Finite Group Databasebeta

← #22596507

Subgroups — #22596507

Order 9088.

416 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 117 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#17985940C212yesmaximal, complemented
2#17985928C212yesmaximal, complemented
3#17985919C211yesmaximal, characteristic, complemented
4#17985917C212yesmaximal
5#17985916C211yesmaximal, characteristic
6#17985912C212yesmaximal, complemented
7#17985910C212yesmaximal, complemented
8#17985890C212yesmaximal, complemented
9#391×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—
20C71⋊#95C2212yes—
21C71⋊#95C2212yes—
22C71⋊#95C2212yes—
23C71⋊#95C2212yes—
24C71⋊#95C2211yescharacteristic
25C71⋊#95C2211yescharacteristic
26C22×Q8×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—42no—
43(C2×D71)⋊C4—42no—
44(C2×D71)⋊C4—42no—
45(C2×D71)⋊C4—42no—
46(C2×D71)⋊C4—42no—
47C2×Q8×C71—22no—
48C2×Q8×C71—41no—
49C2×Q8×C71—41no—
50C22×C4×C71C2311yescharacteristic
51C22×C4×C71C2311yescharacteristic
52C22×C4×C71C2312yes—
53C22×C4×C71C2312yes—
54C22×C4×C71C2311yescharacteristic
55C4⋊C4×C71—41no—
56C42×C71—41no—
57C23×C71C2411yescharacteristic
58C22×D71—22no—
59C22×D71—22no—
60C22×D71—22no—
61C22×D71—22no—
62C22×D71—22no—
63C22×D71—22no—
64C22×D71—22no—
65Q8×C71—44no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71—22no—
69C2×C4×C71—22no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71#4112yes—
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71—22no—
82C2×Dic71—22no—
83C2×Dic71—22no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88D284—44no—
89Dic142—44no—
90C22×C71C2×#4111yescharacteristic
91C22×C71H(5,𝔽2)11yescharacteristic
92C22×C71H(5,𝔽2)11yescharacteristic
93C22×C71H(5,𝔽2)12yes—
94C22×C712−1+412yes—
95C2×D71—42no—
96C2×D71—42no—
97C2×D71—42no—
98C2×D71—42no—
99C2×D71—42no—
100C2×D71—42no—
101C2×D71—42no—
102C4×C71—22no—
103C4×C71—42no—
104C4×C71—41no—
105C4×C71—41no—
106C4×C71—42no—
107Dic71—42no—
108Dic71—42no—
109Dic71—42no—
110C2×C71#53811yescharacteristic
111C2×C71#54212yes—
112C2×C71H(3,𝔽4)11yescharacteristic
113C2×C71#56212yes—
114C2×C71#56311yescharacteristic
115D71—82no—
116#2832—711nomaximal, Sylow
117C71#283211yescharacteristic, complemented, Sylow
118#400—712no—
119#397—712no—
120#395—711no—
121#394—711no—
122#394—712no—
123#393—712no—
124#393—712no—
125#391D7111yescharacteristic, complemented
126#388—712no—
127C22×Q8—711no—
128C22×Q8C2×D7111yescharacteristic
129C22×D4—711no—
130C2×C4⋊C4—712no—
131C2×C4⋊C4—712no—
132C2×C4⋊C4C2×D7111yescharacteristic
133C2×C22⋊C4—712no—
134C2×C22⋊C4—712no—
135C2×C22⋊C4—712no—
136C2×C22⋊C4—712no—
137C2×C22⋊C4—712no—
138C2×C22⋊C4—712no—
139C2×C42C2×D7111yescharacteristic
140#95—712no—
141#95—712no—
142#95—712no—
143#95—712no—
144#95—711no—
145#95—711no—
146#95C2×D7112yes—
147#95C2×D7112yes—
148C24—712no—
149C2×Q8—22no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—2841no—
153C2×Q8—2842no—
154C2×D4—2841no—
155C2×D4—2842no—
156C22×C4—712no—
157C22×C4—712no—
158C22×C4—712no—
159C22×C4C22×D7112yes—
160C22×C4C22×D7112yes—
161C22×C4C22×D7111yescharacteristic
162C22×C4C22×D7111yescharacteristic
163C22×C4C22×D7111yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—2842no—
166C4⋊C4—2842no—
167C22⋊C4—2842no—
168C22⋊C4—2842no—
169C22⋊C4—2842no—
170C22⋊C4—2842no—
171C22⋊C4—2842no—
172C22⋊C4—1424no—
173C42—41no—
174C23—1422no—
175C23—1422no—
176C23—1422no—
177C23—1422no—
178C23—1422no—
179C23—1422no—
180C23—1422no—
181C23C23×D7111yescharacteristic
182Q8—44no—
183Q8—2844no—
184D4—2844no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—1422no—
198C2×C4—1422no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4—1422no—
202C2×C4—1422no—
203C2×C4—1422no—
204C2×C4—1422no—
205C2×C4—1422no—
206C2×C4#1176586512yes—
207C22—2842no—
208C22—2842no—
209C22—2842no—
210C22—2842no—
211C22—2842no—
212C22—2842no—
213C22—2842no—
214C22C2×#1176586511yescharacteristic
215C22#1455617311yescharacteristic
216C22#1455617311yescharacteristic
217C22#1455617412yes—
218C22#1455617612yes—
219C4—22no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—42no—
224C4—2842no—
225C4—2842no—
226C4—2842no—
227C2—5682no—
228C2#1798658711yescharacteristic
229C2#1798662112yes—
230C2#1798674111yescharacteristic
231C2#1798674512yes—
232C2#1798674811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#391×C71C2
hypercenter#391D71
hypocenterC71#2832
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #17985940C2semidirect
2. #17985928C2semidirect
3. #17985919C2semidirect
6. #17985912C2semidirect
7. #17985910C2semidirect
8. #17985890C2semidirect
9. #391×C71C2semidirect
117. C71#2832semidirect
125. #391D71semidirect