Finite Group Databasebeta

← #22594082

Subgroups — #22594082

Order 9088.

632 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14556111C212yesmaximal, complemented
2C2×#14556102C211yesmaximal, characteristic
3#17985929C212yesmaximal, complemented
4#17985921C214yesmaximal, complemented
5#17985916C212yesmaximal, complemented
6#17985878C212yesmaximal, complemented
7#17985859C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C71C211yesmaximal, characteristic, complemented
9C22×D284C2211yescharacteristic
10C22×C22⋊D71C2212yes—
11#14556111—28no—
12#14556102—24no—
13C2×C4⋊Dic71C2211yescharacteristic
14C2×C4⋊Dic71C2212yes—
15C2×Dic71⋊C4C2214yes—
16C2×C22⋊Dic71C2212yes—
17C2×(C2×D71)⋊C4C2214yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2214yes—
20C71⋊#95C2212yes—
21C71⋊#95C2214yes—
22C23×C4×C71C2211yescharacteristic
23C23×C4×C71C2211yescharacteristic
24C23×C4×C71C2211yescharacteristic
25C2×C4⋊C4×C71—24no—
26C2×C4⋊C4×C71C2212yes—
27C2×C4⋊C4×C71C2212yes—
28C23×D71C2312yes—
29C2×D284—22no—
30C2×D284—42no—
31C2×C22⋊D71—24no—
32C2×C22⋊D71—24no—
33C2×C22⋊D71—24no—
34C22×Dic71C2312yes—
35C22×Dic71C2314yes—
36C4⋊Dic71—22no—
37C4⋊Dic71—42no—
38Dic71⋊C4—28no—
39C22⋊Dic71—24no—
40(C2×D71)⋊C4—24no—
41(C2×D71)⋊C4—28no—
42(C2×D71)⋊C4—44no—
43C24×C71C2311yescharacteristic
44C22×C4×C71—22no—
45C22×C4×C71—22no—
46C22×C4×C71—22no—
47C22×C4×C71—42no—
48C22×C4×C71—42no—
49C22×C4×C71D414yes—
50C22×C4×C71D414yescomplemented
51C22×C4×C71C2312yes—
52C22×C4×C71C2311yescharacteristic
53C22×C4×C71C2312yes—
54C22×C4×C71C2311yescharacteristic
55C4⋊C4×C71—24no—
56C4⋊C4×C71—24no—
57C23×C71—22no—
58C23×C71—22no—
59C23×C71C2×D412yes—
60C23×C71C2×D412yes—
61C23×C71#4112yes—
62C23×C71C2411yescharacteristic
63C22×D71—22no—
64C22×D71—22no—
65C22×D71—22no—
66C22×D71—24no—
67C22×D71—24no—
68C2×C4×C71—28no—
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—44no—
76C2×C4×C71—44no—
77C2×C4×C71C2×D412yes—
78C2×C4×C71C2×D412yes—
79C2×C4×C71C2×D414yes—
80C22⋊D71—48no—
81C2×Dic71—24no—
82C2×Dic71—22no—
83C2×Dic71—22no—
84C2×Dic71—22no—
85C2×Dic71—24no—
86C2×Dic71—24no—
87D284—44no—
88C22×C71—22no—
89C22×C71—22no—
90C22×C71—24no—
91C22×C71—24no—
92C22×C71#12114yes—
93C22×C71C22×D411yescharacteristic
94C22×C71C22×D411yescharacteristic
95C22×C71C2×#4111yescharacteristic
96C22×C71H(5,𝔽2)12yes—
97C22×C712−1+412yes—
98C2×D71—42no—
99C2×D71—42no—
100C2×D71—42no—
101C2×D71—44no—
102C2×D71—44no—
103C4×C71—24no—
104C4×C71—22no—
105C4×C71—22no—
106C4×C71—42no—
107Dic71—42no—
108Dic71—44no—
109C2×C71—24no—
110C2×C71C2×#12111yescharacteristic
111C2×C71#53411yescharacteristic
112C2×C71#53611yescharacteristic
113C2×C71#53712yes—
114C2×C71#53912yes—
115D71—82no—
116#2618—711nomaximal, Sylow
