Finite Group Databasebeta

← #11737682

Subgroups — #11737682

Order 896.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90524C211yesmaximal, characteristic
2#90319C211yesmaximal, characteristic, complemented
3C7×#452C211yesmaximal, characteristic, complemented
4#90011C211yesmaximal, characteristic, complemented
5#89961C211yesmaximal, characteristic, complemented
6#89958C211yesmaximal, characteristic, complemented
7#89943C211yesmaximal, characteristic, complemented
8#89942C211yesmaximal, characteristic, complemented
9#89816C211yesmaximal, characteristic, complemented
10#89587C211yesmaximal, characteristic, complemented
11#89585C211yesmaximal, characteristic, complemented
12#89552C211yesmaximal, characteristic, complemented
13#89551C211yesmaximal, characteristic, complemented
14#89499C211yesmaximal, characteristic, complemented
15#89493C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×#1157C2211yescharacteristic
19C2×C7×Q16C2211yescharacteristic
20C2×C7×SD16C2211yescharacteristic
21C7×C22⋊Q8C2211yescharacteristic
22#6509C2211yescharacteristic
23#6509C2211yescharacteristic
24#6509C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26#6502—21no—
27#6502—21no—
28#6501C2211yescharacteristic
29#6501C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31#6493—21no—
32C2×C4⋊Dic7C2211yescharacteristic
33#6487—21no—
34#6487C2211yescharacteristic
35C2×#1131C2211yescharacteristic
36C2×C8×D7C2211yescharacteristic
37#6460C2211yescharacteristic
38#6457—21no—
39#6455C2211yescharacteristic
40#6451—21no—
41#6448—21no—
42#6446—21no—
43C7×#103C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×C22⋊C8C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6406C2211yescharacteristic
50C22⋊(C7⋊C8)C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52(C2×D7)⋊C8C2211yescharacteristic
53#6391C2211yescharacteristic
54#6391C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3201—71nomaximal, Sylow
60C7×#41—41no—
61C7×#41—41no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7D411yescharacteristic
71C2×C22⋊D7D411yescharacteristic
72C22×Dic7—21no—
73Q8×D7—41no—
74Q8×D7—41no—
75#1157—41no—
76#1157—41no—
77C2×D28D411yescharacteristic
78C2×C4×D7—21no—
79C2×C4×D7C2311yescharacteristic
80C2×C4×D7C2311yescharacteristic
81C2×Dic14—21no—
82C2×Dic14D411yescharacteristic
83C2×Dic14C2311yescharacteristic
84C7×Q16—41no—
85C7×SD16—41no—
86C2×C7×C8C2311yescharacteristic
87C2×C7×C8C2311yescharacteristic
88C7×C4⋊C4—21no—
89C7×C4⋊C4C2311yescharacteristic
90C7×C22⋊C4—21no—
91C22⋊Dic7—21no—
92C22⋊Dic7—21no—
93C22⋊Dic7—21no—
94#1144—41no—
95#1142—41no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4—21no—
101C4⋊Dic7—21no—
102C4⋊Dic7—21no—
103C4⋊Dic7C2311yescharacteristic
104C4⋊Dic7C2311yescharacteristic
105C4⋊Dic7C2311yescharacteristic
106Dic7⋊C4—21no—
107Dic7⋊C4—21no—
108Dic7⋊C4—21no—
109Dic7⋊C4—21no—
110C4×Dic7—21no—
111C2×C7⋊C8C2311yescharacteristic
112C2×C7⋊C8C2311yescharacteristic
113#1131—41no—
114C8×D7—41no—
115#537—71no—
116#536—71no—
117#485—71no—
118#484—71no—
119#483—71no—
120#482—71no—
121#470—71no—
122#468—71no—
123#467—71no—
124#465—71no—
125#452—71no—
126#452D711yescharacteristic, complemented
127#451—71no—
128#448—71no—
129#408—71no—
130C23×C7—21no—
131C23×C7C2×D411yescharacteristic
132C22×D7C2×D411yescharacteristic
133C22×D7C2×D411yescharacteristic
134C7×Q8—21no—
135C7×Q8—21no—
136C7×Q8—21no—
137C7×Q8—21no—
138C7×D4—21no—
139C7×D4—21no—
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—42no—
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2411yescharacteristic
149C22⋊D7—41no—
150C22⋊D7—41no—
151C22⋊D7—81no—
152C22⋊D7—81no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—81no—
158C2×Dic7C2×D411yescharacteristic
159C2×Dic7C2×D411yescharacteristic
160D28—41no—
161C4×D7—21no—
162C4×D7—21no—
163C4×D7—21no—
164C4×D7—21no—
165C4×D7—42no—
166Dic14—21no—
167Dic14—21no—
168Dic14—41no—
169Dic14—42no—
170C7×C8—21no—
171C7×C8—21no—
172C7⋊C8—21no—
173C7⋊C8—21no—
174C2×#41—71no—
175C2×#41C2×D711yescharacteristic
176C22×Q8—71no—
177C2×Q16—71no—
