Finite Group Databasebeta

← #396×D71 #22591559

Subgroups — #396×D71 #22591559

Order 9088.

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

normal subgroups only

This group has 908 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
1C2×C4⋊C4×D71C213yesmaximal, complemented
2C2×C22⋊C4×D71C213yesmaximal
3#17985923C211yesmaximal, characteristic, complemented
4#17985922C213yesmaximal, complemented
5#17985921C213yesmaximal, complemented
6#95×D71C211yesmaximal, characteristic, complemented
7#396×C71C211yesmaximal, characteristic, complemented
8C24×D71C2211yescharacteristic
9C22×C4×D71C2213yes—
10C22×C4×D71C2213yes—
11C2×C4⋊Dic71C2213yescomplemented
12C2×Dic71⋊C4C2213yescomplemented
13C2×Dic71⋊C4C2213yescomplemented
14C4⋊C4×D71—212no—
15C2×C22⋊Dic71C2213yes—
16C2×(C2×D71)⋊C4C2213yes—
17C2×(C2×D71)⋊C4C2213yes—
18C22⋊C4×D71—212no—
19C71⋊#95C2213yescomplemented
20C2×C4⋊C4×C71C2213yescomplemented
21C2×C22⋊C4×C71C2213yes—
22#95×C71C2211yescharacteristic, complemented
23C23×D71—26no—
24C23×D71—26no—
25C23×D71Q814yes—
26C23×D71C2311yescharacteristic
27C23×D71C2311yescharacteristic
28C22×Dic71C2313yes—
29C22×Dic71C2313yes—
30C2×C4×D71—26no—
31C2×C4×D71—26no—
32C2×C4×D71—26no—
33C2×C4×D71—26no—
34C2×C4×D71—26no—
35C2×C4×D71D4112yes—
36C4⋊Dic71—26no—
37Dic71⋊C4—26no—
38Dic71⋊C4—26no—
39C22⋊Dic71—26no—
40(C2×D71)⋊C4—26no—
41(C2×D71)⋊C4—26no—
42C24×C71C2311yescharacteristic
43C22×C4×C71C2313yes—
44C22×C4×C71C2313yes—
45C4⋊C4×C71—26no—
46C22⋊C4×C71—26no—
47C23×C71—23no—
48C23×C71—23no—
49C23×C71C2×Q812yes—
50C23×C71C2411yescharacteristic
51C22×D71—23no—
52C22×D71—23no—
53C22×D71—224no—
54C22×D71—412no—
55C22×D71—44no—
56C22×D71C2×D416yes—
57C22×D71C2×Q812yes—
58C22×D71C2×Q812yes—
59C22×D71#4116yes—
60C2×C4×C71—23no—
61C2×C4×C71—23no—
62C2×C4×C71—23no—
63C2×C4×C71—23no—
64C2×C4×C71—23no—
65C2×C4×C71C2×D416yes—
66C2×Dic71—23no—
67C2×Dic71—23no—
68C2×Dic71—23no—
69C2×Dic71—23no—
70C2×Dic71—23no—
71C2×Dic71C2×D416yes—
72C4×D71—224no—
73C4×D71—412no—
74C22×C71—26no—
75C22×C71—43no—
76C22×C71—41no—
77C22×C71C22×D413yes—
78C22×C71C22×Q811yescharacteristic
79C22×C71C2×#4113yes—
80C2×D71—26no—
81C2×D71—43no—
82C2×D71—41no—
83C2×D71—416no—
84C2×D71#121112yes—
85C2×D71C22⋊Q8112yes—
86C2×D71#12614yes—
87C4×C71—26no—
88C4×C71—43no—
89Dic71—26no—
90Dic71—43no—
91C2×C71—42no—
92C2×C71C2×#12113yes—
93C2×C71C2×C22⋊Q813yes—
94C2×C71C2×#12611yescharacteristic
95D71—42no—
96D71#39618yesdirect product factor
97C2×#396—711nomaximal, Sylow
98C71C2×#39611yescharacteristic, complemented, Sylow
99C22×C4⋊C4—713no—
100C22×C22⋊C4—713no—
101#396—713no—
102#396—713no—
103#396—711no—
104#396D7111yescharacteristic, direct product factor
105C2×#95—711no—
106C25—711no—
107C23×C4—713no—
108C23×C4—713no—
109C2×C4⋊C4—713no—
110C2×C4⋊C4—713no—
111C2×C4⋊C4—713no—
112C2×C4⋊C4—14212no—
113C2×C4⋊C4C2×D7113yescomplemented
114C2×C22⋊C4—713no—
115C2×C22⋊C4—713no—
116C2×C22⋊C4—713no—
117C2×C22⋊C4—14212no—
118C2×C22⋊C4C2×D7113yes—
119#95—713no—
120#95C2×D7111yescharacteristic, complemented
121C24—711no—
122C24—714no—
123C24—711no—
124C24—1426no—
125C24—1426no—
126C24C22×D7111yescharacteristic
127C22×C4—713no—
128C22×C4—7112no—
129C22×C4—713no—
130C22×C4—1426no—
131C22×C4—1426no—
132C22×C4—1426no—
133C22×C4—1426no—
134C22×C4—1426no—
135C22×C4C22×D7113yes—
136C22×C4C22×D7113yes—
137C4⋊C4—26no—
138C4⋊C4—1426no—
139C4⋊C4—1426no—
140C4⋊C4—1426no—
141C22⋊C4—26no—
142C22⋊C4—1426no—
143C22⋊C4—1426no—
144C22⋊C4—1426no—
145C23—23no—
146C23—23no—
147C23—28412no—
148C23—2844no—
149C23—716no—
150C23—712no—
151C23—716no—
152C23—712no—
153C23—14224no—
154C23—1423no—
155C23—1423no—
156C23Q8×D7112yes—
157C23C23×D7111yescharacteristic
158C2×C4—23no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—23no—
162C2×C4—23no—
163C2×C4—28412no—
164C2×C4—716no—
165C2×C4—14224no—
166C2×C4—1423no—
167C2×C4—1423no—
168C2×C4—1423no—
169C2×C4—1423no—
170C2×C4—1423no—
171C2×C4D4×D7116yes—
172C22—26no—
173C22—43no—
174C22—41no—
175C22—28416no—
176C22—2843no—
177C22—2841no—
178C22—7112no—
179C22—7112no—
180C22—714no—
181C22—1426no—
182C22C2×D4×D7113yes—
183C22C2×Q8×D7111yescharacteristic
184C22#41×D7113yes—
185C4—26no—
186C4—43no—
187C4—2843no—
188C4—1426no—
189C2—42no—
190C2—2842no—
191C2—718no—
192C2#121×D7113yes—
193C2C22⋊Q8×D7113yes—
194C2#126×D7111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#396×C71C2
hypercenter#396D71
hypocenterC71C2×#396
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D71C2semidirect
3. #17985923C2semidirect
4. #17985922C2semidirect
5. #17985921C2semidirect
6. #95×D71C2semidirect
7. #396×C71C2semidirect
11. C2×C4⋊Dic71C22semidirect
12. C2×Dic71⋊C4C22semidirect
13. C2×Dic71⋊C4C22semidirect
19. C71⋊#95C22semidirect
20. C2×C4⋊C4×C71C22semidirect
22. #95×C71C22semidirect
96. D71#396direct (N × H)
98. C71C2×#396semidirect
104. #396D71direct (N × H)
113. C2×C4⋊C4C2×D71semidirect
120. #95C2×D71semidirect