Finite Group Databasebeta

← #11739000

Subgroups — #11739000

Order 896.

386 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#90323C211yesmaximal, characteristic, complemented
2#90223C211yesmaximal, characteristic
3C7×#485C211yesmaximal, characteristic, complemented
4#89821C211yesmaximal, characteristic, complemented
5#89776C211yesmaximal, characteristic, complemented
6#89658C211yesmaximal, characteristic, complemented
7#89647C211yesmaximal, characteristic, complemented
8#89639C211yesmaximal, characteristic, complemented
9#89624C211yesmaximal, characteristic, complemented
10#89590C211yesmaximal, characteristic, complemented
11#89578C211yesmaximal, characteristic, complemented
12#89574C211yesmaximal, characteristic, complemented
13#89564C211yesmaximal, characteristic, complemented
14#89510C211yesmaximal, characteristic, complemented
15#89501C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C7×#117C2211yescharacteristic
20#6507C2211yescharacteristic
21C2×#1143C2211yescharacteristic
22#6493C2211yescharacteristic
23C4×C22⋊D7C2211yescharacteristic
24C2×#1136C2211yescharacteristic
25C2×#1132C2211yescharacteristic
26C2×#1132C2211yescharacteristic
27C2×#1131C2211yescharacteristic
28#6460C2211yescharacteristic
29#6459—21no—
30#6458C2211yescharacteristic
31#6456C2211yescharacteristic
32#6455C2211yescharacteristic
33#6451—21no—
34#6451C2211yescharacteristic
35#6450C2211yescharacteristic
36#6448—21no—
37#6447—21no—
38(C22⋊D7)⋊C4—21no—
39#6441—21no—
40#6438—21no—
41#6435—21no—
42C7×#107C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46C7×C22⋊C8C2211yescharacteristic
47#6409C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6390C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6383C2211yescharacteristic
58#6381C2211yescharacteristic
59#3329—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—21no—
63C2×C22⋊D7D411yescharacteristic
64C22×Dic7—21no—
65Q8×D7—22no—
66Q8×D7—41no—
67C2×D28—21no—
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7C2311yescharacteristic
72C2×Dic14—22no—
73C2×Dic14—21no—
74C2×Dic14—21no—
75C2×Dic14—21no—
76C2×Dic14—41no—
77C2×Dic14C2311yescharacteristic
78C2×Dic14C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C42×C7—21no—
88C22⋊Dic7D411yescharacteristic
89#1143—41no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4D411yescharacteristic
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106#1136—41no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#1132—22no—
110#1132—41no—
111#1131—41no—
112#555—71no—
113D4×Q8—71no—
114#487—71no—
115#485—71no—
116#485D711yescharacteristic, complemented
117#477—71no—
118#468—71no—
119#468—71no—
120#461—71no—
121#448—71no—
122#448—71no—
123C4×SD16—71no—
124#436—71no—
125#419—71no—
126#419—71no—
127C23×C7C2×D411yescharacteristic
128C22×D7—21no—
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C22⋊D7—22no—
141C22⋊D7—81no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—42no—
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149D28—21no—
150D28—21no—
151D28—81no—
152C4×D7—21no—
153C4×D7—21no—
154C4×D7—42no—
155C4×D7—42no—
156Dic14—21no—
157Dic14—21no—
158Dic14—21no—
159Dic14—42no—
160Dic14—42no—
161Dic14—81no—
162Dic14#4112yes—
163C7×C8—21no—
164C7×C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C22×Q8—71no—
168C22×Q8—71no—
169C2×SD16—71no—
170C2×SD16—71no—
171C2×SD16—71no—
172C2×M16—71no—
173C2×M16—71no—
174C4⋊Q8—71no—
175C4⋊Q8—71no—
176C4⋊Q8—141no—
177#124—141no—
178#124—141no—
179C22⋊Q8—71no—
180C22⋊Q8—141no—
181C22⋊Q8—141no—
182C22⋊Q8C2×D711yescharacteristic
183#121—71no—
184#121—71no—
185#121—141no—
186C4×Q8—71no—
187C4×D4—71no—
188C4×D4—71no—
189C4×D4—141no—
190#117—71no—
191#117—141no—
192#117C2×D711yescharacteristic
193#107—71no—
194#107C2×D711yescharacteristic
195#106C2×D711yescharacteristic
196C4⋊C8—71no—
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⋊C8C2×D711yescharacteristic
209C4×C8—71no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7C22×D411yescharacteristic
214C2×D7—21no—
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—21no—
221C4×C7—41no—
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#13712yes—
233C2×Q8—71no—
234C2×Q8—71no—
235C2×Q8—141no—
236C2×Q8—141no—
237C2×Q8—142no—
238C2×Q8—141no—
239C2×Q8—142no—
240C2×Q8—281no—
241C2×Q8—281no—
242C2×Q8C22×D711yescharacteristic
243C2×D4—71no—
244C2×D4—71no—
245C2×D4—141no—
246C2×D4—141no—
247C22×C4—71no—
248C22×C4—141no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4C22×D711yescharacteristic
252SD16—142no—
253SD16—281no—
254SD16—281no—
255M16—281no—
256M16—281no—
257C2×C8—71no—
258C2×C8—71no—
259C2×C8C22×D711yescharacteristic
260C2×C8C22×D711yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—71no—
263C4⋊C4—71no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4C22×D711yescharacteristic
270C4⋊C4C22×D711yescharacteristic
271C4⋊C4C22×D711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—71no—
275C22⋊C4—71no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C42—21no—
282C42—71no—
283C42—141no—
284C42—141no—
285C2×C7—41no—
286C2×C7#54611yescharacteristic
287C2×C7C2×#13711yescharacteristic
288C2×C7#57711yescharacteristic
289D7—41no—
290D7—81no—
291C23—71no—
292C23—141no—
293C23D4×D711yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—72no—
297Q8—141no—
298Q8—141no—
299Q8—141no—
300Q8—282no—
301Q8—282no—
302Q8—561no—
303D4—142no—
304D4—141no—
305D4—141no—
306D4—561no—
307D4—561no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—71no—
315C2×C4—71no—
316C2×C4—71no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—282no—
323C2×C4—282no—
324C2×C4—282no—
325C2×C4D4×D711yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—21no—
328C8—21no—
329C8—141no—
330C8—141no—
331C7#332911yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—21no—
335C22—141no—
336C22—141no—
337C22—141no—
338C22—281no—
339C22—281no—
340C22—281no—
341C22C2×D4×D711yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—72no—
348C4—141no—
349C4—141no—
350C4—142no—
351C4—281no—
352C4—281no—
353C4D7×#4111yescharacteristic
354C4#655311yescharacteristic
355C2—41no—
356C2—281no—
357C2—561no—
358C2#9034511yescharacteristic
359C2#9046111yescharacteristic
360C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#485C2
socleC22×C7C22×D4
prime radical#90223C2
hypercenter#485D7
hypocenterC7#3329
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #90323C2semidirect
3. C7×#485C2semidirect
4. #89821C2semidirect
5. #89776C2semidirect
6. #89658C2semidirect
7. #89647C2semidirect
8. #89639C2semidirect
9. #89624C2semidirect
10. #89590C2semidirect
11. #89578C2semidirect
12. #89574C2semidirect
13. #89564C2semidirect
14. #89510C2semidirect
15. #89501C2semidirect
116. #485D7semidirect
331. C7#3329semidirect

Maximal subgroups

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

subgroupindex
#903232
#902232
C7×#4852
#898212
#897762
#896582
#896472
#896392
#896242
#895902
#895782
#895742
#895642
#895102
#895012
#33297