Finite Group Databasebeta

← #22588316

Subgroups — #22588316

Order 9088.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#14556175C211yesmaximal, characteristic
2#17986015C212yesmaximal, complemented
3#17986009C212yesmaximal, complemented
4#17985973C211yesmaximal, characteristic, complemented
5#413×C71C211yesmaximal, characteristic, complemented
6#14556175—24no—
7#14556175—24no—
8C2×#11765865C412yescomplemented
9C2×#11765865C2211yescharacteristic
10C2×#11765864—22no—
11C2×#11765864—21no—
12C22×Dic142C412yescomplemented
13C22×Dic142C2211yescharacteristic
14C2×Q8×D71—21no—
15#14556087C2212yes—
16#14556054—24no—
17#14556053—24no—
18#14556043—22no—
19C2×#41×C71C2211yescharacteristic
20C2×M16×C71C2212yes—
21#101×C71—21no—
22#100×C71—21no—
23#11765865—22no—
24#11765865—24no—
25#11765865—24no—
26#11765865D414yes—
27#11765864—216no—
28#11765864—28no—
29C2×Dic142—22no—
30C2×Dic142—21no—
31C2×Dic142—22no—
32C2×Dic142—24no—
33C2×Dic142—24no—
34C2×Dic142C2×C411yescharacteristic
35C2×Dic142C2×C412yes—
36C2×Dic142D414yes—
37Q8×D71—28no—
38C2×D284C2×C411yescharacteristic
39C2×C22⋊D71—22no—
40C2×C22⋊D71—21no—
41C22×Dic71—22no—
42C22×Dic71—21no—
43C2×C4×D71—22no—
44C2×C4×D71—21no—
45C4⋊Dic71—22no—
46C22⋊Dic71—22no—
47C4×Dic71—22no—
48#41×C71—22no—
49#41×C71—22no—
50C2×Q8×C71D411yescharacteristic
51C2×D4×C71D412yes—
52C2×D4×C71D411yescharacteristic
53C22×C4×C71C2×C412yes—
54C22×C4×C71C2311yescharacteristic
55M16×C71—22no—
56M16×C71—22no—
57C2×C8×C71—22no—
58C23×C71C22⋊C412yes—
59C23×C71C2×D411yescharacteristic
60C22×D71—21no—
61Q8×C71—22no—
62D4×C71—24no—
63D4×C71—22no—
64C2×C4×C71—22no—
65C2×C4×C71—22no—
66C2×C4×C71C22⋊C412yes—
67C2×C4×C71C22×C411yescharacteristic
68C2×C4×C71C2×D412yes—
69C2×C4×C71C2×D412yes—
70C2×C4×C71C2×D411yescharacteristic
71C22⋊D71—28no—
72C22⋊D71—24no—
73C2×Dic71—28no—
74C2×Dic71—22no—
75C2×Dic71—21no—
76C2×Dic71—24no—
77C2×Dic71—42no—
78D284—21no—
79D284C22⋊C412yes—
80C4×D71—28no—
81C4×D71—24no—
82Dic142—28no—
83Dic142—22no—
84Dic142—21no—
85Dic142—24no—
86Dic142C22⋊C412yes—
87Dic142C22⋊C414yes—
88C8×C71—42no—
89C22×C71—22no—
90C22×C71—41no—
91C22×C71—42no—
92C22×C71C2×C22⋊C411yescharacteristic
93C22×C71#12012yes—
94C2×D71—22no—
95C2×D71—42no—
96C4×C71—22no—
97C4×C71C2×C22⋊C411yescharacteristic
98C4×C71C2×C22⋊C411yescharacteristic
99C4×C71#12012yes—
100Dic71—22no—
101Dic71—24no—
102Dic71—82no—
103C2×C71—22no—
104C2×C71—21no—
105C2×C71—42no—
106C2×C71#37911yescharacteristic
107D71—42no—
108#1796—711nomaximal, Sylow
109C71#179611yescharacteristic, complemented, Sylow
110C2×2−1+4—711no—
111#421—712no—
112#419—712no—
113#413D7111yescharacteristic, complemented
114#410—711no—
1152−1+4—1424no—
1162−1+4—1424no—
117C2×#41—712no—
118C2×#41—711no—
119C2×#41—1422no—
120C2×#41—1421no—
121C2×#41C2×D7111yescharacteristic
122C22×Q8—712no—
123C22×Q8—711no—
124C22×Q8—1421no—
125C2×M16C2×D7112yes—
126#117—712no—
127#104—1424no—
128#103—1424no—
129#101—21no—
130#100—21no—
131#99—1422no—
132#41—22no—
133#41—22no—
134#41—714no—
135#41—1424no—
136#41—1424no—
137#41—14216no—
138#41—1428no—
139#41—1422no—
140C2×Q8—712no—
141C2×Q8—711no—
142C2×Q8—714no—
143C2×Q8—1424no—
144C2×Q8—1424no—
145C2×Q8—1428no—
146C2×Q8—1422no—
147C2×Q8—1422no—
148C2×Q8—1421no—
149C2×Q8C22⋊D7111yescharacteristic
150C2×D4—711no—
151C2×D4—1422no—
152C2×D4—1421no—
153C2×D4D28412yes—
154C2×D4C22⋊D7111yescharacteristic
155C22×C4—1422no—
156C22×C4—1421no—
157C22×C4—1422no—
158C22×C4—1421no—
159C22×C4C4×D7112yes—
160C22×C4C22×D7111yescharacteristic
161M16—22no—
162M16—22no—
163C2×C8—22no—
164C4⋊C4—1422no—
165C22⋊C4—1422no—
166C42—1422no—
167C23—1421no—
168C23(C2×D71)⋊C412yes—
169C23C2×D28411yescharacteristic
170Q8—22no—
171Q8—714no—
172Q8—712no—
173Q8—1428no—
174Q8—1422no—
175Q8—1421no—
176Q8—1424no—
177D4—24no—
178D4—22no—
179D4—712no—
180D4—1428no—
181D4—1424no—
182D4—1421no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—2842no—
186C2×C4—1428no—
187C2×C4—1428no—
188C2×C4—1421no—
189C2×C4—1422no—
190C2×C4—1424no—
191C2×C4—1424no—
192C2×C4(C2×D71)⋊C412yes—
193C2×C4C2×C4×D7111yescharacteristic
194C2×C4D4×D7112yes—
195C2×C4D4×D7112yes—
196C2×C4C2×C22⋊D7111yescharacteristic
197C8—42no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—2842no—
202C22—1422no—
203C22C2×(C2×D71)⋊C411yescharacteristic
204C22#1455609812yes—
205C4—22no—
206C4—5682no—
207C4—1424no—
208C4—1422no—
209C4C22⋊C4×D7111yescharacteristic
210C4C22⋊C4×D7111yescharacteristic
211C4#1455609612yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—2842no—
216C2#1798585411yescharacteristic
subgroupG/H
centerC2#17985854
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#413×C71C2
socleC2×C71#379
hypercenter#413D71
hypocenterC71#1796
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #17986015C2semidirect
3. #17986009C2semidirect
4. #17985973C2semidirect
5. #413×C71C2semidirect
8. C2×#11765865C4semidirect
12. C22×Dic142C4semidirect
109. C71#1796semidirect
113. #413D71semidirect