Finite Group Databasebeta

← #34068069

Subgroups — #34068069

Order 19328.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#23752838C211yesmaximal, characteristic
2#23752370C212yesmaximal, complemented
3#23752124C211yesmaximal, characteristic
4#23751950C212yesmaximal, complemented
5#323×C151C211yesmaximal, characteristic
6C22×Dic302C2211yescharacteristic
7C2×Dic604—22no—
8#18012718—21no—
9#18012718C2212yes—
10#18012716—22no—
11#18012716—22no—
12C2×C22⋊Dic151—21no—
13C2×C22⋊Dic151—21no—
14#18012664—22no—
15C151⋊#95—21no—
16C151⋊#95C2211yescharacteristic
17C2×C22⋊C4×C151C2211yescharacteristic
18C22⋊C8×C151C2212yes—
19C2×Dic302—21no—
20C2×Dic302—22no—
21C2×Dic302—42no—
22C2×Dic302D412yes—
23C22×Dic151—21no—
24C22×Dic151—21no—
25C22×Dic151—21no—
26C22×Dic151—21no—
27Dic604—44no—
28C4⋊Dic151—21no—
29C4⋊Dic151D412yes—
30Dic151⋊C4—22no—
31Dic151⋊C4—22no—
32C22⋊Dic151—22no—
33C22⋊Dic151—22no—
34C22⋊Dic151—42no—
35C24×C151D411yescharacteristic
36C22×C4×C151D411yescharacteristic
37C22×C4×C151C2311yescharacteristic
38C2×C8×C151—22no—
39C22⋊C4×C151—22no—
40Dic302—24no—
41Dic302—42no—
42Dic302—42no—
43Dic302—81no—
44C2×Dic151—22no—
45C2×Dic151—22no—
46C2×Dic151—22no—
47C2×Dic151—41no—
48C2×Dic151—41no—
49C2×Dic151—41no—
50C2×Dic151—41no—
51C2×Dic151—41no—
52C2×Dic151—81no—
53C2×Dic151—81no—
54C23×C151—21no—
55C23×C151—21no—
56C23×C151—41no—
57C23×C151—41no—
58C23×C151Q1612yescomplemented
59C23×C151C2×D411yescharacteristic
60C2×C4×C151—21no—
61C2×C4×C151—41no—
62C2×C4×C151—41no—
63C2×C4×C151C2×D412yes—
64C8×C151—42no—
65C22×C151—21no—
66C22×C151—21no—
67C22×C151—22no—
68C22×C151—42no—
69C22×C151—41no—
70C22×C151—41no—
71C22×C151—81no—
72C22×C151#12011yescharacteristic
73C22×C151C2×Q1611yescharacteristic
74C22×C151#13711yescharacteristic
75C4×C151—22no—
76C4×C151—41no—
77Dic151—42no—
78Dic151—42no—
79Dic151—42no—
80Dic151—81no—
81C2×C151—21no—
82C2×C151—21no—
83C2×C151—42no—
84C2×C151#45111yescharacteristic
85C2×C151#45511yescharacteristic
86C2×C151B(4,𝔽2)11yescharacteristic
87C151#160211yescharacteristic, complemented, Sylow
88#1602—1511nomaximal, Sylow
89#543—1511no—
90#451—1512no—
91#393—1511no—
92#328—1512no—
93#323D15111yescharacteristic
94C22×Q8—1511no—
95C2×Q16—3022no—
96C22⋊Q8—3022no—
97C22⋊Q8—3022no—
98C22⋊Q8—3021no—
99C22⋊Q8—1512no—
100C2×C22⋊C4—3021no—
101C2×C22⋊C4—3021no—
102C2×C22⋊C4C2×D15111yescharacteristic
103#103—3022no—
104C22⋊C8C2×D15112yes—
105#95—3021no—
106#95—1511no—
107C24D60411yescharacteristic
108C2×Q8—3021no—
109C2×Q8—3022no—
110C2×Q8—6042no—
111C2×Q8—1512no—
112C22×C4—3021no—
113C22×C4—3021no—
114C22×C4—3021no—
115C22×C4—3021no—
116C22×C4D60411yescharacteristic
117C22×C4C22×D15111yescharacteristic
118Q16—6044no—
119C2×C8—22no—
120C4⋊C4—3022no—
121C4⋊C4—3021no—
122C4⋊C4—3022no—
123C4⋊C4—1512no—
124C22⋊C4—22no—
125C22⋊C4—3022no—
126C22⋊C4—3022no—
127C22⋊C4—6042no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic60412yescomplemented
133C23C2×D60411yescharacteristic
134Q8—3024no—
135Q8—6042no—
136Q8—6042no—
137Q8—12081no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—3022no—
142C2×C4—3022no—
143C2×C4—3022no—
144C2×C4—6041no—
145C2×C4—6041no—
146C2×C4—6041no—
147C2×C4—6041no—
148C2×C4—6041no—
149C2×C4—12081no—
150C2×C4—12081no—
151C2×C4D4×D15112yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1801270911yescharacteristic
161C22C2×Dic60411yescharacteristic
162C22#1801277211yescharacteristic
163C4—22no—
164C4—41no—
165C4—6042no—
166C4—6042no—
167C4—6042no—
168C4—12081no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2375237011yescharacteristic
173C2#2375239211yescharacteristic
174C2#2375240011yescharacteristic
subgroupG/H
centerC22#18012709
commutator, cosocleC22×C4×C151C23
FrattiniC22×C4C22×D151
Fitting#323×C151C2
socleC22×C151#120
prime radicalC24×C151D4
hypercenter#323D151
hypocenterC151#1602
LCSG ▷ C22×C4×C151 ▷ C22×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #23752370C2semidirect
4. #23751950C2semidirect
58. C23×C151Q16semidirect
87. C151#1602semidirect
132. C23Dic604semidirect

Maximal subgroups

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

subgroupindex
#237528382
#237523702
#237521242
#237519502
#323×C1512
#1602151