Finite Group Databasebeta

← #22596415

Subgroups — #22596415

Order 9088.

456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#14556123C212yesmaximal, complemented
2#17985933C212yesmaximal, complemented
3#17985923C212yesmaximal, complemented
4#17985921C212yesmaximal, complemented
5#17985921C212yesmaximal, complemented
6#17985917C212yesmaximal
7#17985916C211yesmaximal, characteristic
8#17985868C211yesmaximal, characteristic, complemented
9#398×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11#14556123—28no—
12C2×C4⋊Dic71C2212yes—
13C2×C4⋊Dic71C2211yescharacteristic
14C2×Dic71⋊C4C2212yes—
15C2×Dic71⋊C4C2212yes—
16C2×Dic71⋊C4C2212yes—
17C2×Dic71⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×C4×Dic71C2211yescharacteristic
25C71⋊#95C2211yescharacteristic
26C71⋊#95C2212yes—
27C2×C4⋊C4×C71C2212yes—
28C2×C4⋊C4×C71C2211yescharacteristic
29C2×C4⋊C4×C71C2211yescharacteristic
30#95×C71C2212yes—
31#95×C71C2211yescharacteristic
32C23×D71C2312yes—
33C2×D284—41no—
34C2×D284—42no—
35C22×Dic71C2312yes—
36C22×Dic71C2312yes—
37C22×Dic71C2312yes—
38C4⋊Dic71—24no—
39C4⋊Dic71—41no—
40Dic71⋊C4—24no—
41Dic71⋊C4—42no—
42Dic71⋊C4—42no—
43Dic71⋊C4—42no—
44(C2×D71)⋊C4—24no—
45(C2×D71)⋊C4—24no—
46(C2×D71)⋊C4—24no—
47(C2×D71)⋊C4—42no—
48(C2×D71)⋊C4—42no—
49(C2×D71)⋊C4—42no—
50C4×Dic71D414yescomplemented
51C22×C4×C71C2311yescharacteristic
52C22×C4×C71C2312yes—
53C22×C4×C71C2312yes—
54C22×C4×C71C2311yescharacteristic
55C22×C4×C71C2311yescharacteristic
56C4⋊C4×C71—24no—
57C4⋊C4×C71—41no—
58C4⋊C4×C71—41no—
59C23×C71C2411yescharacteristic
60C22×D71—22no—
61C22×D71—22no—
62C22×D71—22no—
63C22×D71—22no—
64C22×D71—22no—
65C22×D71—22no—
66C22×D71—22no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71—21no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D412yes—
80C2×Dic71—22no—
81C2×Dic71—22no—
82C2×Dic71—22no—
83C2×Dic71—22no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71C2×D414yes—
89D284—44no—
90C22×C71C22×D411yescharacteristic
91C22×C71H(5,𝔽2)11yescharacteristic
92C22×C71H(5,𝔽2)12yes—
93C22×C712−1+411yescharacteristic
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—42no—
105C4×C71—41no—
106C4×C71—41no—
107Dic71—24no—
108Dic71—42no—
109Dic71—42no—
110C2×C71#53711yescharacteristic
111C2×C71#53712yes—
112C2×C71#56011yescharacteristic
113C2×C71#56311yescharacteristic
114C2×C71#56312yes—
115D71—82no—
116#2823—711nomaximal, Sylow
117C71#282311yescharacteristic, complemented, Sylow
118C2×#126—712no—
119#398—712no—
120#398D7111yescharacteristic, complemented
121#396—712no—
122#396—712no—
123#396—712no—
124#394—711no—
125#394—712no—
126#384—711no—
127C22×D4—711no—
128#126—1428no—
129C2×C4⋊C4—712no—
130C2×C4⋊C4—712no—
131C2×C4⋊C4—711no—
132C2×C4⋊C4—712no—
133C2×C4⋊C4—712no—
134C2×C4⋊C4—712no—
135C2×C4⋊C4C2×D7112yes—
136C2×C4⋊C4C2×D7111yescharacteristic
137C2×C4⋊C4C2×D7111yescharacteristic
138C2×C22⋊C4—712no—
139C2×C22⋊C4—712no—
140C2×C22⋊C4—712no—
141C2×C22⋊C4—712no—
142C2×C22⋊C4—712no—
143C2×C22⋊C4—712no—
144C2×C42—711no—
145#95—711no—
146#95—712no—
147#95C2×D7112yes—
148#95C2×D7111yescharacteristic
149C24—712no—
150C2×D4—2841no—
151C2×D4—2842no—
152C22×C4—712no—
153C22×C4—712no—
154C22×C4—712no—
155C22×C4C22×D7111yescharacteristic
156C22×C4C22×D7112yes—
157C22×C4C22×D7112yes—
158C22×C4C22×D7111yescharacteristic
159C22×C4C22×D7111yescharacteristic
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—2842no—
164C4⋊C4—2842no—
165C4⋊C4—2842no—
166C4⋊C4—2841no—
167C4⋊C4—1424no—
168C4⋊C4—1424no—
169C22⋊C4—2842no—
170C22⋊C4—2842no—
171C22⋊C4—2842no—
172C22⋊C4—1424no—
173C22⋊C4—1424no—
174C22⋊C4—1424no—
175C42—714no—
176C23—1422no—
177C23—1422no—
178C23—1422no—
179C23—1422no—
180C23—1422no—
181C23—1422no—
182C23—1422no—
183C23C23×D7111yescharacteristic
184D4—2844no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—714no—
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×C4D4×D7112yes—
207C22—2842no—
208C22—2842no—
209C22—2842no—
210C22—2842no—
211C22—2842no—
212C22—2842no—
213C22—2842no—
214C22C2×D4×D7111yescharacteristic
215C22#1455617311yescharacteristic
216C22#1455617412yes—
217C22#1455617511yescharacteristic
218C22#1455617612yes—
219C4—22no—
220C4—42no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—2842no—
225C4—2842no—
226C4—1424no—
227C2—5682no—
228C2#1798657611yescharacteristic
229C2#1798657912yes—
230C2#1798673911yescharacteristic
231C2#1798674811yescharacteristic
232C2#1798675312yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#398×C71C2
hypercenter#398D71
hypocenterC71#2823
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14556123C2semidirect
2. #17985933C2semidirect
3. #17985923C2semidirect
4. #17985921C2semidirect
5. #17985921C2semidirect
8. #17985868C2semidirect
9. #398×C71C2semidirect
50. C4×Dic71D4semidirect
117. C71#2823semidirect
120. #398D71semidirect