Finite Group Databasebeta

← #22596046

Subgroups — #22596046

Order 9088.

428 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 121 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#17985919C211yesmaximal, characteristic, complemented
2#17985917C212yesmaximal
3#17985916C211yesmaximal, characteristic
4#17985912C212yesmaximal, complemented
5#17985890C212yesmaximal, complemented
6#17985889C212yesmaximal, complemented
7#17985888C212yesmaximal, complemented
8#17985863C212yesmaximal, complemented
9#382×C71C211yesmaximal, characteristic, complemented
10C22×Dic142C2211yescharacteristic
11C22×D284C2211yescharacteristic
12C2×Dic71⋊C4C2212yes—
13C2×Dic71⋊C4C2212yes—
14C2×(C2×D71)⋊C4C2212yes—
15C2×(C2×D71)⋊C4C2212yes—
16C2×(C2×D71)⋊C4C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×C4×Dic71C2212yes—
21C71⋊#95C2211yescharacteristic
22C71⋊#95C2212yes—
23C71⋊#95C2212yes—
24C71⋊#95C2211yescharacteristic
25C71⋊#95C2212yes—
26C2×C4⋊C4×C71C2211yescharacteristic
27C2×C4⋊C4×C71C2211yescharacteristic
28C2×C42×C71C2211yescharacteristic
29#95×C71C2212yes—
30#95×C71C2212yes—
31C23×D71C2312yes—
32C2×Dic142—41no—
33C2×Dic142—42no—
34C2×D284—41no—
35C2×D284—42no—
36C22×Dic71C2312yes—
37C22×Dic71C2312yes—
38C22×Dic71C2312yes—
39Dic71⋊C4—42no—
40Dic71⋊C4—42no—
41(C2×D71)⋊C4—24no—
42(C2×D71)⋊C4—24no—
43(C2×D71)⋊C4—24no—
44(C2×D71)⋊C4—42no—
45(C2×D71)⋊C4—42no—
46(C2×D71)⋊C4—42no—
47C4×Dic71—24no—
48C22×C4×C71C2311yescharacteristic
49C22×C4×C71C2311yescharacteristic
50C22×C4×C71C2312yes—
51C22×C4×C71C2312yes—
52C22×C4×C71C2311yescharacteristic
53C4⋊C4×C71—41no—
54C4⋊C4×C71—41no—
55C42×C71—22no—
56C23×C71C2411yescharacteristic
57C22×D71—22no—
58C22×D71—22no—
59C22×D71—22no—
60C22×D71—22no—
61C22×D71—22no—
62C22×D71—22no—
63C22×D71—22no—
64C2×C4×C71—21no—
65C2×C4×C71—21no—
66C2×C4×C71—22no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71#4112yes—
77C2×Dic71—22no—
78C2×Dic71—22no—
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71—22no—
82C2×Dic71—22no—
83C2×Dic71—22no—
84C2×Dic71—22no—
85C2×Dic71#4114yes—
86D284—44no—
87Dic142—44no—
88C22×C71C2×#4112yes—
89C22×C71C2×#4111yescharacteristic
90C22×C71H(5,𝔽2)11yescharacteristic
91C22×C71H(5,𝔽2)12yes—
92C22×C712−1+411yescharacteristic
93C2×D71—42no—
94C2×D71—42no—
95C2×D71—42no—
96C2×D71—42no—
97C2×D71—42no—
98C2×D71—42no—
99C2×D71—42no—
100C4×C71—22no—
101C4×C71—42no—
102C4×C71—42no—
103C4×C71—41no—
104C4×C71—41no—
105Dic71—24no—
106Dic71—42no—
107Dic71—42no—
108C2×C71#53911yescharacteristic
109C2×C71#53912yes—
110C2×C71#55511yescharacteristic
111C2×C71#55512yes—
112C2×C71H(3,𝔽4)11yescharacteristic
113D71—82no—
114#2785—711nomaximal, Sylow
115C71#278511yescharacteristic, complemented, Sylow
116#395—711no—
117#394—711no—
118#394—712no—
119#393—712no—
120#388—712no—
121#388—712no—
122#388—712no—
123#382—712no—
124#382D7111yescharacteristic, complemented
125C22×Q8—711no—
126C22×D4—711no—
127C2×C4⋊C4—712no—
128C2×C4⋊C4—712no—
129C2×C4⋊C4C2×D7111yescharacteristic
130C2×C4⋊C4C2×D7111yescharacteristic
131C2×C22⋊C4—712no—
132C2×C22⋊C4—712no—
133C2×C22⋊C4—712no—
134C2×C22⋊C4—712no—
135C2×C22⋊C4—712no—
136C2×C22⋊C4—712no—
137C2×C42—712no—
138C2×C42C2×D7111yescharacteristic
139#95—711no—
140#95—712no—
141#95—712no—
142#95—711no—
143#95—712no—
144#95C2×D7112yes—
145#95C2×D7112yes—
146C24—712no—
147C2×Q8—2841no—
148C2×Q8—2842no—
149C2×D4—2841no—
150C2×D4—2842no—
151C22×C4—712no—
152C22×C4—712no—
153C22×C4—712no—
154C22×C4C22×D7112yes—
155C22×C4C22×D7112yes—
156C22×C4C22×D7111yescharacteristic
157C22×C4C22×D7111yescharacteristic
158C22×C4C22×D7111yescharacteristic
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—2842no—
162C4⋊C4—2842no—
163C22⋊C4—2842no—
164C22⋊C4—2842no—
165C22⋊C4—2842no—
166C22⋊C4—1424no—
167C22⋊C4—1424no—
168C22⋊C4—1424no—
169C42—22no—
170C42—1424no—
171C23—1422no—
172C23—1422no—
173C23—1422no—
174C23—1422no—
175C23—1422no—
176C23—1422no—
177C23—1422no—
178C23C23×D7111yescharacteristic
179Q8—2844no—
180D4—2844no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—714no—
194C2×C4—1422no—
195C2×C4—1422no—
196C2×C4—1422no—
197C2×C4—1422no—
198C2×C4—1422no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4—1422no—
202C2×C4#1176586512yes—
203C22—2842no—
204C22—2842no—
205C22—2842no—
206C22—2842no—
207C22—2842no—
208C22—2842no—
209C22—2842no—
210C22#41×D7112yes—
211C22C2×#1176586511yescharacteristic
212C22#1455617311yescharacteristic
213C22#1455617412yes—
214C22#1455617611yescharacteristic
215C4—22no—
216C4—42no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—2842no—
221C4—2842no—
222C4—1424no—
223C2—5682no—
224C2#1798659611yescharacteristic
225C2#1798659712yes—
226C2#1798671111yescharacteristic
227C2#1798671212yes—
228C2#1798674111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#382×C71C2
hypercenter#382D71
hypocenterC71#2785
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #17985919C2semidirect
4. #17985912C2semidirect
5. #17985890C2semidirect
6. #17985889C2semidirect
7. #17985888C2semidirect
8. #17985863C2semidirect
9. #382×C71C2semidirect
115. C71#2785semidirect
124. #382D71semidirect