Finite Group Databasebeta

← #22593337

Subgroups — #22593337

Order 9088.

604 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14556126C211yesmaximal, characteristic, complemented
2C2×#14556117C211yesmaximal, characteristic
3C2×#14556116C211yesmaximal, characteristic
4C2×#14556090C211yesmaximal, characteristic
5#17985928C214yesmaximal, complemented
6#17985904C211yesmaximal, characteristic, complemented
7#17985858C214yesmaximal, complemented
8C4×Dic71⋊C4C211yesmaximal, characteristic, complemented
9#391×C71C211yesmaximal, characteristic, complemented
10C22×Dic142C2211yescharacteristic
11C22×D284C2211yescharacteristic
12C22×C4×D71C2212yes—
13#14556126—24no—
14#14556117—24no—
15#14556116—24no—
16#14556090—24no—
17C2×Dic71⋊C4C2212yes—
18C2×Dic71⋊C4C2212yes—
19C2×Dic71⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2214yes—
21C2×(C2×D71)⋊C4C2214yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×C4×Dic71C2211yescharacteristic
24C2×C4×Dic71C2211yescharacteristic
25C71⋊#95C2212yes—
26C71⋊#95C2214yes—
27C22×Q8×C71C2211yescharacteristic
28C2×C4⋊C4×C71C2211yescharacteristic
29C2×C42×C71C2211yescharacteristic
30#95×C71C2214yes—
31C23×D71C2312yes—
32C2×Dic142—22no—
33C2×Dic142—22no—
34C2×Dic142—22no—
35C2×D284—22no—
36C2×D284—22no—
37C2×D284D414yes—
38C22×Dic71C2312yes—
39C22×Dic71C2312yes—
40C22×Dic71C2311yescharacteristic
41C22×Dic71C2311yescharacteristic
42C2×C4×D71—24no—
43C2×C4×D71—28no—
44Dic71⋊C4—24no—
45Dic71⋊C4—24no—
46Dic71⋊C4—24no—
47(C2×D71)⋊C4—28no—
48(C2×D71)⋊C4—28no—
49(C2×D71)⋊C4—24no—
50C4×Dic71—22no—
51C4×Dic71D414yes—
52C2×Q8×C71—22no—
53C2×Q8×C71—22no—
54C2×Q8×C71—22no—
55C22×C4×C71C2312yes—
56C22×C4×C71C2312yes—
57C22×C4×C71C2312yes—
58C22×C4×C71C2311yescharacteristic
59C4⋊C4×C71—22no—
60C42×C71—22no—
61C23×C71C2411yescharacteristic
62C22×D71—22no—
63C22×D71—24no—
64C22×D71—22no—
65C22×D71—24no—
66C22×D71C2×D414yes—
67Q8×C71—44no—
68C2×C4×C71—22no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—24no—
75C2×C4×C71—22no—
76C2×C4×C71C2×D412yes—
77C2×C4×C71C2×D412yes—
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×Dic71C2×D412yes—
87C2×Dic71C2×D412yes—
88C2×Dic71#4114yes—
89D284—28no—
90C4×D71—48no—
91Dic142—44no—
92C22×C71C22×D411yescharacteristic
93C22×C71C22×D411yescharacteristic
94C22×C71C2×#4112yes—
95C22×C71C2×#4111yescharacteristic
96C22×C712−1+412yes—
97C2×D71—24no—
98C2×D71—28no—
99C2×D71—42no—
100C2×D71—42no—
101C2×D71—44no—
102C4×C71—42no—
103C4×C71—42no—
104C4×C71—42no—
105C4×C71#12114yes—
106Dic71—22no—
107Dic71—42no—
108Dic71—42no—
109Dic71#12414yescomplemented
110C2×C71C2×#12111yescharacteristic
111C2×C71C2×#12411yescharacteristic
112C2×C71#53611yescharacteristic
113C2×C71#54712yes—
114C2×C71#55612yes—
115D71—44no—
116#2547—711nomaximal, Sylow
117C71#254711yescharacteristic, complemented, Sylow
118C2×C4⋊Q8—711no—
119C2×#124—711no—
120C2×#124—711no—
121C2×C4×D4—711no—
122#397—714no—
123#391—711no—
124#391D7111yescharacteristic, complemented
125#380—714no—
126C4×C4⋊C4—711no—
127C22×Q8—711no—
128C22×Q8C2×D7111yescharacteristic
129C22×D4—711no—
130C23×C4—712no—
131C4⋊Q8—1424no—
132#124—1424no—
133#124—1424no—
134C4×D4—1424no—
135C2×C4⋊C4—712no—
136C2×C4⋊C4—712no—
137C2×C4⋊C4—712no—
138C2×C4⋊C4C2×D7111yescharacteristic
139C2×C22⋊C4—712no—
140C2×C22⋊C4—714no—
141C2×C22⋊C4—714no—
142C2×C42—711no—
143C2×C42—711no—
144C2×C42C2×D7111yescharacteristic
145#95—712no—
146#95—714no—
147#95C2×D7114yes—
148C24—712no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—1422no—
153C2×Q8—1422no—
154C2×Q8—1422no—
155C2×D4—714no—
156C2×D4—1422no—
157C2×D4—1422no—
158C22×C4—711no—
159C22×C4—711no—
160C22×C4—712no—
161C22×C4—712no—
162C22×C4—1424no—
163C22×C4—1428no—
164C22×C4C22×D7112yes—
165C22×C4C22×D7112yes—
166C22×C4C22×D7112yes—
167C22×C4C22×D7111yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—1424no—
170C4⋊C4—1424no—
171C4⋊C4—1424no—
172C22⋊C4—1424no—
173C22⋊C4—1428no—
174C22⋊C4—1428no—
175C42—22no—
176C42—714no—
177C42—1422no—
178C23—714no—
179C23—1422no—
180C23—1424no—
181C23—1422no—
182C23—1424no—
183C23C23×D7111yescharacteristic
184Q8—44no—
185Q8—2844no—
186D4—1428no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—2848no—
196C2×C4—712no—
197C2×C4—714no—
198C2×C4—712no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4—1422no—
202C2×C4—1422no—
203C2×C4—1422no—
204C2×C4—1422no—
205C2×C4—1422no—
206C2×C4D4×D7112yes—
207C2×C4D4×D7112yes—
208C2×C4#1176586512yes—
209C22—2842no—
210C22—2844no—
211C22—2842no—
212C22—1428no—
213C22—1424no—
214C22C2×D4×D7111yescharacteristic
215C22C2×D4×D7111yescharacteristic
216C22#41×D7112yes—
217C22C2×#1176586511yescharacteristic
218C22#1455617612yes—
219C4—42no—
220C4—42no—
221C4—42no—
222C4—2842no—
223C4—2842no—
224C4—714no—
225C4—1422no—
226C4#1455610114yes—
227C2—2844no—
228C2C2×#1455610111yescharacteristic
229C2#124×D7111yescharacteristic
230C2#1798657411yescharacteristic
231C2#1798665112yes—
232C2#1798672112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#391×C71C2
hypercenter#391D71
hypocenterC71#2547
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#14556126C2semidirect
5. #17985928C2semidirect
6. #17985904C2semidirect
7. #17985858C2semidirect
8. C4×Dic71⋊C4C2semidirect
9. #391×C71C2semidirect
109. Dic71#124semidirect
117. C71#2547semidirect
124. #391D71semidirect