Finite Group Databasebeta

← #27140353

Subgroups — #27140353

Order 12416.

218 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 59 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#20261264C211yesmaximal, characteristic
2#20260790C211yesmaximal, characteristic, complemented
3#97×D97C211yesmaximal, characteristic, complemented
4#20260476C211yesmaximal, characteristic, complemented
5#20260476C211yesmaximal, characteristic, complemented
6#20260475C211yesmaximal, characteristic, complemented
7#329×C97C211yesmaximal, characteristic, complemented
8C2×C8×D97—21no—
9#15690803—21no—
10#15690802C2211yescharacteristic
11#15690792—21no—
12#15690786C411yescharacteristic
13#15690781C411yescharacteristic
14Q8×Dic97C411yescharacteristic
15D4×Dic97C411yescharacteristic
16#15690770—21no—
17#15690766—21no—
18C42×D97C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21#15690721C2211yescharacteristic
22#15690721C2211yescharacteristic
23#15690720—21no—
24#124×C97C2211yescharacteristic
25#97×C97C2211yescharacteristic
26#97×C97C2211yescharacteristic
27C2×C22⋊D97—21no—
28C22×Dic97—21no—
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C2×#11715895—21no—
33C2×#11715895—21no—
34C8×D97—24no—
35C4⋊Dic97—21no—
36C4⋊Dic97C2×C411yescharacteristic
37C4⋊Dic97C2×C411yescharacteristic
38#13095490—21no—
39#13095490—21no—
40#13095490—21no—
41C22⋊Dic97—21no—
42C22⋊Dic97—21no—
43#13095488—21no—
44C4×Dic97—21no—
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97D411yescharacteristic
47C4×Dic97D411yescharacteristic
48C2×Q8×C97C2×C411yescharacteristic
49C2×D4×C97C2×C411yescharacteristic
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C22⋊C4×C97—21no—
53C22⋊C4×C97—21no—
54C42×C97C2311yescharacteristic
55C23×C97—21no—
56C22×D97C22⋊C411yescharacteristic
57Q8×C97—41no—
58D4×C97—41no—
59C2×C4×C97—21no—
60C2×C4×C97C22×C411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C22⋊D97—42no—
64C2×Dic97—21no—
65C2×Dic97—21no—
66C2×Dic97—42no—
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C22⋊C411yescharacteristic
69C2×Dic97C22⋊C411yescharacteristic
70C4×D97—22no—
71C4×D97—41no—
72C4×D97—41no—
73C8×C97—22no—
74C8×C97—41no—
75#11715895—22no—
76#11715895—41no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97C2×C22⋊C411yescharacteristic
81C2×D97—21no—
82C2×D97—21no—
83C2×D97#10412yes—
84C4×C97—21no—
85C4×C97—21no—
86C4×C97—21no—
87C4×C97—41no—
88Dic97—21no—
89Dic97—21no—
90Dic97—21no—
91Dic97—41no—
92Dic97—41no—
93C2×C97—81no—
94C2×C97#41311yescharacteristic
95C2×C97C2×#10411yescharacteristic
96C2×C97#42111yescharacteristic
97D97—22no—
98#1532—971nomaximal, Sylow
99C97#153211yescharacteristic, complemented, Sylow
100#529—971no—
101#432—971no—
102C2×#97—971no—
103#330—971no—
104#330—971no—
105#329—971no—
106#329D9711yescharacteristic, complemented
107C22×C8—1941no—
108#126—1941no—
109#125—971no—
110#124C2×D9711yescharacteristic
111#123—1941no—
112C22⋊Q8—971no—
113#121—971no—
114C4×Q8—971no—
115C4×D4—971no—
116C4×D4—1941no—
117#117—1941no—
118C2×C42—971no—
119C4⋊C8—1941no—
120C22⋊C8—1941no—
121#97—971no—
122#97—971no—
123#97—1941no—
124#97C2×D9711yescharacteristic
125#97C2×D9711yescharacteristic
126C2×Q8C4×D9711yescharacteristic
127C2×D4—1941no—
128C2×D4C4×D9711yescharacteristic
129C22×C4—971no—
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—1941no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—1944no—
136C2×C8—1941no—
137C2×C8—1941no—
138C4⋊C4—971no—
139C4⋊C4—971no—
140C4⋊C4—1941no—
141C4⋊C4—1941no—
142C4⋊C4—1941no—
143C4⋊C4—1941no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—1941no—
147C22⋊C4—1941no—
148C22⋊C4—1941no—
149C42—971no—
150C42—971no—
151C42—971no—
152C42—1941no—
153C42C22×D9711yescharacteristic
154C23—21no—
155C23—971no—
156Q8—41no—
157D4—41no—
158D4—3882no—
159C2×C4—21no—
160C2×C4—971no—
161C2×C4—971no—
162C2×C4—971no—
163C2×C4—3881no—
164C2×C4—3881no—
165C2×C4—3882no—
166C2×C4—1942no—
167C2×C4—1941no—
168C2×C4—1941no—
169C2×C4C2×C4×D9711yescharacteristic
170C2×C4D4×D9711yescharacteristic
171C2×C4D4×D9711yescharacteristic
172C8—22no—
173C8—41no—
174C8—3881no—
175C8—1942no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—972no—
180C22—1941no—
181C22—1941no—
182C22C22⋊C4×D9711yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—3881no—
188C4—3881no—
189C4—1941no—
190C4—1941no—
191C4—1941no—
192C2—81no—
193C2—1942no—
194C2#2026071611yescharacteristic
195C2#104×D9711yescharacteristic
196C2#2026075011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#329×C97C2
socleC22×C97C2×C22⋊C4
prime radical#15690781C4
cosocleC42×C97C23
hypercenter#329D97
hypocenterC97#1532
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20260790C2semidirect
3. #97×D97C2semidirect
4. #20260476C2semidirect
5. #20260476C2semidirect
6. #20260475C2semidirect
7. #329×C97C2semidirect
99. C97#1532semidirect
106. #329D97semidirect

Maximal subgroups

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

subgroupindex
#202612642
#202607902
#97×D972
#202604762
#202604752
#329×C972
#153297