Finite Group Databasebeta

← #22599113

Subgroups — #22599113

Order 9088.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#14556104C211yesmaximal, characteristic
2C22⋊Q8×D71C211yesmaximal, characteristic
3#17986335C211yesmaximal, characteristic, complemented
4#17986333C212yesmaximal, complemented
5#17986332C211yesmaximal, characteristic, complemented
6#17986239C212yesmaximal, complemented
7#17986237C212yesmaximal, complemented
8#17986155C211yesmaximal, characteristic, complemented
9#17986153C212yesmaximal, complemented
10C22⋊C8×D71C211yesmaximal, characteristic
11#451×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C2×#11765855C2212yescomplemented
15C2×C8×D71C2212yes—
16#14556108C2211yescharacteristic
17#14556107C2211yescharacteristic
18#14556106—21no—
19#14556105—21no—
20#14556104—21no—
21#14556104—21no—
22#14556104C2212yes—
23#14556104C2211yescharacteristic
24#14556104C2212yes—
25C2×C4⋊Dic71C2211yescharacteristic
26C2×Dic71⋊C4—21no—
27C4⋊C4×D71—21no—
28C4⋊C4×D71C2211yescharacteristic
29C2×(C2×D71)⋊C4—21no—
30C22⋊C4×D71—21no—
31#14556062C2212yescomplemented
32#14556061C2212yescomplemented
33#14556053C2212yescomplemented
34#14556052C2212yescomplemented
35#14556052C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic
37(C2×D71)⋊C8C2212yes—
38C22×Q8×C71C2211yescharacteristic
39C2×Q16×C71C2212yescomplemented
40C22⋊Q8×C71C2211yescharacteristic
41#103×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic
43C23×D71D411yescharacteristic
44C2×Dic142—21no—
45C2×Dic142C2311yescharacteristic
46Q8×D71—42no—
47C22×Dic71—21no—
48C22×Dic71D411yescharacteristic
49C2×C4×D71—21no—
50C2×C4×D71—21no—
51C2×C4×D71—42no—
52C2×C4×D71D411yescharacteristic
53C2×C4×D71D411yescharacteristic
54C2×C4×D71C2311yescharacteristic
55C2×C4×D71C2311yescharacteristic
56#11765855—42no—
57C2×C71⋊C8C2312yes—
58C8×D71—42no—
59C4⋊Dic71—21no—
60C4⋊Dic71—21no—
61C4⋊Dic71C2311yescharacteristic
62C4⋊Dic71C2312yes—
63Dic71⋊C4—21no—
64Dic71⋊C4—21no—
65Dic71⋊C4—21no—
66Dic71⋊C4—22no—
67Dic71⋊C4—22no—
68C22⋊Dic71—21no—
69(C2×D71)⋊C4—21no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—22no—
72(C2×D71)⋊C4—22no—
73C2×Q8×C71—21no—
74C2×Q8×C71—42no—
75C2×Q8×C71C2311yescharacteristic
76C2×Q8×C71C2312yes—
77C22×C4×C71—21no—
78C22×C4×C71C2311yescharacteristic
79Q16×C71—42no—
80C2×C8×C71C2312yes—
81C4⋊C4×C71—21no—
82C4⋊C4×C71C2311yescharacteristic
83C22⋊C4×C71—21no—
84C23×C71C2×D411yescharacteristic
85C22×D71—41no—
86C22×D71—41no—
87C22×D71Q1614yescomplemented
88C22×D71C2×D411yescharacteristic
89C22×D71C2×D411yescharacteristic
90Q8×C71—22no—
91Q8×C71—22no—
92Q8×C71—22no—
93Q8×C71—42no—
94C2×C4×C71—22no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—42no—
99C2×C4×C71C2×D411yescharacteristic
100C2×C4×C71C2411yescharacteristic
101C2×Dic71—22no—
102C2×Dic71—21no—
103C2×Dic71—21no—
104C2×Dic71—41no—
105C2×Dic71—42no—
106C2×Dic71C2×D411yescharacteristic
107C2×Dic71C2×D411yescharacteristic
108C4×D71—22no—
109C4×D71—22no—
110C4×D71—41no—
111C4×D71—41no—
112C4×D71—42no—
113C4×D71—42no—
114Dic142—22no—
115Dic142—42no—
116C8×C71—22no—
117C71⋊C8—22no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71C2×Q1612yes—
121C22×C71C22×D411yescharacteristic
122C2×D71—21no—
123C2×D71—21no—
124C2×D71—21no—
125C2×D71—21no—
126C2×D71—44no—
127C2×D71C2×Q1612yes—
128C2×D71C2×Q1612yes—
129C4×C71—21no—
130C4×C71—41no—
131C4×C71—42no—
132C4×C71—41no—
133C4×C71H(5,𝔽2)12yes—
134Dic71—21no—
135Dic71—21no—
136Dic71—41no—
137Dic71—41no—
138Dic71—42no—
139C2×C71—22no—
140C2×C71#53411yescharacteristic
