Finite Group Databasebeta

← #393×D71 #22591538

Subgroups — #393×D71 #22591538

Order 9088.

1036 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 203 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
1C22×Q8×D71C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D71C213yesmaximal
3#17985912C213yesmaximal, complemented
4#17985911C213yesmaximal, complemented
5#17985910C211yesmaximal, characteristic, complemented
6#95×D71C213yesmaximal, complemented
7#393×C71C211yesmaximal, characteristic, complemented
8C24×D71C2211yescharacteristic
9C22×Dic142C2213yescomplemented
10C2×Q8×D71—212no—
11C22×C4×D71C2213yes—
12C22×C4×D71C2213yes—
13C2×C22⋊Dic71C2213yes—
14C2×(C2×D71)⋊C4C2213yes—
15C2×(C2×D71)⋊C4C2213yes—
16C22⋊C4×D71—212no—
17C71⋊#95C2213yescomplemented
18C71⋊#95C2216yescomplemented
19C22×Q8×C71C2211yescharacteristic, complemented
20C2×C22⋊C4×C71C2213yes—
21#95×C71C2213yescomplemented
22C23×D71—26no—
23C23×D71—26no—
24C23×D71Q814yescomplemented
25C23×D71C2311yescharacteristic
26C23×D71C2311yescharacteristic
27C2×Dic142—26no—
28C2×Dic142—212no—
29Q8×D71—232no—
30C22×Dic71C2313yes—
31C22×Dic71C2313yes—
32C2×C4×D71—212no—
33C2×C4×D71—26no—
34C2×C4×D71—212no—
35C2×C4×D71D4112yes—
36C22⋊Dic71—26no—
37(C2×D71)⋊C4—26no—
38(C2×D71)⋊C4—26no—
39C24×C71C2311yescharacteristic
40C2×Q8×C71—26no—
41C22×C4×C71C2313yes—
42C22×C4×C71C2313yes—
43C22⋊C4×C71—26no—
44C23×C71—23no—
45C23×C71—23no—
46C23×C71C2×Q812yescomplemented
47C23×C71C2411yescharacteristic
48C22×D71—23no—
49C22×D71—23no—
50C22×D71—224no—
51C22×D71—44no—
52C22×D71—412no—
53C22×D71C2×D416yes—
54C22×D71C2×Q812yes—
55C22×D71C2×Q812yescomplemented
56C22×D71#4116yes—
57Q8×C71—28no—
58C2×C4×C71—26no—
59C2×C4×C71—23no—
60C2×C4×C71—26no—
61C2×C4×C71C2×D416yes—
62C2×Dic71—26no—
63C2×Dic71—23no—
64C2×Dic71—26no—
65C2×Dic71C2×D416yes—
66C4×D71—224no—
67C4×D71—412no—
68Dic142—224no—
69C22×C71—26no—
70C22×C71—41no—
71C22×C71—43no—
72C22×C71C22×D413yes—
73C22×C71C22×Q811yescharacteristic
74C22×C71C2×#4113yes—
75C2×D71—26no—
76C2×D71—41no—
77C2×D71—43no—
78C2×D71—416no—
79C2×D71#12014yes—
80C2×D71C22⋊Q8112yes—
81C2×D71#124112yes—
82C4×C71—26no—
83C4×C71—43no—
84Dic71—26no—
85Dic71—43no—
86C2×C71—42no—
87C2×C71C2×#12011yescharacteristic
88C2×C71C2×C22⋊Q813yes—
89C2×C71C2×#12413yes—
90D71—42no—
91D71#39318yesdirect product factor
92C2×#393—711nomaximal, Sylow
93C71C2×#39311yescharacteristic, complemented, Sylow
94C23×Q8—711no—
95C22×C22⋊C4—713no—
96#393—711no—
97#393—713no—
98#393—713no—
99#393D7111yescharacteristic, direct product factor
100C2×#95—713no—
101C25—711no—
102C22×Q8—713no—
103C22×Q8—14212no—
104C22×Q8C2×D7111yescharacteristic, complemented
105C23×C4—713no—
106C23×C4—713no—
107C2×C22⋊C4—713no—
108C2×C22⋊C4—713no—
109C2×C22⋊C4—713no—
110C2×C22⋊C4—14212no—
111C2×C22⋊C4C2×D7113yes—
112#95—716no—
113#95—713no—
114#95C2×D7113yescomplemented
115C24—711no—
116C24—714no—
117C24—711no—
118C24—1426no—
119C24—1426no—
120C24C22×D7111yescharacteristic
121C2×Q8—26no—
122C2×Q8—14212no—
123C2×Q8—14232no—
124C2×Q8—1426no—
125C22×C4—713no—
126C22×C4—7112no—
127C22×C4—713no—
128C22×C4—1426no—
129C22×C4—14212no—
130C22×C4—14212no—
131C22×C4C22×D7113yes—
132C22×C4C22×D7113yes—
133C22⋊C4—26no—
134C22⋊C4—1426no—
135C22⋊C4—1426no—
136C22⋊C4—1426no—
137C23—23no—
138C23—23no—
139C23—2844no—
140C23—28412no—
141C23—716no—
142C23—712no—
143C23—716no—
144C23—712no—
145C23—14224no—
146C23—1423no—
147C23—1423no—
148C23Q8×D7112yescomplemented
149C23C23×D7111yescharacteristic
150Q8—28no—
151Q8—14224no—
152C2×C4—26no—
153C2×C4—23no—
154C2×C4—26no—
155C2×C4—28412no—
156C2×C4—716no—
157C2×C4—14224no—
158C2×C4—1423no—
159C2×C4—1426no—
160C2×C4—1426no—
161C2×C4D4×D7116yes—
162C22—26no—
163C22—41no—
164C22—43no—
165C22—28416no—
166C22—2841no—
167C22—2843no—
168C22—714no—
169C22—7112no—
170C22—7112no—
171C22—1426no—
172C22C2×D4×D7113yes—
173C22C2×Q8×D7111yescharacteristic
174C22#41×D7113yes—
175C4—26no—
176C4—43no—
177C4—2843no—
178C4—1426no—
179C2—42no—
180C2—2842no—
181C2—718no—
182C2#120×D7111yescharacteristic
183C2C22⋊Q8×D7113yes—
184C2#124×D7113yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#393×C71C2
hypercenter#393D71
hypocenterC71C2×#393
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D71C2semidirect
3. #17985912C2semidirect
4. #17985911C2semidirect
5. #17985910C2semidirect
6. #95×D71C2semidirect
7. #393×C71C2semidirect
9. C22×Dic142C22semidirect
17. C71⋊#95C22semidirect
18. C71⋊#95C22semidirect
19. C22×Q8×C71C22semidirect
21. #95×C71C22semidirect
24. C23×D71Q8semidirect
46. C23×C71C2×Q8semidirect
55. C22×D71C2×Q8semidirect
91. D71#393direct (N × H)
93. C71C2×#393semidirect
99. #393D71direct (N × H)
104. C22×Q8C2×D71semidirect
114. #95C2×D71semidirect
148. C23Q8×D71semidirect