Finite Group Databasebeta

← #27139733

Subgroups — #27139733

Order 12416.

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

normal subgroups only

This group has 222 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#20260502C211yesmaximal, characteristic, complemented
2#20260459C211yesmaximal, characteristic, complemented
3#392×C97C211yesmaximal, characteristic
4C2×C22⋊Dic97—21no—
5C22⋊#11715895—21no—
6C22×D4×C97—21no—
7C22×D4×C97C411yescharacteristic, complemented
8C2×C22⋊C4×C97—21no—
9C2×C22⋊C4×C97C2211yescharacteristic
10#95×C97C411yescharacteristic, complemented
11C22×Dic97—41no—
12C2×#11715895—41no—
13C22⋊Dic97—22no—
14C22⋊Dic97—41no—
15C24×C97—21no—
16C24×C97D411yescharacteristic
17C2×D4×C97—41no—
18C2×D4×C97—42no—
19C2×D4×C97—42no—
20C2×D4×C97—41no—
21C2×D4×C97—41no—
22C2×D4×C97—42no—
23C22×C4×C97—21no—
24C22×C4×C97C2×C411yescharacteristic
25C22×C4×C97D411yescharacteristic
26C22⋊C4×C97—41no—
27C22⋊C4×C97—42no—
28C23×C97—21no—
29C23×C97—21no—
30C23×C97—22no—
31C23×C97—41no—
32C23×C97—41no—
33C23×C97—41no—
34C23×C97—41no—
35C23×C97—41no—
36C23×C97—41no—
37C23×C97—41no—
38C23×C97—41no—
39C23×C97C22⋊C411yescharacteristic
40C23×C97D811yescharacteristic
41C23×C97SD1611yescharacteristic
42D4×C97—42no—
43D4×C97—44no—
44D4×C97—44no—
45D4×C97—42no—
46C2×C4×C97—22no—
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97—41no—
50C2×C4×C97—41no—
51C2×C4×C97—41no—
52C2×C4×C97—41no—
53C2×Dic97—42no—
54C2×Dic97—81no—
55C2×Dic97—81no—
56#11715895—81no—
57C22×C97—21no—
58C22×C97—21no—
59C22×C97—21no—
60C22×C97—21no—
61C22×C97—41no—
62C22×C97—41no—
63C22×C97—42no—
64C22×C97—42no—
65C22×C97—41no—
66C22×C97—41no—
67C22×C97—42no—
68C22×C97—81no—
69C22×C97—81no—
70C22×C97—81no—
71C22×C97—81no—
72C22×C97—81no—
73C22×C97#9911yescharacteristic
74C22×C97#10211yescharacteristic
75C22×C97#10411yescharacteristic
76C4×C97—42no—
77C4×C97—41no—
78C4×C97—41no—
79Dic97—82no—
80C2×C97—21no—
81C2×C97—21no—
82C2×C97—41no—
83C2×C97—41no—
84C2×C97—82no—
85C2×C97#32711yescharacteristic
86C2×C97AΣL(1,𝔽16)11yescharacteristic
87C2×C97#35311yescharacteristic
88#1343—971nomaximal, Sylow
89C97#134311yescharacteristic, complemented, Sylow
90#392D9711yescharacteristic
91#342—971no—
92#323—971no—
93C22×D4—21no—
94C22×D4Dic9711yescharacteristic, complemented
95C2×C22⋊C4—21no—
96C2×C22⋊C4—1941no—
97C2×C22⋊C4C2×D9711yescharacteristic
98C22⋊C8—1941no—
99#95Dic9711yescharacteristic, complemented
100C24—21no—
101C24C22⋊D9711yescharacteristic
102C2×D4—41no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—42no—
108C22×C4—21no—
109C22×C4—3881no—
110C22×C4C2×Dic9711yescharacteristic
111C22×C4C22⋊D9711yescharacteristic
112C2×C8—3881no—
113C22⋊C4—41no—
114C22⋊C4—42no—
115C22⋊C4—3881no—
116C22⋊C4—1942no—
117C23—21no—
118C23—21no—
119C23—22no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23C22⋊Dic9711yescharacteristic
129C23#1309549611yescharacteristic
130C23#1309549911yescharacteristic
131D4—42no—
132D4—44no—
133D4—44no—
134D4—42no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—7761no—
143C2×C4—7761no—
144C2×C4—3882no—
145C8—7761no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—42no—
153C22—42no—
154C22—41no—
155C22—41no—
156C22—42no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22#1569072511yescharacteristic
163C22#1569073111yescharacteristic
164C22#1569073711yescharacteristic
165C4—42no—
166C4—41no—
167C4—41no—
168C4—7762no—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—82no—
174C2#2026046911yescharacteristic
175C2#2026052211yescharacteristic
176C2#2026052811yescharacteristic
subgroupG/H
centerC22#15690725
commutatorC22×C4×C97C2×C4
FrattiniC2×C22⋊C4C2×D97
Fitting, prime radical#392×C97C2
socleC22×C97#99
cosocleC2×C22⋊C4×C97C22
hypercenter#392D97
hypocenterC97#1343
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #20260502C2semidirect
2. #20260459C2semidirect
7. C22×D4×C97C4semidirect
10. #95×C97C4semidirect
89. C97#1343semidirect
94. C22×D4Dic97semidirect
99. #95Dic97semidirect

Maximal subgroups

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

subgroupindex
#202605022
#202604592
#392×C972
#134397