Finite Group Databasebeta

← #22588841

Subgroups — #22588841

Order 9088.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#17986797C211yesmaximal, characteristic
2#17986016C211yesmaximal, characteristic, complemented
3C2×#14556056C211yesmaximal, characteristic, complemented
4#17985972C211yesmaximal, characteristic, complemented
5#17985769C211yesmaximal, characteristic, complemented
6#17985759C211yesmaximal, characteristic, complemented
7C4×#41×C71C211yesmaximal, characteristic, complemented
8C2×#11765865C2211yescharacteristic
9#14556159—21no—
10#14556158C412yescomplemented
11#14556154—21no—
12#14556153C412yescomplemented
13#14556148—22no—
14C2×#11765849C2211yescharacteristic
15C2×#11765849C2211yescharacteristic
16#14556087C2211yescharacteristic
17#14556056—21no—
18#14556056—21no—
19#14556055—22no—
20#14556042—22no—
21C2×#41×C71C2211yescharacteristic
22C4×Q8×C71—21no—
23C4×D4×C71—21no—
24C4×D4×C71—22no—
25#117×C71—21no—
26#117×C71C2211yescharacteristic
27C2×C42×C71—21no—
28C2×C42×C71C2211yescharacteristic
29#11765865—21no—
30#11765865—21no—
31#11765865—21no—
32#11765865—21no—
33C2×Dic142C2×C411yescharacteristic, complemented
34C2×D284C2×C411yescharacteristic, complemented
35C2×C22⋊D71—21no—
36C2×C4×D71—21no—
37#11765855—22no—
38#11765853—22no—
39#11765852—22no—
40#11765850—22no—
41#11765849—21no—
42#11765849—21no—
43#11765849—21no—
44#11765849C2×C412yescomplemented
45C2×C71⋊C8—21no—
46C2×C71⋊C8—21no—
47C4⋊Dic71—21no—
48C22⋊Dic71—21no—
49C4×Dic71—21no—
50#41×C71—21no—
51#41×C71—21no—
52#41×C71C2×C412yes—
53#41×C71D412yes—
54C2×Q8×C71D411yescharacteristic
55C2×D4×C71—21no—
56C2×D4×C71D411yescharacteristic
57C22×C4×C71—21no—
58C22×C4×C71—22no—
59C22×C4×C71—21no—
60C22×C4×C71C2311yescharacteristic
61C4⋊C4×C71—22no—
62C4⋊C4×C71—21no—
63C22⋊C4×C71—22no—
64C22⋊C4×C71D412yes—
65C42×C71—22no—
66C42×C71—21no—
67C42×C71—21no—
68C42×C71D411yescharacteristic
69C42×C71D411yescharacteristic
70C23×C71—21no—
71C23×C71C2×D411yescharacteristic
72C22×D71—21no—
73Q8×C71—21no—
74Q8×C71C22⋊C412yes—
75D4×C71—22no—
76D4×C71—22no—
77D4×C71—21no—
78D4×C71C22⋊C412yes—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—21no—
85C2×C4×C71—42no—
86C2×C4×C71—44no—
87C2×C4×C71C22×C411yescharacteristic
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71C2×D411yescharacteristic
91C2×C4×C71#4111yescharacteristic
92C2×C4×C71#4111yescharacteristic
93C22⋊D71—41no—
94C22⋊D71—41no—
95C2×Dic71—21no—
96C2×Dic71—41no—
97D284—21no—
98D284—21no—
99C4×D71—41no—
100C4×D71—41no—
101Dic142—21no—
102Dic142—21no—
103C71⋊C8—22no—
104C71⋊C8—41no—
105C22×C71—22no—
106C22×C71—41no—
107C22×C71—42no—
108C22×C71C4×D411yescharacteristic
109C22×C71#12011yescharacteristic
110C22×C71#12111yescharacteristic
111C2×D71—41no—
112C2×D71—81no—
113C4×C71—22no—
114C4×C71—22no—
115C4×C71—42no—
116C4×C71—41no—
117C4×C71C2×C22⋊C411yescharacteristic
118C4×C71C4×D411yescharacteristic
119C4×C71#12111yescharacteristic
120C4×C71#12311yescharacteristic
121Dic71—41no—
122Dic71—81no—
123C2×C71—21no—
124C2×C71—21no—
125C2×C71—21no—
126C2×C71—42no—
