Finite Group Databasebeta

← #11736342

Subgroups — #11736342

Order 896.

404 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#90525C211yesmaximal, characteristic, complemented
2#90205C211yesmaximal, characteristic, complemented
3C7×#417C211yesmaximal, characteristic, complemented
4#90010C211yesmaximal, characteristic, complemented
5#89907C211yesmaximal, characteristic, complemented
6#89899C211yesmaximal, characteristic, complemented
7#89739C211yesmaximal, characteristic, complemented
8#89589C211yesmaximal, characteristic, complemented
9#89583C211yesmaximal, characteristic, complemented
10#89576C211yesmaximal, characteristic, complemented
11#89571C211yesmaximal, characteristic, complemented
12#89558C211yesmaximal, characteristic, complemented
13#89551C211yesmaximal, characteristic, complemented
14#89547C211yesmaximal, characteristic, complemented
15#89536C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×#1161C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1143C2211yescharacteristic
22#6503—21no—
23#6503C2211yescharacteristic
24#6501—21no—
25#6501C2211yescharacteristic
26C2×#1141C2211yescharacteristic
27#6493—21no—
28#6493—21no—
29#6493C2211yescharacteristic
30#6493C2211yescharacteristic
31C4×C22⋊D7—21no—
32C4×C22⋊D7—21no—
33C4×C22⋊D7C2211yescharacteristic
34C2×D56C2211yescharacteristic
35C2×C8×D7C2211yescharacteristic
36#6461—21no—
37#6458C2211yescharacteristic
38#6457—21no—
39#6456C2211yescharacteristic
40#6455C2211yescharacteristic
41#6453C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52#6391C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3309—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—41no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72C2×C22⋊D7—21no—
73C2×C22⋊D7D411yescharacteristic
74#1161—22no—
75#1161—41no—
76C2×D28—21no—
77C2×D28—21no—
78C2×D28C2311yescharacteristic
79C2×D28C2311yescharacteristic
80C2×C4×D7—21no—
81C2×C4×D7—21no—
82C2×C4×D7C2311yescharacteristic
83C7×M16—41no—
84C2×C7×C8C2311yescharacteristic
85C2×C7×C8C2311yescharacteristic
86C7×C4⋊C4—21no—
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C22⋊Dic7—21no—
90C22⋊Dic7D411yescharacteristic
91#1143—22no—
92#1141—41no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4D411yescharacteristic
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
103Dic7⋊C4—21no—
104Dic7⋊C4D411yescharacteristic
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110D56—41no—
111C8×D7—41no—
112#553—71no—
113#550—71no—
114#490—71no—
115#482—71no—
116#480—71no—
117#479—71no—
118#469—71no—
119#466—71no—
120#459—71no—
121#449—71no—
122#449—71no—
123#440—71no—
124#435—71no—
125#417D711yescharacteristic, complemented
126#416—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C22×D7C2×D411yescharacteristic
132C7×Q8—21no—
133C7×Q8#4112yes—
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—42no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—42no—
143C2×C4×C7—81no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—41no—
147C22⋊D7—42no—
148C22⋊D7—42no—
149C22⋊D7—81no—
150C22⋊D7—81no—
151C22⋊D7—81no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7C2×D411yescharacteristic
155C2×Dic7C2×D411yescharacteristic
156C2×Dic7C2×D411yescharacteristic
157D28—21no—
158D28—21no—
159D28—21no—
160D28—21no—
161D28—42no—
162D28—81no—
163C4×D7—21no—
164C4×D7—21no—
165C4×D7—42no—
166C4×D7—81no—
167C7×C8—21no—
168C7×C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—71no—
172C2×#41C2×D711yescharacteristic
173C2×SD16—71no—
174C2×D8—71no—
175C2×D8—71no—
176C2×M16C2×D711yescharacteristic
177C22×C8—71no—
178#127—71no—
179#127—141no—
180#126—141no—
181#125—71no—
182#123—141no—
183#121—71no—
184#121—71no—
185#121—71no—
186#121—71no—
187#121—141no—
188#121—141no—
