Finite Group Databasebeta

← #27139663

Subgroups — #27139663

Order 12416.

294 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 37 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#20260590C211yesmaximal, characteristic
2#20260459C211yesmaximal, characteristic, complemented
3#323×C97C211yesmaximal, characteristic, complemented
4C24×D97C411yescharacteristic
5C2×C22⋊Dic97C411yescharacteristic
6C2×#13095488—21no—
7C2×#13095488—21no—
8C22⋊#11715895—21no—
9C2×C22⋊C4×C97C2211yescharacteristic
10C22⋊C8×C97—21no—
11C23×D97—21no—
12C23×D97—21no—
13C23×D97—42no—
14C23×D97—42no—
15C23×D97—41no—
16C23×D97—41no—
17C23×D97C812yescomplemented
18C22×Dic97—21no—
19C2×#11715895—41no—
20C22⋊Dic97—41no—
21#13095488—42no—
22#13095488—42no—
23C24×C97C2×C411yescharacteristic
24C22×C4×C97D411yescharacteristic
25C22×C4×C97D411yescharacteristic
26C2×C8×C97—41no—
27C22⋊C4×C97—41no—
28C23×C97—21no—
29C23×C97—21no—
30C23×C97—41no—
31C23×C97—41no—
32C23×C97C22⋊C411yescharacteristic
33C23×C97C2×C811yescharacteristic
34C23×C97M1611yescharacteristic
35C22×D97—22no—
36C22×D97—22no—
37C22×D97—22no—
38C22×D97—44no—
39C22×D97—44no—
40C22×D97—41no—
41C22×D97—41no—
42C22×D97—42no—
43C22×D97—48no—
44C22×D97—42no—
45C22×D97—42no—
46C22×D97—41no—
47C22×D97—41no—
48C22×D97—42no—
49C22×D97—42no—
50C22×D97—42no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×Dic97—41no—
56C2×Dic97—41no—
57C2×Dic97—41no—
58C8×C97—81no—
59#11715895—81no—
60C22×C97—21no—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97—41no—
65C22×C97—41no—
66C22×C97—42no—
67C22×C97—41no—
68C22×C97—41no—
69C22×C97C22⋊C811yescharacteristic
70C22×C97#9911yescharacteristic
71C22×C97#10011yescharacteristic
72C2×D97—24no—
73C2×D97—42no—
74C2×D97—42no—
75C2×D97—42no—
76C2×D97—44no—
77C2×D97—44no—
78C2×D97—44no—
79C2×D97—42no—
80C2×D97—42no—
81C2×D97—42no—
82C2×D97—44no—
83C4×C97—41no—
84C4×C97—41no—
85Dic97—81no—
86C2×C97—21no—
87C2×C97—21no—
88C2×C97—41no—
89C2×C97—41no—
90C2×C97#32311yescharacteristic
91C2×C97AΣL(1,𝔽16)11yescharacteristic
92C2×C97AΣL(1,𝔽16)11yescharacteristic
93D97—44no—
94#1316—971nomaximal, Sylow
95C97#131611yescharacteristic, complemented, Sylow
96#379—971no—
97#323—971no—
98#323D9711yescharacteristic, complemented
99C25—971no—
100C2×C22⋊C4—971no—
101C2×C22⋊C4—1941no—
102C2×C22⋊C4—1941no—
103C2×C22⋊C4C2×D9711yescharacteristic
104C22⋊C8—21no—
105C22⋊C8—1941no—
106C24—972no—
107C24—3881no—
108C24—3881no—
109C24—3882no—
110C24—3882no—
111C24—1941no—
112C24—1941no—
113C24C4×D9711yescharacteristic
114C22×C4—1941no—
115C22×C4D38811yescharacteristic
116C22×C4C22⋊D9711yescharacteristic
117C2×C8—41no—
118C2×C8—3881no—
119C22⋊C4—41no—
120C22⋊C4—3882no—
121C22⋊C4—3882no—
122C22⋊C4—3881no—
123C23—21no—
124C23—21no—
125C23—41no—
126C23—41no—
127C23—3882no—
128C23—3881no—
129C23—3881no—
130C23—3882no—
131C23—3882no—
132C23—3882no—
133C23—3888no—
134C23—3882no—
135C23—3882no—
136C23—3884no—
137C23—3884no—
138C23—3881no—
139C23—3881no—
140C23—1942no—
141C23—1942no—
142C23—1942no—
143C23#1309548811yescharacteristic
144C23C8×D9711yescharacteristic
145C23#1309549411yescharacteristic
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—3881no—
151C2×C4—3881no—
152C2×C4—3881no—
153C8—81no—
154C8—7761no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—41no—
160C22—41no—
161C22—42no—
162C22—41no—
163C22—41no—
164C22—3882no—
165C22—3882no—
166C22—3882no—
167C22—3884no—
168C22—3884no—
169C22—3884no—
170C22—3884no—
171C22—3882no—
172C22—3882no—
173C22—3882no—
174C22—1944no—
175C22#1569072211yescharacteristic
176C22#1569072411yescharacteristic
177C22#1569072611yescharacteristic
178C4—41no—
179C4—41no—
180C4—7761no—
181C2—21no—
182C2—21no—
183C2—41no—
184C2—41no—
185C2—3884no—
186C2#2026045811yescharacteristic
187C2#2026052111yescharacteristic
188C2#2026052311yescharacteristic
subgroupG/H
centerC22#15690724
commutatorC23×C97C2×C8
FrattiniC2×C22⋊C4C2×D97
Fitting#323×C97C2
socleC22×C97#99
prime radicalC24×D97C4
cosocleC2×C22⋊C4×C97C22
hypercenter#323D97
hypocenterC97#1316
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20260459C2semidirect
3. #323×C97C2semidirect
17. C23×D97C8semidirect
95. C97#1316semidirect
98. #323D97semidirect

Maximal subgroups

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

subgroupindex
#202605902
#202604592
#323×C972
#131697