Finite Group Databasebeta

← #11737898

Subgroups — #11737898

Order 896.

388 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#90418C211yesmaximal, characteristic, complemented
2#90249C211yesmaximal, characteristic
3C7×#461C211yesmaximal, characteristic, complemented
4#89946C211yesmaximal, characteristic, complemented
5#89943C211yesmaximal, characteristic, complemented
6#89942C211yesmaximal, characteristic, complemented
7#89942C211yesmaximal, characteristic, complemented
8#89854C211yesmaximal, characteristic, complemented
9#89790C211yesmaximal, characteristic, complemented
10#89608C211yesmaximal, characteristic, complemented
11#89601C211yesmaximal, characteristic, complemented
12#89587C211yesmaximal, characteristic, complemented
13#89581C211yesmaximal, characteristic, complemented
14#89574C211yesmaximal, characteristic, complemented
15D7×#103C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C7×C4⋊Q8C2211yescharacteristic
21C4×C7×D4C2211yescharacteristic
22#6509C2211yescharacteristic
23#6509C2211yescharacteristic
24#6501C2211yescharacteristic
25#6501C2211yescharacteristic
26C2×#1142C2211yescharacteristic
27#6492—21no—
28C4×C22⋊D7—21no—
29C2×#1131C2211yescharacteristic
30C2×C8×D7C2211yescharacteristic
31#6459—21no—
32#6459—21no—
33#6456—21no—
34#6455C2211yescharacteristic
35D7×C4⋊C4C2211yescharacteristic
36#6451—21no—
37#6447—21no—
38#6442—21no—
39C4×D28C2211yescharacteristic
40#6433C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6406C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52#6392C2211yescharacteristic
53#6391C2211yescharacteristic
54#6390C2211yescharacteristic
55#6390C2211yescharacteristic
56#6384C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3320—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66Q8×D7—22no—
67Q8×D7—22no—
68Q8×D7—41no—
69Q8×D7—41no—
70C2×D28D411yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×C4×D7C2311yescharacteristic
75C2×C4×D7C2311yescharacteristic
76C2×Dic14—21no—
77C2×Dic14—21no—
78C2×Dic14C2311yescharacteristic
79C7×SD16—41no—
80C7×SD16—41no—
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4—21no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4C2311yescharacteristic
86C7×C22⋊C4—21no—
87C42×C7C2311yescharacteristic
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90#1142—22no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4D411yescharacteristic
95(C2×D7)⋊C4D411yescharacteristic
96C4⋊Dic7D411yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1131—41no—
111C8×D7—41no—
112#553—71no—
113D4×Q8—71no—
114#491—71no—
115#485—71no—
116#484—71no—
117#479—71no—
118#468—71no—
119#465—71no—
120#461D711yescharacteristic, complemented
121#448—71no—
122#448—71no—
123#440—71no—
124#435—71no—
125#419—71no—
126C2×#103—71no—
127C23×C7—21no—
128C22×D7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×D4—21no—
135C7×D4—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—81no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2411yescharacteristic
143C22⋊D7—42no—
144C22⋊D7—81no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151D28—22no—
152C4×D7—21no—
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—21no—
156C4×D7—42no—
157C4×D7—42no—
158C4×D7—81no—
159Dic14—21no—
160Dic14—42no—
161Dic14—42no—
162Dic14#4112yes—
163C7×C8—21no—
164C7×C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C22×Q8—71no—
168C22×Q8—71no—
169C2×SD16—71no—
170C2×SD16C2×D711yescharacteristic
171C2×SD16C2×D711yescharacteristic
172C2×M16—71no—
173C22×C8—71no—
174C4⋊Q8—141no—
175C4⋊Q8C2×D711yescharacteristic
176#126—141no—
177#125—71no—
178#123—141no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183#121—71no—
184#121—71no—
185#121—141no—
186C4×Q8—71no—
187C4×D4—71no—
188C4×D4—141no—
189C4×D4—141no—
190C4×D4C2×D711yescharacteristic
191#117—71no—
192C2×C4⋊C4—71no—
193#107—71no—
194#107—71no—
195#106—71no—
196C4⋊C8C2×D711yescharacteristic
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#103C2×D711yescharacteristic
205#103C2×D711yescharacteristic
206#102—71no—
207C22⋊C8—71no—
208C22⋊C8—71no—
209#97—71no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7C22×D411yescharacteristic
214C2×D7—21no—
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—21no—
218C2×D7—21no—
219C2×D7#13712yes—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7C2×#4111yescharacteristic
226C4×C7H(5,𝔽2)11yescharacteristic
227Dic7—22no—
228Dic7—21no—
229Dic7—21no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233C2×Q8—71no—
234C2×Q8—141no—
235C2×Q8—141no—
236C2×Q8—142no—
237C2×Q8—142no—
238C2×Q8—281no—
239C2×Q8—281no—
240C2×Q8C22×D711yescharacteristic
241C2×Q8C22×D711yescharacteristic
242C2×D4—71no—
243C2×D4—141no—
244C2×D4—141no—
245C2×D4C22×D711yescharacteristic
246C22×C4—21no—
247C22×C4—71no—
248C22×C4—71no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4—141no—
252SD16—41no—
253SD16—41no—
254SD16—142no—
255M16—281no—
256C2×C8—71no—
257C2×C8—71no—
258C2×C8—281no—
259C2×C8C22×D711yescharacteristic
260C2×C8C22×D711yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—71no—
264C4⋊C4—71no—
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4C22×D711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—71no—
276C22⋊C4—71no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C42—141no—
283C42—141no—
284C42C22×D711yescharacteristic
285C2×C7—81no—
286C2×C7#54611yescharacteristic
287C2×C7C2×#13711yescharacteristic
288C2×C7#57711yescharacteristic
289D7—22no—
290D7—41no—
291C23—21no—
292C23—71no—
293C23—71no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—72no—
299Q8—141no—
300Q8—282no—
301Q8—282no—
302D4—21no—
303D4—21no—
304D4—142no—
305D4—282no—
306D4—561no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—81no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—282no—
322C2×C4—282no—
323C2×C4—561no—
324C2×C4D4×D711yescharacteristic
325C2×C4D4×D711yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—21no—
328C8—21no—
329C8—141no—
330C8—141no—
331C7#332011yescharacteristic, complemented, Sylow
332C22—41no—
333C22—41no—
334C22—41no—
335C22—72no—
336C22—141no—
337C22—141no—
338C22—141no—
339C22—141no—
340C22—141no—
341C22C2×D4×D711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—141no—
348C4—141no—
349C4—142no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4#654811yescharacteristic
354C4D7×#4111yescharacteristic
355C2—81no—
356C2—142no—
357C2—281no—
358C2#9030311yescharacteristic
359C2#9045311yescharacteristic
360C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#461C2
socleC22×C7C22×D4
prime radical#90249C2
hypercenter#461D7
hypocenterC7#3320
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #90418C2semidirect
3. C7×#461C2semidirect
4. #89946C2semidirect
5. #89943C2semidirect
6. #89942C2semidirect
7. #89942C2semidirect
8. #89854C2semidirect
9. #89790C2semidirect
10. #89608C2semidirect
11. #89601C2semidirect
12. #89587C2semidirect
13. #89581C2semidirect
14. #89574C2semidirect
15. D7×#103C2semidirect
120. #461D7semidirect
331. C7#3320semidirect

Maximal subgroups

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

subgroupindex
#904182
#902492
C7×#4612
#899462
#899432
#899422
#898542
#897902
#896082
#896012
#895872
#895812
#895742
D7×#1032
#33207