Finite Group Databasebeta

← #22588859

Subgroups — #22588859

Order 9088.

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

normal subgroups only

This group has 312 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#17986765C211yesmaximal, characteristic
2#17986573C211yesmaximal, characteristic
3#17986008C211yesmaximal, characteristic, complemented
4#17985972C211yesmaximal, characteristic
5#17985756C211yesmaximal, characteristic, complemented
6#17985755C211yesmaximal, characteristic, complemented
7#419×C71C211yesmaximal, characteristic, complemented
8C2×#11765865C2211yescharacteristic
9#14556136—22no—
10#14556134—22no—
11C2×#11765849C2211yescharacteristic
12M16×D71—22no—
13#14556126C412yescomplemented
14#14556117C412yescomplemented
15#14556111—22no—
16#14556104—22no—
17#14556087C2211yescharacteristic
18#14556087C2211yescharacteristic
19#14556052—22no—
20#14556042—22no—
21C22×Q8×C71C2211yescharacteristic
22C2×M16×C71C2211yescharacteristic
23#117×C71C2211yescharacteristic
24#103×C71—22no—
25#11765865—21no—
26#11765865—21no—
27#11765865—21no—
28#11765865—21no—
29C2×Dic142C2×C411yescharacteristic
30C2×D284C2×C411yescharacteristic
31C2×C22⋊D71D412yes—
32C2×C4×D71D412yes—
33#11765849—21no—
34#11765849—21no—
35#11765848—24no—
36C2×C71⋊C8D412yes—
37C8×D71—24no—
38C4⋊Dic71—21no—
39C4⋊Dic71—21no—
40Dic71⋊C4—24no—
41C22⋊Dic71—21no—
42C22⋊Dic71—21no—
43(C2×D71)⋊C4—24no—
44C4×Dic71—21no—
45C4×Dic71C2×C412yes—
46C2×Q8×C71—21no—
47C2×Q8×C71—21no—
48C2×Q8×C71—21no—
49C2×Q8×C71—21no—
50C2×Q8×C71—21no—
51C2×Q8×C71C2×C412yes—
52C22×C4×C71—21no—
53C22×C4×C71C2311yescharacteristic
54M16×C71—21no—
55M16×C71—21no—
56C2×C8×C71D412yes—
57C4⋊C4×C71—21no—
58C22⋊C4×C71—21no—
59C42×C71—21no—
60C23×C71C2×D411yescharacteristic
61C22×D71C22⋊C412yes—
62Q8×C71—22no—
63Q8×C71—22no—
64Q8×C71—41no—
65Q8×C71—41no—
66C2×C4×C71—22no—
67C2×C4×C71—41no—
68C2×C4×C71—41no—
69C2×C4×C71—41no—
70C2×C4×C71C22×C411yescharacteristic
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71#4111yescharacteristic
75C2×C4×C71#4111yescharacteristic
76C22⋊D71—22no—
77C22⋊D71—22no—
78C2×Dic71—22no—
79C2×Dic71—41no—
80C2×Dic71C22⋊C412yes—
81D284—21no—
82D284—21no—
83C4×D71—22no—
84C4×D71—22no—
85Dic142—21no—
86Dic142—21no—
87C8×C71—22no—
88C71⋊C8—22no—
89C22×C71—41no—
90C22×C71C2×C22⋊C411yescharacteristic
91C22×C71#12111yescharacteristic
92C22×C71#12111yescharacteristic
93C2×D71—22no—
94C2×D71—42no—
95C4×C71—42no—
96C4×C71—81no—
97C4×C71C4×D411yescharacteristic
98C4×C71C4×D411yescharacteristic
99C4×C71#12011yescharacteristic
100C4×C71#12311yescharacteristic
101Dic71—22no—
102Dic71—42no—
103Dic71—81no—
104C2×C71—21no—
105C2×C71—21no—
106C2×C71—21no—
107C2×C71#38611yescharacteristic
108D71—42no—
109#1909—711nomaximal, Sylow
110C71#190911yescharacteristic, complemented, Sylow
111#568—711no—
112#536—711no—
113#419—711no—
114#419D7111yescharacteristic, complemented
115#410—711no—
116H(3,ℤ/4)—711no—
117H(3,ℤ/4)—711no—
118C2×#41—711no—
119C22×Q8C2×D7111yescharacteristic
120#131—1422no—
121#131—1422no—
122C2×M16—711no—
123C2×M16—1422no—
124C2×M16C2×D7111yescharacteristic
125C4⋊Q8—712no—
126#124—712no—
127#123—1422no—
128C22⋊Q8—1422no—
129#117—711no—
130#117—711no—
131#117C2×D7111yescharacteristic
132#103—22no—
133#103—1422no—
134#99—1422no—
135#41—1421no—
136#41—1421no—
137#41—1421no—
138#41—1421no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—711no—
145C2×Q8C4×D7112yes—
146C2×D4—712no—
147C2×D4—711no—
148C22×C4—21no—
149C22×C4—712no—
150C22×C4C22×D7111yescharacteristic
151M16—21no—
152M16—21no—
153M16—1424no—
154M16—1421no—
155M16—1421no—
156C2×C8—712no—
157C2×C8—1424no—
158C2×C8C22⋊D7112yes—
159C4⋊C4—21no—
160C4⋊C4—1424no—
161C4⋊C4—1421no—
162C4⋊C4—1421no—
163C22⋊C4—21no—
164C22⋊C4—1424no—
165C22⋊C4—1421no—
166C22⋊C4—1421no—
167C42—21no—
168C42—712no—
169C42—1421no—
170C23—712no—
171C23D4×D7111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—1421no—
177Q8—1421no—
178D4—1422no—
179D4—1422no—
180D4—1421no—
181D4—1421no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—2841no—
187C2×C4—712no—
188C2×C4—1422no—
189C2×C4—1422no—
190C2×C4—1422no—
191C2×C4C2×C4×D7111yescharacteristic
192C2×C4D4×D7111yescharacteristic
193C2×C4D4×D7111yescharacteristic
194C2×C4C2×C22⋊D7111yescharacteristic
195C2×C4#1176586411yescharacteristic
196C2×C4#1176586511yescharacteristic
197C8—22no—
198C8—1422no—
199C22—41no—
200C22—2842no—
201C22—1422no—
202C22C22⋊C4×D7111yescharacteristic
203C22#1455609911yescharacteristic
204C22#1455610111yescharacteristic
205C4—42no—
206C4—81no—
207C4—2842no—
208C4—5681no—
209C4—1422no—
210C4(C22⋊D71)⋊C411yescharacteristic
211C4C4×C22⋊D7111yescharacteristic
212C4#1455609611yescharacteristic
213C4#1455611011yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—2842no—
218C2#1798587511yescharacteristic
subgroupG/H
centerC2#17985875
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#419×C71C2
socleC2×C71#386
hypercenter#419D71
hypocenterC71#1909
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #17986008C2semidirect
5. #17985756C2semidirect
6. #17985755C2semidirect
7. #419×C71C2semidirect
13. #14556126C4semidirect
14. #14556117C4semidirect
110. C71#1909semidirect
114. #419D71semidirect