Finite Group Databasebeta

← #22594123

Subgroups — #22594123

Order 9088.

526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556111C211yesmaximal, characteristic
2C22×C4⋊Dic71C211yesmaximal, characteristic, complemented
3#17985939C212yesmaximal, complemented
4#17985936C212yesmaximal, complemented
5#17985931C212yesmaximal, complemented
6#17985926C212yesmaximal, complemented
7#17985876C212yesmaximal
8#17985874C212yesmaximal, complemented
9#381×C71C211yesmaximal, characteristic, complemented
10C22×C22⋊D71C2211yescharacteristic
11C23×Dic71C2212yes—
12#14556111—24no—
13C2×C4⋊Dic71—24no—
14C2×C4⋊Dic71C2212yes—
15C2×C4⋊Dic71C2212yes—
16C2×C4⋊Dic71C2214yes—
17C2×Dic71⋊C4C2211yescharacteristic
18C2×Dic71⋊C4C2211yescharacteristic
19C2×C22⋊Dic71C2211yescharacteristic
20C2×(C2×D71)⋊C4C2211yescharacteristic
21C2×(C2×D71)⋊C4C2211yescharacteristic
22C2×(C2×D71)⋊C4C2214yes—
23C71⋊#95C2214yes—
24C71⋊#95C2214yes—
25C23×C4×C71C2211yescharacteristic
26C2×C22⋊C4×C71C2212yes—
27#95×C71C2212yes—
28#95×C71C2212yes—
29C23×D71C2311yescharacteristic
30C2×C22⋊D71—22no—
31C2×C22⋊D71—24no—
32C22×Dic71—24no—
33C22×Dic71—44no—
34C22×Dic71C2314yes—
35C22×Dic71C2311yescharacteristic
36C22×Dic71C2311yescharacteristic
37C22×Dic71C2311yescharacteristic
38C4⋊Dic71—24no—
39C4⋊Dic71—24no—
40C4⋊Dic71—44no—
41Dic71⋊C4—22no—
42Dic71⋊C4—22no—
43C22⋊Dic71—22no—
44(C2×D71)⋊C4—22no—
45(C2×D71)⋊C4—22no—
46(C2×D71)⋊C4—44no—
47C24×C71C2311yescharacteristic
48C22×C4×C71—24no—
49C22×C4×C71D414yes—
50C22×C4×C71C2314yes—
51C22×C4×C71C2311yescharacteristic
52C22×C4×C71C2311yescharacteristic
53C22⋊C4×C71—24no—
54C23×C71—22no—
55C23×C71—22no—
56C23×C71C2×D412yes—
57C23×C71#4114yes—
58C23×C71C2411yescharacteristic
59C22×D71—21no—
60C22×D71—22no—
61C22×D71—22no—
62C22×D71—22no—
63C2×C4×C71—28no—
64C2×C4×C71—22no—
65C2×C4×C71—22no—
66C2×C4×C71—24no—
67C2×C4×C71—24no—
68C2×C4×C71—24no—
69C2×C4×C71C2×D412yes—
70C2×C4×C71C2×D412yes—
71C22⋊D71—44no—
72C2×Dic71—21no—
73C2×Dic71—21no—
74C2×Dic71—21no—
75C2×Dic71—24no—
76C2×Dic71—22no—
77C2×Dic71—24no—
78C2×Dic71—22no—
79C2×Dic71—24no—
80C2×Dic71—22no—
81C2×Dic71—48no—
82C22×C71—22no—
83C22×C71—22no—
84C22×C71—28no—
85C22×C71#12314yes—
86C22×C71C22×D411yescharacteristic
87C22×C71C2×#4112yes—
88C22×C71H(5,𝔽2)12yes—
89C22×C712−1+412yes—
90C2×D71—41no—
91C2×D71—41no—
92C2×D71—41no—
93C2×D71—44no—
94C4×C71—22no—
95C4×C71—22no—
96C4×C71—44no—
97Dic71—41no—
98Dic71—41no—
99Dic71—44no—
100Dic71—41no—
101C2×C71—24no—
102C2×C71C2×#12311yescharacteristic
103C2×C71#53511yescharacteristic
104C2×C71#53611yescharacteristic
105C2×C71#53914yes—
106D71—81no—
107#2623—711nomaximal, Sylow
108C71#262311yescharacteristic, complemented, Sylow
109C2×#123—711no—
110C22×C4⋊C4—711no—
111#400—712no—
112#399—712no—
113#398—712no—
114#397—712no—
115#386—712no—
116#385—712no—
117#381D7111yescharacteristic, complemented
118C22×D4—711no—
119C23×C4—712no—
120C23×C4C2×D7111yescharacteristic
121#123—1424no—
122C2×C4⋊C4—714no—
123C2×C4⋊C4—712no—
124C2×C4⋊C4—712no—
125C2×C4⋊C4—711no—
126C2×C4⋊C4—711no—
127C2×C4⋊C4—1424no—
128C2×C22⋊C4—714no—
129C2×C22⋊C4—711no—
130C2×C22⋊C4—711no—
131C2×C22⋊C4—711no—
132C2×C22⋊C4C2×D7112yes—
133#95—714no—
134#95—714no—
135#95C2×D7112yes—
136#95C2×D7112yes—
137C24—711no—
138C24C22×D7111yescharacteristic
139C2×D4—1422no—
140C2×D4—1424no—
141C22×C4—24no—
142C22×C4—2844no—
143C22×C4—711no—
144C22×C4—711no—
145C22×C4—714no—
146C22×C4—711no—
147C22×C4—1424no—
148C22×C4D28414yes—
149C22×C4C22×D7114yes—
150C22×C4C22×D7111yescharacteristic
151C22×C4C22×D7111yescharacteristic
152C4⋊C4—2844no—
153C4⋊C4—1422no—
154C4⋊C4—1422no—
155C4⋊C4—1424no—
156C4⋊C4—1424no—
157C22⋊C4—24no—
158C22⋊C4—2844no—
159C22⋊C4—1422no—
160C22⋊C4—1422no—
161C22⋊C4—1422no—
162C23—22no—
163C23—22no—
164C23—1421no—
165C23—1422no—
166C23—1422no—
167C23—1422no—
168C23C2×D28412yes—
169C23#1176586414yes—
170C23C23×D7111yescharacteristic
171D4—2844no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—28no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—2848no—
179C2×C4—1421no—
180C2×C4—1421no—
181C2×C4—1424no—
182C2×C4—1421no—
183C2×C4—1422no—
184C2×C4—1424no—
185C2×C4—1422no—
186C2×C4—1424no—
187C2×C4—1422no—
188C2×C4C2×D28412yes—
189C2×C4C2×D28412yes—
190C22—22no—
191C22—22no—
192C22—28no—
193C22—2841no—
194C22—2841no—
195C22—2844no—
196C22—2841no—
197C22#1455611214yes—
198C22C22×D28411yescharacteristic
199C22C2×#1176586412yes—
200C22#1455617412yes—
201C22#1455617512yes—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—2844no—
206C4—2841no—
207C4—2841no—
208C4—2841no—
209C2—24no—
210C2—5681no—
211C2C2×#1455611211yescharacteristic
212C2#1798656811yescharacteristic
213C2#1798657511yescharacteristic
214C2#1798659314yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#381×C71C2
hypercenter#381D71
hypocenterC71#2623
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic71C2semidirect
3. #17985939C2semidirect
4. #17985936C2semidirect
5. #17985931C2semidirect
6. #17985926C2semidirect
8. #17985874C2semidirect
9. #381×C71C2semidirect
108. C71#2623semidirect
117. #381D71semidirect