Finite Group Databasebeta

← #22598703

Subgroups — #22598703

Order 9088.

372 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986630C211yesmaximal, characteristic, complemented
2#17986573C211yesmaximal, characteristic
3#17986366C212yesmaximal, complemented
4#17986360C212yesmaximal, complemented
5#17986355C212yesmaximal, complemented
6#17986348C212yesmaximal, complemented
7#17986008C211yesmaximal, characteristic, complemented
8#17985998C211yesmaximal, characteristic
9#17985996C211yesmaximal, characteristic
10#17985954C211yesmaximal, characteristic, complemented
11C2×#103×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C2×#11765849C2211yescharacteristic
14#14556127—21no—
15#14556127C2212yes—
16#14556126C2212yes—
17#14556119—21no—
18#14556119C2212yes—
19#14556117C2212yes—
20#14556111—22no—
21#14556107—22no—
22#14556104—21no—
23#14556104—21no—
24#14556087C2211yescharacteristic
25#14556087C2211yescharacteristic
26#14556053C2211yescharacteristic
27#14556053C2211yescharacteristic
28#14556052C2212yes—
29#14556052C2212yes—
30#14556051C2212yes—
31#14556050C2211yescharacteristic
32#14556050C2211yescharacteristic
33#14556048C2212yes—
34#14556038C2212yes—
35C8×Dic71C2212yes—
36C22×Q8×C71C2211yescharacteristic
37C22×C8×C71C2211yescharacteristic
38C2×C4⋊C4×C71C2211yescharacteristic
39#103×C71C2212yes—
40#103×C71C2212yes—
41#11765865—22no—
42#11765865—22no—
43C2×Dic142—22no—
44C2×Dic142C2311yescharacteristic
45C2×D284C2311yescharacteristic
46C2×C22⋊D71—21no—
47C2×C4×D71—21no—
48#11765849—22no—
49C2×C71⋊C8C2312yes—
50C4⋊Dic71—22no—
51C4⋊Dic71D411yescharacteristic
52C4⋊Dic71D411yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2311yescharacteristic
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59C22⋊Dic71—21no—
60C22⋊Dic71D412yes—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63C4×Dic71—21no—
64C4×Dic71C2312yes—
65C2×Q8×C71—21no—
66C2×Q8×C71—22no—
67C2×Q8×C71—22no—
68C2×Q8×C71C2312yes—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71C2311yescharacteristic
72C2×C8×C71—22no—
73C2×C8×C71C2311yescharacteristic
74C2×C8×C71C2311yescharacteristic
75C4⋊C4×C71—21no—
76C4⋊C4×C71C2312yes—
77C23×C71C2×D411yescharacteristic
78C22×D71—21no—
79Q8×C71—22no—
80Q8×C71—22no—
81Q8×C71—42no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—42no—
85C2×C4×C71—81no—
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71#4112yes—
88C2×C4×C71#4112yes—
89C2×C4×C71C2411yescharacteristic
90C22⋊D71—42no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—21no—
94C2×Dic71C2×D412yes—
95C2×Dic71C2×D412yes—
96D284—21no—
97D284—21no—
98C4×D71—42no—
99Dic142—21no—
100Dic142—21no—
101Dic142—82no—
102C8×C71—21no—
103C8×C71—21no—
104C71⋊C8—22no—
105C22×C71—21no—
106C22×C71—21no—
107C22×C71#12412yes—
108C22×C71C22×D411yescharacteristic
109C2×D71—41no—
110C2×D71—41no—
111C2×D71—41no—
112C4×C71—42no—
113C4×C71—42no—
114C4×C71#12412yes—
115C4×C71C2×#4111yescharacteristic
116C4×C71C2×#4111yescharacteristic
117Dic71—22no—
118Dic71—22no—
119Dic71—41no—
