Finite Group Databasebeta

← #11736349

Subgroups — #11736349

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
1#90522C211yesmaximal, characteristic, complemented
2#90205C211yesmaximal, characteristic, complemented
3C7×#417C211yesmaximal, characteristic, complemented
4#90010C211yesmaximal, characteristic, complemented
5#89906C211yesmaximal, characteristic, complemented
6#89900C211yesmaximal, characteristic, complemented
7#89743C211yesmaximal, characteristic, complemented
8#89578C211yesmaximal, characteristic, complemented
9#89574C211yesmaximal, characteristic, complemented
10#89568C211yesmaximal, characteristic, complemented
11#89560C211yesmaximal, characteristic, complemented
12#89559C211yesmaximal, characteristic, complemented
13#89556C211yesmaximal, characteristic, complemented
14#89554C211yesmaximal, characteristic, complemented
15#89540C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1143C2211yescharacteristic
22#6501—21no—
23#6501C2211yescharacteristic
24#6499—21no—
25#6499C2211yescharacteristic
26C2×#1141C2211yescharacteristic
27#6493—21no—
28#6493C2211yescharacteristic
29C4×C22⋊D7—21no—
30C4×C22⋊D7—21no—
31C4×C22⋊D7C2211yescharacteristic
32#6487—21no—
33#6487C2211yescharacteristic
34C2×#1132C2211yescharacteristic
35C2×#1131C2211yescharacteristic
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(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic
52#6390C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54C8×Dic7C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6382C2211yescharacteristic
58#6382C2211yescharacteristic
59#3326—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⋊D7D411yescharacteristic
71Q8×D7—22no—
72Q8×D7—41no—
73C2×D28—21no—
74C2×D28C2311yescharacteristic
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7C2311yescharacteristic
78C2×Dic14—21no—
79C2×Dic14C2311yescharacteristic
80C7×M16—41no—
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4—21no—
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
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)⋊C4D411yescharacteristic
98C4⋊Dic7—21no—
99C4⋊Dic7—21no—
100C4⋊Dic7C2311yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4D411yescharacteristic
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1132—41no—
111#1131—41no—
112#553—71no—
113#548—71no—
114#488—71no—
115#483—71no—
116#482—71no—
117#479—71no—
118#469—71no—
119#468—71no—
120#463—71no—
121#449—71no—
122#448—71no—
123C4×SD16—71no—
124#436—71no—
125#419—71no—
126#417D711yescharacteristic, complemented
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7C2×D411yescharacteristic
131C7×Q8—21no—
132C7×Q8#4112yes—
133C7×D4—21no—
134C7×D4—21no—
135C7×D4—42no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—42no—
142C2×C4×C7—81no—
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2411yescharacteristic
145C22⋊D7—41no—
146C22⋊D7—42no—
147C22⋊D7—42no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154D28—21no—
155D28—21no—
156D28—81no—
157C4×D7—21no—
158C4×D7—21no—
159C4×D7—42no—
160C4×D7—81no—
161Dic14—21no—
162Dic14—21no—
163Dic14—42no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41C2×D711yescharacteristic
169C22×Q8—71no—
170C2×SD16—71no—
171C2×SD16—71no—
172C2×D8—71no—
173C2×M16—71no—
174C2×M16C2×D711yescharacteristic
175#126—141no—
176#125—71no—
177#124—71no—
178#124—141no—
179#123—141no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182#121—71no—
183#121—71no—
184#121—71no—
185#121—141no—
186#121—141no—
187C4×Q8—71no—
188C4×D4—71no—
189C4×D4—71no—
190C4×D4—141no—
191C4×D4—141no—
192#117—71no—
193C2×C4⋊C4C2×D711yescharacteristic
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#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204#102—71no—
205#102—71no—
206#102C2×D711yescharacteristic
207#102C2×D711yescharacteristic
208C22⋊C8—71no—
209C22⋊C8—71no—
210C4×C8—71no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7C22×D411yescharacteristic
218C2×D7—21no—
219C2×D7—21no—
220C2×D7—21no—
221C2×D7—41no—
222C2×D7—41no—
223C2×D7—41no—
224C4×C7—22no—
225C4×C7—21no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7C2×#4111yescharacteristic
229C4×C7H(5,𝔽2)11yescharacteristic
230Dic7—21no—
231Dic7—21no—
232Dic7—21no—
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236#41—22no—
237#41—41no—
238C2×Q8—71no—
239C2×Q8—141no—
240C2×Q8—142no—
241C2×Q8—281no—
242C2×Q8C22×D711yescharacteristic
243C2×D4—21no—
244C2×D4—71no—
245C2×D4—71no—
246C2×D4—141no—
247C2×D4—141no—
248C2×D4—141no—
249C2×D4C22×D711yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—71no—
253C22×C4—141no—
254C22×C4—141no—
255C22×C4C22×D711yescharacteristic
256SD16—142no—
257SD16—281no—
258D8—281no—
259M16—41no—
260M16—281no—
261C2×C8—71no—
262C2×C8—71no—
263C2×C8C22×D711yescharacteristic
264C2×C8C22×D711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—71no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4C22×D711yescharacteristic
274C4⋊C4C22×D711yescharacteristic
275C22⋊C4—71no—
276C22⋊C4—71no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C42—71no—
286C42—141no—
287C42—141no—
288C2×C7—41no—
289C2×C7—81no—
290C2×C7#54611yescharacteristic
291C2×C7#57511yescharacteristic
292C2×C7#57711yescharacteristic
293D7—41no—
294D7—81no—
295C23—21no—
296C23—71no—
297C23—141no—
298C23D4×D711yescharacteristic
299Q8—21no—
300Q8—141no—
301Q8—141no—
302Q8—282no—
303Q8#115712yes—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—141no—
308D4—141no—
309D4—281no—
310D4—282no—
311D4—282no—
312D4—561no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—71no—
320C2×C4—71no—
321C2×C4—71no—
322C2×C4—81no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—282no—
329C2×C4—561no—
330C2×C4D4×D711yescharacteristic
331C2×C4C23×D711yescharacteristic
332C8—21no—
333C8—21no—
334C8—141no—
335C8—141no—
336C7#332611yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—141no—
344C22—141no—
345C22—141no—
346C22—281no—
347C22—281no—
348C22—281no—
349C22C2×D4×D711yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—141no—
355C4—141no—
356C4—141no—
357C4—281no—
358C4—281no—
359C4—281no—
360C4C2×#115711yescharacteristic
361C4#654811yescharacteristic
362C2—41no—
363C2—81no—
364C2—281no—
365C2—561no—
366C2#9018111yescharacteristic
367C2#9046611yescharacteristic
368C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
hypercenter#417D7
hypocenterC7#3326
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. #90522C2semidirect
2. #90205C2semidirect
3. C7×#417C2semidirect
4. #90010C2semidirect
5. #89906C2semidirect
6. #89900C2semidirect
7. #89743C2semidirect
8. #89578C2semidirect
9. #89574C2semidirect
10. #89568C2semidirect
11. #89560C2semidirect
12. #89559C2semidirect
13. #89556C2semidirect
14. #89554C2semidirect
15. #89540C2semidirect
126. #417D7semidirect
336. C7#3326semidirect

Maximal subgroups

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

subgroupindex
#905222
#902052
C7×#4172
#900102
#899062
#899002
#897432
#895782
#895742
#895682
#895602
#895592
#895562
#895542
#895402
#33267