Finite Group Databasebeta

← #27143377

Subgroups — #27143377

Order 12416.

264 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 73 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#20260719C211yesmaximal, characteristic, complemented
2#20260709C211yesmaximal, characteristic, complemented
3#20260525C212yesmaximal, complemented
4#20260522C212yesmaximal, complemented
5#525×C97C211yesmaximal, characteristic
6C2×#13095495—21no—
7#15690769—21no—
8#15690729—21no—
9#15690727C2212yes—
10#15690725—21no—
11#15690725C2212yes—
12C2×#41×C97C2211yescharacteristic
13C23×C4×C97C411yescharacteristic, complemented
14#123×C97C411yescharacteristic, complemented
15#123×C97C2211yescharacteristic
16C22⋊Q8×C97—21no—
17#121×C97—21no—
18#120×C97C411yescharacteristic, complemented
19#120×C97C2211yescharacteristic
20C4×D4×C97—21no—
21C4×D4×C97—21no—
22#117×C97C411yescharacteristic, complemented
23#13095495—22no—
24#13095495—41no—
25C2×#11715895—41no—
26C4⋊Dic97—41no—
27C22⋊Dic97—22no—
28C4×Dic97—41no—
29C24×C97C2×C411yescharacteristic
30#41×C97—41no—
31#41×C97—41no—
32C2×Q8×C97D411yescharacteristic
33C2×D4×C97—21no—
34C2×D4×C97—21no—
35C2×D4×C97C2×C411yescharacteristic
36C2×D4×C97D411yescharacteristic
37C2×D4×C97C2311yescharacteristic
38C22×C4×C97—21no—
39C22×C4×C97—21no—
40C22×C4×C97—41no—
41C22×C4×C97—41no—
42C22×C4×C97C2×C411yescharacteristic
43C22×C4×C97C2×C411yescharacteristic
44C22×C4×C97D411yescharacteristic
45C22×C4×C97D411yescharacteristic
46C4⋊C4×C97—21no—
47C4⋊C4×C97—21no—
48C4⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C22⋊C4×C97—21no—
53C22⋊C4×C97C2×C411yescharacteristic
54C22⋊C4×C97C2×C411yescharacteristic
55C42×C97—21no—
56C23×C97—21no—
57C23×C97—41no—
58C23×C97—42no—
59C23×C97C22⋊C411yescharacteristic
60C23×C97C22×C411yescharacteristic
61C23×C97C2×D411yescharacteristic
62Q8×C97—41no—
63D4×C97—41no—
64D4×C97—41no—
65D4×C97—41no—
66D4×C97—42no—
67D4×C97—42no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—21no—
73C2×C4×C97—21no—
74C2×C4×C97—21no—
75C2×C4×C97—21no—
76C2×C4×C97—21no—
77C2×C4×C97—42no—
78C2×C4×C97—41no—
79C2×C4×C97—41no—
80C2×C4×C97—41no—
81C2×C4×C97C22⋊C411yescharacteristic
82C2×C4×C97C22⋊C411yescharacteristic
83C2×C4×C97C22⋊C411yescharacteristic
84C2×C4×C97C2×D411yescharacteristic
85C2×Dic97—42no—
86#11715895—42no—
87C22×C97—22no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—21no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97—41no—
94C22×C97—41no—
95C22×C97—42no—
96C22×C97—42no—
97C22×C97—41no—
98C22×C97C2×C22⋊C411yescharacteristic
99C4×C97—21no—
100C4×C97—21no—
101C4×C97—21no—
102C4×C97—41no—
103C4×C97—41no—
104C4×C97—41no—
105C4×C97#9911yescharacteristic
106C4×C97#9911yescharacteristic
107Dic97—82no—
108C2×C97—21no—
109C2×C97—41no—
110C2×C97—41no—
111C2×C97—42no—
112C2×C97—41no—
113C2×C97C2×#9911yescharacteristic
114#2120—971nomaximal, Sylow
115C97#212011yescharacteristic, complemented, Sylow
116#525D9711yescharacteristic
117#413—971no—
118#410—971no—
119#352—972no—
120AΣL(1,𝔽16)—972no—
121C2×#41C2×D9711yescharacteristic
122C23×C4Dic9711yescharacteristic, complemented
123C2×M16—1941no—
124#123Dic9711yescharacteristic, complemented
125#123C2×D9711yescharacteristic
126C22⋊Q8—21no—
127#121—21no—
128#120Dic9711yescharacteristic, complemented
129#120C2×D9711yescharacteristic
130C4×D4—21no—
131C4×D4—21no—
132#117—1941no—
133#117Dic9711yescharacteristic, complemented
134#101—1941no—
135#100—972no—
136#99—972no—
137#99—1941no—
138C24C2×Dic9711yescharacteristic
139#41—41no—
140#41—41no—
141C2×Q8C22⋊D9711yescharacteristic
142C2×D4—21no—
143C2×D4—21no—
144C2×D4C2×Dic9711yescharacteristic
145C2×D4C22⋊D9711yescharacteristic
146C2×D4C22×D9711yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4C2×Dic9711yescharacteristic
152C22×C4C2×Dic9711yescharacteristic
153C22×C4C22⋊D9711yescharacteristic
154C22×C4C22⋊D9711yescharacteristic
155M16—3881no—
156M16—1942no—
157C2×C8—3881no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—3881no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—1942no—
167C22⋊C4C2×Dic9711yescharacteristic
168C22⋊C4C2×Dic9711yescharacteristic
169C42—21no—
170C42—3881no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23C22⋊Dic9711yescharacteristic
175C23C22×Dic9711yescharacteristic
176C23C2×C22⋊D9711yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—42no—
182D4—42no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—3882no—
197C2×C4C22⋊Dic9711yescharacteristic
198C2×C4C22⋊Dic9711yescharacteristic
199C2×C4C22⋊Dic9711yescharacteristic
200C2×C4C2×C22⋊D9711yescharacteristic
201C8—3882no—
202C22—22no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22C2×C22⋊Dic9711yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—7762no—
221C4#1569072511yescharacteristic
222C4#1569072511yescharacteristic
223C2—21no—
224C2—41no—
225C2—42no—
226C2—41no—
227C2—41no—
228C2C2×#1569072511yescharacteristic
subgroupG/H
centerC4#15690725
commutatorC23×C97C22×C4
FrattiniC2×D4C22×D97
Fitting, prime radical#525×C97C2
socleC2×C97C2×#99
cosocleC2×D4×C97C23
hypercenter#525D97
hypocenterC97#2120
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #20260719C2semidirect
2. #20260709C2semidirect
3. #20260525C2semidirect
4. #20260522C2semidirect
13. C23×C4×C97C4semidirect
14. #123×C97C4semidirect
18. #120×C97C4semidirect
22. #117×C97C4semidirect
115. C97#2120semidirect
122. C23×C4Dic97semidirect
124. #123Dic97semidirect
128. #120Dic97semidirect
133. #117Dic97semidirect

Maximal subgroups

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

subgroupindex
#202607192
#202607092
#202605252
#202605222
#525×C972
#212097