Finite Group Databasebeta

← #22599363

Subgroups — #22599363

Order 9088.

386 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊Q8×D71C211yesmaximal, characteristic, complemented
2#17986528C211yesmaximal, characteristic
3#17986394C211yesmaximal, characteristic, complemented
4#17986368C211yesmaximal, characteristic, complemented
5#17986367C211yesmaximal, characteristic, complemented
6#17986207C212yesmaximal, complemented
7#17986202C212yesmaximal, complemented
8#17986153C212yesmaximal, complemented
9#17986136C212yesmaximal
10#17986054C211yesmaximal, characteristic, complemented
11#497×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2212yes—
13C2×#11765855C2212yes—
14C2×#11765853C2212yes—
15#14556128C2211yescharacteristic
16#14556127—21no—
17#14556127—21no—
18#14556126C2212yes—
19#14556121—21no—
20#14556120C2211yescharacteristic
21#14556110—21no—
22#14556104C2211yescharacteristic
23#14556099C2211yescharacteristic
24Q8×Dic71C2211yescharacteristic
25D4×Dic71C2211yescharacteristic
26(C22⋊D71)⋊C4—21no—
27#14556084—21no—
28C4⋊C4×D71—21no—
29C4⋊C4×D71—21no—
30C42×D71C2211yescharacteristic
31Dic71⋊C8C2212yes—
32#14556053C2212yes—
33#14556053C2212yes—
34#14556052C2212yes—
35#14556052C2212yes—
36(C2×D71)⋊C8C2212yes—
37#14556039C2211yescharacteristic
38C2×Q16×C71C2212yes—
39C2×SD16×C71C2212yes—
40C4⋊Q8×C71C2211yescharacteristic
41#124×C71C2211yescharacteristic
42#97×C71C2211yescharacteristic
43C2×Dic142—22no—
44C2×Dic142C2312yes—
45Q8×D71—42no—
46Q8×D71—42no—
47C2×C22⋊D71—21no—
48C22×Dic71—21no—
49C2×C4×D71—22no—
50C2×C4×D71D411yescharacteristic
51C2×C4×D71D411yescharacteristic
52C2×C4×D71C2311yescharacteristic
53#11765855—42no—
54#11765853—42no—
55C2×C71⋊C8C2312yes—
56C4⋊Dic71—21no—
57C4⋊Dic71—21no—
58C4⋊Dic71—21no—
59C4⋊Dic71C2311yescharacteristic
60C4⋊Dic71C2311yescharacteristic
61Dic71⋊C4—21no—
62Dic71⋊C4—21no—
63Dic71⋊C4—22no—
64Dic71⋊C4—22no—
65Dic71⋊C4—21no—
66C22⋊Dic71—21no—
67C22⋊Dic71—21no—
68(C2×D71)⋊C4—21no—
69C4×Dic71—21no—
70C4×Dic71D411yescharacteristic
71C4×Dic71D411yescharacteristic
72C4×Dic71C2311yescharacteristic
73C2×Q8×C71C2312yes—
74C2×Q8×C71C2311yescharacteristic
75C2×D4×C71C2311yescharacteristic
76Q16×C71—42no—
77SD16×C71—42no—
78C2×C8×C71C2312yes—
79C4⋊C4×C71—21no—
80C4⋊C4×C71—21no—
81C22⋊C4×C71—21no—
82C22⋊C4×C71—21no—
83C42×C71C2311yescharacteristic
84C23×C71—21no—
85C22×D71C2×D411yescharacteristic
86Q8×C71—22no—
87Q8×C71—22no—
88Q8×C71—22no—
89D4×C71—22no—
90C2×C4×C71—21no—
91C2×C4×C71—22no—
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2411yescharacteristic
95C22⋊D71—42no—
96C2×Dic71—21no—
97C2×Dic71—21no—
98C2×Dic71—22no—
99C2×Dic71—42no—
100C2×Dic71C2×D411yescharacteristic
101C2×Dic71C2×D411yescharacteristic
102C2×Dic71C2×D411yescharacteristic
103C4×D71—22no—
104C4×D71—22no—
105C4×D71—22no—
106C4×D71—44no—
107Dic142—22no—
108Dic142—22no—
109Dic142—44no—
110C8×C71—22no—
111C71⋊C8—22no—
112C22×C71—41no—
113C22×C71—41no—
114C22×C71—41no—
115C22×C71C22×D411yescharacteristic
116C2×D71—21no—
117C2×D71—21no—
118C2×D71#13712yes—
119C4×C71—21no—
120C4×C71—21no—
121C4×C71—42no—
122C4×C71—41no—
123C4×C71H(5,𝔽2)12yes—
124Dic71—21no—
