Finite Group Databasebeta

← #27140364

Subgroups — #27140364

Order 12416.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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#20260790C211yesmaximal, characteristic, complemented
4#20260476C211yesmaximal, characteristic, complemented
5#20260476C211yesmaximal, characteristic, complemented
6#20260475C211yesmaximal, characteristic, complemented
7#329×C97C211yesmaximal, characteristic, complemented
8#15690803—21no—
9#15690802C2211yescharacteristic
10#15690792—21no—
11#15690786C411yescharacteristic
12#15690781C411yescharacteristic
13Q8×Dic97C411yescharacteristic
14D4×Dic97C411yescharacteristic
15#15690770—21no—
16#15690766—21no—
17C42×D97C2211yescharacteristic
18#15690739—21no—
19#15690739—21no—
20#15690722—21no—
21#15690722—21no—
22#15690721C2211yescharacteristic
23#15690721C2211yescharacteristic
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—
34C4⋊Dic97—21no—
35C4⋊Dic97C2×C411yescharacteristic
36C4⋊Dic97C2×C411yescharacteristic
37#13095490—21no—
38#13095490—21no—
39#13095490—21no—
40C22⋊Dic97—21no—
41C22⋊Dic97—21no—
42#13095488—21no—
43C4×Dic97—21no—
44C4×Dic97C2×C411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97D411yescharacteristic
47C2×Q8×C97C2×C411yescharacteristic
48C2×D4×C97C2×C411yescharacteristic
49C2×C8×C97—21no—
50C2×C8×C97—21no—
51C22⋊C4×C97—21no—
52C22⋊C4×C97—21no—
53C42×C97C2311yescharacteristic
54C23×C97—21no—
55C22×D97C22⋊C411yescharacteristic
56Q8×C97—41no—
57D4×C97—41no—
58C2×C4×C97—21no—
59C2×C4×C97C22×C411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C22⋊D97—42no—
63C2×Dic97—21no—
64C2×Dic97—21no—
65C2×Dic97—42no—
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C22⋊C411yescharacteristic
69C4×D97—41no—
70C4×D97—41no—
71C4×D97—41no—
72C8×C97—41no—
73C8×C97—41no—
74#11715895—41no—
75#11715895—41no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97C2×C22⋊C411yescharacteristic
80C2×D97—21no—
81C2×D97—21no—
82C2×D97—21no—
83C4×C97—21no—
84C4×C97—21no—
85C4×C97—21no—
86C4×C97—41no—
87Dic97—21no—
88Dic97—21no—
89Dic97—21no—
90Dic97—41no—
91Dic97—41no—
92C2×C97—81no—
93C2×C97#41311yescharacteristic
94C2×C97#42111yescharacteristic
95C2×C97#42111yescharacteristic
96D97—41no—
97#1535—971nomaximal, Sylow
98C97#153511yescharacteristic, complemented, Sylow
99#529—971no—
100#432—971no—
101#432—971no—
102#330—971no—
103#330—971no—
104#329—971no—
105#329D9711yescharacteristic, complemented
106#126—1941no—
107#125—971no—
108#124C2×D9711yescharacteristic
109#123—1941no—
110C22⋊Q8—971no—
111#121—971no—
112C4×Q8—971no—
113C4×D4—971no—
114C4×D4—1941no—
115#117—1941no—
116C2×C42—971no—
117C4⋊C8—1941no—
118C4⋊C8—1941no—
119C22⋊C8—1941no—
120C22⋊C8—1941no—
121#97—971no—
122#97—971no—
123#97C2×D9711yescharacteristic
124#97C2×D9711yescharacteristic
125C2×Q8C4×D9711yescharacteristic
126C2×D4—1941no—
127C2×D4C4×D9711yescharacteristic
128C22×C4—971no—
129C22×C4—971no—
130C22×C4—971no—
131C22×C4—1941no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—1941no—
135C2×C8—1941no—
136C4⋊C4—971no—
137C4⋊C4—971no—
138C4⋊C4—1941no—
139C4⋊C4—1941no—
140C4⋊C4—1941no—
141C4⋊C4—1941no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—1941no—
145C22⋊C4—1941no—
146C22⋊C4—1941no—
147C42—971no—
148C42—971no—
149C42—971no—
150C42—1941no—
151C42C22×D9711yescharacteristic
152C23—21no—
153C23—971no—
154Q8—41no—
155D4—41no—
156D4—3882no—
157C2×C4—21no—
158C2×C4—971no—
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—3881no—
162C2×C4—3881no—
163C2×C4—3881no—
164C2×C4—3882no—
165C2×C4—1941no—
166C2×C4—1941no—
167C2×C4C2×C4×D9711yescharacteristic
168C2×C4D4×D9711yescharacteristic
169C2×C4D4×D9711yescharacteristic
170C8—41no—
171C8—41no—
172C8—3881no—
173C8—3881no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—1941no—
178C22—1941no—
179C22—1941no—
180C22C22⋊C4×D9711yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—41no—
185C4—3881no—
186C4—3881no—
187C4—1941no—
188C4—1941no—
189C4—1941no—
190C2—81no—
191C2—3881no—
192C2#2026071611yescharacteristic
193C2#2026075011yescharacteristic
194C2#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#1535
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. #20260790C2semidirect
4. #20260476C2semidirect
5. #20260476C2semidirect
6. #20260475C2semidirect
7. #329×C97C2semidirect
98. C97#1535semidirect
105. #329D97semidirect

Maximal subgroups

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

subgroupindex
#202612642
#202607902
#202604762
#202604752
#329×C972
#153597