Finite Group Databasebeta

← #27157617

Subgroups — #27157617

Order 12416.

272 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 76 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×#15690881C211yesmaximal, characteristic, complemented
2C2×#15690880C211yesmaximal, characteristic, complemented
3#20261706C212yesmaximal
4#20261611C212yesmaximal, complemented
5C2×M16×D97C211yesmaximal, characteristic
6C2×#13095521C2212yes—
7C2×#13095521C2212yes—
8#15690881—22no—
9#15690880—22no—
10#15690867—22no—
11C22⋊#11715897—22no—
12C22×C4×D97C2211yescharacteristic
13C2×#13095495C411yescharacteristic
14C2×#13095494C411yescharacteristic
15C2×#13095494C411yescharacteristic
16M16×D97—22no—
17M16×D97—22no—
18C2×C8×D97C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20C2×M16×C97C411yescharacteristic
21#13095521—22no—
22#13095521—22no—
23#13095521—22no—
24#13095521—22no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27C23×D97Q811yescharacteristic
28C22×Dic97D411yescharacteristic
29C2×C4×D97—21no—
30C2×C4×D97—21no—
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97Q811yescharacteristic
36C2×C4×D97C2311yescharacteristic
37#13095495—22no—
38#13095494—22no—
39#13095494—22no—
40C2×#11715895C2×C411yescharacteristic
41C2×#11715895C2×C411yescharacteristic
42C8×D97—21no—
43C8×D97—21no—
44C8×D97D412yes—
45C8×D97D412yes—
46C22×C4×C97C2×C411yescharacteristic
47M16×C97—22no—
48C2×C8×C97C2×C411yescharacteristic
49C2×C8×C97C2×C411yescharacteristic
50C23×C97C4⋊C411yescharacteristic
51C22×D97—21no—
52C22×D97—21no—
53C22×D97—42no—
54C22×D97C4⋊C411yescharacteristic
55C22×D97C2×Q811yescharacteristic
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97C4⋊C411yescharacteristic
59C2×C4×C97C22×C411yescharacteristic
60C2×Dic97—21no—
61C2×Dic97—21no—
62C2×Dic97C4⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C4×D97—21no—
65C4×D97—21no—
66C4×D97—21no—
67C4×D97—21no—
68C4×D97—21no—
69C4×D97—21no—
70C4×D97C2×D411yescharacteristic
71C4×D97C2×D411yescharacteristic
72C4×D97#4111yescharacteristic
73C4×D97#4111yescharacteristic
74#11715897—42no—
75#11715897—42no—
76C8×C97—21no—
77C8×C97C22⋊C412yes—
78#11715895—21no—
79#11715895C22⋊C412yes—
80C22×C97—21no—
81C22×C97—22no—
82C22×C97C2×C4⋊C411yescharacteristic
83C2×D97—22no—
84C2×D97—21no—
85C2×D97—22no—
86C2×D97—42no—
87C2×D97—42no—
88C2×D97C22⋊Q811yescharacteristic
89C2×D97C22⋊Q811yescharacteristic
90C4×C97—21no—
91C4×C97C2×C22⋊C411yescharacteristic
92C4×C97#11711yescharacteristic
93Dic97—21no—
94Dic97#12111yescharacteristic
95Dic97#12111yescharacteristic
96C2×C97—41no—
97C2×C97#38011yescharacteristic
98C2×C97#43012yes—
99D97—21no—
100D97—21no—
101D97—41no—
102#1811—971nomaximal, Sylow
103C97#181111yescharacteristic, complemented, Sylow
104C22×M16—971no—
105C2×#108—971no—
106C2×#108—971no—
107#407—972no—
108#341—972no—
109C23×C4—971no—
110C2×M16—971no—
111C2×M16—971no—
112C2×M16—971no—
113C2×M16—972no—
114C2×M16—972no—
115C2×M16—1942no—
116C2×M16—1942no—
117C2×M16#8760411yescharacteristic
118C22×C8—971no—
119C22×C8—971no—
120#108—1942no—
121#108—1942no—
122C22⋊C8—1942no—
123C22⋊C8—1942no—
124C24—971no—
125C22×C4—971no—
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—971no—
129C22×C4—971no—
130C22×C4—1941no—
131C22×C4—1941no—
132C22×C4—1941no—
133C22×C4—1941no—
134C22×C4C2×#8760411yescharacteristic
135M16—22no—
136M16—1942no—
137M16—1942no—
138M16—1942no—
139M16—1942no—
140M16—1942no—
141M16—1942no—
142M16—1942no—
143C2×C8—971no—
144C2×C8—972no—
145C2×C8—972no—
146C2×C8—971no—
147C2×C8—1941no—
148C2×C8—1941no—
149C2×C8—1942no—
150C2×C8—1942no—
151C2×C8C2×#8760411yescharacteristic
152C2×C8C2×#8760411yescharacteristic
153C23—971no—
154C23—971no—
155C23—3882no—
156C23—1941no—
157C23—1941no—
158C23C4⋊#8760411yescharacteristic
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—971no—
162C2×C4—971no—
163C2×C4—971no—
164C2×C4—971no—
165C2×C4—971no—
166C2×C4—971no—
167C2×C4—1941no—
168C2×C4—1941no—
169C2×C4—1941no—
170C2×C4—1941no—
171C2×C4—1941no—
172C2×C4—1941no—
173C2×C4—1941no—
174C2×C4—1941no—
175C2×C4C4⋊#8760411yescharacteristic
176C2×C4C22×#8760411yescharacteristic
177C8—21no—
178C8—972no—
179C8—3882no—
180C8—3882no—
181C8—1941no—
182C8C22⋊#8760412yes—
183C22—21no—
184C22—22no—
185C22—971no—
186C22—971no—
187C22—3882no—
188C22—3882no—
189C22—1941no—
190C22—1942no—
191C22—1942no—
192C22C2×C4⋊#8760411yescharacteristic
193C4—21no—
194C4—971no—
195C4—971no—
196C4—1941no—
197C4C2×C22⋊#8760411yescharacteristic
198C4#1569088911yescharacteristic
199C2—41no—
200C2—3881no—
201C2—1941no—
202C2—1941no—
203C2#2026167711yescharacteristic
204C2#2026175012yes—
subgroupG/H
centerC22C2×C4⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×M16×C97C4
socleC22×C97C2×C4⋊C4
prime radicalC23×D97Q8
cosocleC2×C4×D97C23
hypercenterC2×M16#87604
hypocenterC97#1811
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#15690881C2semidirect
2. C2×#15690880C2semidirect
4. #20261611C2semidirect
103. C97#1811semidirect

Maximal subgroups

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

subgroupindex
C2×#156908812
C2×#156908802
#202617062
#202616112
C2×M16×D972
#181197