Finite Group Databasebeta

← #396×D97 #27144955

Subgroups — #396×D97 #27144955

Order 12416.

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×D97C213yesmaximal, complemented
2C2×C22⋊C4×D97C213yesmaximal
3#20260659C211yesmaximal, characteristic, complemented
4#20260658C213yesmaximal, complemented
5#20260657C213yesmaximal, complemented
6#95×D97C211yesmaximal, characteristic, complemented
7#396×C97C211yesmaximal, characteristic, complemented
8C24×D97C2211yescharacteristic
9C22×C4×D97C2213yes—
10C22×C4×D97C2213yes—
11C2×C4⋊Dic97C2213yescomplemented
12C2×#13095490C2213yescomplemented
13C2×#13095490C2213yescomplemented
14C4⋊C4×D97—212no—
15C2×C22⋊Dic97C2213yes—
16C2×#13095488C2213yes—
17C2×#13095488C2213yes—
18C22⋊C4×D97—212no—
19#15690717C2213yescomplemented
20C2×C4⋊C4×C97C2213yescomplemented
21C2×C22⋊C4×C97C2213yes—
22#95×C97C2211yescharacteristic, complemented
23C23×D97—26no—
24C23×D97—26no—
25C23×D97Q814yes—
26C23×D97C2311yescharacteristic
27C23×D97C2311yescharacteristic
28C22×Dic97C2313yes—
29C22×Dic97C2313yes—
30C2×C4×D97—26no—
31C2×C4×D97—26no—
32C2×C4×D97—26no—
33C2×C4×D97—26no—
34C2×C4×D97—26no—
35C2×C4×D97D4112yes—
36C4⋊Dic97—26no—
37#13095490—26no—
38#13095490—26no—
39C22⋊Dic97—26no—
40#13095488—26no—
41#13095488—26no—
42C24×C97C2311yescharacteristic
43C22×C4×C97C2313yes—
44C22×C4×C97C2313yes—
45C4⋊C4×C97—26no—
46C22⋊C4×C97—26no—
47C23×C97—23no—
48C23×C97—23no—
49C23×C97C2×Q812yes—
50C23×C97C2411yescharacteristic
51C22×D97—23no—
52C22×D97—23no—
53C22×D97—224no—
54C22×D97—412no—
55C22×D97—44no—
56C22×D97C2×D416yes—
57C22×D97C2×Q812yes—
58C22×D97C2×Q812yes—
59C22×D97#4116yes—
60C2×C4×C97—23no—
61C2×C4×C97—23no—
62C2×C4×C97—23no—
63C2×C4×C97—23no—
64C2×C4×C97—23no—
65C2×C4×C97C2×D416yes—
66C2×Dic97—23no—
67C2×Dic97—23no—
68C2×Dic97—23no—
69C2×Dic97—23no—
70C2×Dic97—23no—
71C2×Dic97C2×D416yes—
72C4×D97—224no—
73C4×D97—412no—
74C22×C97—26no—
75C22×C97—43no—
76C22×C97—41no—
77C22×C97C22×D413yes—
78C22×C97C22×Q811yescharacteristic
79C22×C97C2×#4113yes—
80C2×D97—26no—
81C2×D97—43no—
82C2×D97—41no—
83C2×D97—416no—
84C2×D97#121112yes—
85C2×D97C22⋊Q8112yes—
86C2×D97#12614yes—
87C4×C97—26no—
88C4×C97—43no—
89Dic97—26no—
90Dic97—43no—
91C2×C97—42no—
92C2×C97C2×#12113yes—
93C2×C97C2×C22⋊Q813yes—
94C2×C97C2×#12611yescharacteristic
95D97—42no—
96D97#39618yesdirect product factor
97C2×#396—971nomaximal, Sylow
98C97C2×#39611yescharacteristic, complemented, Sylow
99C22×C4⋊C4—973no—
100C22×C22⋊C4—973no—
101#396—973no—
102#396—973no—
103#396—971no—
104#396D9711yescharacteristic, direct product factor
105C2×#95—971no—
106C25—971no—
107C23×C4—973no—
108C23×C4—973no—
109C2×C4⋊C4—973no—
110C2×C4⋊C4—973no—
111C2×C4⋊C4—973no—
112C2×C4⋊C4—19412no—
113C2×C4⋊C4C2×D9713yescomplemented
114C2×C22⋊C4—973no—
115C2×C22⋊C4—973no—
116C2×C22⋊C4—973no—
117C2×C22⋊C4—19412no—
118C2×C22⋊C4C2×D9713yes—
119#95—973no—
120#95C2×D9711yescharacteristic, complemented
121C24—971no—
122C24—974no—
123C24—971no—
124C24—1946no—
125C24—1946no—
126C24C22×D9711yescharacteristic
127C22×C4—973no—
128C22×C4—9712no—
129C22×C4—973no—
130C22×C4—1946no—
131C22×C4—1946no—
132C22×C4—1946no—
133C22×C4—1946no—
134C22×C4—1946no—
135C22×C4C22×D9713yes—
136C22×C4C22×D9713yes—
137C4⋊C4—26no—
138C4⋊C4—1946no—
139C4⋊C4—1946no—
140C4⋊C4—1946no—
141C22⋊C4—26no—
142C22⋊C4—1946no—
143C22⋊C4—1946no—
144C22⋊C4—1946no—
145C23—23no—
146C23—23no—
147C23—976no—
148C23—972no—
149C23—976no—
150C23—972no—
151C23—38812no—
152C23—3884no—
153C23—19424no—
154C23—1943no—
155C23—1943no—
156C23Q8×D9712yes—
157C23C23×D9711yescharacteristic
158C2×C4—23no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—23no—
162C2×C4—23no—
163C2×C4—976no—
164C2×C4—38812no—
165C2×C4—19424no—
166C2×C4—1943no—
167C2×C4—1943no—
168C2×C4—1943no—
169C2×C4—1943no—
170C2×C4—1943no—
171C2×C4D4×D9716yes—
172C22—26no—
173C22—43no—
174C22—41no—
175C22—9712no—
176C22—9712no—
177C22—974no—
178C22—38816no—
179C22—3883no—
180C22—3881no—
181C22—1946no—
182C22C2×D4×D9713yes—
183C22C2×Q8×D9711yescharacteristic
184C22#41×D9713yes—
185C4—26no—
186C4—43no—
187C4—3883no—
188C4—1946no—
189C2—42no—
190C2—978no—
191C2—3882no—
192C2#121×D9713yes—
193C2C22⋊Q8×D9713yes—
194C2#126×D9711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#396×C97C2
prime radicalC24×D97C22
hypercenter#396D97
hypocenterC97C2×#396
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D97C2semidirect
3. #20260659C2semidirect
4. #20260658C2semidirect
5. #20260657C2semidirect
6. #95×D97C2semidirect
7. #396×C97C2semidirect
11. C2×C4⋊Dic97C22semidirect
12. C2×#13095490C22semidirect
13. C2×#13095490C22semidirect
19. #15690717C22semidirect
20. C2×C4⋊C4×C97C22semidirect
22. #95×C97C22semidirect
96. D97#396direct (N × H)
98. C97C2×#396semidirect
104. #396D97direct (N × H)
113. C2×C4⋊C4C2×D97semidirect
120. #95C2×D97semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4⋊C4×D972
C2×C22⋊C4×D972
#202606592
#202606582
#202606572
#95×D972
#396×C972
C2×#39697