Finite Group Databasebeta

← #22596597

Subgroups — #22596597

Order 9088.

396 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 133 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#17985941C212yesmaximal, complemented
2#17985935C211yesmaximal, characteristic
3#17985933C212yesmaximal, complemented
4#17985931C212yesmaximal, complemented
5#17985922C212yesmaximal
6#17985920C212yesmaximal, complemented
7#17985911C211yesmaximal, characteristic
8#17985859C212yesmaximal
9#381×C71C211yesmaximal, characteristic
10C22×Dic142C2211yescharacteristic
11C2×C4⋊Dic71C2212yes—
12C2×C4⋊Dic71C2212yes—
13C2×C4⋊Dic71C2212yes—
14C2×C4⋊Dic71C2211yescharacteristic
15C2×Dic71⋊C4C2212yes—
16C2×Dic71⋊C4C2212yes—
17C2×Dic71⋊C4C2212yes—
18C2×C22⋊Dic71C2212yes—
19C2×C22⋊Dic71C2212yes—
20C71⋊#95C2212yes—
21C71⋊#95C2212yes—
22C71⋊#95C2212yes—
23C71⋊#95C2211yescharacteristic
24C71⋊#95C2212yes—
25C71⋊#95C2211yescharacteristic
26C23×C4×C71C2211yescharacteristic
27C2×C22⋊C4×C71C2211yescharacteristic
28C2×C22⋊C4×C71C2211yescharacteristic
29#95×C71C2212yes—
30#95×C71C2212yes—
31C2×Dic142—42no—
32C2×Dic142—41no—
33C22×Dic71C2312yes—
34C22×Dic71C2312yes—
35C22×Dic71C2312yes—
36C22×Dic71C2312yes—
37C4⋊Dic71—42no—
38C4⋊Dic71—42no—
39C4⋊Dic71—41no—
40C4⋊Dic71—42no—
41Dic71⋊C4—42no—
42Dic71⋊C4—42no—
43Dic71⋊C4—42no—
44C22⋊Dic71—42no—
45C22⋊Dic71—42no—
46C24×C71C2311yescharacteristic
47C22×C4×C71—41no—
48C22×C4×C71—41no—
49C22×C4×C71Q814yescomplemented
50C22×C4×C71C2311yescharacteristic
51C22×C4×C71C2311yescharacteristic
52C22×C4×C71C2312yes—
53C22×C4×C71C2312yes—
54C22⋊C4×C71—41no—
55C22⋊C4×C71—41no—
56C23×C71—21no—
57C23×C71—21no—
58C23×C71—22no—
59C23×C71—22no—
60C23×C71C2×Q812yes—
61C23×C71C2411yescharacteristic
62C2×C4×C71—22no—
63C2×C4×C71—22no—
64C2×C4×C71—22no—
65C2×C4×C71—21no—
66C2×C4×C71—21no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—44no—
72C2×C4×C71C2×Q814yes—
73C2×Dic71—22no—
74C2×Dic71—22no—
75C2×Dic71—22no—
76C2×Dic71—22no—
77C2×Dic71—22no—
78C2×Dic71—22no—
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71—22no—
82C2×Dic71—22no—
83C2×Dic71—22no—
84C2×Dic71—22no—
85Dic142—44no—
86C22×C71—22no—
87C22×C71—24no—
88C22×C71—41no—
89C22×C71—42no—
90C22×C71—41no—
91C22×C71C22×Q811yescharacteristic
92C22×C71H(5,𝔽2)11yescharacteristic
93C22×C71H(5,𝔽2)11yescharacteristic
94C22×C71H(5,𝔽2)12yes—
95C22×C712−1+412yes—
96C4×C71—24no—
97C4×C71—41no—
98C4×C71—42no—
99C4×C71—41no—
100Dic71—42no—
101Dic71—42no—
102Dic71—42no—
103Dic71—42no—
104C2×C71—42no—
105C2×C71#54311yescharacteristic
106C2×C71#54412yes—
107C2×C71#56011yescharacteristic
108C2×C71#56212yes—
109C2×C71#56311yescharacteristic
110#2841—711nomaximal, Sylow
111C71#284111yescharacteristic, complemented, Sylow
112#400—712no—
113#399—711no—
114#398—712no—
115#398—712no—
116#396—712no—
117#395—712no—
118#393—711no—
119#381D7111yescharacteristic
120#380—712no—
121C22×Q8—711no—
122C23×C4C2×D7111yescharacteristic
123C2×C4⋊C4—712no—
124C2×C4⋊C4—712no—
125C2×C4⋊C4—712no—
126C2×C4⋊C4—712no—
127C2×C4⋊C4—712no—
128C2×C4⋊C4—712no—
129C2×C4⋊C4—711no—
130C2×C22⋊C4—712no—
131C2×C22⋊C4—712no—
132C2×C22⋊C4C2×D7111yescharacteristic
133C2×C22⋊C4C2×D7111yescharacteristic
134#95—712no—
135#95—712no—
136#95—711no—
137#95—712no—
138#95—712no—
139#95—711no—
140#95C2×D7112yes—
141#95C2×D7112yes—
142C24C22×D7111yescharacteristic
143C2×Q8—2841no—
144C2×Q8—2842no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—712no—
148C22×C4—712no—
149C22×C4—712no—
150C22×C4—712no—
151C22×C4Dic14214yescomplemented
152C22×C4C22×D7111yescharacteristic
153C22×C4C22×D7112yes—
154C22×C4C22×D7111yescharacteristic
155C22×C4C22×D7112yes—
156C4⋊C4—2842no—
157C4⋊C4—2842no—
158C4⋊C4—2842no—
159C4⋊C4—2842no—
160C4⋊C4—2842no—
161C4⋊C4—2841no—
162C4⋊C4—2842no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C22⋊C4—2842no—
166C22⋊C4—2842no—
167C23—21no—
168C23—21no—
169C23—22no—
170C23—22no—
171C23C2×Dic14212yes—
172C23C23×D7111yescharacteristic
173Q8—2844no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—1422no—
185C2×C4—1422no—
186C2×C4—1422no—
187C2×C4—1422no—
188C2×C4—1422no—
189C2×C4—1422no—
190C2×C4—1422no—
191C2×C4—1422no—
192C2×C4—1422no—
193C2×C4—1422no—
194C2×C4—1422no—
195C2×C4—1422no—
196C2×C4C2×Dic14214yes—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22×Dic14211yescharacteristic
203C22#1455617311yescharacteristic
204C22#1455617311yescharacteristic
205C22#1455617412yes—
206C22#1455617512yes—
207C4—24no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—2842no—
212C4—2842no—
213C4—2842no—
214C4—2842no—
215C2—42no—
216C2#1798662511yescharacteristic
217C2#1798662712yes—
218C2#1798673911yescharacteristic
219C2#1798674612yes—
220C2#1798675211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#381×C71C2
hypercenter#381D71
hypocenterC71#2841
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. #17985941C2semidirect
3. #17985933C2semidirect
4. #17985931C2semidirect
6. #17985920C2semidirect
49. C22×C4×C71Q8semidirect
111. C71#2841semidirect
151. C22×C4Dic142semidirect