Finite Group Databasebeta

← #22588882

Subgroups — #22588882

Order 9088.

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

normal subgroups only

This group has 380 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#17986558C211yesmaximal, characteristic
2C2×#14556043C211yesmaximal, characteristic, complemented
3C2×#14556042C212yesmaximal, complemented
4#17985765C212yesmaximal, complemented
5#518×C71C211yesmaximal, characteristic, complemented
6C22×C22⋊D71C2211yescharacteristic
7#14556111—21no—
8#14556109C412yescomplemented
9#14556100—22no—
10#14556097—21no—
11#14556096C412yescomplemented
12C2×C22⋊Dic71C2212yes—
13C2×C22⋊Dic71C2211yescharacteristic
14#14556043—22no—
15#14556042—24no—
16C22×D4×C71C2211yescharacteristic
17C4×D4×C71—24no—
18#117×C71—21no—
19C2×C22⋊C4×C71—21no—
20C2×C22⋊C4×C71C2212yes—
21C23×D71C2×C411yescharacteristic, complemented
22C2×C22⋊D71—22no—
23C2×C22⋊D71—21no—
24C2×C22⋊D71—21no—
25C2×C22⋊D71—22no—
26C2×C22⋊D71—22no—
27C22×Dic71—21no—
28C22×Dic71—22no—
29C22×Dic71C2×C411yescharacteristic, complemented
30Dic71⋊C4—22no—
31C22⋊Dic71—22no—
32C22⋊Dic71—21no—
33C22⋊Dic71—22no—
34C22⋊Dic71—22no—
35C22⋊Dic71C2×C412yescomplemented
36(C2×D71)⋊C4—22no—
37C24×C71D411yescharacteristic
38C24×C71C2311yescharacteristic
39C2×D4×C71—22no—
40C2×D4×C71—22no—
41C2×D4×C71C2×C412yes—
42C2×D4×C71D412yes—
43C22×C4×C71—22no—
44C22×C4×C71—22no—
45C22×C4×C71D411yescharacteristic
46C4⋊C4×C71—22no—
47C22⋊C4×C71—22no—
48C22⋊C4×C71—22no—
49C22⋊C4×C71D414yescomplemented
50C42×C71—22no—
51C23×C71—21no—
52C23×C71—21no—
53C23×C71—81no—
54C23×C71—81no—
55C23×C71C22⋊C412yes—
56C23×C71C22×C411yescharacteristic
57C23×C71C2×D411yescharacteristic
58C23×C71C2×D412yes—
59C23×C71C2×D411yescharacteristic
60C23×C71#4112yes—
61C22×D71—21no—
62C22×D71—22no—
63C22×D71—81no—
64D4×C71—24no—
65D4×C71—24no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71—24no—
69C2×C4×C71—82no—
70C2×C4×C71—82no—
71C2×C4×C71C22⋊C412yes—
72C22⋊D71—41no—
73C22⋊D71—42no—
74C22⋊D71—41no—
75C22⋊D71—82no—
76C2×Dic71—22no—
77C2×Dic71—21no—
78C2×Dic71—22no—
79C2×Dic71—42no—
80C2×Dic71—81no—
81C2×Dic71—82no—
82C22×C71—22no—
83C22×C71—41no—
84C22×C71—42no—
85C22×C71—42no—
86C22×C71—41no—
87C22×C71—41no—
88C22×C71—42no—
89C22×C71—42no—
90C22×C71—81no—
91C22×C71C2×C22⋊C411yescharacteristic
92C22×C71C4×D412yes—
93C22×C71#12011yescharacteristic
94C22×C71#12112yes—
95C22×C71#12311yescharacteristic
96C2×D71—42no—
97C2×D71—41no—
98C2×D71—82no—
99C4×C71—22no—
100C4×C71—44no—
101Dic71—42no—
102Dic71—41no—
103Dic71—82no—
104C2×C71—21no—
105C2×C71—22no—
106C2×C71—21no—
107C2×C71—22no—
108C2×C71—21no—
109C2×C71—42no—
110C2×C71#38611yescharacteristic
111D71—81no—
112#1913—711nomaximal, Sylow
113C71#191311yescharacteristic, complemented, Sylow
114#534—711no—
115#518D7111yescharacteristic, complemented
116C2×#99—711no—
117C2×#99—712no—
118#342—712no—
119C22×D4—711no—
120C22×D4C2×D7111yescharacteristic
121#123—712no—
122#123—1421no—
123#121—1422no—
124#120—712no—
125#120—1421no—
126C4×D4—24no—
127#117—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—711no—
130C2×C22⋊C4—712no—
131C2×C22⋊C4C2×D7112yes—
132#99—1422no—
133#99—1424no—
134C24—711no—
135C24C22⋊D7111yescharacteristic
136C24C22×D7111yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—1422no—
140C2×D4—1421no—
141C2×D4—1421no—
142C2×D4—1422no—
143C2×D4—1422no—
144C2×D4C4×D7112yes—
145C2×D4D28412yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—711no—
149C22×C4—1422no—
150C22×C4—1421no—
151C22×C4C22⋊D7111yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—1422no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—712no—
157C22⋊C4—1422no—
158C22⋊C4—1422no—
159C22⋊C4—1422no—
160C22⋊C4—1421no—
161C22⋊C4—1422no—
162C22⋊C4C22⋊D7114yescomplemented
163C42—22no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—5681no—
169C23—1421no—
170C23—1422no—
171C23(C2×D71)⋊C412yes—
172C23C2×C4×D7111yescharacteristic
173C23C2×C22⋊D7111yescharacteristic
174C23C2×C22⋊D7112yes—
175C23C2×D28411yescharacteristic
176C23#1176586512yes—
177D4—24no—
178D4—24no—
179D4—2841no—
180D4—2842no—
181D4—2841no—
182D4—5682no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—2842no—
189C2×C4—5681no—
190C2×C4—5682no—
191C2×C4—1422no—
192C2×C4—1421no—
193C2×C4—1422no—
194C2×C4(C2×D71)⋊C412yes—
195C22—22no—
196C22—42no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—2842no—
205C22—2841no—
206C22—5682no—
207C22C2×(C2×D71)⋊C411yescharacteristic
208C22C4×C22⋊D7112yes—
209C22#1455609711yescharacteristic
210C22#1455610212yes—
211C22#1455611111yescharacteristic
212C4—22no—
213C4—44no—
214C4—2842no—
215C4—2841no—
216C4—5682no—
217C2—21no—
218C2—21no—
219C2—22no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—5681no—
224C2#1798587811yescharacteristic
subgroupG/H
centerC2#17985878
commutatorC23×C71C22×C4
FrattiniC24C22×D71
Fitting#518×C71C2
socleC2×C71#386
hypercenter#518D71
hypocenterC71#1913
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#14556043C2semidirect
3. C2×#14556042C2semidirect
4. #17985765C2semidirect
5. #518×C71C2semidirect
8. #14556109C4semidirect
11. #14556096C4semidirect
21. C23×D71C2×C4semidirect
29. C22×Dic71C2×C4semidirect
35. C22⋊Dic71C2×C4semidirect
49. C22⋊C4×C71D4semidirect
113. C71#1913semidirect
115. #518D71semidirect
162. C22⋊C4C22⋊D71semidirect