Finite Group Databasebeta

← #22596620

Subgroups — #22596620

Order 9088.

1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C24×Dic71C211yesmaximal, characteristic, complemented
2C2×#14556100C213yesmaximal, complemented
3C2×#14556097C211yesmaximal, characteristic, complemented
4#17985879C213yesmaximal, complemented
5#17985876C213yesmaximal, complemented
6#17985857C213yesmaximal, complemented
7C2×#120×C71C211yesmaximal, characteristic, complemented
8C22×C22⋊D71C2213yes—
9C22×C22⋊D71C2211yescharacteristic, complemented
10C23×Dic71—212no—
11C23×Dic71C2213yes—
12C23×Dic71C2213yes—
13#14556100—212no—
14#14556097—24no—
15C2×Dic71⋊C4C2213yescomplemented
16C2×C22⋊Dic71C2213yes—
17C2×C22⋊Dic71C2213yescomplemented
18C2×(C2×D71)⋊C4C2213yescomplemented
19C71⋊#95C2216yescomplemented
20C25×C71C2211yescharacteristic
21C22×D4×C71C2213yes—
22#120×C71—24no—
23C2×C22⋊C4×C71C2213yes—
24C23×D71C2311yescharacteristic
25C2×C22⋊D71—26no—
26C2×C22⋊D71—26no—
27C2×C22⋊D71—26no—
28C2×C22⋊D71—26no—
29C22×Dic71—232no—
30C22×Dic71—26no—
31C22×Dic71—212no—
32C22×Dic71—212no—
33C22×Dic71D4112yescomplemented
34C22×Dic71C2313yes—
35C22×Dic71C2313yes—
36C22×Dic71C2311yescharacteristic
37Dic71⋊C4—26no—
38C22⋊Dic71—26no—
39C22⋊Dic71—26no—
40(C2×D71)⋊C4—26no—
41C24×C71—26no—
42C24×C71—26no—
43C24×C71D414yescomplemented
44C24×C71C2311yescharacteristic
45C24×C71C2313yes—
46C2×D4×C71—26no—
47C2×D4×C71—26no—
48C2×D4×C71—26no—
49C22×C4×C71C2313yes—
50C22⋊C4×C71—26no—
51C23×C71—224no—
52C23×C71—28no—
53C23×C71—224no—
54C23×C71—26no—
55C23×C71—26no—
56C23×C71—21no—
57C23×C71—23no—
58C23×C71—23no—
59C23×C71C2×D416yes—
60C23×C71C2×D416yes—
61C23×C71#4116yes—
62C23×C71C2411yescharacteristic
63C22×D71—23no—
64C22×D71—21no—
65C22×D71—23no—
66D4×C71—412no—
67C2×C4×C71—23no—
68C2×C4×C71—23no—
69C2×C4×C71—23no—
70C22⋊D71—412no—
71C22⋊D71—44no—
72C2×Dic71—224no—
73C2×Dic71—26no—
74C2×Dic71—224no—
75C2×Dic71—23no—
76C2×Dic71—23no—
77C2×Dic71—23no—
78C2×Dic71C2×D416yes—
79C2×Dic71C2×D416yes—
80C22×C71—232no—
81C22×C71—26no—
82C22×C71—212no—
83C22×C71—26no—
84C22×C71—212no—
85C22×C71—41no—
86C22×C71—43no—
87C22×C71—43no—
88C22×C71#121112yescomplemented
89C22×C71C22×D413yes—
90C22×C71C22×D411yescharacteristic
91C22×C71C2×#4113yes—
92C2×D71—41no—
93C2×D71—43no—
94C2×D71—43no—
95C4×C71—43no—
96Dic71—26no—
97Dic71—43no—
98Dic71#12014yescomplemented
99C2×C71—212no—
100C2×C71—81no—
101C2×C71C2×#12011yescharacteristic
102C2×C71C2×#12113yes—
103C2×C71#52513yes—
104D71—81no—
105#2847—711nomaximal, Sylow
106C71#284711yescharacteristic, complemented, Sylow