127C2×C71#38611yescharacteristic
128D71—81no—
129#1906—711nomaximal, Sylow
130C71#190611yescharacteristic, complemented, Sylow
131#575—711no—
132C4×#41D7111yescharacteristic, complemented
133#421—711no—
134C2×#104—711no—
135#410—711no—
136#344—711no—
137#339—711no—
138C2×#41—711no—
139C2×#41C2×D7111yescharacteristic
140#137—712no—
141#137—1421no—
142AGL(1,ℤ/8)—712no—
143AGL(1,ℤ/8)—1421no—
144#135—1422no—
145C2×M16—711no—
146C2×M16—711no—
147C4×Q8—21no—
148C4×D4—21no—
149C4×D4—22no—
150#117—21no—
151#117—711no—
152#117C2×D7111yescharacteristic
153C2×C42—21no—
154C2×C42C2×D7111yescharacteristic
155#104—1422no—
156#104—1421no—
157#104—1421no—
158#99—1422no—
159#41—21no—
160#41—21no—
161#41—1421no—
162#41—1421no—
163#41—1421no—
164#41—1421no—
165#41C4×D7112yes—
166#41D28412yes—
167C2×Q8—711no—
168C2×Q8C22⋊D7111yescharacteristic
169C2×D4—21no—
170C2×D4—711no—
171C2×D4—1421no—
172C2×D4C22⋊D7111yescharacteristic
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—1421no—
177C22×C4C22×D7111yescharacteristic
178Q16—1422no—
179SD16—1422no—
180SD16—1422no—
181D8—1422no—
182M16—712no—
183M16—1421no—
184M16—1421no—
185M16—1421no—
186C2×C8—1421no—
187C2×C8—1421no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—1421no—
191C22⋊C4—22no—
192C22⋊C4—1421no—
193C22⋊C4C22⋊D7112yes—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—1421no—
198C42C22⋊D7111yescharacteristic
199C42C22⋊D7111yescharacteristic
200C23—21no—
201C23—1421no—
202C23C2×C22⋊D7111yescharacteristic
203Q8—21no—
204Q8—1421no—
205Q8—1421no—
206Q8(C2×D71)⋊C412yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—2841no—
211D4—2841no—
212D4—1421no—
213D4—1421no—
214D4(C2×D71)⋊C412yes—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—2841no—
224C2×C4—2841no—
225C2×C4—2841no—
226C2×C4—1421no—
227C2×C4C2×C4×D7111yescharacteristic
228C2×C4C2×C22⋊D7111yescharacteristic
229C2×C4C2×C22⋊D7111yescharacteristic
230C2×C4C2×D28411yescharacteristic
231C2×C4#1176586511yescharacteristic
232C2×C4#1176586511yescharacteristic
233C8—2841no—
234C8—1422no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—2841no—
239C22—5681no—
240C22C4×C22⋊D7111yescharacteristic
241C22#1455609711yescharacteristic
242C22#1455610211yescharacteristic
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—2841no—
248C4—5681no—
249C4C2×(C2×D71)⋊C411yescharacteristic
250C4C4×C22⋊D7111yescharacteristic
251C4#1455610211yescharacteristic
252C4#1455611111yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—5681no—
258C2#1798587811yescharacteristic
subgroupG/H
centerC4#14556102
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
FittingC4×#41×C71C2
socleC2×C71#386
hypercenterC4×#41D71
hypocenterC71#1906
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #17986016C2semidirect
3. C2×#14556056C2semidirect
4. #17985972C2semidirect
5. #17985769C2semidirect
6. #17985759C2semidirect
7. C4×#41×C71C2semidirect
10. #14556158C4semidirect
12. #14556153C4semidirect
33. C2×Dic142C2×C4semidirect
34. C2×D284C2×C4semidirect
44. #11765849C2×C4semidirect
130. C71#1906semidirect
132. C4×#41D71semidirect