×Q8—21no—
129C7×Q8#4112yes—
130C7×D4—21no—
131C7×D4—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—81no—
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2411yescharacteristic
139C22⋊D7—81no—
140C22⋊D7—81no—
141C22⋊D7—81no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147D28—24no—
148D28—21no—
149D28—21no—
150D28—21no—
151D28—81no—
152D28#4112yes—
153C4×D7—24no—
154C4×D7—21no—
155C4×D7—21no—
156C4×D7—81no—
157C4×D7—81no—
158C4×D7—81no—
159C4×D7—81no—
160C7×C8—21no—
161C7×C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C2×#41—71no—
165C2×SD16—71no—
166C2×SD16C2×D711yescharacteristic
167C2×D8—71no—
168C2×D8—71no—
169#127—71no—
170#125—71no—
171#123—141no—
172#123—141no—
173#121—71no—
174#121—71no—
175#121—141no—
176#121—141no—
177C4×Q8—71no—
178C4×Q8C2×D711yescharacteristic
179C4×D4—71no—
180C4×D4—71no—
181C4×D4—71no—
182C4×D4—141no—
183C4×D4—141no—
184C4×D4C2×D711yescharacteristic
185#117—71no—
186#117—141no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—141no—
189#107—71no—
190#107C2×D711yescharacteristic
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194C4⋊C8—71no—
195#103—71no—
196#103C2×D711yescharacteristic
197#102—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102C2×D711yescharacteristic
203C22⋊C8—71no—
204C22⋊C8—71no—
205#97C2×D711yescharacteristic
206C4×C8—71no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7C22×D411yescharacteristic
211C2×D7—21no—
212C2×D7—22no—
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—22no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7C2×#4111yescharacteristic
226C4×C7C2×#4111yescharacteristic
227Dic7—21no—
228Dic7—21no—
229Dic7—21no—
230Dic7—41no—
231Dic7—41no—
232#41—142no—
233#41—142no—
234C2×Q8C22×D711yescharacteristic
235C2×D4—71no—
236C2×D4—71no—
237C2×D4—141no—
238C2×D4—141no—
239C2×D4—141no—
240C2×D4—141no—
241C2×D4—141no—
242C2×D4C22×D711yescharacteristic
243C22×C4—21no—
244C22×C4—71no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—141no—
249C22×C4—141no—
250SD16—22no—
251SD16—142no—
252D8—142no—
253D8—142no—
254C2×C8—71no—
255C2×C8—71no—
256C2×C8C22×D711yescharacteristic
257C2×C8C22×D711yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—71no—
262C4⋊C4—71no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4C22×D711yescharacteristic
268C4⋊C4C22×D711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—71no—
271C22⋊C4—71no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C42—21no—
278C42—71no—
279C42—141no—
280C42C22×D711yescharacteristic
281C2×C7—81no—
282C2×C7#52511yescharacteristic
283C2×C7#57611yescharacteristic
284C2×C7#57711yescharacteristic
285D7—41no—
286D7—42no—
287D7—81no—
288C23—21no—
289C23—71no—
290C23—141no—
291C23—141no—
292Q8—21no—
293Q8#115712yes—
294D4—21no—
295D4—21no—
296D4—72no—
297D4—144no—
298D4—141no—
299D4—141no—
300D4—141no—
301D4—561no—
302D4—561no—
303D4—561no—
304D4—561no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—71no—
309C2×C4—71no—
310C2×C4—71no—
311C2×C4—81no—
312C2×C4—144no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—561no—
318C2×C4—561no—
319C2×C4—561no—
320C2×C4—561no—
321C2×C4D4×D711yescharacteristic
322C2×C4D4×D711yescharacteristic
323C2×C4C23×D711yescharacteristic
324C8—21no—
325C8—21no—
326C8—141no—
327C8—141no—
328C7#311911yescharacteristic, complemented, Sylow
329C22—41no—
330C22—41no—
331C22—41no—
332C22—141no—
333C22—141no—
334C22—142no—
335C22—141no—
336C22—281no—
337C22—281no—
338C22—281no—
339C22—281no—
340C22—281no—
341C22C2×D4×D711yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—141no—
348C4—141no—
349C4—141no—
350C4—281no—
351C4—281no—
352C4C2×#115711yescharacteristic
353C4D7×#4111yescharacteristic
354C2—81no—
355C2—281no—
356C2—282no—
357C2—561no—
358C2#9023811yescharacteristic
359C2#9046511yescharacteristic
360C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
hypercenter#440D7
hypocenterC7#3119
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #90265C2semidirect
2. #90253C2semidirect
3. C7×#440C2semidirect
4. #89944C2semidirect
5. #89935C2semidirect
6. C4×#1143C2semidirect
7. C4×#1141C2semidirect
8. #89663C2semidirect
9. #89654C2semidirect
10. #89583C2semidirect
11. #89571C2semidirect
12. #89558C2semidirect
13. #89549C2semidirect
14. #89483C2semidirect
15. #89481C2semidirect
120. #440D7semidirect
328. C7#3119semidirect

Maximal subgroups

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

subgroupindex
#902652
#902532
C7×#4402
#899442
#899352
C4×#11432
C4×#11412
#896632
#896542
#895832
#895712
#895582
#895492
#894832
#894812
#31197