125Dic71—21no—
126Dic71—42no—
127Dic71—41no—
128Dic71—41no—
129Dic71#13712yes—
130C2×C71—81no—
131C2×C71#53511yescharacteristic
132C2×C71C2×#13711yescharacteristic
133C2×C71C2×#13711yescharacteristic
134D71—22no—
135#3218—711nomaximal, Sylow
136C71#321811yescharacteristic, complemented, Sylow
137C2×C4⋊Q8—711no—
138#529—711no—
139#497—711no—
140#497D7111yescharacteristic, complemented
141#491—711no—
142#491—711no—
143#462—712no—
144#461—712no—
145#451—712no—
146#448—712no—
147#432—711no—
148C22×Q8—712no—
149C2×Q16—712no—
150C2×Q16C2×D7112yes—
151C2×SD16—712no—
152C2×SD16C2×D7112yes—
153C4⋊Q8—712no—
154C4⋊Q8—711no—
155C4⋊Q8—1421no—
156C4⋊Q8—1421no—
157C4⋊Q8C2×D7111yescharacteristic
158#126—1421no—
159#125—711no—
160#124C2×D7111yescharacteristic
161#123—1421no—
162C22⋊Q8—711no—
163#121—711no—
164C4×Q8—711no—
165C4×D4—711no—
166C4×D4—1421no—
167#117—1421no—
168C2×C4⋊C4—1421no—
169C2×C4⋊C4—1421no—
170C2×C42—711no—
171C4⋊C8—712no—
172#103—712no—
173#103—712no—
174#103—712no—
175#103—712no—
176C22⋊C8—712no—
177#97—711no—
178#97C2×D7111yescharacteristic
179C2×Q8—2842no—
180C2×Q8—2842no—
181C2×Q8—712no—
182C2×Q8—1422no—
183C2×Q8C22×D7112yes—
184C2×Q8C22×D7111yescharacteristic
185C2×D4—1421no—
186C2×D4C22×D7111yescharacteristic
187C22×C4—711no—
188C22×C4—711no—
189C22×C4—711no—
190C22×C4—1422no—
191C22×C4—1421no—
192Q16—42no—
193Q16—2842no—
194SD16—42no—
195SD16—2842no—
196C2×C8—712no—
197C2×C8C22×D7112yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—711no—
201C4⋊C4—711no—
202C4⋊C4—1422no—
203C4⋊C4—1422no—
204C4⋊C4—1421no—
205C4⋊C4—1421no—
206C4⋊C4—1421no—
207C4⋊C4—1421no—
208C4⋊C4—1421no—
209C4⋊C4—1421no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—1421no—
213C22⋊C4—1421no—
214C22⋊C4—1421no—
215C42—711no—
216C42—711no—
217C42—711no—
218C42—1421no—
219C42C22×D7111yescharacteristic
220C23—21no—
221C23—711no—
222Q8—22no—
223Q8—22no—
224Q8—22no—
225Q8—2844no—
226Q8—1422no—
227Q8—1422no—
228D4—22no—
229D4—2842no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—2844no—
233C2×C4—2842no—
234C2×C4—711no—
235C2×C4—711no—
236C2×C4—711no—
237C2×C4—1422no—
238C2×C4—1422no—
239C2×C4—1421no—
240C2×C4—1421no—
241C2×C4—1422no—
242C2×C4—1422no—
243C2×C4D4×D7111yescharacteristic
244C2×C4D4×D7111yescharacteristic
245C2×C4C23×D7111yescharacteristic
246C8—22no—
247C8—1422no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—712no—
252C22—1421no—
253C22—1421no—
254C22C2×D4×D7111yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—41no—
259C4—2842no—
260C4—2841no—
261C4—2841no—
262C4—712no—
263C4—1421no—
264C4—1421no—
265C4#1455617312yes—
266C2—81no—
267C2—1422no—
268C2#1798656211yescharacteristic
269C2#137×D7111yescharacteristic
270C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#497×C71C2
socleC22×C71C22×D4
hypercenter#497D71
hypocenterC71#3218
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D71C2semidirect
3. #17986394C2semidirect
4. #17986368C2semidirect
5. #17986367C2semidirect
6. #17986207C2semidirect
7. #17986202C2semidirect
8. #17986153C2semidirect
10. #17986054C2semidirect
11. #497×C71C2semidirect
136. C71#3218semidirect
140. #497D71semidirect