Finite Group Databasebeta

← #22600271

Subgroups — #22600271

Order 9088.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#17986714C211yesmaximal, characteristic, complemented
2#17986652C211yesmaximal, characteristic, complemented
3#17986349C211yesmaximal, characteristic, complemented
4#17986338C212yesmaximal, complemented
5#17986290C211yesmaximal, characteristic, complemented
6#17986225C212yesmaximal, complemented
7#17986143C212yesmaximal, complemented
8C4×#11765850C211yesmaximal, characteristic, complemented
9#17986064C211yesmaximal, characteristic, complemented
10#17985994C212yesmaximal, complemented
11#461×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2212yes—
13C2×#11765850C2211yescharacteristic
14C2×C8×D71C2212yes—
15#14556128C2211yescharacteristic
16#14556114—22no—
17#14556113—22no—
18#14556104C2212yes—
19#14556103—21no—
20#14556102—21no—
21#14556099C2212yes—
22C4×D284C2211yescharacteristic
23C4×D284C2211yescharacteristic
24#14556087—21no—
25#14556086C2212yes—
26C4⋊C4×D71—21no—
27#14556061C2211yescharacteristic
28#14556060C2212yes—
29#14556052C2212yes—
30#14556052C2212yes—
31#14556050C2212yes—
32#14556049C2211yescharacteristic
33#14556048C2211yescharacteristic
34(C2×D71)⋊C8C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C2×SD16×C71C2212yes—
37C4⋊Q8×C71C2211yescharacteristic
38C4×D4×C71C2211yescharacteristic
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41#11765866—24no—
42#11765866—42no—
43C2×Dic142—21no—
44C2×D284—22no—
45C2×D284D411yescharacteristic
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—22no—
48C2×C4×D71—21no—
49C2×C4×D71—22no—
50C2×C4×D71C2312yes—
51#11765850—22no—
52C2×C71⋊C8C2311yescharacteristic
53C2×C71⋊C8C2311yescharacteristic
54C8×D71—42no—
55C4⋊Dic71—21no—
56C4⋊Dic71D411yescharacteristic
57C4⋊Dic71C2311yescharacteristic
58C4⋊Dic71C2312yes—
59Dic71⋊C4—22no—
60C22⋊Dic71—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—21no—
64(C2×D71)⋊C4D412yes—
65C4×Dic71—22no—
66C2×Q8×C71C2312yes—
67C2×D4×C71C2311yescharacteristic
68C22×C4×C71—21no—
69SD16×C71—42no—
70C2×C8×C71C2312yes—
71C4⋊C4×C71—21no—
72C4⋊C4×C71—21no—
73C4⋊C4×C71C2311yescharacteristic
74C22⋊C4×C71—21no—
75C42×C71C2311yescharacteristic
76C23×C71—21no—
77C22×D71—21no—
78C22×D71C2×D412yes—
79Q8×C71—22no—
80Q8×C71—22no—
81D4×C71—21no—
82D4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71—42no—
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2411yescharacteristic
89C22⋊D71—82no—
90C2×Dic71—22no—
91C2×Dic71—21no—
92C2×Dic71C2×D412yes—
93D284—22no—
94D284—21no—
95D284—44no—
96D284#4112yes—
97C4×D71—22no—
98C4×D71—22no—
99C4×D71—44no—
100C4×D71—42no—
101Dic142—81no—
102C8×C71—22no—
103C71⋊C8—21no—
104C71⋊C8—21no—
105C22×C71—41no—
106C22×C71—41no—
107C22×C71—41no—
108C22×C71C22×D411yescharacteristic
109C2×D71—22no—
110C2×D71—22no—
111C2×D71—22no—
112C2×D71—22no—
113C2×D71—41no—
114C2×D71—41no—
115C4×C71—21no—
116C4×C71—42no—
117C4×C71—41no—
118C4×C71#13512yes—
119C4×C71C2×#4111yescharacteristic
120C4×C71H(5,𝔽2)11yescharacteristic
121Dic71—22no—
122Dic71—41no—
