Finite Group Databasebeta

← #27158910

Subgroups — #27158910

Order 12416.

236 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 69 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#20261854C212yesmaximal, complemented
2#20261834C212yesmaximal, complemented
3#20261833C212yesmaximal, complemented
4C2×D97⋊M16C211yesmaximal, characteristic
5#15690907—22no—
6#15690906C2214yescomplemented
7#15690905—22no—
8C2×#13095521C411yescharacteristic
9C2×#13095521C2211yescharacteristic
10D97⋊M16—22no—
11D97⋊M16—22no—
12C2×#13095520C411yescharacteristic
13C2×#13095520C2211yescharacteristic
14C2×#13095519C411yescharacteristic
15C2×#13095519C2211yescharacteristic
16C22×C4×D97C411yescharacteristic
17#13095523—24no—
18#13095521—22no—
19#13095521—22no—
20#13095520—22no—
21#13095520—22no—
22#13095519—21no—
23#13095519—21no—
24#13095519D412yes—
25#13095519D412yes—
26C2×#11715897C2×C411yescharacteristic
27C2×#11715897C2×C411yescharacteristic
28C2×#11715897C2×C411yescharacteristic
29C2×#11715897C2311yescharacteristic
30C23×D97C811yescharacteristic
31C22×Dic97C2×C411yescharacteristic
32C2×C4×D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97C811yescharacteristic
37C2×C4×D97C811yescharacteristic
38C2×C4×D97C2×C411yescharacteristic
39C2×C4×D97C2×C411yescharacteristic
40C22×C4×C97C811yescharacteristic
41C23×C97C2×C811yescharacteristic
42C22×D97—21no—
43C22×D97—21no—
44C22×D97—42no—
45C22×D97C2×C811yescharacteristic
46C22×D97C2×C811yescharacteristic
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97C2×C811yescharacteristic
50C2×C4×C97C2×C811yescharacteristic
51C2×Dic97—21no—
52C2×Dic97—21no—
53C2×Dic97C2×C811yescharacteristic
54C2×Dic97C22×C411yescharacteristic
55C4×D97—21no—
56C4×D97—21no—
57C4×D97—21no—
58C4×D97—21no—
59C4×D97—21no—
60C4×D97—21no—
61C4×D97C22⋊C411yescharacteristic
62C4×D97C22⋊C411yescharacteristic
63C4×D97M1611yescharacteristic
64C4×D97M1611yescharacteristic
65#11715897—21no—
66#11715897—21no—
67#11715897C22⋊C412yes—
68#11715897C2×D412yes—
69C22×C97—21no—
70C22×C97—41no—
71C22×C97C22×C811yescharacteristic
72C2×D97—22no—
73C2×D97—21no—
74C2×D97—42no—
75C2×D97—42no—
76C2×D97—41no—
77C2×D97C22⋊C811yescharacteristic
78C2×D97C22⋊C811yescharacteristic
79C4×C97—21no—
80C4×C97C22⋊C811yescharacteristic
81C4×C97C22⋊C811yescharacteristic
82Dic97—21no—
83Dic97C2×C22⋊C411yescharacteristic
84Dic97C2×M1611yescharacteristic
85C2×C97—21no—
86C2×C97—41no—
87C2×C97C2×C22⋊C811yescharacteristic
88D97—21no—
89D97—41no—
90D97#34912yescomplemented
91C2×#349—971nomaximal, Sylow
92C97C2×#34911yescharacteristic, complemented, Sylow
93C22×M16—971no—
94C2×M32—972no—
95#349—972no—
96#349—972no—
97C23×C4—971no—
98C2×M16—971no—
99C2×M16—971no—
100C2×M16—971no—
101C2×M16—971no—
102C2×M16—1942no—
103C2×M16—1942no—
104C22×C8—971no—
105C22×C8—971no—
106M32—974no—
107M32—1942no—
108C2×C16—1942no—
109C24—971no—
110C22×C4—971no—
111C22×C4—971no—
112C22×C4—971no—
113C22×C4—971no—
114C22×C4—971no—
115C22×C4—1941no—
116C22×C4—1941no—
117C22×C4—1941no—
118C22×C4—1941no—
119C22×C4#1171590611yescharacteristic
120M16—1942no—
121M16—1942no—
122M16—1942no—
123M16—1942no—
124C2×C8—971no—
125C2×C8—971no—
126C2×C8—972no—
127C2×C8—972no—
128C2×C8—971no—
129C2×C8—971no—
130C2×C8—1941no—
131C2×C8—1941no—
132C16—1944no—
133C23—971no—
134C23—971no—
135C23—3882no—
136C23—1941no—
137C23—1941no—
138C23C2×#1171590611yescharacteristic
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—971no—
142C2×C4—971no—
143C2×C4—971no—
144C2×C4—971no—
145C2×C4—971no—
146C2×C4—971no—
147C2×C4—1941no—
148C2×C4—1941no—
149C2×C4—1941no—
150C2×C4—1941no—
151C2×C4—1941no—
152C2×C4—1941no—
153C2×C4—1941no—
154C2×C4—1941no—
155C2×C4C2×#1171590611yescharacteristic
156C2×C4C2×#1171590611yescharacteristic
157C8—972no—
158C8—972no—
159C8—1941no—
160C8—1941no—
161C22—21no—
162C22—41no—
163C22—971no—
164C22—971no—
165C22—3881no—
166C22—3882no—
167C22—3882no—
168C22—1942no—
169C22—1941no—
170C22C22×#1171590611yescharacteristic
171C4—21no—
172C4—971no—
173C4—971no—
174C4—1941no—
175C4C22⋊#1171590611yescharacteristic
176C4C22⋊#1171590611yescharacteristic
177C2—21no—
178C2—41no—
179C2—972no—
180C2—3881no—
181C2—1941no—
182C2C2×C22⋊#1171590611yescharacteristic
subgroupG/H
centerC4C22⋊#11715906
commutatorC22×C97C22×C8
FrattiniC22C22×#11715906
FittingC22×C4×C97C8
socleC2×C97C2×C22⋊C8
prime radicalC23×D97C8
cosocleC2×#11715897C23
hypercenterC22×C4#11715906
hypocenterC97C2×#349
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #20261854C2semidirect
2. #20261834C2semidirect
3. #20261833C2semidirect
6. #15690906C22semidirect
90. D97#349semidirect
92. C97C2×#349semidirect

Maximal subgroups

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

subgroupindex
#202618542
#202618342
#202618332
C2×D97⋊M162
C2×#34997