141C2×C71C22×Q1611yescharacteristic
142C2×C71#57511yescharacteristic
143D71—22no—
144D71—22no—
145#3189—711nomaximal, Sylow
146C71#318911yescharacteristic, complemented, Sylow
147C2×C22⋊Q8—711no—
148C2×C22⋊Q8—711no—
149#484—711no—
150#484—712no—
151#484—711no—
152#467—712no—
153#467—712no—
154#451—711no—
155#451—712no—
156#451D7111yescharacteristic, complemented
157C2×C22⋊C8—711no—
158C22×Q8—711no—
159C22×Q8C2×D7111yescharacteristic
160C23×C4—711no—
161C2×Q16—712no—
162C2×Q16C2×D7112yescomplemented
163C22×C8—712no—
164C22⋊Q8—712no—
165C22⋊Q8—711no—
166C22⋊Q8—712no—
167C22⋊Q8—711no—
168C22⋊Q8—711no—
169C22⋊Q8—1421no—
170C22⋊Q8—1421no—
171C22⋊Q8—1421no—
172C22⋊Q8—1421no—
173C22⋊Q8C2×D7111yescharacteristic
174C2×C4⋊C4—711no—
175C2×C4⋊C4—711no—
176C2×C4⋊C4—1421no—
177C2×C4⋊C4—1421no—
178C2×C22⋊C4—1421no—
179C2×C22⋊C4—1421no—
180#107—712no—
181#107—712no—
182#103—712no—
183#103—712no—
184#103—712no—
185#103C2×D7112yescomplemented
186C22⋊C8—711no—
187C22⋊C8—712no—
188C22⋊C8C2×D7111yescharacteristic
189C24—711no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—2842no—
193C2×Q8—711no—
194C2×Q8—1421no—
195C2×Q8C22×D7111yescharacteristic
196C2×Q8C22×D7112yes—
197C22×C4—21no—
198C22×C4—2842no—
199C22×C4—711no—
200C22×C4—711no—
201C22×C4—711no—
202C22×C4—711no—
203C22×C4—711no—
204C22×C4—1421no—
205C22×C4—1421no—
206C22×C4—1421no—
207C22×C4C22×D7111yescharacteristic
208Q16—42no—
209Q16—2842no—
210C2×C8—2842no—
211C2×C8—712no—
212C2×C8C22×D7112yes—
213C4⋊C4—21no—
214C4⋊C4—711no—
215C4⋊C4—712no—
216C4⋊C4—1421no—
217C4⋊C4—1421no—
218C4⋊C4—1422no—
219C4⋊C4—1422no—
220C4⋊C4—1421no—
221C4⋊C4—1421no—
222C4⋊C4—1421no—
223C4⋊C4C22×D7111yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—1421no—
226C22⋊C4—1421no—
227C22⋊C4—1422no—
228C22⋊C4—1422no—
229C22⋊C4—1421no—
230C23—2841no—
231C23—2841no—
232C23—714no—
233C23—711no—
234C23—711no—
235C23D4×D7111yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—22no—
239Q8—42no—
240Q8—2842no—
241Q8—1422no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—2841no—
248C2×C4—2841no—
249C2×C4—2842no—
250C2×C4—2842no—
251C2×C4—2842no—
252C2×C4—2841no—
253C2×C4—711no—
254C2×C4—711no—
255C2×C4—1422no—
256C2×C4—1421no—
257C2×C4—1421no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4D4×D7111yescharacteristic
261C2×C4C23×D7111yescharacteristic
262C8—22no—
263C8—1422no—
264C22—21no—
265C22—21no—
266C22—2844no—
267C22—712no—
268C22—712no—
269C22—1421no—
270C22—1421no—
271C22—1421no—
272C22—1421no—
273C22Q16×D7112yes—
274C22C2×D4×D7111yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—2841no—
280C4—2842no—
281C4—2841no—
282C4—1421no—
283C4—1421no—
284C4#1455617312yes—
285C2—22no—
286C2—1422no—
287C2—1422no—
288C2#1798655911yescharacteristic
289C2C2×Q16×D7111yescharacteristic
290C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#451×C71C2
socleC22×C71C22×D4
hypercenter#451D71
hypocenterC71#3189
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #17986335C2semidirect
4. #17986333C2semidirect
5. #17986332C2semidirect
6. #17986239C2semidirect
7. #17986237C2semidirect
8. #17986155C2semidirect
9. #17986153C2semidirect
11. #451×C71C2semidirect
14. C2×#11765855C22semidirect
31. #14556062C22semidirect
32. #14556061C22semidirect
33. #14556053C22semidirect
34. #14556052C22semidirect
35. #14556052C22semidirect
39. C2×Q16×C71C22semidirect
41. #103×C71C22semidirect
87. C22×D71Q16semidirect
146. C71#3189semidirect
156. #451D71semidirect
162. C2×Q16C2×D71semidirect
185. #103C2×D71semidirect