Finite Group Databasebeta

← #27140067

Subgroups — #27140067

Order 12416.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C2×#15690766C211yesmaximal, characteristic
2#20260789C212yesmaximal, complemented
3#20260460C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic
5#323×C97C211yesmaximal, characteristic
6C23×Dic97C411yescharacteristic
7#15690766—22no—
8#15690766—22no—
9C2×#13095490C411yescharacteristic
10C2×#13095490C2211yescharacteristic
11C2×C22⋊Dic97C411yescharacteristic
12C2×C4×Dic97C411yescharacteristic
13C2×C4×Dic97C2211yescharacteristic
14#15690739—22no—
15C22⋊#11715895C2212yes—
16#15690720—22no—
17C2×C22⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19C22×Dic97—22no—
20C22×Dic97—21no—
21C22×Dic97—21no—
22C22×Dic97—22no—
23C22×Dic97C2×C411yescharacteristic
24C22×Dic97C2×C411yescharacteristic
25C22×Dic97C2×C411yescharacteristic
26C22×Dic97C2×C411yescharacteristic
27C2×#11715895—22no—
28#13095490—22no—
29#13095490—22no—
30C22⋊Dic97—22no—
31C4×Dic97—21no—
32C4×Dic97—21no—
33C4×Dic97D412yes—
34C4×Dic97D412yes—
35C24×C97C2×C411yescharacteristic
36C22×C4×C97C2×C411yescharacteristic
37C22×C4×C97C2311yescharacteristic
38C2×C8×C97—22no—
39C22⋊C4×C97—22no—
40C23×C97—21no—
41C23×C97—21no—
42C23×C97—41no—
43C23×C97—41no—
44C23×C97M1612yes—
45C23×C97C22×C411yescharacteristic
46C2×C4×C97—21no—
47C2×C4×C97—41no—
48C2×C4×C97—41no—
49C2×C4×C97C2×D412yes—
50C2×Dic97—24no—
51C2×Dic97—24no—
52C2×Dic97—21no—
53C2×Dic97—21no—
54C2×Dic97—21no—
55C2×Dic97—21no—
56C2×Dic97—21no—
57C2×Dic97—41no—
58C2×Dic97—41no—
59C2×Dic97C22⋊C411yescharacteristic
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C22⋊C412yes—
62C2×Dic97M1612yes—
63C8×C97—42no—
64#11715895—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97—22no—
68C22×C97—42no—
69C22×C97—42no—
70C22×C97—41no—
71C22×C97—41no—
72C22×C97C2×C22⋊C411yescharacteristic
73C22×C97C2×M1611yescharacteristic
74C22×C97C2×M1611yescharacteristic
75C4×C97—22no—
76C4×C97—41no—
77Dic97—21no—
78Dic97—21no—
79Dic97—22no—
80Dic97—22no—
81Dic97—41no—
82C2×C97—21no—
83C2×C97—21no—
84C2×C97—42no—
85C2×C97#40711yescharacteristic
86C2×C97#41011yescharacteristic
87C2×C97#41311yescharacteristic
88#1464—971nomaximal, Sylow
89C97#146411yescharacteristic, complemented, Sylow
90C2×#117—971no—
91#432—972no—
92#324—972no—
93#323—971no—
94#323D9711yescharacteristic
95C23×C4—971no—
96#117—1942no—
97#117—1942no—
98C2×C4⋊C4—971no—
99C2×C4⋊C4—971no—
100C2×C22⋊C4—971no—
101C2×C22⋊C4C2×D9711yescharacteristic
102C2×C42—971no—
103C2×C42—971no—
104C4⋊C8—1942no—
105C22⋊C8—972no—
106C22⋊C8C2×D9712yes—
107#97—1942no—
108C24C4×D9711yescharacteristic
109C22×C4—971no—
110C22×C4—971no—
111C22×C4—971no—
112C22×C4—971no—
113C22×C4—1941no—
114C22×C4—1941no—
115C22×C4—1942no—
116C22×C4—1942no—
117C22×C4C4×D9711yescharacteristic
118C22×C4C22×D9711yescharacteristic
119C2×C8—22no—
120C2×C8—1942no—
121C4⋊C4—1942no—
122C4⋊C4—1942no—
123C22⋊C4—22no—
124C22⋊C4—1942no—
125C42—972no—
126C42—972no—
127C42—1941no—
128C42—1941no—
129C23—21no—
130C23—21no—
131C23—41no—
132C23—41no—
133C23#1309549412yes—
134C23C2×C4×D9711yescharacteristic
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—971no—
139C2×C4—971no—
140C2×C4—972no—
141C2×C4—972no—
142C2×C4—3881no—
143C2×C4—3881no—
144C2×C4—1941no—
145C2×C4—1944no—
146C2×C4—1944no—
147C2×C4—1941no—
148C2×C4—1941no—
149C2×C4—1941no—
150C2×C4—1941no—
151C2×C4D4×D9712yes—
152C8—42no—
153C8—3882no—
154C22—21no—
155C22—21no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—41no—
161C22C22⋊C4×D9711yescharacteristic
162C22M16×D9711yescharacteristic
163C22C2×#1309549411yescharacteristic
164C4—22no—
165C4—41no—
166C4—3881no—
167C4—1941no—
168C4—1941no—
169C4—1942no—
170C4—1942no—
171C2—21no—
172C2—21no—
173C2—42no—
174C2#2026069711yescharacteristic
175C2#2026070711yescharacteristic
176C2#2026071611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#323×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×C4×C97C23
hypercenter#323D97
hypocenterC97#1464
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20260789C2semidirect
3. #20260460C2semidirect
89. C97#1464semidirect

Maximal subgroups

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

subgroupindex
C2×#156907662
#202607892
#202604602
#202604592
#323×C972
#146497