Finite Group Databasebeta

← #27143386

Subgroups — #27143386

Order 12416.

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

normal subgroups only

This group has 266 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#20260522C214yesmaximal, complemented
4#535×C97C211yesmaximal, characteristic
5C2×#13095495—21no—
6#15690769—21no—
7#15690729—21no—
8#15690727C2212yes—
9#15690725—21no—
10#15690725C2212yes—
11C2×#41×C97C2211yescharacteristic
12C22×D4×C97C411yescharacteristic, complemented
13#127×C97—21no—
14#124×C97—21no—
15#121×C97—21no—
16#121×C97—21no—
17#120×C97C412yescomplemented
18#120×C97C2212yes—
19#117×C97C411yescharacteristic, complemented
20#13095495—22no—
21#13095495—41no—
22C2×#11715895—41no—
23C4⋊Dic97—41no—
24C22⋊Dic97—22no—
25C4×Dic97—41no—
26C24×C97C2×C412yes—
27#41×C97—41no—
28#41×C97—41no—
29C2×Q8×C97D411yescharacteristic
30C2×D4×C97—22no—
31C2×D4×C97—22no—
32C2×D4×C97—21no—
33C2×D4×C97—21no—
34C2×D4×C97—41no—
35C2×D4×C97—41no—
36C2×D4×C97C2×C411yescharacteristic
37C2×D4×C97D411yescharacteristic
38C2×D4×C97C2311yescharacteristic
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D411yescharacteristic
42C4⋊C4×C97—21no—
43C22⋊C4×C97—22no—
44C22⋊C4×C97—22no—
45C22⋊C4×C97C2×C412yes—
46C42×C97—21no—
47C23×C97—22no—
48C23×C97—42no—
49C23×C97—82no—
50C23×C97C22⋊C411yescharacteristic
51C23×C97C22×C411yescharacteristic
52C23×C97C2×D411yescharacteristic
53Q8×C97—41no—
54D4×C97—41no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—42no—
58D4×C97—41no—
59D4×C97—41no—
60D4×C97—82no—
61D4×C97—82no—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—22no—
69C2×C4×C97C22⋊C411yescharacteristic
70C2×C4×C97C22⋊C411yescharacteristic
71C2×C4×C97C22⋊C411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×Dic97—42no—
74#11715895—42no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97—42no—
82C22×C97—42no—
83C22×C97—42no—
84C22×C97—84no—
85C22×C97C2×C22⋊C411yescharacteristic
86C4×C97—21no—
87C4×C97—21no—
88C4×C97—21no—
89C4×C97—42no—
90C4×C97#9911yescharacteristic
91C4×C97#9911yescharacteristic
92Dic97—82no—
93C2×C97—21no—
94C2×C97—41no—
95C2×C97—41no—
96C2×C97—41no—
97C2×C97—82no—
98C2×C97C2×#9911yescharacteristic
99#2122—971nomaximal, Sylow
100C97#212211yescharacteristic, complemented, Sylow
101#535D9711yescharacteristic
102#413—971no—
103#410—971no—
104AΣL(1,𝔽16)—974no—
105C2×#41C2×D9711yescharacteristic
106C22×D4Dic9711yescharacteristic, complemented
107C2×M16—1941no—
108#127—21no—
109#124—21no—
110#121—21no—
111#121—21no—
112#120Dic9712yescomplemented
113#120C2×D9712yes—
114#117—1941no—
115#117Dic9711yescharacteristic, complemented
116#101—1941no—
117#100—972no—
118#99—972no—
119#99—1941no—
120C24C2×Dic9712yes—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D9711yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4C2×Dic9711yescharacteristic
131C2×D4C22⋊D9711yescharacteristic
132C2×D4C22×D9711yescharacteristic
133C22×C4C2×Dic9711yescharacteristic
134C22×C4C22⋊D9711yescharacteristic
135C22×C4C22⋊D9711yescharacteristic
136M16—3881no—
137M16—1942no—
138C2×C8—3881no—
139C4⋊C4—21no—
140C4⋊C4—3881no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—1942no—
144C22⋊C4C2×Dic9712yes—
145C42—21no—
146C42—3881no—
147C23—22no—
148C23—42no—
149C23—82no—
150C23C22⋊Dic9711yescharacteristic
151C23C22×Dic9711yescharacteristic
152C23C2×C22⋊D9711yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—42no—
158D4—41no—
159D4—41no—
160D4—82no—
161D4—82no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—3882no—
170C2×C4C22⋊Dic9711yescharacteristic
171C2×C4C22⋊Dic9711yescharacteristic
172C2×C4C22⋊Dic9711yescharacteristic
173C2×C4C2×C22⋊D9711yescharacteristic
174C8—3882no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22C2×C22⋊Dic9711yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—42no—
190C4—7762no—
191C4#1569072511yescharacteristic
192C4#1569072511yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—82no—
198C2C2×#1569072511yescharacteristic
subgroupG/H
centerC2C2×#15690725
commutatorC23×C97C22×C4
FrattiniC2×D4C22×D97
Fitting, prime radical#535×C97C2
socleC2×C97C2×#99
cosocleC2×D4×C97C23
hypercenter#535D97
hypocenterC97#2122
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #20260719C2semidirect
2. #20260709C2semidirect
3. #20260522C2semidirect
12. C22×D4×C97C4semidirect
17. #120×C97C4semidirect
19. #117×C97C4semidirect
100. C97#2122semidirect
106. C22×D4Dic97semidirect
112. #120Dic97semidirect
115. #117Dic97semidirect

Maximal subgroups

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

subgroupindex
#202607192
#202607092
#202605222
#535×C972
#212297