Finite Group Databasebeta

← #27142121

Subgroups — #27142121

Order 12416.

360 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15690799C211yesmaximal, characteristic
2#20261209C211yesmaximal, characteristic
3#20260744C211yesmaximal, characteristic, complemented
4C2×#15690726C211yesmaximal, characteristic
5#20260496C211yesmaximal, characteristic, complemented
6#20260494C211yesmaximal, characteristic, complemented
7#419×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690799—22no—
11#15690799C414yescomplemented
12C4×D388—22no—
13#15690772—22no—
14#15690769C2211yescharacteristic
15C2×#13095488—22no—
16C22⋊C4×D97—22no—
17C2×C4×Dic97C2211yescharacteristic
18#15690734—22no—
19#15690726—22no—
20C22×Q8×C97C2211yescharacteristic
21C2×M16×C97C2211yescharacteristic
22#117×C97C2211yescharacteristic
23#103×C97—22no—
24C23×D97D412yes—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388C2×C412yes—
30C2×D388D412yes—
31C22×Dic97—21no—
32C2×C4×D97—24no—
33#13095495—21no—
34#13095495—21no—
35C2×#11715895—21no—
36C4⋊Dic97D412yes—
37C22⋊Dic97—21no—
38#13095488—24no—
39#13095488—28no—
40C4×Dic97—21no—
41C4×Dic97—21no—
42C4×Dic97C2×C412yescomplemented
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97C2×C412yescomplemented
49C22×C4×C97—21no—
50C22×C4×C97C2311yescharacteristic
51M16×C97—21no—
52M16×C97—21no—
53C2×C8×C97—21no—
54C4⋊C4×C97D412yes—
55C22⋊C4×C97—21no—
56C42×C97—21no—
57C23×C97C2×D411yescharacteristic
58C22×D97—22no—
59C22×D97—22no—
60C22×D97—44no—
61C22×D97C22⋊C414yes—
62Q8×C97—22no—
63Q8×C97—22no—
64Q8×C97—41no—
65Q8×C97—41no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97C22×C411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97#4111yescharacteristic
75C2×C4×C97#4111yescharacteristic
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—42no—
79D388—22no—
80D388—22no—
81D388—22no—
82D388—22no—
83C4×D97—84no—
84C8×C97—41no—
85#11715895—41no—
86C22×C97—22no—
87C22×C97C2×C22⋊C411yescharacteristic
88C22×C97#12011yescharacteristic
89C22×C97#12311yescharacteristic
90C2×D97—24no—
91C2×D97—44no—
92C2×D97—44no—
93C2×D97—82no—
94C4×C97—42no—
95C4×C97—42no—
96C4×C97C4×D411yescharacteristic
97C4×C97C4×D411yescharacteristic
98C4×C97#12111yescharacteristic
99C4×C97#12111yescharacteristic
100Dic97—42no—
101Dic97—42no—
102C2×C97—21no—
103C2×C97—21no—
104C2×C97#38611yescharacteristic
105C2×C97#45512yes—
106D97—44no—
107#1885—971nomaximal, Sylow
108C97#188511yescharacteristic, complemented, Sylow
109C2×#124—971no—
110#518—971no—
111#419—971no—
112#419D9711yescharacteristic, complemented
113C2×#100—971no—
114#339—971no—
115#339—971no—
116C22×Q8C2×D9711yescharacteristic
117C22×D4—971no—
118C2×M16—971no—
119C2×M16C2×D9711yescharacteristic
120#124—974no—
121#124—1942no—
122C4×D4—1942no—
123C4×D4—1942no—
124#117—971no—
125#117C2×D9711yescharacteristic
126C2×C22⋊C4—1942no—
127C2×C22⋊C4—1942no—
128C2×C42—971no—
129#103—22no—
130#103—1942no—
131#100—1942no—
132C24—972no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8C4×D9712yescomplemented
139C2×D4—972no—
140C2×D4—972no—
141C2×D4—1941no—
142C2×D4—1941no—
143C2×D4—1941no—
144C2×D4—1941no—
145C22×C4—21no—
146C22×C4—1944no—
147C22×C4—1941no—
148C22×C4C22×D9711yescharacteristic
149M16—21no—
150M16—21no—
151M16—1941no—
152M16—1941no—
153C2×C8—21no—
154C2×C8—1941no—
155C4⋊C4—972no—
156C4⋊C4C22⋊D9712yes—
157C22⋊C4—21no—
158C22⋊C4—1948no—
159C22⋊C4—1944no—
160C22⋊C4—1941no—
161C42—21no—
162C42—972no—
163C42—1941no—
164C42—1941no—
165C23—974no—
166C23—3884no—
167C23—1942no—
168C23—1942no—
169C23D4×D9711yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—41no—
173Q8—41no—
174D4—1942no—
175D4—1942no—
176D4—1942no—
177D4—1942no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—7764no—
183C2×C4—3882no—
184C2×C4—1942no—
185C2×C4—1942no—
186C2×C4C2×C4×D9711yescharacteristic
187C2×C4D4×D9711yescharacteristic
188C2×C4D4×D9711yescharacteristic
189C2×C4C2×C22⋊D9711yescharacteristic
190C2×C4#1309551011yescharacteristic
191C2×C4#1309551111yescharacteristic
192C8—41no—
193C8—3881no—
194C22—22no—
195C22—7762no—
196C22—3884no—
197C22—3884no—
198C22—1944no—
199C22C22⋊C4×D9711yescharacteristic
200C22#1569077811yescharacteristic
201C22#1569079211yescharacteristic
202C4—42no—
203C4—42no—
204C4—3882no—
205C4—3882no—
206C4#1569077011yescharacteristic
207C4C4×C22⋊D9711yescharacteristic
208C4#1569078111yescharacteristic
209C4#1569078311yescharacteristic
210C2—21no—
211C2—21no—
212C2—3884no—
213C2#2026061111yescharacteristic
214C2#2026091112yes—
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#419×C97C2
socleC22×C97#120
prime radicalC22×D388C22
cosocleC22×C4×C97C23
hypercenter#419D97
hypocenterC97#1885
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #20260744C2semidirect
5. #20260496C2semidirect
6. #20260494C2semidirect
7. #419×C97C2semidirect
11. #15690799C4semidirect
42. C4×Dic97C2×C4semidirect
48. C2×Q8×C97C2×C4semidirect
108. C97#1885semidirect
112. #419D97semidirect
138. C2×Q8C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907992
#202612092
#202607442
C2×#156907262
#202604962
#202604942
#419×C972
#188597