178C2×Q16C2×D711yescharacteristic
179C2×SD16—71no—
180C2×SD16C2×D711yescharacteristic
181C2×M16—71no—
182C22×C8—71no—
183C4⋊Q8—141no—
184#124—141no—
185#123—141no—
186#123—141no—
187C22⋊Q8—71no—
188C22⋊Q8—71no—
189C22⋊Q8—71no—
190C22⋊Q8—71no—
191C22⋊Q8—71no—
192C22⋊Q8—141no—
193C22⋊Q8C2×D711yescharacteristic
194#121—71no—
195#121—71no—
196#121—141no—
197#121—141no—
198#121—141no—
199#117—71no—
200C2×C4⋊C4—71no—
201#107—71no—
202#107—71no—
203#106—71no—
204#106—71no—
205#103—71no—
206#103—71no—
207#103—71no—
208#103—71no—
209#103—71no—
210#103C2×D711yescharacteristic
211#102—71no—
212#102C2×D711yescharacteristic
213C22⋊C8—71no—
214C22⋊C8—71no—
215C22⋊C8—71no—
216C22⋊C8C2×D711yescharacteristic
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7C22×D411yescharacteristic
224C2×D7—21no—
225C2×D7—21no—
226C2×D7—21no—
227C2×D7—21no—
228C2×D7—21no—
229C2×D7—21no—
230C4×C7—21no—
231C4×C7—41no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7H(5,𝔽2)11yescharacteristic
235C4×C7H(5,𝔽2)11yescharacteristic
236Dic7—21no—
237Dic7—21no—
238Dic7—41no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242#41—41no—
243#41—41no—
244#41—281no—
245#41—281no—
246C2×Q8—71no—
247C2×Q8—71no—
248C2×Q8—141no—
249C2×Q8—281no—
250C2×Q8—281no—
251C2×Q8C22×D711yescharacteristic
252C2×Q8C22×D711yescharacteristic
253C2×D4—21no—
254C2×D4—71no—
255C2×D4—71no—
256C2×D4—71no—
257C2×D4—141no—
258C2×D4—141no—
259C2×D4C22×D711yescharacteristic
260C22×C4—21no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—141no—
264C22×C4—141no—
265C22×C4C22×D711yescharacteristic
266Q16—41no—
267Q16—281no—
268SD16—41no—
269SD16—281no—
270M16—281no—
271C2×C8—71no—
272C2×C8—71no—
273C2×C8—281no—
274C2×C8C22×D711yescharacteristic
275C2×C8C22×D711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—71no—
278C4⋊C4—71no—
279C4⋊C4—71no—
280C4⋊C4—141no—
281C4⋊C4—141no—
282C4⋊C4—141no—
283C4⋊C4—141no—
284C4⋊C4—141no—
285C4⋊C4—141no—
286C4⋊C4C22×D711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C22⋊C4—141no—
292C22⋊C4—141no—
293C22⋊C4—141no—
294C22⋊C4—141no—
295C22⋊C4—141no—
296C42—141no—
297C2×C7—41no—
298C2×C7—81no—
299C2×C7#53411yescharacteristic
300C2×C7#57711yescharacteristic
301C2×C7#57811yescharacteristic
302D7—41no—
303D7—41no—
304C23—21no—
305C23—71no—
306C23—71no—
307C23D4×D711yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—141no—
313Q8—141no—
314Q8—281no—
315Q8—282no—
316D4—21no—
317D4—21no—
318D4—42no—
319D4—281no—
320D4—281no—
321D4—281no—
322D4—561no—
323D4—561no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—71no—
331C2×C4—71no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—282no—
341C2×C4—561no—
342C2×C4D4×D711yescharacteristic
343C2×C4C23×D711yescharacteristic
344C8—21no—
345C8—21no—
346C8—141no—
347C8—141no—
348C7#320111yescharacteristic, complemented, Sylow
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—141no—
356C22—141no—
357C22—141no—
358C22—141no—
359C22—141no—
360C22—141no—
361C22C2×D4×D711yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—141no—
367C4—141no—
368C4—281no—
369C4—281no—
370C4—281no—
371C4—281no—
372C4#654811yescharacteristic
373C4#654811yescharacteristic
374C2—41no—
375C2—81no—
376C2—281no—
377C2—281no—
378C2#9028011yescharacteristic
379C2#9046111yescharacteristic
380C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#452C2
socleC22×C7C22×D4
prime radical#90524C2
hypercenter#452D7
hypocenterC7#3201
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #90319C2semidirect
3. C7×#452C2semidirect
4. #90011C2semidirect
5. #89961C2semidirect
6. #89958C2semidirect
7. #89943C2semidirect
8. #89942C2semidirect
9. #89816C2semidirect
10. #89587C2semidirect
11. #89585C2semidirect
12. #89552C2semidirect
13. #89551C2semidirect
14. #89499C2semidirect
15. #89493C2semidirect
126. #452D7semidirect
348. C7#3201semidirect

Maximal subgroups

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

subgroupindex
#905242
#903192
C7×#4522
#900112
#899612
#899582
#899432
#899422
#898162
#895872
#895852
#895522
#895512
#894992
#894932
#32017