Finite Group Databasebeta

← #11736956

Subgroups — #11736956

Order 896.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90232C211yesmaximal, characteristic
2D4×Dic14C211yesmaximal, characteristic
3C7×#435C211yesmaximal, characteristic
4#89909C211yesmaximal, characteristic, complemented
5#89892C211yesmaximal, characteristic, complemented
6#89888C211yesmaximal, characteristic, complemented
7C2×#6390C211yesmaximal, characteristic, complemented
8#89621C211yesmaximal, characteristic, complemented
9#89601C211yesmaximal, characteristic, complemented
10#89511C211yesmaximal, characteristic, complemented
11#89511C211yesmaximal, characteristic, complemented
12#89495C211yesmaximal, characteristic, complemented
13#89493C211yesmaximal, characteristic, complemented
14#89488C211yesmaximal, characteristic, complemented
15#89475C211yesmaximal, characteristic, complemented
16C22×Dic14C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C22×C7×C8C2211yescharacteristic
20C4×C7×D4C2211yescharacteristic
21#6498—21no—
22D4×Dic7—21no—
23#6489C2211yescharacteristic
24C2×C4⋊Dic7C2211yescharacteristic
25#6487C2211yescharacteristic
26#6487C2211yescharacteristic
27#6487C2211yescharacteristic
28#6487C2211yescharacteristic
29C2×Dic28C2211yescharacteristic
30C2×Dic28C2211yescharacteristic
31C2×Dic28C2211yescharacteristic
32#6452—21no—
33#6451—21no—
34#6442—21no—
35#6441—21no—
36#6441—21no—
37#6440—21no—
38#6433C2211yescharacteristic
39#6432C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45C7×#97C2211yescharacteristic
46#6392C2211yescharacteristic
47#6391C2211yescharacteristic
48#6391C2211yescharacteristic
49#6390C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3312—71nomaximal, Sylow
60C2×C7×D4D411yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×Dic7—21no—
64C22×Dic7—21no—
65C22×Dic7—21no—
66C2×Dic14—22no—
67C2×Dic14—22no—
68C2×Dic14—21no—
69C2×Dic14—21no—
70C2×Dic14—41no—
71C2×Dic14—41no—
72C2×Dic14C2311yescharacteristic
73C2×Dic14C2311yescharacteristic
74C2×Dic14C2311yescharacteristic
75C2×Dic14C2311yescharacteristic
76C7×M16—41no—
77C2×C7×C8—41no—
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4D411yescharacteristic
83C7×C22⋊C4D411yescharacteristic
84C7×C22⋊C4D411yescharacteristic
85C42×C7C2311yescharacteristic
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7—21no—
109Dic28—22no—
110Dic28—41no—
111Dic28—41no—
112#552—71no—
113D4×Q8—71no—
114#491—71no—
115#485—71no—
116#483—71no—
117#479—71no—
118#470—71no—
119#467—71no—
120#462—71no—
121#451—71no—
122#451—71no—
123#441—71no—
124#435D711yescharacteristic
125#419—71no—
126C2×#103—71no—
127C23×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C7×D4—22no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2411yescharacteristic
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—42no—
150Dic14—21no—
151Dic14—21no—
152Dic14—21no—
153Dic14—21no—
154Dic14—21no—
155Dic14—21no—
156Dic14—21no—
157Dic14—42no—
158Dic14—42no—
159Dic14#4112yes—
160C7×C8—21no—
161C7×C8—21no—
162C7×C8—21no—
163C7×C8—21no—
164C22×Q8—71no—
165C22×Q8—71no—
166C2×Q16—71no—
167C2×Q16—71no—
168C2×Q16—71no—
169C2×M16C2×D711yescharacteristic
170C22×C8C2×D711yescharacteristic
171C4⋊Q8—71no—
172C4⋊Q8—141no—
173#126—141no—
174#125—71no—
175#125—141no—
176#123—141no—
177C22⋊Q8—71no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183C4×Q8—71no—
184C4×Q8—71no—
185C4×D4—141no—
186C4×D4C2×D711yescharacteristic
187#117—71no—
188#117—141no—
189C2×C4⋊C4—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8C2×D711yescharacteristic
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204C22⋊C8C2×D711yescharacteristic
205C22⋊C8C2×D711yescharacteristic
206#97C2×D711yescharacteristic
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7#13712yes—
213C22×C7C22×D411yescharacteristic
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—21no—
218C4×C7C2×#4111yescharacteristic
219C4×C7H(5,𝔽2)11yescharacteristic
220Dic7—22no—
221Dic7—41no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228C2×Q8—71no—
229C2×Q8—71no—
230C2×Q8—71no—
231C2×Q8—71no—
232C2×Q8—142no—
233C2×Q8—142no—
234C2×Q8—141no—
235C2×Q8—141no—
236C2×Q8—281no—
237C2×Q8—281no—
238C2×D4D2811yescharacteristic
239C22×C4—141no—
240C22×C4—141no—
241C22×C4—141no—
242C22×C4C22×D711yescharacteristic
243C22×C4C22×D711yescharacteristic
244Q16—142no—
245Q16—281no—
246Q16—281no—
247M16—41no—
248C2×C8—41no—
249C2×C8C22×D711yescharacteristic
250C2×C8C22×D711yescharacteristic
251C2×C8C22×D711yescharacteristic
252C2×C8C22×D711yescharacteristic
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4D2811yescharacteristic
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4D2811yescharacteristic
275C22⋊C4D2811yescharacteristic
276C42—141no—
277C42—141no—
278C42—141no—
279C42C22×D711yescharacteristic
280C2×C7—22no—
281C2×C7—41no—
282C2×C7#54611yescharacteristic
283C2×C7C2×#13711yescharacteristic
284C2×C7#57811yescharacteristic
285C23C2×D2811yescharacteristic
286C23C2×D2811yescharacteristic
287Q8—72no—
288Q8—141no—
289Q8—141no—
290Q8—141no—
291Q8—141no—
292Q8—141no—
293Q8—141no—
294Q8—141no—
295Q8—282no—
296Q8—282no—
297D4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—282no—
311C2×C4—282no—
312C2×C4—282no—
313C2×C4C2×D2811yescharacteristic
314C2×C4C2×D2811yescharacteristic
315C2×C4C2×D2811yescharacteristic
316C2×C4C2×D2811yescharacteristic
317C2×C4C23×D711yescharacteristic
318C8—21no—
319C8—21no—
320C8—21no—
321C8—21no—
322C7#331211yescharacteristic, complemented, Sylow
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#647212yes—
329C22C22×D2811yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—142no—
335C4—281no—
336C4—281no—
337C4—281no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4—281no—
342C4#654811yescharacteristic
343C4D7×#4111yescharacteristic
344C2—22no—
345C2—41no—
346C2#9024511yescharacteristic
347C2C2×#647211yescharacteristic
348C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
prime radicalC2×C7×D4D4
hypercenter#435D7
hypocenterC7#3312
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
4. #89909C2semidirect
5. #89892C2semidirect
6. #89888C2semidirect
7. C2×#6390C2semidirect
8. #89621C2semidirect
9. #89601C2semidirect
10. #89511C2semidirect
11. #89511C2semidirect
12. #89495C2semidirect
13. #89493C2semidirect
14. #89488C2semidirect
15. #89475C2semidirect
322. C7#3312semidirect

Maximal subgroups

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

subgroupindex
#902322
D4×Dic142
C7×#4352
#899092
#898922
#898882
C2×#63902
#896212
#896012
#895112
#894952
#894932
#894882
#894752
#33127