Finite Group Databasebeta

← #22598305

Subgroups — #22598305

Order 9088.

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

normal subgroups only

This group has 386 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#17986573C211yesmaximal, characteristic
2#17986480C211yesmaximal, characteristic
3#17986339C212yesmaximal, complemented
4#17986338C212yesmaximal, complemented
5#17986333C212yesmaximal, complemented
6#17986328C212yesmaximal, complemented
7#17986039C211yesmaximal, characteristic, complemented
8#17986037C211yesmaximal, characteristic, complemented
9#17986008C211yesmaximal, characteristic, complemented
10#17985965C211yesmaximal, characteristic
11C2×#103×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C2×#11765849C2211yescharacteristic
14#14556126—21no—
15#14556117—21no—
16#14556111—22no—
17#14556104C2212yes—
18#14556104C2212yes—
19#14556093—21no—
20#14556090—21no—
21C4×C22⋊D71—22no—
22#14556087C2211yescharacteristic
23#14556087C2211yescharacteristic
24#14556086C2212yes—
25C4⋊C4×D71C2212yes—
26#14556062C2211yescharacteristic
27#14556062C2211yescharacteristic
28#14556061C2212yes—
29#14556060C2211yescharacteristic
30#14556060C2211yescharacteristic
31#14556059C2212yes—
32#14556052C2212yes—
33#14556052C2212yes—
34(C2×D71)⋊C8C2212yes—
35(C2×D71)⋊C8C2212yes—
36C22×Q8×C71C2211yescharacteristic
37C22×C8×C71C2211yescharacteristic
38C2×C4⋊C4×C71C2211yescharacteristic
39#103×C71C2212yes—
40#103×C71C2212yes—
41#11765865—22no—
42#11765865—22no—
43C2×Dic142D411yescharacteristic
44C2×D284D411yescharacteristic
45C2×C22⋊D71D412yes—
46C2×C4×D71—22no—
47C2×C4×D71C2312yes—
48#11765849—22no—
49C2×C71⋊C8C2312yes—
50C4⋊Dic71C2311yescharacteristic
51C4⋊Dic71C2311yescharacteristic
52C4⋊Dic71C2311yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57C22⋊Dic71—21no—
58C22⋊Dic71—21no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—22no—
62C4×Dic71—21no—
63C4×Dic71—21no—
64C4×Dic71—22no—
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×D71C2×D412yes—
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—24no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—21no—
94C2×Dic71—21no—
95C2×Dic71C2×D412yes—
96D284—22no—
97C4×D71—22no—
98C4×D71—22no—
99C4×D71—82no—
100Dic142—22no—
101C8×C71—21no—
102C8×C71—21no—
103C71⋊C8—22no—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71#12312yes—
107C22×C71C22×D411yescharacteristic
108C2×D71—22no—
109C2×D71—22no—
110C2×D71—22no—
111C4×C71—42no—
112C4×C71—42no—
113C4×C71#12312yes—
114C4×C71C2×#4111yescharacteristic
115C4×C71C2×#4111yescharacteristic
116Dic71—22no—
117Dic71—41no—
118Dic71—41no—
119Dic71—41no—
120Dic71—41no—
121C2×C71—22no—
122C2×C71C2×#12311yescharacteristic
123C2×C71#57711yescharacteristic
124C2×C71#57811yescharacteristic
125D71—42no—
126#3097—711nomaximal, Sylow
127C71#309711yescharacteristic, complemented, Sylow
128#536—711no—
129#520—711no—
130#485—712no—
131#485—712no—
132#484—712no—
133#483—712no—
134#428—711no—
135#427—711no—
136#419—711no—
137C2×#103D7111yescharacteristic, complemented
138#408—711no—
139C2×#41—711no—
140C22×Q8C2×D7111yescharacteristic
141C2×M16—711no—
142C22×C8C2×D7111yescharacteristic
143C4⋊Q8—1421no—
144#124—1421no—
145#123—1422no—
146C22⋊Q8—712no—
147C22⋊Q8—712no—
148C4×Q8—1421no—
149C4×D4—1421no—
150C4×D4—1422no—
151#117—711no—
152#117—711no—
153#117—712no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4C2×D7111yescharacteristic
156#107—711no—
157#107—711no—
158#107—712no—
159#106—711no—
160#106—711no—
161#106—712no—
162#103—712no—
163#103—712no—
164#103C2×D7112yes—
165#103C2×D7112yes—
166C22⋊C8—712no—
167C22⋊C8—712no—
168#41—1422no—
169#41—1422no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—711no—
174C2×Q8C22×D7112yes—
175C2×D4—712no—
176C2×D4—711no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—712no—
180C22×C4—1422no—
181C22×C4C22×D7111yescharacteristic
182M16—1422no—
183C2×C8—22no—
184C2×C8—712no—
185C2×C8C22×D7111yescharacteristic
186C2×C8C22×D7111yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—711no—
189C4⋊C4—711no—
190C4⋊C4—711no—
191C4⋊C4—711no—
192C4⋊C4—1422no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C4⋊C4C22×D7112yes—
196C22⋊C4—1422no—
197C22⋊C4—1422no—
198C22⋊C4—1422no—
199C22⋊C4—1421no—
200C22⋊C4—1421no—
201C42—1422no—
202C42—1421no—
203C42—1421no—
204C23—712no—
205C23D4×D7111yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—42no—
209Q8—1422no—
210D4—1424no—
211D4—1422no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—5682no—
217C2×C4—712no—
218C2×C4—1421no—
219C2×C4—1421no—
220C2×C4—1421no—
221C2×C4—1421no—
222C2×C4—1422no—
223C2×C4—1422no—
224C2×C4D4×D7111yescharacteristic
225C2×C4#1176586412yes—
226C2×C4#1176586512yes—
227C2×C4C23×D7111yescharacteristic
228C8—21no—
229C8—21no—
230C8—1422no—
231C22—21no—
232C22—21no—
233C22—1422no—
234C22—1422no—
235C22—1422no—
236C22#1455611012yes—
237C22C2×D4×D7111yescharacteristic
238C4—42no—
239C4—42no—
240C4—2841no—
241C4—2841no—
242C4—2841no—
243C4—2841no—
244C4—1422no—
245C4#1455611012yes—
246C4C2×#1176586411yescharacteristic
247C4C2×#1176586511yescharacteristic
248C2—22no—
249C2—2842no—
250C2C2×#1455611011yescharacteristic
251C2#1798680511yescharacteristic
252C2#1798681211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#103×C71C2
socleC22×C71C22×D4
hypercenterC2×#103D71
hypocenterC71#3097
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
3. #17986339C2semidirect
4. #17986338C2semidirect
5. #17986333C2semidirect
6. #17986328C2semidirect
7. #17986039C2semidirect
8. #17986037C2semidirect
9. #17986008C2semidirect
11. C2×#103×C71C2semidirect
127. C71#3097semidirect
137. C2×#103D71semidirect