189#121—141no—
190C4×Q8—71no—
191C4×D4—71no—
192C4×D4—71no—
193C4×D4—141no—
194C4×D4—141no—
195#117—71no—
196C2×C4⋊C4C2×D711yescharacteristic
197#107—71no—
198#107—71no—
199#106—71no—
200C4⋊C8—71no—
201#103—71no—
202#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208#102—71no—
209#102C2×D711yescharacteristic
210#102C2×D711yescharacteristic
211C22⋊C8—71no—
212C22⋊C8—71no—
213#97—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7C22×D411yescharacteristic
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—21no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C4×C7—22no—
231C4×C7—21no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7C2×#4111yescharacteristic
235C4×C7H(5,𝔽2)11yescharacteristic
236Dic7—21no—
237Dic7—21no—
238Dic7—21no—
239Dic7—41no—
240Dic7—41no—
241#41—22no—
242#41—41no—
243#41—142no—
244#41—281no—
245C2×Q8C22×D711yescharacteristic
246C2×D4—21no—
247C2×D4—71no—
248C2×D4—71no—
249C2×D4—71no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4—141no—
256C2×D4—141no—
257C2×D4C22×D711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—71no—
261C22×C4—141no—
262C22×C4—141no—
263C22×C4C22×D711yescharacteristic
264SD16—142no—
265D8—281no—
266D8—281no—
267M16—41no—
268C2×C8—71no—
269C2×C8—71no—
270C2×C8—281no—
271C2×C8C22×D711yescharacteristic
272C2×C8C22×D711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—71no—
275C4⋊C4—71no—
276C4⋊C4—71no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4—141no—
280C4⋊C4C22×D711yescharacteristic
281C4⋊C4C22×D711yescharacteristic
282C22⋊C4—71no—
283C22⋊C4—71no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C42—71no—
291C42—141no—
292C42—141no—
293C2×C7—41no—
294C2×C7—81no—
295C2×C7#54611yescharacteristic
296C2×C7#57511yescharacteristic
297C2×C7#57611yescharacteristic
298D7—41no—
299D7—81no—
300D7—81no—
301C23—21no—
302C23—71no—
303C23—141no—
304C23—141no—
305C23D4×D711yescharacteristic
306Q8—21no—
307Q8#115712yes—
308D4—21no—
309D4—21no—
310D4—42no—
311D4—141no—
312D4—141no—
313D4—141no—
314D4—141no—
315D4—282no—
316D4—281no—
317D4—282no—
318D4—282no—
319D4—561no—
320D4—561no—
321D4—561no—
322D4—561no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—71no—
330C2×C4—71no—
331C2×C4—71no—
332C2×C4—81no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—282no—
338C2×C4—561no—
339C2×C4D4×D711yescharacteristic
340C2×C4C23×D711yescharacteristic
341C8—21no—
342C8—21no—
343C8—141no—
344C8—141no—
345C7#330911yescharacteristic, complemented, Sylow
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—141no—
353C22—141no—
354C22—141no—
355C22—281no—
356C22—281no—
357C22—281no—
358C22—281no—
359C22—281no—
360C22—281no—
361C22C2×D4×D711yescharacteristic
362C4—21no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—141no—
367C4—141no—
368C4—141no—
369C4—281no—
370C4—281no—
371C4C2×#115711yescharacteristic
372C4#654811yescharacteristic
373C2—41no—
374C2—81no—
375C2—281no—
376C2—561no—
377C2—561no—
378C2#9018111yescharacteristic
379C2#9046511yescharacteristic
380C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
hypercenter#417D7
hypocenterC7#3309
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #90525C2semidirect
2. #90205C2semidirect
3. C7×#417C2semidirect
4. #90010C2semidirect
5. #89907C2semidirect
6. #89899C2semidirect
7. #89739C2semidirect
8. #89589C2semidirect
9. #89583C2semidirect
10. #89576C2semidirect
11. #89571C2semidirect
12. #89558C2semidirect
13. #89551C2semidirect
14. #89547C2semidirect
15. #89536C2semidirect
125. #417D7semidirect
345. C7#3309semidirect

Maximal subgroups

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

subgroupindex
#905252
#902052
C7×#4172
#900102
#899072
#898992
#897392
#895892
#895832
#895762
#895712
#895582
#895512
#895472
#895362
#33097