Finite Group Databasebeta

← #11737282

Subgroups — #11737282

Order 896.

366 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90271C211yesmaximal, characteristic
2#90269C211yesmaximal, characteristic, complemented
3C7×#441C211yesmaximal, characteristic, complemented
4#89962C211yesmaximal, characteristic, complemented
5#89960C211yesmaximal, characteristic, complemented
6#89802C211yesmaximal, characteristic, complemented
7#89798C211yesmaximal, characteristic, complemented
8#89636C211yesmaximal, characteristic
9#89628C211yesmaximal, characteristic, complemented
10#89585C211yesmaximal, characteristic, complemented
11#89579C211yesmaximal, characteristic
12#89569C211yesmaximal, characteristic, complemented
13#89567C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15D7×#97C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19#6510C2211yescharacteristic
20#6509C2211yescharacteristic
21#6509C2211yescharacteristic
22#6507C2211yescharacteristic
23C2×#1144C2211yescharacteristic
24C2×#1143C2211yescharacteristic
25C2×#1132C2211yescharacteristic
26C2×C8×D7C2211yescharacteristic
27C2×C8×D7C2211yescharacteristic
28#6461—21no—
29#6460C2211yescharacteristic
30#6459—21no—
31#6458C2211yescharacteristic
32#6457—21no—
33#6453C2211yescharacteristic
34#6452—21no—
35#6451C2211yescharacteristic
36C4×D28—21no—
37C4×D28C2211yescharacteristic
38#6435—21no—
39#6435—21no—
40C42×D7C2211yescharacteristic
41C4×Dic14—21no—
42C4×Dic14C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46C7×#97C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6395C2211yescharacteristic
50#6392C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53#6389C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3181—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×D28—21no—
63C2×D28C2311yescharacteristic
64C2×C4×D7—21no—
65C2×C4×D7D411yescharacteristic
66C2×C4×D7D411yescharacteristic
67C2×C4×D7C2311yescharacteristic
68C2×Dic14—21no—
69C2×Dic14C2311yescharacteristic
70C7×Q16—22no—
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C42×C7—21no—
78C42×C7—21no—
79C42×C7C2311yescharacteristic
80#1144—22no—
81#1143—22no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7D411yescharacteristic
102C4×Dic7D411yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#1132—22no—
107C8×D7—22no—
108C8×D7—22no—
109#533—71no—
110#529—71no—
111#501—71no—
112#497—71no—
113#491—71no—
114#489—71no—
115#467—71no—
116#467—71no—
117#465—71no—
118#465—71no—
119#441—71no—
120#441D711yescharacteristic, complemented
121#440—71no—
122#440—71no—
123C2×#97—71no—
124C22×D7—21no—
125C22×D7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—21no—
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×C7C2411yescharacteristic
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143D28—21no—
144D28—21no—
145D28—42no—
146C4×D7—22no—
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—41no—
150C4×D7—42no—
151Dic14—21no—
152Dic14—21no—
153Dic14—42no—
154C7×C8—21no—
155C7×C8#4112yes—
156C7⋊C8—21no—
157C7⋊C8#4112yes—
158C2×Q16—71no—
159C2×Q16C2×D711yescharacteristic
160C2×SD16—71no—
161C2×SD16—71no—
162C22×C8—71no—
163C22×C8—71no—
164C4⋊Q8—71no—
165C4⋊Q8—71no—
166#126—141no—
167#125—71no—
168#125—141no—
169#124—71no—
170#123—141no—
171C22⋊Q8—71no—
172C22⋊Q8—71no—
173C22⋊Q8—71no—
174C22⋊Q8—141no—
175#121—71no—
176C4×Q8—71no—
177C4×Q8—141no—
178C4×Q8C2×D711yescharacteristic
179C4×Q8C2×D711yescharacteristic
180C4×D4—71no—
181C4×D4—141no—
182#117—141no—
183#117—141no—
184C2×C42—71no—
185#107—71no—
186#107—71no—
187#106—71no—
188#106C2×D711yescharacteristic
189#103—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193#103C2×D711yescharacteristic
194#103C2×D711yescharacteristic
195#102—71no—
196#102—71no—
197#97—71no—
198#97—71no—
199#97—71no—
200#97C2×D711yescharacteristic
201C22×C7C22×D411yescharacteristic
202C2×D7—21no—
203C2×D7—21no—
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C2×D7#13712yes—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7C2×#4111yescharacteristic
215C4×C7C2×#4111yescharacteristic
216Dic7—21no—
217Dic7—21no—
218Dic7—21no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222C2×Q8—71no—
223C2×Q8—141no—
224C2×Q8C22×D711yescharacteristic
225C2×Q8C22×D711yescharacteristic
226C2×D4—71no—
227C2×D4—141no—
228C22×C4—71no—
229C22×C4—71no—
230C22×C4—71no—
231C22×C4—141no—
232Q16—22no—
233Q16—142no—
234SD16—142no—
235SD16—142no—
236C2×C8—71no—
237C2×C8—71no—
238C2×C8—142no—
239C2×C8—142no—
240C2×C8C22×D711yescharacteristic
241C2×C8C22×D711yescharacteristic
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—71no—
245C4⋊C4—71no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4C22×D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C42—21no—
265C42—21no—
266C42—71no—
267C42—71no—
268C42—71no—
269C42—141no—
270C42C22×D711yescharacteristic
271C2×C7#53211yescharacteristic
272C2×C7C2×#13711yescharacteristic
273C2×C7#57511yescharacteristic
274D7—22no—
275D7—81no—
276C23—71no—
277C23—141no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—141no—
283Q8—141no—
284Q8—282no—
285D4—141no—
286D4—141no—
287D4—282no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—71no—
293C2×C4—71no—
294C2×C4—71no—
295C2×C4—142no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—281no—
302C2×C4—282no—
303C2×C4D4×D711yescharacteristic
304C2×C4D4×D711yescharacteristic
305C2×C4C23×D711yescharacteristic
306C8—21no—
307C8—72no—
308C8—141no—
309C8#115712yes—
310C7#318111yescharacteristic, complemented, Sylow
311C22—72no—
312C22—141no—
313C22—141no—
314C22—281no—
315C22—281no—
316C22—281no—
317C22C2×D4×D711yescharacteristic
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—141no—
325C4—141no—
326C4—141no—
327C4—281no—
328C4—281no—
329C4—281no—
330C4C2×#115711yescharacteristic
331C4D7×#4111yescharacteristic
332C2—142no—
333C2—561no—
334C2#9023911yescharacteristic
335C2D7×#13711yescharacteristic
336C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#441C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#441D7
hypocenterC7#3181
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
2. #90269C2semidirect
3. C7×#441C2semidirect
4. #89962C2semidirect
5. #89960C2semidirect
6. #89802C2semidirect
7. #89798C2semidirect
9. #89628C2semidirect
10. #89585C2semidirect
12. #89569C2semidirect
13. #89567C2semidirect
14. #89482C2semidirect
15. D7×#97C2semidirect
120. #441D7semidirect
310. C7#3181semidirect

Maximal subgroups

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

subgroupindex
#902712
#902692
C7×#4412
#899622
#899602
#898022
#897982
#896362
#896282
#895852
#895792
#895692
#895672
#894822
D7×#972
#31817