Finite Group Databasebeta

← #27157782

Subgroups — #27157782

Order 12416.

362 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 66 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, complemented
2#20261733C212yesmaximal, complemented
3D97⋊#100C211yesmaximal, characteristic, complemented
4#20261602C212yesmaximal, complemented
5#20261298C211yesmaximal, characteristic
6D97⋊M16C2212yes—
7D4×#87604—24no—
8#15690889C2212yes—
9C2×C22⋊#87604—22no—
10#15690874—24no—
11#15690870—22no—
12C2×#13095510C2211yescharacteristic
13C2×D4×D97C2211yescharacteristic
14#15690806C411yescharacteristic, complemented
15#15690798C411yescharacteristic, complemented
16#15690797C411yescharacteristic, complemented
17#15690782—22no—
18#15690778—22no—
19#15690767C2211yescharacteristic
20#127×C97C411yescharacteristic, complemented
21C22×#87604—24no—
22#13095521—22no—
23#13095520—22no—
24#13095519—22no—
25C4⋊#87604—22no—
26C22⋊#87604—24no—
27C22⋊#87604—22no—
28C4×#87604D414yescomplemented
29C23×D97D412yes—
30#13095510—22no—
31#13095510—22no—
32C2×Dic194C2×C411yescharacteristic, complemented
33C2×D388C2×C411yescharacteristic
34C2×C22⋊D97—21no—
35C2×C22⋊D97—22no—
36C2×C22⋊D97D412yes—
37D4×D97—22no—
38D4×D97—22no—
39C22×Dic97—21no—
40C2×C4×D97C2311yescharacteristic
41#13095490—21no—
42C22⋊Dic97—22no—
43#13095488—22no—
44#13095488—21no—
45C4×Dic97C2×C411yescharacteristic, complemented
46C2×D4×C97—22no—
47C2×D4×C97C2×C411yescharacteristic
48C2×D4×C97C2×C411yescharacteristic, complemented
49C42×C97C2×C411yescharacteristic, complemented
50C23×C97—21no—
51C23×C97C22⋊C412yes—
52C22×D97—22no—
53C22×D97—22no—
54C22×D97—82no—
55C22×D97C22⋊C412yes—
56C22×D97C2×D411yescharacteristic
57C2×#87604—24no—
58C2×#87604—84no—
59D4×C97—22no—
60D4×C97—22no—
61D4×C97—82no—
62C2×C4×C97—21no—
63C2×C4×C97C22×C411yescharacteristic
64C22⋊D97—24no—
65C22⋊D97—41no—
66C22⋊D97—41no—
67C22⋊D97—82no—
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97—42no—
71C2×Dic97C2×D411yescharacteristic
72D388—22no—
73C4×D97C2×D412yes—
74C4×D97#4112yes—
75Dic194—22no—
76#11715897—42no—
77C22×C97—22no—
78C22×C97—42no—
79C22×C97—42no—
80C22×C97—41no—
81C22×C97C2×C22⋊C411yescharacteristic
82C2×D97—22no—
83C2×D97—41no—
84C2×D97—44no—
85C2×D97—42no—
86C2×D97—82no—
87C2×D97#12011yescharacteristic
88C2×D97#12311yescharacteristic
89#87604—44no—
90C4×C97—41no—
91C4×C97C4×D412yes—
92Dic97—41no—
93Dic97—41no—
94Dic97#12112yes—
95C2×C97—21no—
96C2×C97—42no—
97C2×C97—81no—
98C2×C97#38611yescharacteristic
99D97—21no—
100D97—21no—
101D97—42no—
102#1911—971nomaximal, Sylow
103C97#191111yescharacteristic, complemented, Sylow
104#535—971no—
105#518—971no—
106#421—972no—
107C2×#100—971no—
108H(3,ℤ/4)—972no—
109C2×#41—971no—
110C22×D4—971no—
111C2×M16—972no—
112#127—971no—
113#127#8760411yescharacteristic, complemented
114#124—971no—
115#124—971no—
116#121—1942no—
117#120—1942no—
118C4×D4—1944no—
119#117—972no—
120#117—971no—
121C2×C22⋊C4—1942no—
122#104—1944no—
123#100—1942no—
124C24—972no—
125#41—1942no—
126#41—1942no—
127C2×Q8—971no—
128C2×D4—22no—
129C2×D4—972no—
130C2×D4—971no—
131C2×D4—1942no—
132C2×D4—1942no—
133C2×D4—1942no—
134C2×D4—1941no—
135C2×D4C2×#8760411yescharacteristic
136C2×D4C2×#8760411yescharacteristic, complemented
137C22×C4—971no—
138C22×C4—1944no—
139C22×C4—1941no—
140M16—1942no—
141M16—1942no—
142C2×C8—1942no—
143C4⋊C4—1941no—
144C4⋊C4—1942no—
145C22⋊C4—1941no—
146C22⋊C4—1942no—
147C22⋊C4—1942no—
148C22⋊C4—1944no—
149C22⋊C4—1942no—
150C42—974no—
151C42—971no—
152C42C2×#8760411yescharacteristic, complemented
153C23—21no—
154C23—7762no—
155C23—971no—
156C23—972no—
157C23—1942no—
158C23—1942no—
159C23C22⋊#8760412yes—
160Q8—1942no—
161D4—22no—
162D4—22no—
163D4—82no—
164D4—7762no—
165D4—3881no—
166D4—3881no—
167D4—1944no—
168D4—1942no—
169C2×C4—21no—
170C2×C4—7764no—
171C2×C4—971no—
172C2×C4—972no—
173C2×C4—972no—
174C2×C4—3882no—
175C2×C4—1944no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4C22×#8760411yescharacteristic
179C8—3882no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—7762no—
185C22—971no—
186C22—971no—
187C22—3882no—
188C22—3881no—
189C22—3884no—
190C22—1942no—
191C22C2×C22⋊#8760411yescharacteristic
192C4—41no—
193C4—972no—
194C4—3881no—
195C4—3884no—
196C4—3881no—
197C4D4×#8760412yes—
198C2—21no—
199C2—42no—
200C2—81no—
201C2—3882no—
202C2—1941no—
203C2—1941no—
204C2#2026168311yescharacteristic
subgroupG/H
centerC2#20261683
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC2×C97#386
prime radical#20261298C2
cosocleC2×C4×D97C23
hypercenter#127#87604
hypocenterC97#1911
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261802C2semidirect
2. #20261733C2semidirect
3. D97⋊#100C2semidirect
4. #20261602C2semidirect
14. #15690806C4semidirect
15. #15690798C4semidirect
16. #15690797C4semidirect
20. #127×C97C4semidirect
28. C4×#87604D4semidirect
32. C2×Dic194C2×C4semidirect
45. C4×Dic97C2×C4semidirect
48. C2×D4×C97C2×C4semidirect
49. C42×C97C2×C4semidirect
103. C97#1911semidirect
113. #127#87604semidirect
136. C2×D4C2×#87604semidirect
152. C42C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202618022
#202617332
D97⋊#1002
#202616022
#202612982
#191197