Finite Group Databasebeta

← #11738028

Subgroups — #11738028

Order 896.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90372C211yesmaximal, characteristic
2#90260C211yesmaximal, characteristic, complemented
3C7×#464C211yesmaximal, characteristic, complemented
4#89957C211yesmaximal, characteristic, complemented
5#89956C211yesmaximal, characteristic, complemented
6#89936C211yesmaximal, characteristic, complemented
7#89935C211yesmaximal, characteristic
8#89828C211yesmaximal, characteristic, complemented
9C4×#1143C211yesmaximal, characteristic, complemented
10#89619C211yesmaximal, characteristic, complemented
11#89603C211yesmaximal, characteristic, complemented
12#89568C211yesmaximal, characteristic, complemented
13#89567C211yesmaximal, characteristic, complemented
14#89536C211yesmaximal, characteristic
15#89533C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C7×#124C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20#6510C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic
24C2×#1143C2211yescharacteristic
25#6503C2211yescharacteristic
26D4×Dic7C2211yescharacteristic
27#6453—21no—
28#6453C2211yescharacteristic
29#6451—21no—
30#6451C2211yescharacteristic
31#6449—21no—
32#6448—21no—
33#6446—21no—
34(C22⋊D7)⋊C4—21no—
35C4×D28C2211yescharacteristic
36#6433—21no—
37#6433C2211yescharacteristic
38C4×Dic14—21no—
39C4×Dic14C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6389C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3371—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66C2×D28C2311yescharacteristic
67C2×C4×D7—21no—
68C2×Dic14—21no—
69C2×Dic14D411yescharacteristic
70C7×Q16—22no—
71C7×SD16—22no—
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C7×C22⋊C4—21no—
77C7×C22⋊C4—21no—
78C42×C7—21no—
79C42×C7C2311yescharacteristic
80C22⋊Dic7—21no—
81#1143—22no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7D411yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98Dic7⋊C4D411yescharacteristic
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#559—71no—
107#557—71no—
108#490—71no—
109#489—71no—
110#488—71no—
111#479—71no—
112#478—71no—
113#476—71no—
114#464—71no—
115#464D711yescharacteristic, complemented
116#460—71no—
117#446—71no—
118#441—71no—
119#440—71no—
120C4×SD16—71no—
121C23×C7—21no—
122C22×D7—21no—
123C7×Q8—21no—
124C7×Q8—21no—
125C7×Q8—21no—
126C7×Q8#4112yes—
127C7×D4—21no—
128C7×D4—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2411yescharacteristic
135C22⋊D7—42no—
136C22⋊D7—81no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—81no—
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145D28—21no—
146D28—21no—
147C4×D7—81no—
148Dic14—41no—
149Dic14—42no—
150C7×C8—21no—
151C7×C8—21no—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C2×Q16C2×D711yescharacteristic
155C2×SD16—71no—
156C2×SD16C2×D711yescharacteristic
157C4⋊Q8—71no—
158C4⋊Q8—141no—
159#127—71no—
160#125—71no—
161#125—71no—
162#125—141no—
163#125—141no—
164#124—71no—
165#124—141no—
166#124C2×D711yescharacteristic
167#123—141no—
168#123—141no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—141no—
174C4×Q8C2×D711yescharacteristic
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—141no—
178#107—71no—
179#106—71no—
180#106—71no—
181C4⋊C8—71no—
182C4⋊C8—71no—
183C4⋊C8C2×D711yescharacteristic
184#103—71no—
185#103—71no—
186#103—71no—
187#103C2×D711yescharacteristic
188#102—71no—
189#102—71no—
190#102—71no—
191#102—71no—
192#102—71no—
193#102C2×D711yescharacteristic
194#97—71no—
195#97—71no—
196C4×C8—71no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7C22×D411yescharacteristic
201C2×D7—41no—
202C2×D7—41no—
203C2×D7—41no—
204C4×C7—22no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7C2×#4111yescharacteristic
210C4×C72−1+411yescharacteristic
211Dic7—21no—
212Dic7—21no—
213Dic7—21no—
214Dic7—21no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218C2×Q8—71no—
219C2×Q8—141no—
220C2×Q8C22×D711yescharacteristic
221C2×Q8C22×D711yescharacteristic
222C2×D4—71no—
223C2×D4—141no—
224C2×D4—141no—
225C2×D4C22×D711yescharacteristic
226C22×C4—141no—
227C22×C4—141no—
228Q16—22no—
229SD16—22no—
230SD16—142no—
231C2×C8—71no—
232C2×C8—71no—
233C2×C8C22×D711yescharacteristic
234C2×C8C22×D711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—71no—
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⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C42—21no—
258C42—71no—
259C42—71no—
260C42—141no—
261C42—141no—
262C42—141no—
263C42C22×D711yescharacteristic
264C2×C7—81no—
265C2×C7#54811yescharacteristic
266C2×C7#57511yescharacteristic
267C2×C7#57711yescharacteristic
268D7—81no—
269C23—21no—
270C23—141no—
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—281no—
275Q8—282no—
276Q8#115912yes—
277D4—21no—
278D4—21no—
279D4—141no—
280D4—141no—
281D4—282no—
282D4—561no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—71no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—71no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—561no—
294C2×C4—561no—
295C2×C4D4×D711yescharacteristic
296C2×C4D4×D711yescharacteristic
297C2×C4C23×D711yescharacteristic
298C8—21no—
299C8—21no—
300C8—141no—
301C8—141no—
302C7#337111yescharacteristic, complemented, Sylow
303C22—41no—
304C22—41no—
305C22—41no—
306C22—281no—
307C22—281no—
308C22—281no—
309C22C2×D4×D711yescharacteristic
310C4—21no—
311C4—21no—
312C4—22no—
313C4—41no—
314C4—41no—
315C4—141no—
316C4—141no—
317C4—141no—
318C4—141no—
319C4—281no—
320C4—281no—
321C4—281no—
322C4C2×#115911yescharacteristic
323C4#655411yescharacteristic
324C2—81no—
325C2—561no—
326C2#9028911yescharacteristic
327C2#9045911yescharacteristic
328C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#464C2
socleC22×C7C22×D4
prime radical#6503C22
hypercenter#464D7
hypocenterC7#3371
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #90260C2semidirect
3. C7×#464C2semidirect
4. #89957C2semidirect
5. #89956C2semidirect
6. #89936C2semidirect
8. #89828C2semidirect
9. C4×#1143C2semidirect
10. #89619C2semidirect
11. #89603C2semidirect
12. #89568C2semidirect
13. #89567C2semidirect
15. #89533C2semidirect
115. #464D7semidirect
302. C7#3371semidirect

Maximal subgroups

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

subgroupindex
#903722
#902602
C7×#4642
#899572
#899562
#899362
#899352
#898282
C4×#11432
#896192
#896032
#895682
#895672
#895362
#895332
#33717