Finite Group Databasebeta

← #22588702

Subgroups — #22588702

Order 9088.

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

normal subgroups only

This group has 310 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#17986015C211yesmaximal, characteristic, complemented
3#17985995C211yesmaximal, characteristic, complemented
4#17985967C211yesmaximal, characteristic, complemented
5#17985770C211yesmaximal, characteristic, complemented
6#17985763C211yesmaximal, characteristic, complemented
7#568×C71C211yesmaximal, characteristic, complemented
8C2×#11765865C2211yescharacteristic
9#14556159—21no—
10#14556158C412yescomplemented
11#14556154—21no—
12#14556153C412yescomplemented
13#14556148—22no—
14C2×#11765849C2211yescharacteristic
15C22×C71⋊C8C2211yescharacteristic
16#14556087C2211yescharacteristic
17#14556054—22no—
18#14556052—21no—
19#14556049—21no—
20(C2×D71)⋊C8—22no—
21C2×#41×C71C2211yescharacteristic
22#131×C71—22no—
23#131×C71—22no—
24C2×M16×C71—22no—
25C2×M16×C71C2211yescharacteristic
26C2×M16×C71C2211yescharacteristic
27#11765865—21no—
28#11765865—21no—
29#11765865—21no—
30#11765865—21no—
31C2×Dic142C2×C411yescharacteristic
32C2×D284C2×C411yescharacteristic
33C2×C22⋊D71—21no—
34C2×C4×D71—21no—
35#11765855—22no—
36#11765853—22no—
37#11765852—22no—
38#11765850—22no—
39#11765849—21no—
40#11765849C2×C412yes—
41C2×C71⋊C8—21no—
42C2×C71⋊C8—21no—
43C2×C71⋊C8—21no—
44C2×C71⋊C8—21no—
45C4⋊Dic71—21no—
46C22⋊Dic71—21no—
47C4×Dic71—21no—
48#41×C71—21no—
49#41×C71—21no—
50#41×C71C2×C412yes—
51#41×C71D412yes—
52C2×Q8×C71D411yescharacteristic
53C2×D4×C71—21no—
54C2×D4×C71D411yescharacteristic
55C22×C4×C71—21no—
56C22×C4×C71C2311yescharacteristic
57M16×C71—24no—
58M16×C71—21no—
59M16×C71—21no—
60M16×C71—21no—
61M16×C71D412yes—
62C2×C8×C71—24no—
63C2×C8×C71—21no—
64C2×C8×C71D412yes—
65C23×C71—21no—
66C23×C71#4111yescharacteristic
67C22×D71—21no—
68Q8×C71—21no—
69Q8×C71C22⋊C412yes—
70D4×C71—22no—
71D4×C71—22no—
72D4×C71—21no—
73D4×C71C22⋊C412yes—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71C22×C411yescharacteristic
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71#4111yescharacteristic
83C22⋊D71—41no—
84C22⋊D71—41no—
85C2×Dic71—21no—
86C2×Dic71—41no—
87D284—21no—
88D284—21no—
89C4×D71—41no—
90C4×D71—41no—
91Dic142—21no—
92Dic142—21no—
93C8×C71—22no—
94C8×C71—22no—
95C71⋊C8—22no—
96C71⋊C8—41no—
97C22×C71—22no—
98C22×C71—41no—
99C22×C71—81no—
100C22×C71C4×D411yescharacteristic
101C22×C71#12111yescharacteristic
102C22×C71#12311yescharacteristic
103C2×D71—42no—
104C2×D71—41no—
105C4×C71—22no—
106C4×C71C2×C22⋊C411yescharacteristic
107C4×C71C4×D411yescharacteristic
108C4×C71#12011yescharacteristic
109C4×C71#12111yescharacteristic
110Dic71—41no—
111Dic71—81no—
112C2×C71—21no—
113C2×C71—21no—
114C2×C71—21no—
115C2×C71—42no—
116C2×C71#38611yescharacteristic
117D71—81no—
118#1881—711nomaximal, Sylow
119C71#188111yescharacteristic, complemented, Sylow
120#575—711no—
121#568D7111yescharacteristic, complemented
122#421—711no—
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—22no—
135#131—22no—
136C2×M16—22no—
137C2×M16—711no—
138C2×M16C2×D7111yescharacteristic
139C2×M16C2×D7111yescharacteristic
140C22×C8—711no—
141#117—711no—
142#104—1422no—
143#103—1421no—
144#102—1421no—
145C22⋊C8—1422no—
146#41—21no—
147#41—21no—
148#41—1421no—
149#41—1421no—
150#41—1421no—
151#41—1421no—
152#41C4×D7112yes—
153#41C22⋊D7112yes—
154C2×Q8—711no—
155C2×Q8D28411yescharacteristic
156C2×D4—21no—
157C2×D4—711no—
158C2×D4—1421no—
159C2×D4D28411yescharacteristic
160C22×C4—21no—
161C22×C4—1421no—
162C22×C4C22×D7111yescharacteristic
163Q16—1422no—
164SD16—1422no—
165SD16—1422no—
166D8—1422no—
167M16—24no—
168M16—21no—
169M16—21no—
170M16—21no—
171M16—712no—
172M16—1421no—
173M16C22⋊D7112yes—
174C2×C8—24no—
175C2×C8—21no—
176C2×C8—1421no—
177C2×C8—1421no—
178C2×C8—1421no—
179C2×C8—1421no—
180C2×C8C22⋊D7112yes—
181C4⋊C4—1421no—
182C22⋊C4—1421no—
183C42—1421no—
184C23—21no—
185C23—1421no—
186C23#1176586511yescharacteristic
187Q8—21no—
188Q8—1421no—
189Q8—1421no—
190Q8(C2×D71)⋊C412yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—2841no—
195D4—2841no—
196D4—1421no—
197D4—1421no—
198D4(C2×D71)⋊C412yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—2841no—
203C2×C4—2841no—
204C2×C4—2841no—
205C2×C4—1421no—
206C2×C4C2×C4×D7111yescharacteristic
207C2×C4C2×C22⋊D7111yescharacteristic
208C2×C4C2×C22⋊D7111yescharacteristic
209C2×C4C2×C22⋊D7111yescharacteristic
210C2×C4C2×D28411yescharacteristic
211C2×C4#1176586511yescharacteristic
212C8—22no—
213C8—22no—
214C8—2841no—
215C8—1422no—
216C22—22no—
217C22—41no—
218C22—81no—
219C22—2842no—
220C22—2841no—
221C22C4×C22⋊D7111yescharacteristic
222C22#1455610211yescharacteristic
223C22#1455611111yescharacteristic
224C4—22no—
225C4—2841no—
226C4—5681no—
227C4C2×(C2×D71)⋊C411yescharacteristic
228C4C4×C22⋊D7111yescharacteristic
229C4#1455609711yescharacteristic
230C4#1455610211yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—5681no—
236C2#1798587811yescharacteristic
subgroupG/H
centerC4#14556097
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#568×C71C2
socleC2×C71#386
hypercenter#568D71
hypocenterC71#1881
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #17986015C2semidirect
3. #17985995C2semidirect
4. #17985967C2semidirect
5. #17985770C2semidirect
6. #17985763C2semidirect
7. #568×C71C2semidirect
10. #14556158C4semidirect
12. #14556153C4semidirect
119. C71#1881semidirect
121. #568D71semidirect