107C24×C4—711no—
108C2×#121—713no—
109C2×#120—711no—
110C2×#120D7111yescharacteristic, complemented
111#386—713no—
112#386—713no—
113#380—713no—
114C25C2×D7111yescharacteristic
115C22×D4—713no—
116C22×D4—711no—
117C22×D4C2×D7113yes—
118C23×C4—713no—
119C23×C4—713no—
120C23×C4—14212no—
121#121—14212no—
122#120—24no—
123#120—1424no—
124C2×C4⋊C4—713no—
125C2×C22⋊C4—713no—
126C2×C22⋊C4—713no—
127C2×C22⋊C4—713no—
128C2×C22⋊C4C2×D7113yes—
129#95—716no—
130C24—26no—
131C24—26no—
132C24—711no—
133C24C22⋊D7114yescomplemented
134C24C22×D7111yescharacteristic
135C24C22×D7113yes—
136C2×D4—26no—
137C2×D4—26no—
138C2×D4—26no—
139C2×D4—1426no—
140C2×D4—1426no—
141C2×D4—1426no—
142C2×D4—1426no—
143C22×C4—711no—
144C22×C4—7112no—
145C22×C4—713no—
146C22×C4—713no—
147C22×C4—14212no—
148C22×C4—14212no—
149C22×C4—14232no—
150C22×C4—1426no—
151C22×C4C22×D7113yes—
152C4⋊C4—1426no—
153C22⋊C4—26no—
154C22⋊C4—1426no—
155C22⋊C4—1426no—
156C22⋊C4—1426no—
157C23—28no—
158C23—224no—
159C23—26no—
160C23—26no—
161C23—224no—
162C23—23no—
163C23—21no—
164C23—23no—
165C23—1423no—
166C23—1421no—
167C23—1423no—
168C23D4×D7116yes—
169C23C2×C22⋊D7116yes—
170C23#1176586416yes—
171C23C23×D7111yescharacteristic
172D4—412no—
173D4—28412no—
174D4—2844no—
175C2×C4—23no—
176C2×C4—23no—
177C2×C4—23no—
178C2×C4—716no—
179C2×C4—716no—
180C2×C4—1423no—
181C2×C4—1423no—
182C2×C4—1423no—
183C2×C4—14224no—
184C2×C4—14224no—
185C2×C4—1426no—
186C22—232no—
187C22—26no—
188C22—212no—
189C22—26no—
190C22—212no—
191C22—41no—
192C22—43no—
193C22—43no—
194C22—2841no—
195C22—2843no—
196C22—2843no—
197C22#14556100112yescomplemented
198C22C2×D4×D7113yes—
199C22C22×C22⋊D7111yescharacteristic
200C22C2×#1176586413yes—
201C4—43no—
202C4—2843no—
203C4—714no—
204C4—1426no—
205C2—212no—
206C2—81no—
207C2—5681no—
208C2#120×D7111yescharacteristic
209C2C2×#1455610013yes—
210C2#1798650513yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#120×C71C2
hypercenterC2×#120D71
hypocenterC71#2847
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic71C2semidirect
2. C2×#14556100C2semidirect
3. C2×#14556097C2semidirect
4. #17985879C2semidirect
5. #17985876C2semidirect
6. #17985857C2semidirect
7. C2×#120×C71C2semidirect
9. C22×C22⋊D71C22semidirect
15. C2×Dic71⋊C4C22semidirect
17. C2×C22⋊Dic71C22semidirect
18. C2×(C2×D71)⋊C4C22semidirect
19. C71⋊#95C22semidirect
33. C22×Dic71D4semidirect
43. C24×C71D4semidirect
88. C22×C71#121semidirect
98. Dic71#120semidirect
106. C71#2847semidirect
110. C2×#120D71semidirect
133. C24C22⋊D71semidirect
197. C22#14556100semidirect