Finite Group Databasebeta

← #11741069

Subgroups — #11741069

Order 896.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#554C211yesmaximal, characteristic
2#90232C211yesmaximal, characteristic
3#89866C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89824C211yesmaximal, characteristic, complemented
6#89823C211yesmaximal, characteristic, complemented
7#89817C211yesmaximal, characteristic, complemented
8#89815C211yesmaximal, characteristic, complemented
9#89802C211yesmaximal, characteristic, complemented
10#89794C211yesmaximal, characteristic, complemented
11#89783C211yesmaximal, characteristic
12#89743C211yesmaximal, characteristic, complemented
13#89742C211yesmaximal, characteristic, complemented
14#89736C211yesmaximal, characteristic, complemented
15C2×#6382C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#125—21no—
18C7×C22⋊Q8—21no—
19C7×C22⋊Q8—21no—
20C7×C22⋊Q8C2211yescharacteristic
21C7×C22⋊Q8C2211yescharacteristic
22C4×C7×Q8C2211yescharacteristic
23C4×C7×D4—21no—
24C4×C7×D4C2211yescharacteristic
25C2×C7×C4⋊C4C2211yescharacteristic
26C2×C7×C4⋊C4C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28#6498—21no—
29#6489C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31#6487C2211yescharacteristic
32#6487C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6452—21no—
36#6442—21no—
37#6440—21no—
38#6433C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40#6409C2211yescharacteristic
41#6409C2211yescharacteristic
42#6409C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6384C2211yescharacteristic
46#6384C2211yescharacteristic
47#6384C2211yescharacteristic
48#6382C2211yescharacteristic
49#6382C2211yescharacteristic
50#6382C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6382C2211yescharacteristic
54#6381C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3344—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4D411yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×Dic7—21no—
68C2×Dic14C2311yescharacteristic
69C7×C4⋊C4—21no—
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4D411yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C7×C22⋊C4D411yescharacteristic
82C7×C22⋊C4D411yescharacteristic
83C42×C7—21no—
84C42×C7C2311yescharacteristic
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88#1144—22no—
89C4⋊Dic7—21no—
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100#1136—22no—
101C2×C7⋊C8—22no—
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#554D711yescharacteristic
107#552—71no—
108#501—71no—
109#485—71no—
110#484—71no—
111#483—71no—
112#483—71no—
113#479—71no—
114#475—71no—
115#470—71no—
116#467—71no—
117#441—71no—
118#435—71no—
119#428—71no—
120C2×#106—71no—
121C23×C7C2×D411yescharacteristic
122C23×C7C2×D411yescharacteristic
123C7×Q8—21no—
124C7×Q8—21no—
125C7×Q8—81no—
126C7×D4—22no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—81no—
136C2×C4×C7—81no—
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—42no—
147Dic14—21no—
148Dic14—21no—
149C7⋊C8—21no—
150C7⋊C8—21no—
151C7⋊C8—21no—
152C7⋊C8#4112yes—
153C2×Q16—71no—
154C2×M16—71no—
155C22×C8—71no—
156C4⋊Q8C2×D711yescharacteristic
157#126—141no—
158#125—21no—
159#125—71no—
160#125—141no—
161#123—141no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166C22⋊Q8C2×D711yescharacteristic
167C22⋊Q8C2×D711yescharacteristic
168C4×Q8—71no—
169C4×Q8C2×D711yescharacteristic
170C4×D4—21no—
171C4×D4C2×D711yescharacteristic
172#117—71no—
173#117—141no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4C2×D711yescharacteristic
176C2×C4⋊C4C2×D711yescharacteristic
177#107—71no—
178#107—71no—
179#107—71no—
180#106—71no—
181#106—71no—
182#106—71no—
183#106—71no—
184#106—71no—
185#106—71no—
186C4⋊C8—71no—
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193C22⋊C8—71no—
194C22⋊C8—71no—
195#97—71no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7#13712yes—
202C22×C7C22×D411yescharacteristic
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7C2×#4111yescharacteristic
212C4×C72−1+411yescharacteristic
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217C2×Q8—21no—
218C2×Q8—71no—
219C2×Q8C22×D711yescharacteristic
220C2×D4C22⋊D711yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—141no—
224C22×C4C22×D711yescharacteristic
225C22×C4C22×D711yescharacteristic
226Q16—142no—
227M16—142no—
228C2×C8—71no—
229C2×C8—71no—
230C2×C8—71no—
231C2×C8—71no—
232C2×C8—142no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4C22⋊D711yescharacteristic
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C4⋊C4C22×D711yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4C22⋊D711yescharacteristic
258C22⋊C4C22⋊D711yescharacteristic
259C42—21no—
260C42—141no—
261C42—141no—
262C42C22×D711yescharacteristic
263C2×C7—22no—
264C2×C7—41no—
265C2×C7#54711yescharacteristic
266C2×C7C2×#13711yescharacteristic
267C2×C7#57711yescharacteristic
268C23C2×C22⋊D711yescharacteristic
269C23C2×C22⋊D711yescharacteristic
270Q8—21no—
271Q8—21no—
272Q8—81no—
273Q8—141no—
274Q8—141no—
275D4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—282no—
291C2×C4C2×C22⋊D711yescharacteristic
292C2×C4C2×C22⋊D711yescharacteristic
293C2×C4C2×C22⋊D711yescharacteristic
294C2×C4C2×C22⋊D711yescharacteristic
295C2×C4C23×D711yescharacteristic
296C8—72no—
297C8—141no—
298C8—141no—
299C8—141no—
300C7#334411yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22#650512yes—
307C22C22×C22⋊D711yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—281no—
317C4—281no—
318C4—281no—
319C4—281no—
320C4D7×#4111yescharacteristic
321C4#655411yescharacteristic
322C2—22no—
323C2—41no—
324C2C2×#650511yescharacteristic
325C2#9051511yescharacteristic
326C2#9052711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#554C2
socleC22×C7C22×D4
prime radicalC2×C7×D4D4
hypercenter#554D7
hypocenterC7#3344
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
3. #89866C2semidirect
4. #89852C2semidirect
5. #89824C2semidirect
6. #89823C2semidirect
7. #89817C2semidirect
8. #89815C2semidirect
9. #89802C2semidirect
10. #89794C2semidirect
12. #89743C2semidirect
13. #89742C2semidirect
14. #89736C2semidirect
15. C2×#6382C2semidirect
300. C7#3344semidirect

Maximal subgroups

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

subgroupindex
C7×#5542
#902322
#898662
#898522
#898242
#898232
#898172
#898152
#898022
#897942
#897832
#897432
#897422
#897362
C2×#63822
#33447