Finite Group Databasebeta

← #27141842

Subgroups — #27141842

Order 12416.

296 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 81 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#20261210C211yesmaximal, characteristic
2C2×#15690725C211yesmaximal, characteristic
3#20260502C212yesmaximal, complemented
4#20260501C212yesmaximal, complemented
5C2×#99×C97C211yesmaximal, characteristic
6D4×Dic97—22no—
7D4×Dic97—22no—
8#15690769C412yescomplemented
9C2×C22⋊Dic97C412yescomplemented
10C2×C22⋊Dic97C2212yes—
11C2×C22⋊Dic97C2211yescharacteristic
12C2×C22⋊Dic97C2211yescharacteristic
13#15690725—22no—
14C22×D4×C97C2211yescharacteristic
15C2×C22⋊C4×C97C2212yes—
16#99×C97—22no—
17C22×Dic97—22no—
18C22×Dic97—21no—
19C22×Dic97—21no—
20C22×Dic97—22no—
21C4⋊Dic97—22no—
22C22⋊Dic97—22no—
23C22⋊Dic97—21no—
24C22⋊Dic97—21no—
25C22⋊Dic97—22no—
26C22⋊Dic97—22no—
27C22⋊Dic97C2×C412yes—
28C22⋊Dic97C2×C412yes—
29C4×Dic97—22no—
30C24×C97C2×C411yescharacteristic
31C24×C97C2311yescharacteristic
32C2×D4×C97—22no—
33C2×D4×C97—22no—
34C2×D4×C97D412yes—
35C2×D4×C97D412yes—
36C22×C4×C97—22no—
37C22×C4×C97C2×C411yescharacteristic
38C22⋊C4×C97—22no—
39C22⋊C4×C97—22no—
40C23×C97—21no—
41C23×C97—21no—
42C23×C97—81no—
43C23×C97—81no—
44C23×C97C22⋊C412yes—
45C23×C97C22×C411yescharacteristic
46C23×C97C2×D411yescharacteristic
47C23×C97C2×D411yescharacteristic
48C23×C97#4112yes—
49C23×C97#4112yes—
50D4×C97—24no—
51D4×C97—24no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—42no—
55C2×C4×C97—82no—
56C2×C4×C97C22⋊C412yes—
57C2×Dic97—22no—
58C2×Dic97—22no—
59C2×Dic97—42no—
60C2×Dic97—81no—
61C2×Dic97—81no—
62C2×Dic97—82no—
63C2×Dic97—82no—
64C22×C97—22no—
65C22×C97—41no—
66C22×C97—41no—
67C22×C97—42no—
68C22×C97—42no—
69C22×C97—41no—
70C22×C97—42no—
71C22×C97—81no—
72C22×C97—81no—
73C22×C97C2×C22⋊C411yescharacteristic
74C22×C97#11711yescharacteristic
75C22×C97#11711yescharacteristic
76C22×C97#12312yes—
77C22×C97#12312yes—
78C4×C97—22no—
79C4×C97—82no—
80Dic97—42no—
81Dic97—42no—
82Dic97—82no—
83C2×C97—21no—
84C2×C97—22no—
85C2×C97—22no—
86C2×C97—21no—
87C2×C97—21no—
88C2×C97—42no—
89C2×C97#38111yescharacteristic
90#1828—971nomaximal, Sylow
91C97#182811yescharacteristic, complemented, Sylow
92#518—971no—
93C2×#99—971no—
94C2×#99D9711yescharacteristic
95#342—972no—
96#342—972no—
97C22×D4C2×D9711yescharacteristic
98C4×D4—1942no—
99C4×D4—1942no—
100#117—972no—
101C2×C22⋊C4—972no—
102C2×C22⋊C4—971no—
103C2×C22⋊C4—971no—
104C2×C22⋊C4—972no—
105C2×C22⋊C4C2×D9712yes—
106#99—22no—
107#99—1942no—
108C24C4×D9711yescharacteristic
109C24C22×D9711yescharacteristic
110C2×D4—22no—
111C2×D4—22no—
112C2×D4D38812yes—
113C2×D4C22⋊D9712yes—
114C22×C4—22no—
115C22×C4—1942no—
116C22×C4—1942no—
117C22×C4—1941no—
118C22×C4—1941no—
119C22×C4C4×D9711yescharacteristic
120C4⋊C4—1942no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—972no—
124C22⋊C4—972no—
125C22⋊C4—1942no—
126C22⋊C4—1942no—
127C22⋊C4—1942no—
128C22⋊C4—1941no—
129C22⋊C4—1941no—
130C42—1942no—
131C23—21no—
132C23—21no—
133C23—81no—
134C23—81no—
135C23#1309548812yes—
136C23C2×C4×D9711yescharacteristic
137C23C2×C22⋊D9711yescharacteristic
138C23C2×D38811yescharacteristic
139C23#1309551012yes—
140C23#1309551012yes—
141D4—24no—
142D4—24no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—82no—
147C2×C4—7762no—
148C2×C4—7761no—
149C2×C4—7761no—
150C2×C4—7762no—
151C2×C4—3882no—
152C2×C4—1942no—
153C2×C4—1942no—
154C2×C4#1309548812yes—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—42no—
160C22—41no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22C2×#1309548811yescharacteristic
165C22#1569076611yescharacteristic
166C22#1569076611yescharacteristic
167C22#1569079112yes—
168C22#1569079412yes—
169C4—22no—
170C4—82no—
171C4—7762no—
172C4—3882no—
173C4—3882no—
174C2—22no—
175C2—22no—
176C2—21no—
177C2—21no—
178C2—21no—
179C2—42no—
180C2#2026059611yescharacteristic
subgroupG/H
centerC2#20260596
commutatorC23×C97C22×C4
FrattiniC24C22×D97
FittingC2×#99×C97C2
socleC2×C97#381
prime radicalC22×D4×C97C22
cosocleC24×C97C23
hypercenterC2×#99D97
hypocenterC97#1828
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #20260502C2semidirect
4. #20260501C2semidirect
8. #15690769C4semidirect
9. C2×C22⋊Dic97C4semidirect
91. C97#1828semidirect

Maximal subgroups

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

subgroupindex
#202612102
C2×#156907252
#202605022
#202605012
C2×#99×C972
#182897