Finite Group Databasebeta

← #27157244

Subgroups — #27157244

Order 12416.

344 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261753C211yesmaximal, characteristic, complemented
2#20261740C211yesmaximal, characteristic, complemented
3#20261593C214yesmaximal, complemented
4#127×D97C211yesmaximal, characteristic
5C2×#13095521—21no—
6#15690877—21no—
7#15690877C2212yescomplemented
8#15690877C2212yescomplemented
9#15690867—21no—
10C4×#11715897—21no—
11C2×D4×D97—21no—
12C2×D4×D97—21no—
13C2×D4×D97C412yes—
14#15690807C411yescharacteristic
15#15690806—21no—
16#15690806—21no—
17#15690806C2212yes—
18C42×D97C2211yescharacteristic
19#127×C97C411yescharacteristic
20#13095521—42no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C23×D97—22no—
24C2×D388—21no—
25C2×D388—21no—
26C2×D388C2×C412yes—
27C2×C22⋊D97—22no—
28C2×C22⋊D97—22no—
29C2×C22⋊D97—22no—
30D4×D97—44no—
31D4×D97—44no—
32D4×D97—42no—
33D4×D97—42no—
34C2×C4×D97C2×C411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C4×Dic97D411yescharacteristic
38C4×Dic97D411yescharacteristic
39C4×Dic97C2311yescharacteristic
40C2×D4×C97—21no—
41C2×D4×C97—21no—
42C2×D4×C97C2×C412yes—
43C42×C97C2×C411yescharacteristic
44C23×C97—22no—
45C22×D97—22no—
46C22×D97—44no—
47C22×D97—44no—
48C22×D97—44no—
49C22×D97C22⋊C411yescharacteristic
50D4×C97—42no—
51D4×C97—42no—
52D4×C97—42no—
53C2×C4×C97C22⋊C411yescharacteristic
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22×C411yescharacteristic
56C22⋊D97—44no—
57C22⋊D97—42no—
58C22⋊D97—42no—
59C22⋊D97—44no—
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C2×D411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63D388—42no—
64D388—42no—
65D388—42no—
66C4×D97—22no—
67C4×D97—22no—
68C4×D97D812yes—
69C4×D97SD1612yes—
70#11715897—42no—
71#11715897—42no—
72C22×C97—42no—
73C22×C97—42no—
74C22×C97—42no—
75C22×C97C2×C22⋊C411yescharacteristic
76C2×D97—21no—
77C2×D97—21no—
78C2×D97—42no—
79C2×D97—42no—
80C2×D97—42no—
81C2×D97—88no—
82C2×D97#10212yes—
83C4×C97—21no—
84C4×C97—21no—
85C4×C97#10212yes—
86Dic97#10012yes—
87Dic97C2×D811yescharacteristic
88Dic97C2×SD1611yescharacteristic
89Dic97AGL(1,ℤ/8)12yes—
90C2×C97—82no—
91C2×C97C2×#10011yescharacteristic
92C2×C97C2×#10211yescharacteristic
93C2×C97#41811yescharacteristic
94D97—22no—
95D97—82no—
96#1545—971nomaximal, Sylow
97C97#154511yescharacteristic, complemented, Sylow
98C2×#127—971no—
99#431—971no—
100#423—971no—
101#331—974no—
102C22×D4—972no—
103C22×D4—1941no—
104C22×D4—1941no—
105C2×M16—1941no—
106#127—972no—
107#127—971no—
108#127—1941no—
109#127—1941no—
110#127#8760411yescharacteristic
111C2×C42—971no—
112C4⋊C8—972no—
113C4⋊C8—972no—
114C4⋊C8—1941no—
115C22⋊C8—1941no—
116C4×C8—1941no—
117C24—1942no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—972no—
121C2×D4—3882no—
122C2×D4—3882no—
123C2×D4—3884no—
124C2×D4—3884no—
125C2×D4—1942no—
126C2×D4—1942no—
127C2×D4—1942no—
128C2×D4—1941no—
129C2×D4—1941no—
130C2×D4C2×#8760412yes—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—971no—
134M16—3882no—
135C2×C8—1942no—
136C2×C8—1942no—
137C42—971no—
138C42—971no—
139C42—971no—
140C42C2×#8760411yescharacteristic
141C23—22no—
142C23—971no—
143C23—3884no—
144C23—3884no—
145C23—3884no—
146C23—1942no—
147D4—42no—
148D4—42no—
149D4—42no—
150D4—3882no—
151D4—3882no—
152D4—3882no—
153D4—3882no—
154D4—3884no—
155D4—3884no—
156D4—3882no—
157C2×C4—971no—
158C2×C4—971no—
159C2×C4—971no—
160C2×C4—972no—
161C2×C4—972no—
162C2×C4—1942no—
163C2×C4—1942no—
164C2×C4C22⋊#8760411yescharacteristic
165C2×C4C22⋊#8760411yescharacteristic
166C2×C4C22×#8760411yescharacteristic
167C8—3882no—
168C8—3882no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—7768no—
173C22—972no—
174C22—3882no—
175C22—3882no—
176C22—3882no—
177C22—1941no—
178C22—1941no—
179C22C2×C22⋊#8760411yescharacteristic
180C4—21no—
181C4—21no—
182C4—972no—
183C4—971no—
184C4—971no—
185C4—972no—
186C4#1569087212yes—
187C2—82no—
188C2—7762no—
189C2—1942no—
190C2D97⋊#10011yescharacteristic
191C2C2×#1569087211yescharacteristic
192C2#2026172711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC22×C97C2×C22⋊C4
prime radical#127×D97C2
cosocleC4×Dic97C23
hypercenter#127#87604
hypocenterC97#1545
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261753C2semidirect
2. #20261740C2semidirect
3. #20261593C2semidirect
7. #15690877C22semidirect
8. #15690877C22semidirect
97. C97#1545semidirect

Maximal subgroups

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

subgroupindex
#202617532
#202617402
#202615932
#127×D972
#154597