Finite Group Databasebeta

← #11739126

Subgroups — #11739126

Order 896.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90420C211yesmaximal, characteristic, complemented
2#90366C211yesmaximal, characteristic
3C7×#489C211yesmaximal, characteristic, complemented
4#89865C211yesmaximal, characteristic, complemented
5#89828C211yesmaximal, characteristic, complemented
6#89574C211yesmaximal, characteristic, complemented
7D7×#103C211yesmaximal, characteristic, complemented
8#89569C211yesmaximal, characteristic, complemented
9#89564C211yesmaximal, characteristic, complemented
10#89544C211yesmaximal, characteristic
11#89543C211yesmaximal, characteristic
12#89540C211yesmaximal, characteristic, complemented
13#89537C211yesmaximal, characteristic, complemented
14#89480C211yesmaximal, characteristic, complemented
15#89475C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C7×#124C2211yescharacteristic
20#6509—21no—
21#6507—21no—
22#6507C2211yescharacteristic
23#6499C2211yescharacteristic
24C2×#1131C2211yescharacteristic
25C2×C8×D7C2211yescharacteristic
26#6459—21no—
27#6455C2211yescharacteristic
28D7×C4⋊C4C2211yescharacteristic
29#6453C2211yescharacteristic
30#6452—21no—
31#6451C2211yescharacteristic
32#6446—21no—
33#6446—21no—
34#6441—21no—
35#6441—21no—
36#6435C2211yescharacteristic
37#6433C2211yescharacteristic
38#6432C2211yescharacteristic
39C7×#103C2211yescharacteristic
40C7×#103C2211yescharacteristic
41C7×#102C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6384C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3137—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C2×C22⋊D7—21no—
64C22×Dic7—21no—
65Q8×D7—22no—
66Q8×D7—22no—
67#1159—22no—
68#1159—22no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7C2311yescharacteristic
72C2×Dic14—21no—
73C2×Dic14—21no—
74C2×Dic14C2311yescharacteristic
75C2×Dic14C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C7×C4⋊C4—21no—
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C42×C7C2311yescharacteristic
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4D411yescharacteristic
88(C2×D7)⋊C4D411yescharacteristic
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4D411yescharacteristic
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—22no—
108C8×D7—22no—
109#544—71no—
110#536—71no—
111#491—71no—
112#491—71no—
113#489—71no—
114#489D711yescharacteristic, complemented
115#488—71no—
116#488—71no—
117#487—71no—
118#487—71no—
119#419—71no—
120#417—71no—
121#416—71no—
122C2×#103—71no—
123#405—71no—
124C23×C7—21no—
125C22×D7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—81no—
129C7×D4—21no—
130C7×D4—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C22⋊D7—42no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—42no—
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146C4×D7—42no—
147C4×D7—81no—
148C4×D7#4112yes—
149C4×D7#4112yes—
150Dic14—21no—
151Dic14—21no—
152Dic14—21no—
153Dic14—21no—
154Dic14—42no—
155Dic14—81no—
156C7×C8—21no—
157C7×C8—21no—
158C7⋊C8—21no—
159C7⋊C8—21no—
160C2×#41—71no—
161C22×Q8—71no—
162C2×M16—71no—
163C22×C8—71no—
164C4⋊Q8—71no—
165C4⋊Q8—71no—
166C4⋊Q8—71no—
167C4⋊Q8—141no—
168C4⋊Q8C2×D711yescharacteristic
169#125—71no—
170#125—71no—
171#125—141no—
172#124—71no—
173#124C2×D711yescharacteristic
174#123—141no—
175#123—141no—
176C22⋊Q8—141no—
177C22⋊Q8—141no—
178C22⋊Q8—141no—
179C22⋊Q8—141no—
180#117—71no—
181#117—71no—
182C2×C4⋊C4—71no—
183#103—71no—
184#103—71no—
185#103—71no—
186#103—71no—
187#103—71no—
188#103—71no—
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#102C2×D711yescharacteristic
198#102C2×D711yescharacteristic
199#97—71no—
200#97C2×D711yescharacteristic
201C4×C8—71no—
202C4×C8—71no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7C22×D411yescharacteristic
207C2×D7—21no—
208C2×D7—21no—
209C2×D7#12412yes—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7C2×#4111yescharacteristic
216C4×C7C2×#4111yescharacteristic
217Dic7—21no—
218Dic7—21no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223Dic7#12412yes—
224#41—142no—
225#41—142no—
226C2×Q8—21no—
227C2×Q8—71no—
228C2×Q8—71no—
229C2×Q8—141no—
230C2×Q8—141no—
231C2×Q8—142no—
232C2×Q8—142no—
233C2×Q8C22×D711yescharacteristic
234C2×D4—141no—
235C2×D4C22×D711yescharacteristic
236C22×C4—71no—
237C22×C4—141no—
238C22×C4—141no—
239C22×C4—141no—
240M16—142no—
241C2×C8—71no—
242C2×C8—71no—
243C2×C8—142no—
244C2×C8C22×D711yescharacteristic
245C2×C8C22×D711yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4C22×D711yescharacteristic
262C4⋊C4C22×D711yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—71no—
266C22⋊C4—71no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C42—71no—
271C42—141no—
272C42C22×D711yescharacteristic
273C2×C7—81no—
274C2×C7C2×#12411yescharacteristic
275C2×C7#57711yescharacteristic
276C2×C7#57811yescharacteristic
277D7—22no—
278C23—21no—
279C23—71no—
280Q8—21no—
281Q8—21no—
282Q8—81no—
283Q8—141no—
284Q8—141no—
285Q8—141no—
286Q8—141no—
287Q8—282no—
288Q8—561no—
289D4—21no—
290D4—21no—
291D4—282no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—71no—
296C2×C4—72no—
297C2×C4—72no—
298C2×C4—71no—
299C2×C4—71no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—282no—
305C2×C4—282no—
306C2×C4—561no—
307C2×C4D4×D711yescharacteristic
308C2×C4D4×D711yescharacteristic
309C2×C4C23×D711yescharacteristic
310C8—21no—
311C8—21no—
312C8—141no—
313C8—141no—
314C7#313711yescharacteristic, complemented, Sylow
315C22—41no—
316C22—41no—
317C22—41no—
318C22—72no—
319C22—141no—
320C22—141no—
321C22C2×D4×D711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—72no—
328C4—141no—
329C4—141no—
330C4—281no—
331C4—281no—
332C4—281no—
333C4—281no—
334C4D7×#4111yescharacteristic
335C4D7×#4111yescharacteristic
336C2—81no—
337C2—142no—
338C2D7×#12411yescharacteristic
339C2#9046611yescharacteristic
340C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#489C2
socleC22×C7C22×D4
prime radicalC2×#1159C22
hypercenter#489D7
hypocenterC7#3137
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #90420C2semidirect
3. C7×#489C2semidirect
4. #89865C2semidirect
5. #89828C2semidirect
6. #89574C2semidirect
7. D7×#103C2semidirect
8. #89569C2semidirect
9. #89564C2semidirect
12. #89540C2semidirect
13. #89537C2semidirect
14. #89480C2semidirect
15. #89475C2semidirect
114. #489D7semidirect
314. C7#3137semidirect

Maximal subgroups

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

subgroupindex
#904202
#903662
C7×#4892
#898652
#898282
#895742
D7×#1032
#895692
#895642
#895442
#895432
#895402
#895372
#894802
#894752
#31377