Finite Group Databasebeta

← #22589982

Subgroups — #22589982

Order 9088.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#17986506C211yesmaximal, characteristic
2#17985973C211yesmaximal, characteristic, complemented
3#17985790C212yesmaximal, complemented
4#17985787C212yesmaximal, complemented
5#413×C71C211yesmaximal, characteristic, complemented
6C22×C4×D71C411yescharacteristic, complemented
7#14556109C411yescharacteristic, complemented
8#14556109C2211yescharacteristic
9#14556104—21no—
10#14556099—21no—
11#14556096C411yescharacteristic, complemented
12#14556096C2211yescharacteristic
13C4×C22⋊D71—21no—
14C4×C22⋊D71—21no—
15#14556087—21no—
16#14556087C411yescharacteristic, complemented
17#14556043—21no—
18#14556043C2212yes—
19C2×#41×C71C2211yescharacteristic
20C2×M16×C71—21no—
21#101×C71—21no—
22#100×C71C2212yes—
23C23×D71C2×C411yescharacteristic
24C2×C22⋊D71—21no—
25C2×C22⋊D71—21no—
26C22×Dic71C2×C411yescharacteristic
27C2×C4×D71—21no—
28C2×C4×D71—21no—
29C2×C4×D71—41no—
30C2×C4×D71—41no—
31C4⋊Dic71—21no—
32C4⋊Dic71—41no—
33Dic71⋊C4—21no—
34Dic71⋊C4—21no—
35C22⋊Dic71—21no—
36C22⋊Dic71—21no—
37C22⋊Dic71—22no—
38C22⋊Dic71C2×C411yescharacteristic
39C22⋊Dic71C2×C411yescharacteristic
40(C2×D71)⋊C4—21no—
41(C2×D71)⋊C4—21no—
42C4×Dic71—21no—
43C4×Dic71—41no—
44#41×C71—41no—
45#41×C71—41no—
46C2×Q8×C71D411yescharacteristic
47C2×D4×C71C2×C411yescharacteristic
48C2×D4×C71D411yescharacteristic
49C2×D4×C71C2311yescharacteristic
50C22×C4×C71C2×C411yescharacteristic
51C22×C4×C71D411yescharacteristic
52C22×C4×C71D411yescharacteristic
53M16×C71—22no—
54M16×C71—41no—
55C2×C8×C71—41no—
56C23×C71C22⋊C411yescharacteristic
57C23×C71C22×C411yescharacteristic
58C23×C71C2×D411yescharacteristic
59C22×D71—21no—
60C22×D71—42no—
61C22×D71—41no—
62Q8×C71—41no—
63D4×C71—41no—
64D4×C71—41no—
65D4×C71—41no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71C22⋊C411yescharacteristic
73C2×C4×C71C22⋊C411yescharacteristic
74C2×C4×C71C22⋊C411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C22⋊D71—42no—
77C22⋊D71—42no—
78C2×Dic71—21no—
79C2×Dic71—21no—
80C2×Dic71—21no—
81C2×Dic71—41no—
82C2×Dic71—42no—
83C4×D71—42no—
84C4×D71—41no—
85C4×D71—41no—
86C8×C71—42no—
87C22×C71—21no—
88C22×C71—21no—
89C22×C71—21no—
90C22×C71—41no—
91C22×C71—41no—
92C22×C71—41no—
93C22×C71C2×C22⋊C411yescharacteristic
94C2×D71—22no—
95C2×D71—41no—
96C2×D71—42no—
97C2×D71—41no—
98C2×D71—42no—
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—21no—
102C4×C71#9911yescharacteristic
103C4×C71#9911yescharacteristic
104Dic71—41no—
105Dic71—41no—
106Dic71—41no—
107Dic71—82no—
108C2×C71—21no—
109C2×C71—41no—
110C2×C71—41no—
111C2×C71—41no—
112C2×C71C2×#9911yescharacteristic
113D71—42no—
114#2120—711nomaximal, Sylow
115C71#212011yescharacteristic, complemented, Sylow
116#525—711no—
117#413D7111yescharacteristic, complemented
118#410—711no—
119#352—712no—
120AΣL(1,𝔽16)—712no—
121C2×#41C2×D7111yescharacteristic
122C23×C4—711no—
123C2×M16—21no—
124#123—711no—
125#123—711no—
126C22⋊Q8—1421no—
127#121—1421no—
128#120—711no—
129#120—711no—
130C4×D4—1421no—
131C4×D4—1421no—
132#117—711no—
133#117—1421no—
134#101—21no—
135#100C2×D7112yes—
136#99—712no—
137#99—1421no—
138C24—711no—
139#41—41no—
140#41—41no—
141C2×Q8C22⋊D7111yescharacteristic
142C2×D4—1421no—
143C2×D4—1421no—
144C2×D4C4×D7111yescharacteristic
145C2×D4D28411yescharacteristic
146C2×D4C22×D7111yescharacteristic
147C22×C4—2841no—
148C22×C4—2841no—
149C22×C4—711no—
150C22×C4—1421no—
151C22×C4—1421no—
152C22×C4C4×D7111yescharacteristic
153C22×C4D28411yescharacteristic
154C22×C4C22⋊D7111yescharacteristic
155M16—22no—
156M16—41no—
157C2×C8—41no—
158C4⋊C4—2841no—
159C4⋊C4—1421no—
160C4⋊C4—1421no—
161C4⋊C4—1421no—
162C22⋊C4—711no—
163C22⋊C4—711no—
164C22⋊C4—1422no—
165C22⋊C4—1421no—
166C22⋊C4—1421no—
167C22⋊C4—1421no—
168C22⋊C4—1421no—
169C42—2841no—
170C42—1421no—
171C23—2841no—
172C23—2842no—
173C23—1421no—
174C23(C2×D71)⋊C411yescharacteristic
175C23C2×C4×D7111yescharacteristic
176C23C2×D28411yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—2842no—
182D4—2842no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—2842no—
190C2×C4—2841no—
191C2×C4—2841no—
192C2×C4—2841no—
193C2×C4—2842no—
194C2×C4—1421no—
195C2×C4—1421no—
196C2×C4—1421no—
197C2×C4(C2×D71)⋊C411yescharacteristic
198C2×C4(C2×D71)⋊C411yescharacteristic
199C2×C4(C2×D71)⋊C411yescharacteristic
200C2×C4C2×C22⋊D7111yescharacteristic
201C8—42no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—2841no—
209C22—2842no—
210C22—2842no—
211C22—2841no—
212C22—1422no—
213C22C2×(C2×D71)⋊C411yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—2841no—
218C4—2841no—
219C4—2841no—
220C4—5682no—
221C4#1455604211yescharacteristic
222C4#1455604211yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—2842no—
228C2C2×#1455604211yescharacteristic
subgroupG/H
centerC4#14556042
commutatorC23×C71C22×C4
FrattiniC2×D4C22×D71
Fitting#413×C71C2
socleC2×C71C2×#99
hypercenter#413D71
hypocenterC71#2120
LCSG ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #17985973C2semidirect
3. #17985790C2semidirect
4. #17985787C2semidirect
5. #413×C71C2semidirect
6. C22×C4×D71C4semidirect
7. #14556109C4semidirect
11. #14556096C4semidirect
16. #14556087C4semidirect
115. C71#2120semidirect
117. #413D71semidirect