117C71#261811yescharacteristic, complemented, Sylow
118C2×#123—712no—
119C2×#121—711no—
120C22×C4⋊C4D7111yescharacteristic, complemented
121#398—712no—
122#396—714no—
123#394—712no—
124#386—712no—
125#380—711no—
126C22×D4—712no—
127C22×D4—711no—
128C23×C4C2×D7111yescharacteristic
129C23×C4C2×D7111yescharacteristic
130C23×C4C2×D7111yescharacteristic
131#123—1428no—
132#121—1424no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—712no—
135C2×C4⋊C4—711no—
136C2×C4⋊C4—714no—
137C2×C4⋊C4C2×D7112yes—
138C2×C4⋊C4C2×D7112yes—
139C2×C22⋊C4—714no—
140C2×C22⋊C4—712no—
141C2×C22⋊C4—712no—
142C2×C22⋊C4—714no—
143#95—712no—
144#95—714no—
145C24—712no—
146C24C22×D7111yescharacteristic
147C2×D4—2842no—
148C2×D4—1424no—
149C2×D4—1424no—
150C2×D4—1424no—
151C2×D4—1422no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—714no—
158C22×C4—712no—
159C22×C4D28414yescomplemented
160C22×C4C22⋊D7114yes—
161C22×C4C22×D7112yes—
162C22×C4C22×D7111yescharacteristic
163C22×C4C22×D7111yescharacteristic
164C22×C4C22×D7112yes—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—2842no—
168C4⋊C4—1422no—
169C4⋊C4—1428no—
170C22⋊C4—2844no—
171C22⋊C4—1424no—
172C22⋊C4—1424no—
173C22⋊C4—1428no—
174C23—22no—
175C23—22no—
176C23—1422no—
177C23—1422no—
178C23—1422no—
179C23—1424no—
180C23—1424no—
181C23C2×C22⋊D7112yes—
182C23C2×D28412yes—
183C23#1176586512yes—
184C23C23×D7111yescharacteristic
185D4—2844no—
186D4—2848no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—28no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—1422no—
197C2×C4—1424no—
198C2×C4—1422no—
199C2×C4—1422no—
200C2×C4—1424no—
201C2×C4—1424no—
202C2×C4C2×C22⋊D7112yes—
203C2×C4C2×C22⋊D7112yes—
204C2×C4C2×D28414yes—
205C22—22no—
206C22—24no—
207C22—24no—
208C22—22no—
209C22—2842no—
210C22—2844no—
211C22—2844no—
212C22—2842no—
213C22—2842no—
214C22#1455610214yes—
215C22C22×C22⋊D7111yescharacteristic
216C22C22×D28411yescharacteristic
217C22C2×#1176586511yescharacteristic
218C22#1455617312yes—
219C22#1455617612yes—
220C4—24no—
221C4—22no—
222C4—22no—
223C4—42no—
224C4—2844no—
225C4—2842no—
226C2—24no—
227C2—5682no—
228C2C2×#1455610211yescharacteristic
229C2#1798655811yescharacteristic
230C2#1798657311yescharacteristic
231C2#1798658012yes—
232C2#1798659612yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC22×C4⋊C4×C71C2
hypercenterC22×C4⋊C4D71
hypocenterC71#2618
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14556111C2semidirect
3. #17985929C2semidirect
4. #17985921C2semidirect
5. #17985916C2semidirect
6. #17985878C2semidirect
7. #17985859C2semidirect
8. C22×C4⋊C4×C71C2semidirect
50. C22×C4×C71D4semidirect
117. C71#2618semidirect
120. C22×C4⋊C4D71semidirect
159. C22×C4D284semidirect