Finite Group Databasebeta

← #27144921

Subgroups — #27144921

Order 12416.

408 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15690802C211yesmaximal, characteristic
2#20260632C212yesmaximal
3#20260630C212yesmaximal
4#20260622C212yesmaximal
5#20260620C212yesmaximal
6#20260618C212yesmaximal
7#20260616C212yesmaximal
8#20260601C211yesmaximal, characteristic
9#382×C97C211yesmaximal, characteristic
10#15690802C418yescomplemented
11C2×C4⋊Dic97C2211yescharacteristic
12C2×C4⋊Dic97C2211yescharacteristic
13C2×C4⋊Dic97C2211yescharacteristic
14C2×C4⋊Dic97C2211yescharacteristic
15C2×#13095490C2212yes—
16C2×#13095490C2212yes—
17C2×#13095490C2212yes—
18C2×#13095490C2212yes—
19C2×C4×Dic97C2212yes—
20C2×C4×Dic97C2212yes—
21#15690717C2212yes—
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2212yes—
26#15690717C2212yes—
27C2×C4⋊C4×C97C2211yescharacteristic
28C2×C4⋊C4×C97C2211yescharacteristic
29C2×C42×C97C2211yescharacteristic
30#95×C97C2212yes—
31#95×C97C2212yes—
32C22×Dic97C2312yes—
33C22×Dic97C2312yes—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C4⋊Dic97—41no—
37C4⋊Dic97—41no—
38C4⋊Dic97C2×C414yes—
39C4⋊Dic97C2×C414yes—
40#13095490—42no—
41#13095490—42no—
42#13095490C2×C418yes—
43#13095490C2×C418yes—
44C4×Dic97—42no—
45C4×Dic97—42no—
46C22×C4×C97C2312yes—
47C22×C4×C97C2312yes—
48C22×C4×C97C2311yescharacteristic
49C22×C4×C97C2311yescharacteristic
50C22×C4×C97C2311yescharacteristic
51C4⋊C4×C97—41no—
52C4⋊C4×C97—41no—
53C42×C97C2×C414yes—
54C23×C97C2411yescharacteristic
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97C22×C412yes—
68C2×C4×C97C22×C412yes—
69C2×C4×C97C22×C412yes—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97C22×C414yes—
81C2×Dic97C22×C414yes—
82C22×C97C23×C411yescharacteristic
83C22×C97H(5,𝔽2)11yescharacteristic
84C22×C97H(5,𝔽2)11yescharacteristic
85C22×C972−1+411yescharacteristic
86C22×C972−1+411yescharacteristic
87C22×C972−1+411yescharacteristic
88C22×C972−1+411yescharacteristic
89C4×C97—22no—
90C4×C97—22no—
91C4×C97—22no—
92C4×C97—42no—
93C4×C97—42no—
94Dic97—24no—
95Dic97—24no—
96Dic97—42no—
97Dic97—42no—
98C2×C97#51911yescharacteristic
99C2×C97#52011yescharacteristic
100C2×C97#52011yescharacteristic
101C2×C97#56211yescharacteristic
102C2×C97#56311yescharacteristic
103C2×C97#56311yescharacteristic
104C2×C97#56411yescharacteristic
105#2382—971nomaximal, Sylow
106C97#238211yescharacteristic, complemented, Sylow
107C2×#125—971no—
108#389—972no—
109#389—972no—
110#387—972no—
111#387—972no—
112#387—972no—
113#387—972no—
114#382—971no—
115#382D9711yescharacteristic
116#125—978no—
117C2×C4⋊C4—972no—
118C2×C4⋊C4—972no—
119C2×C4⋊C4—971no—
120C2×C4⋊C4—971no—
121C2×C4⋊C4—972no—
122C2×C4⋊C4—972no—
123C2×C4⋊C4—971no—
124C2×C4⋊C4—971no—
125C2×C4⋊C4C2×D9711yescharacteristic
126C2×C4⋊C4C2×D9711yescharacteristic
127C2×C42—972no—
128C2×C42—972no—
129C2×C42C2×D9711yescharacteristic
130#95—972no—
131#95—972no—
132#95—972no—
133#95—972no—
134#95—972no—
135#95—972no—
136#95C2×D9712yes—
137#95C2×D9712yes—
138C22×C4—972no—
139C22×C4—972no—
140C22×C4—972no—
141C22×C4—972no—
142C22×C4C22×D9712yes—
143C22×C4C22×D9712yes—
144C22×C4C22×D9711yescharacteristic
145C22×C4C22×D9711yescharacteristic
146C22×C4C22×D9711yescharacteristic
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—978no—
150C4⋊C4—978no—
151C4⋊C4—974no—
152C4⋊C4—974no—
153C4⋊C4—3882no—
154C4⋊C4—3882no—
155C4⋊C4—3881no—
156C4⋊C4—3881no—
157C42—3882no—
158C42—3882no—
159C42C4×D9714yes—
160C23C23×D9711yescharacteristic
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—974no—
174C2×C4—974no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4—1942no—
178C2×C4—1942no—
179C2×C4—1942no—
180C2×C4—1942no—
181C2×C4—1942no—
182C2×C4—1942no—
183C2×C4—1942no—
184C2×C4—1942no—
185C2×C4C2×C4×D9712yes—
186C2×C4C2×C4×D9712yes—
187C2×C4C2×C4×D9712yes—
188C22C22×C4×D9711yescharacteristic
189C22#1569085511yescharacteristic
190C22#1569085611yescharacteristic
191C22#1569085711yescharacteristic
192C22#1569085711yescharacteristic
193C22#1569085711yescharacteristic
194C22#1569085811yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—3882no—
201C4—3882no—
202C4—1944no—
203C4—1944no—
204C2#2026121211yescharacteristic
205C2#2026121511yescharacteristic
206C2#2026121611yescharacteristic
207C2#2026147911yescharacteristic
208C2#2026148511yescharacteristic
209C2#2026148911yescharacteristic
210C2C97⋊#56411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#382×C97C2
socle, prime radical, cosocleC23×C97C24
hypercenter#382D97
hypocenterC97#2382
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
10. #15690802C4semidirect
106. C97#2382semidirect

Maximal subgroups

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

subgroupindex
C2×#156908022
#202606322
#202606302
#202606222
#202606202
#202606182
#202606162
#202606012
#382×C972
#238297