Finite Group Databasebeta

← #22588699

Subgroups — #22588699

Order 9088.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#17986765C211yesmaximal, characteristic, complemented
3#17985993C211yesmaximal, characteristic, complemented
4#17985966C211yesmaximal, characteristic, complemented
5#17985769C211yesmaximal, characteristic, complemented
6#17985763C211yesmaximal, characteristic, complemented
7#421×C71C211yesmaximal, characteristic, complemented
8C2×#11765865C2211yescharacteristic
9#14556159C412yescomplemented
10#14556158—21no—
11#14556154C412yescomplemented
12#14556153—21no—
13#14556148—22no—
14#14556136—22no—
15#14556134—22no—
16C2×#11765849C2211yescharacteristic
17C2×#11765849C2211yescharacteristic
18M16×D71—22no—
19C22×C71⋊C8C2211yescharacteristic
20#14556051—21no—
21#14556048—21no—
22C22⋊(C71⋊C8)—22no—
23C2×#41×C71C2211yescharacteristic
24C2×M16×C71C2211yescharacteristic
25#117×C71C2211yescharacteristic
26#104×C71—22no—
27#11765865—21no—
28#11765865—21no—
29#11765865C2×C412yes—
30#11765865D412yes—
31C2×Dic142D411yescharacteristic
32C2×D284D411yescharacteristic
33C2×C22⋊D71—21no—
34C2×C4×D71—21no—
35#11765855—22no—
36#11765853—22no—
37#11765852—22no—
38#11765850—22no—
39#11765849—21no—
40#11765849—21no—
41#11765849—21no—
42#11765849C2×C412yes—
43#11765848—24no—
44C2×C71⋊C8—21no—
45C2×C71⋊C8—21no—
46C2×C71⋊C8—21no—
47C2×C71⋊C8—21no—
48C2×C71⋊C8D412yes—
49C8×D71—24no—
50#41×C71—21no—
51#41×C71—21no—
52#41×C71—21no—
53#41×C71—21no—
54C2×Q8×C71C2×C411yescharacteristic
55C2×D4×C71—21no—
56C2×D4×C71C2×C411yescharacteristic
57C22×C4×C71—21no—
58C22×C4×C71C2311yescharacteristic
59M16×C71—21no—
60M16×C71D412yes—
61C2×C8×C71—21no—
62C4⋊C4×C71—21no—
63C22⋊C4×C71—21no—
64C42×C71—21no—
65C23×C71—21no—
66C23×C71#4111yescharacteristic
67C22×D71—21no—
68Q8×C71—21no—
69Q8×C71—21no—
70D4×C71—21no—
71D4×C71—21no—
72D4×C71—41no—
73D4×C71—41no—
74C2×C4×C71—21no—
75C2×C4×C71—41no—
76C2×C4×C71—41no—
77C2×C4×C71—41no—
78C2×C4×C71C22×C411yescharacteristic
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71#4111yescharacteristic
84C22⋊D71—22no—
85C22⋊D71—22no—
86C2×Dic71—21no—
87D284—21no—
88D284C22⋊C412yes—
89C4×D71—22no—
90C4×D71—22no—
91Dic142—21no—
92Dic142C22⋊C412yes—
93C8×C71—22no—
94C71⋊C8—22no—
95C71⋊C8—22no—
96C71⋊C8—41no—
97C22×C71—41no—
98C22×C71—42no—
99C22×C71—41no—
100C22×C71C4×D411yescharacteristic
101C22×C71#12111yescharacteristic
102C22×C71#12311yescharacteristic
103C2×D71—22no—
104C2×D71—81no—
105C4×C71—41no—
106C4×C71—81no—
107C4×C71C2×C22⋊C411yescharacteristic
108C4×C71C4×D411yescharacteristic
109C4×C71#12011yescharacteristic
110C4×C71#12111yescharacteristic
111Dic71—22no—
112C2×C71—21no—
113C2×C71—21no—
114C2×C71—21no—
115C2×C71—81no—
116C2×C71#38611yescharacteristic
117D71—42no—
118#1881—711nomaximal, Sylow
119C71#188111yescharacteristic, complemented, Sylow
120#575—711no—
121#568—711no—
122#421D7111yescharacteristic, complemented
123#416—711no—
124#408—711no—
125#344—711no—
126#341—711no—
127C2×#41—711no—
128C2×#41C2×D7111yescharacteristic
129#137—712no—
130#137—1421no—
131AGL(1,ℤ/8)—712no—
132AGL(1,ℤ/8)—1421no—
133#135—1422no—
134#131—1422no—
135#131—1422no—
136C2×M16—711no—
137C2×M16—711no—
138C2×M16—1422no—
139C2×M16C2×D7111yescharacteristic
140C22×C8—711no—
141#117C2×D7111yescharacteristic
142#104—22no—
143#103—1421no—
144#102—1421no—
145C22⋊C8—1422no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—712no—
151#41—712no—
152#41—1421no—
153#41—1421no—
154C2×Q8—711no—
155C2×Q8C4×D7111yescharacteristic
156C2×D4—21no—
157C2×D4—711no—
158C2×D4—1421no—
159C2×D4C4×D7111yescharacteristic
160C22×C4—21no—
161C22×C4—1421no—
162C22×C4C22×D7111yescharacteristic
163Q16—1422no—
164SD16—1422no—
165SD16—1422no—
166D8—1422no—
167M16—21no—
168M16—712no—
169M16—1424no—
170M16—1421no—
171M16—1421no—
172M16—1421no—
173M16C22⋊D7112yes—
174C2×C8—21no—
175C2×C8—712no—
176C2×C8—1424no—
177C2×C8—1421no—
178C2×C8—1421no—
179C2×C8—1421no—
180C2×C8—1421no—
181C4⋊C4—21no—
182C22⋊C4—21no—
183C42—21no—
184C23—21no—
185C23—1421no—
186C23#1176586411yescharacteristic
187Q8—21no—
188Q8—21no—
189Q8—712no—
190Q8—1421no—
191D4—21no—
192D4—21no—
193D4—41no—
194D4—41no—
195D4—712no—
196D4—1422no—
197D4—1422no—
198D4—1421no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—1421no—
204C2×C4—1422no—
205C2×C4—1422no—
206C2×C4C2×C4×D7111yescharacteristic
207C2×C4D4×D7111yescharacteristic
208C2×C4D4×D7111yescharacteristic
209C2×C4D4×D7111yescharacteristic
210C2×C4C2×C22⋊D7111yescharacteristic
211C2×C4#1176586511yescharacteristic
212C8—22no—
213C8—2841no—
214C8—1422no—
215C8—1422no—
216C22—41no—
217C22—42no—
218C22—41no—
219C22—5681no—
220C22—1422no—
221C22(C22⋊D71)⋊C411yescharacteristic
222C22#1455609911yescharacteristic
223C22#1455611011yescharacteristic
224C4—41no—
225C4—81no—
226C4—1422no—
227C4C22⋊C4×D7111yescharacteristic
228C4C4×C22⋊D7111yescharacteristic
229C4#1455609611yescharacteristic
230C4#1455610111yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—2842no—
236C2#1798587511yescharacteristic
subgroupG/H
centerC4#14556096
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#421×C71C2
socleC2×C71#386
hypercenter#421D71
hypocenterC71#1881
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #17986765C2semidirect
3. #17985993C2semidirect
4. #17985966C2semidirect
5. #17985769C2semidirect
6. #17985763C2semidirect
7. #421×C71C2semidirect
9. #14556159C4semidirect
11. #14556154C4semidirect
119. C71#1881semidirect
122. #421D71semidirect