123Dic71—42no—
124C2×C71—81no—
125C2×C71#54611yescharacteristic
126C2×C71C2×#13511yescharacteristic
127C2×C71#57711yescharacteristic
128D71—42no—
129D71—42no—
130#3303—711nomaximal, Sylow
131C71#330311yescharacteristic, complemented, Sylow
132#555—711no—
133#547—711no—
134#487—711no—
135#485—712no—
136#476—711no—
137#465—712no—
138#461D7111yescharacteristic, complemented
139#449—712no—
140C4×D8—711no—
141C8×D4—711no—
142#416—712no—
143C2×#41—712no—
144C2×SD16C2×D7112yes—
145C2×D8—711no—
146C22×C8—712no—
147C4⋊Q8—711no—
148C4⋊Q8C2×D7111yescharacteristic
149#124—1422no—
150#123—1422no—
151C22⋊Q8—712no—
152#121—712no—
153#121—1421no—
154#121—1421no—
155C4×D4—711no—
156C4×D4—711no—
157C4×D4C2×D7111yescharacteristic
158#117—712no—
159#117—1421no—
160C2×C4⋊C4—1421no—
161#107—711no—
162#106—712no—
163C4⋊C8C2×D7111yescharacteristic
164#103—712no—
165#103—712no—
166#103C2×D7112yes—
167#102—711no—
168#102—712no—
169#102—711no—
170C22⋊C8—712no—
171C4×C8—711no—
172#41—2842no—
173#41—1424no—
174C2×Q8—1421no—
175C2×Q8C22×D7112yes—
176C2×D4—711no—
177C2×D4—711no—
178C2×D4—1422no—
179C2×D4—1422no—
180C2×D4C22×D7111yescharacteristic
181C22×C4—21no—
182C22×C4—712no—
183C22×C4—1422no—
184C22×C4—1421no—
185SD16—42no—
186D8—1422no—
187C2×C8—2842no—
188C2×C8—711no—
189C2×C8—711no—
190C2×C8C22×D7112yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—711no—
194C4⋊C4—712no—
195C4⋊C4—711no—
196C4⋊C4—1422no—
197C4⋊C4—1421no—
198C4⋊C4C22×D7111yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—712no—
201C22⋊C4—1422no—
202C22⋊C4—1422no—
203C22⋊C4—1422no—
204C22⋊C4—1421no—
205C42—1422no—
206C42C22×D7111yescharacteristic
207C23—21no—
208C23—712no—
209C23—1421no—
210Q8—22no—
211Q8—22no—
212Q8—5681no—
213D4—21no—
214D4—21no—
215D4—2844no—
216D4—5682no—
217D4—712no—
218D4—1421no—
219D4—1422no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—2844no—
224C2×C4—2842no—
225C2×C4—712no—
226C2×C4—1422no—
227C2×C4—1421no—
228C2×C4—1422no—
229C2×C4—1422no—
230C2×C4D4×D7111yescharacteristic
231C2×C4D4×D7111yescharacteristic
232C2×C4C23×D7111yescharacteristic
233C8—22no—
234C8—1421no—
235C8—1421no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—2841no—
240C22—2841no—
241C22—1422no—
242C22—1422no—
243C22—1422no—
244C22—1422no—
245C22C2×D4×D7111yescharacteristic
246C4—21no—
247C4—42no—
248C4—41no—
249C4—2841no—
250C4—2842no—
251C4—1422no—
252C4#1455614912yes—
253C4#41×D7111yescharacteristic
254C4#1455617311yescharacteristic
255C2—81no—
256C2—2842no—
257C2—2842no—
258C2#1798664611yescharacteristic
259C2C2×#1455614911yescharacteristic
260C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#461×C71C2
socleC22×C71C22×D4
hypercenter#461D71
hypocenterC71#3303
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #17986714C2semidirect
2. #17986652C2semidirect
3. #17986349C2semidirect
4. #17986338C2semidirect
5. #17986290C2semidirect
6. #17986225C2semidirect
7. #17986143C2semidirect
8. C4×#11765850C2semidirect
9. #17986064C2semidirect
10. #17985994C2semidirect
11. #461×C71C2semidirect
131. C71#3303semidirect
138. #461D71semidirect