Finite Group Databasebeta

← #27144317

Subgroups — #27144317

Order 12416.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C23×C22⋊D97C211yesmaximal, characteristic
2C2×#15690766C212yesmaximal, complemented
3#20260635C218yesmaximal, complemented
4#20260591C211yesmaximal, characteristic, complemented
5#20260590C212yesmaximal, complemented
6#379×C97C211yesmaximal, characteristic, complemented
7C24×D97C2211yescharacteristic
8C22×C22⋊D97—28no—
9C22×C22⋊D97C418yescomplemented
10C22×C22⋊D97C2214yes—
11C23×Dic97C2211yescharacteristic
12#15690766—28no—
13C2×#13095490C2214yescomplemented
14C2×C22⋊Dic97C2212yes—
15C2×C22⋊Dic97C2214yescomplemented
16C2×#13095488C2218yescomplemented
17C2×C4×Dic97C2214yescomplemented
18C25×C97C2211yescharacteristic
19C2×C22⋊C4×C97C2212yes—
20C2×C22⋊C4×C97C2214yescomplemented
21C23×D97—24no—
22C23×D97—44no—
23C23×D97C2×C414yes—
24C23×D97C2312yes—
25C2×C22⋊D97—232no—
26C2×C22⋊D97—28no—
27C2×C22⋊D97—28no—
28C2×C22⋊D97C2×C4116yes—
29C22×Dic97C2×C414yes—
30C22×Dic97D418yes—
31C22×Dic97C2314yes—
32C22×Dic97C2312yes—
33#13095490—28no—
34C22⋊Dic97—24no—
35C22⋊Dic97—44no—
36#13095488—48no—
37C4×Dic97—28no—
38C24×C97—24no—
39C24×C97—44no—
40C24×C97C2×C414yes—
41C24×C97C2312yes—
42C24×C97C2311yescharacteristic
43C22×C4×C97C2314yes—
44C22⋊C4×C97—24no—
45C22⋊C4×C97—44no—
46C23×C97—24no—
47C23×C97—28no—
48C23×C97—24no—
49C23×C97—24no—
50C23×C97—24no—
51C23×C97—48no—
52C23×C97—416no—
53C23×C97C22×C414yes—
54C23×C97C22×C412yes—
55C23×C97C2×D414yes—
56C23×C97C2411yescharacteristic
57C22×D97—28no—
58C22×D97—24no—
59C22×D97—28no—
60C22×D97—48no—
61C22×D97—416no—
62C22×D97C22×C414yes—
63C2×C4×C97—24no—
64C2×C4×C97—28no—
65C22⋊D97—232no—
66C2×Dic97—24no—
67C2×Dic97—28no—
68C2×Dic97C22⋊C4116yescomplemented
69C2×Dic97C22×C414yes—
70C2×Dic97C2×D414yes—
71C2×Dic97C2×D414yes—
72C22×C97—24no—
73C22×C97—28no—
74C22×C97—24no—
75C22×C97—24no—
76C22×C97—24no—
77C22×C97—416no—
78C22×C97—48no—
79C22×C97C2×C22⋊C414yes—
80C22×C97C23×C411yescharacteristic
81C22×C97C22×D412yes—
82C22×C97H(5,𝔽2)14yes—
83C2×D97—24no—
84C2×D97—28no—
85C2×D97—416no—
86C2×D97—48no—
87C4×C97—44no—
88Dic97—44no—
89Dic97C2×C22⋊C418yes—
90C2×C97—24no—
91C2×C97—44no—
92C2×C97C22×C22⋊C411yescharacteristic
93C2×C97#51812yes—
94C2×C97#53514yes—
95D97—44no—
96#2309—971nomaximal, Sylow
97C97#230911yescharacteristic, complemented, Sylow
98C23×D4—971no—
99C2×#117—972no—
100#390—978no—
101#379—971no—
102#379—972no—
103#379D9711yescharacteristic, complemented
104C25—971no—
105C25C2×D9711yescharacteristic
106C22×D4—978no—
107C22×D4—974no—
108C22×D4—1948no—
109C23×C4—971no—
110#117—1948no—
111C2×C4⋊C4—974no—
112C2×C22⋊C4—974no—
113C2×C22⋊C4—978no—
114C2×C22⋊C4—972no—
115C2×C22⋊C4C2×D9712yes—
116C2×C22⋊C4C2×D9714yescomplemented
117C2×C42—974no—
118C24—24no—
119C24—44no—
120C24—974no—
121C24—972no—
122C24—3884no—
123C24—1944no—
124C24C4×D9714yes—
125C24C22×D9712yes—
126C24C22×D9711yescharacteristic
127C2×D4—9716no—
128C2×D4—19432no—
129C2×D4—1948no—
130C2×D4—1948no—
131C22×C4—974no—
132C22×C4—978no—
133C22×C4—972no—
134C22×C4—974no—
135C22×C4C22×D9714yes—
136C4⋊C4—1948no—
137C22⋊C4—24no—
138C22⋊C4—44no—
139C22⋊C4—3888no—
140C22⋊C4—3884no—
141C22⋊C4—1944no—
142C42—1948no—
143C23—24no—
144C23—28no—
145C23—24no—
146C23—24no—
147C23—24no—
148C23—48no—
149C23—416no—
150C23—974no—
151C23—38816no—
152C23—3888no—
153C23—1948no—
154C23—1944no—
155C23—1948no—
156C23C2×C4×D9714yes—
157C23C2×C4×D9712yes—
158C23D4×D9714yes—
159C23C23×D9711yescharacteristic
160D4—19432no—
161C2×C4—24no—
162C2×C4—28no—
163C2×C4—9716no—
164C2×C4—974no—
165C2×C4—974no—
166C2×C4—974no—
167C2×C4—1944no—
168C2×C4—1948no—
169C22—24no—
170C22—28no—
171C22—24no—
172C22—24no—
173C22—24no—
174C22—416no—
175C22—48no—
176C22—38816no—
177C22—3888no—
178C22—1944no—
179C22—1948no—
180C22C22⋊C4×D9714yes—
181C22C22×C4×D9711yescharacteristic
182C22C2×D4×D9712yes—
183C22#1569085514yes—
184C4—44no—
185C4—978no—
186C4—3884no—
187C2—24no—
188C2—44no—
189C2—3884no—
190C2C2×C22⋊C4×D9711yescharacteristic
191C2#2026120812yes—
192C2#2026130014yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#379×C97C2
socle, cosocleC23×C97C24
prime radicalC23×C22⋊D97C2
hypercenter#379D97
hypocenterC97#2309
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#15690766C2semidirect
3. #20260635C2semidirect
4. #20260591C2semidirect
5. #20260590C2semidirect
6. #379×C97C2semidirect
9. C22×C22⋊D97C4semidirect
13. C2×#13095490C22semidirect
15. C2×C22⋊Dic97C22semidirect
16. C2×#13095488C22semidirect
17. C2×C4×Dic97C22semidirect
20. C2×C22⋊C4×C97C22semidirect
68. C2×Dic97C22⋊C4semidirect
97. C97#2309semidirect
103. #379D97semidirect
116. C2×C22⋊C4C2×D97semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D972
C2×#156907662
#202606352
#202605912
#202605902
#379×C972
#230997