Finite Group Databasebeta

← #27150076

Subgroups — #27150076

Order 12416.

564 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 177 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×#15690789C211yesmaximal, characteristic
2C22×C4×Dic97C211yesmaximal, characteristic
3#20260640C212yesmaximal, complemented
4#20260634C212yesmaximal, complemented
5#20260620C214yesmaximal, complemented
6#20260610C212yesmaximal
7#20260596C212yesmaximal
8#380×C97C211yesmaximal, characteristic
9C22×Dic194C2211yescharacteristic
10C23×Dic97C2211yescharacteristic
11C23×Dic97C2211yescharacteristic
12#15690789—24no—
13C2×C4⋊Dic97C2214yes—
14C2×C4⋊Dic97C2211yescharacteristic
15C2×#13095490C2212yes—
16C2×C22⋊Dic97C2212yes—
17C2×C4×Dic97—24no—
18C2×C4×Dic97C2212yes—
19C2×C4×Dic97C2212yes—
20#15690717C2214yes—
21#15690717C2214yes—
22#15690717C2214yes—
23C23×C4×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2212yes—
25C2×C22⋊C4×C97C2212yes—
26#95×C97C2212yes—
27C2×Dic194—22no—
28C2×Dic194—24no—
29C22×Dic97—22no—
30C22×Dic97—22no—
31C22×Dic97—22no—
32C22×Dic97—24no—
33C22×Dic97Q814yescomplemented
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C22×Dic97C2312yes—
37C22×Dic97C2312yes—
38C4⋊Dic97—22no—
39C4⋊Dic97—28no—
40#13095490—24no—
41C22⋊Dic97—24no—
42C4×Dic97—24no—
43C4×Dic97—24no—
44C24×C97C2311yescharacteristic
45C22×C4×C97—24no—
46C22×C4×C97D414yes—
47C22×C4×C97C2314yes—
48C22×C4×C97C2311yescharacteristic
49C22×C4×C97C2311yescharacteristic
50C4⋊C4×C97—24no—
51C22⋊C4×C97—24no—
52C23×C97—22no—
53C23×C97—22no—
54C23×C97C2×D412yes—
55C23×C97C2×Q812yes—
56C23×C97#4112yes—
57C23×C97C2411yescharacteristic
58C2×C4×C97—22no—
59C2×C4×C97—28no—
60C2×C4×C97—24no—
61C2×C4×C97—24no—
62C2×C4×C97—24no—
63C2×C4×C97C2×D412yes—
64C2×C4×C97C2×D412yes—
65C2×C4×C97#4114yes—
66C2×Dic97—28no—
67C2×Dic97—28no—
68C2×Dic97—22no—
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—24no—
75C2×Dic97—24no—
76C2×Dic97C2×Q814yes—
77C2×Dic97#4114yes—
78Dic194—44no—
79C22×C97—22no—
80C22×C97—22no—
81C22×C97—24no—
82C22×C97—24no—
83C22×C97C22⋊Q814yes—
84C22×C97C22×D411yescharacteristic
85C22×C97C22×Q811yescharacteristic
86C22×C97C2×#4112yes—
87C22×C97C2×#4111yescharacteristic
88C22×C97C2×#4112yes—
89C4×C97—22no—
90C4×C97—44no—
91C4×C97#12314yes—
92Dic97—24no—
93Dic97—24no—
94Dic97—42no—
95Dic97—42no—
96C2×C97—24no—
97C2×C97C2×C22⋊Q811yescharacteristic
98C2×C97C2×#12311yescharacteristic
99C2×C97#52511yescharacteristic
100C2×C97#52912yes—
101C2×C97#53312yes—
102#2855—971nomaximal, Sylow
103C97#285511yescharacteristic, complemented, Sylow
104C2×C22⋊Q8—971no—
105C22×C42—971no—
106#391—972no—
107#389—972no—
108#387—974no—
109#385—972no—
110#381—972no—
111#380D9711yescharacteristic
112C22×Q8—971no—
113C23×C4—971no—
114C23×C4—971no—
115C23×C4C2×D9711yescharacteristic
116C22⋊Q8—1944no—
117C2×C4⋊C4—972no—
118C2×C4⋊C4—971no—
119C2×C4⋊C4—974no—
120C2×C4⋊C4C2×D9712yes—
121C2×C22⋊C4—972no—
122C2×C22⋊C4C2×D9712yes—
123C2×C42—972no—
124C2×C42—972no—
125C2×C42—1944no—
126#95—974no—
127#95—974no—
128#95—974no—
129#95C2×D9712yes—
130C24C22×D9711yescharacteristic
131C2×Q8—1942no—
132C2×Q8—1944no—
133C22×C4—24no—
134C22×C4—974no—
135C22×C4—972no—
136C22×C4—972no—
137C22×C4—972no—
138C22×C4—972no—
139C22×C4—1944no—
140C22×C4—1942no—
141C22×C4—1942no—
142C22×C4—1942no—
143C22×C4D38814yes—
144C22×C4C22×D9714yes—
145C22×C4C22×D9711yescharacteristic
146C22×C4C22×D9711yescharacteristic
147C4⋊C4—24no—
148C4⋊C4—1948no—
149C4⋊C4—1944no—
150C4⋊C4—1942no—
151C22⋊C4—24no—
152C22⋊C4—1944no—
153C42—1944no—
154C42—1944no—
155C23—22no—
156C23—22no—
157C23C2×D38812yes—
158C23Q8×D9712yes—
159C23#1309551012yes—
160C23C23×D9711yescharacteristic
161Q8—3884no—
162C2×C4—22no—
163C2×C4—28no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—974no—
168C2×C4—974no—
169C2×C4—1942no—
170C2×C4—1942no—
171C2×C4—1944no—
172C2×C4—1944no—
173C2×C4—1948no—
174C2×C4—1948no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4—1942no—
178C2×C4—1942no—
179C2×C4C2×D38812yes—
180C2×C4C2×D38812yes—
181C2×C4#1309551014yes—
182C22—22no—
183C22—22no—
184C22—24no—
185C22—24no—
186C22#1569079014yes—
187C22C22×D38811yescharacteristic
188C22C2×Q8×D9711yescharacteristic
189C22#41×D9712yes—
190C22C2×#1309551011yescharacteristic
191C22C2×#1309551012yes—
192C4—22no—
193C4—44no—
194C4—3882no—
195C4—3882no—
196C4—1944no—
197C4—1944no—
198C4#1569079414yes—
199C2—24no—
200C2C2×#1569079011yescharacteristic
201C2C2×#1569079411yescharacteristic
202C2#2026124711yescharacteristic
203C2#2026126612yes—
204C2#2026128712yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#380×C97C2
prime radicalC24×C97C23
hypercenter#380D97
hypocenterC97#2855
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20260640C2semidirect
4. #20260634C2semidirect
5. #20260620C2semidirect
33. C22×Dic97Q8semidirect
103. C97#2855semidirect

Maximal subgroups

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

subgroupindex
C2×#156907892
C22×C4×Dic972
#202606402
#202606342
#202606202
#202606102
#202605962
#380×C972
#285597