120Dic71—41no—
121Dic71—41no—
122C2×C71—22no—
123C2×C71C2×#12411yescharacteristic
124C2×C71#57711yescharacteristic
125C2×C71#57811yescharacteristic
126D71—81no—
127#3137—711nomaximal, Sylow
128C71#313711yescharacteristic, complemented, Sylow
129#544—711no—
130#536—711no—
131#491—712no—
132#489—712no—
133#488—712no—
134#487—712no—
135#419—711no—
136#417—711no—
137#416—711no—
138C2×#103D7111yescharacteristic, complemented
139#405—711no—
140C2×#41—711no—
141C22×Q8C2×D7111yescharacteristic
142C2×M16—711no—
143C22×C8C2×D7111yescharacteristic
144C4⋊Q8—712no—
145C4⋊Q8—712no—
146C4⋊Q8—1421no—
147#125—712no—
148#125—1421no—
149#124—712no—
150#123—1422no—
151C22⋊Q8—1422no—
152C22⋊Q8—1421no—
153C22⋊Q8—1421no—
154#117—711no—
155#117—711no—
156C2×C4⋊C4C2×D7111yescharacteristic
157#103—712no—
158#103—712no—
159#103—711no—
160#103—711no—
161#103—712no—
162#103C2×D7112yes—
163#103C2×D7112yes—
164#102—711no—
165#102—711no—
166#102—712no—
167#97—712no—
168C4×C8—712no—
169#41—1422no—
170#41—1422no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—711no—
175C2×Q8—1422no—
176C2×Q8C22×D7112yes—
177C2×D4—711no—
178C2×D4—1421no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—1421no—
182C22×C4C22×D7111yescharacteristic
183M16—1422no—
184C2×C8—22no—
185C2×C8—712no—
186C2×C8C22×D7111yescharacteristic
187C2×C8C22×D7111yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—711no—
190C4⋊C4—711no—
191C4⋊C4—711no—
192C4⋊C4—711no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C4⋊C4—1422no—
196C4⋊C4—1422no—
197C4⋊C4—1422no—
198C4⋊C4C22×D7112yes—
199C22⋊C4—712no—
200C22⋊C4—1422no—
201C22⋊C4—1422no—
202C22⋊C4—1421no—
203C42—712no—
204C42—1421no—
205C23—1421no—
206C23D4×D7111yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—42no—
210Q8—5682no—
211Q8—1421no—
212Q8—1421no—
213D4—2842no—
214D4—1421no—
215D4—1421no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—2842no—
221C2×C4—712no—
222C2×C4—712no—
223C2×C4—1421no—
224C2×C4—1421no—
225C2×C4—1421no—
226C2×C4D4×D7111yescharacteristic
227C2×C4#1176586412yes—
228C2×C4#1176586512yes—
229C2×C4C23×D7111yescharacteristic
230C8—21no—
231C8—21no—
232C8—1422no—
233C22—21no—
234C22—21no—
235C22—2841no—
236C22—2841no—
237C22—2841no—
238C22#1455611412yes—
239C22C2×D4×D7111yescharacteristic
240C4—42no—
241C4—42no—
242C4—2841no—
243C4—2841no—
244C4—2841no—
245C4—1422no—
246C4—1422no—
247C4#1455611412yes—
248C4C2×#1176586411yescharacteristic
249C4C2×#1176586511yescharacteristic
250C2—22no—
251C2—5681no—
252C2C2×#1455611411yescharacteristic
253C2#1798680511yescharacteristic
254C2#1798681211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#103×C71C2
socleC22×C71C22×D4
hypercenterC2×#103D71
hypocenterC71#3137
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. #17986630C2semidirect
3. #17986366C2semidirect
4. #17986360C2semidirect
5. #17986355C2semidirect
6. #17986348C2semidirect
7. #17986008C2semidirect
10. #17985954C2semidirect
11. C2×#103×C71C2semidirect
128. C71#3137semidirect
138. C2×#103D71semidirect