Finite Group Databasebeta

← #27157741

Subgroups — #27157741

Order 12416.

360 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 68 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
1#20261802C211yesmaximal, characteristic
2#20261728C212yesmaximal, complemented
3D97⋊#100C211yesmaximal, characteristic
4#20261606C212yesmaximal, complemented
5#124×D97C211yesmaximal, characteristic
6D97⋊M16C2212yes—
7D4×#87604—24no—
8#15690889C2212yes—
9C2×C22⋊#87604—22no—
10#15690873—24no—
11#15690870—22no—
12C2×Q8×D97C2211yescharacteristic
13C2×D4×D97C2211yescharacteristic
14#15690799C411yescharacteristic, complemented
15#15690798C411yescharacteristic, complemented
16#15690797C411yescharacteristic, complemented
17#15690796—22no—
18C22⋊C4×D97—22no—
19C42×D97C2211yescharacteristic
20#124×C97C411yescharacteristic, complemented
21C22×#87604—24no—
22#13095521—22no—
23#13095520—22no—
24#13095519—22no—
25C4⋊#87604D414yes—
26C22⋊#87604—24no—
27C22⋊#87604—22no—
28C4×#87604—22no—
29C23×D97D412yes—
30C2×Dic194—21no—
31C2×Dic194C2×C411yescharacteristic, complemented
32Q8×D97—22no—
33Q8×D97—22no—
34C2×D388C2×C411yescharacteristic
35C2×C22⋊D97D412yes—
36D4×D97—22no—
37D4×D97—22no—
38C2×C4×D97—21no—
39C2×C4×D97—21no—
40C2×C4×D97C2311yescharacteristic
41C22⋊Dic97—22no—
42#13095488—22no—
43#13095488—22no—
44C4×Dic97—21no—
45C4×Dic97C2×C411yescharacteristic, complemented
46C2×Q8×C97C2×C411yescharacteristic, complemented
47C2×D4×C97C2×C411yescharacteristic
48C22⋊C4×C97—22no—
49C42×C97C2×C411yescharacteristic, complemented
50C23×C97C22⋊C412yes—
51C22×D97—22no—
52C22×D97—22no—
53C22×D97—44no—
54C22×D97C22⋊C412yes—
55C22×D97C2×D411yescharacteristic
56C2×#87604—24no—
57C2×#87604—84no—
58Q8×C97—22no—
59D4×C97—22no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97C22×C411yescharacteristic
63C22⋊D97—24no—
64C2×Dic97—21no—
65C2×Dic97—21no—
66C2×Dic97C2×D411yescharacteristic
67D388—22no—
68C4×D97—42no—
69C4×D97—41no—
70C4×D97—41no—
71C4×D97C2×D412yes—
72C4×D97#4112yes—
73Dic194—22no—
74Dic194—42no—
75#11715897—42no—
76C22×C97—22no—
77C22×C97—42no—
78C22×C97C2×C22⋊C411yescharacteristic
79C2×D97—22no—
80C2×D97—22no—
81C2×D97—44no—
82C2×D97—42no—
83C2×D97—82no—
84C2×D97#12011yescharacteristic
85C2×D97#12311yescharacteristic
86#87604—44no—
87C4×C97—41no—
88C4×C97—41no—
89C4×C97C4×D412yes—
90Dic97—41no—
91Dic97—41no—
92Dic97#12112yes—
93C2×C97—21no—
94C2×C97—42no—
95C2×C97#38611yescharacteristic
96D97—21no—
97D97—42no—
98D97#45512yes—
99#1885—971nomaximal, Sylow
100C97#188511yescharacteristic, complemented, Sylow
101C2×#124—971no—
102#518—971no—
103#419—972no—
104C2×#100—971no—
105#339—972no—
106C22×Q8—971no—
107C22×D4—971no—
108C2×M16—972no—
109#124—971no—
110#124—971no—
111#124—971no—
112#124—1942no—
113#124#8760411yescharacteristic, complemented
114C4×D4—1944no—
115#117—972no—
116C2×C22⋊C4—1942no—
117C2×C22⋊C4—1942no—
118C2×C42—971no—
119#103—1944no—
120#100—1942no—
121C24—972no—
122C2×Q8—971no—
123C2×Q8—1941no—
124C2×Q8—1942no—
125C2×Q8—1942no—
126C2×Q8C2×#8760411yescharacteristic, complemented
127C2×D4—972no—
128C2×D4—971no—
129C2×D4—1942no—
130C2×D4—1942no—
131C2×D4C2×#8760411yescharacteristic
132C22×C4—971no—
133C22×C4—1941no—
134C22×C4—1941no—
135C22×C4—1944no—
136M16—1942no—
137M16—1942no—
138C2×C8—1942no—
139C4⋊C4—974no—
140C22⋊C4—22no—
141C22⋊C4—1942no—
142C22⋊C4—1942no—
143C22⋊C4—1942no—
144C22⋊C4—1944no—
145C22⋊C4—1942no—
146C42—971no—
147C42—1941no—
148C42—1942no—
149C42C2×#8760411yescharacteristic, complemented
150C23—971no—
151C23—972no—
152C23—3884no—
153C23—1942no—
154C23—1942no—
155C23C22⋊#8760412yes—
156Q8—22no—
157Q8—3882no—
158Q8—1942no—
159D4—22no—
160D4—1944no—
161D4—1942no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—7764no—
165C2×C4—971no—
166C2×C4—972no—
167C2×C4—972no—
168C2×C4—3881no—
169C2×C4—3881no—
170C2×C4—3882no—
171C2×C4—1944no—
172C2×C4—1941no—
173C2×C4—1941no—
174C2×C4C22×#8760411yescharacteristic
175C8—3882no—
176C22—22no—
177C22—42no—
178C22—7762no—
179C22—971no—
180C22—971no—
181C22—3882no—
182C22—3884no—
183C22—1942no—
184C22—1942no—
185C22C2×C22⋊#8760411yescharacteristic
186C4—41no—
187C4—41no—
188C4—972no—
189C4—3881no—
190C4—3884no—
191C4—3881no—
192C4D4×#8760412yes—
193C2—21no—
194C2—42no—
195C2—972no—
196C2—3882no—
197C2—1941no—
198C2#2026168311yescharacteristic
subgroupG/H
centerC2#20261683
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC2×C97#386
prime radicalC2×D4×D97C22
cosocleC2×C4×D97C23
hypercenter#124#87604
hypocenterC97#1885
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #20261728C2semidirect
4. #20261606C2semidirect
14. #15690799C4semidirect
15. #15690798C4semidirect
16. #15690797C4semidirect
20. #124×C97C4semidirect
31. C2×Dic194C2×C4semidirect
45. C4×Dic97C2×C4semidirect
46. C2×Q8×C97C2×C4semidirect
49. C42×C97C2×C4semidirect
100. C97#1885semidirect
113. #124#87604semidirect
126. C2×Q8C2×#87604semidirect
149. C42C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202618022
#202617282
D97⋊#1002
#202616062
#124×D972
#188597