Finite Group Databasebeta

← #37571479

Subgroups — #37571479

Order 24448.

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#27133289C211yesmaximal, characteristic
2#27132821C212yesmaximal, complemented
3#27132575C211yesmaximal, characteristic
4#27132401C212yesmaximal, complemented
5#323×C191C211yesmaximal, characteristic
6C22×Dic382C2211yescharacteristic
7C2×Dic764—22no—
8#20259301—21no—
9#20259301C2212yes—
10#20259299—22no—
11#20259299—22no—
12C2×C22⋊Dic191—21no—
13C2×C22⋊Dic191—21no—
14#20259247—22no—
15C191⋊#95—21no—
16C191⋊#95C2211yescharacteristic
17C2×C22⋊C4×C191C2211yescharacteristic
18C22⋊C8×C191C2212yes—
19C2×Dic382—21no—
20C2×Dic382—22no—
21C2×Dic382—42no—
22C2×Dic382D412yes—
23C22×Dic191—21no—
24C22×Dic191—21no—
25C22×Dic191—21no—
26C22×Dic191—21no—
27Dic764—44no—
28C4⋊Dic191—21no—
29C4⋊Dic191D412yes—
30Dic191⋊C4—22no—
31Dic191⋊C4—22no—
32C22⋊Dic191—22no—
33C22⋊Dic191—22no—
34C22⋊Dic191—42no—
35C24×C191D411yescharacteristic
36C22×C4×C191D411yescharacteristic
37C22×C4×C191C2311yescharacteristic
38C2×C8×C191—22no—
39C22⋊C4×C191—22no—
40Dic382—24no—
41Dic382—42no—
42Dic382—42no—
43Dic382—81no—
44C2×Dic191—22no—
45C2×Dic191—22no—
46C2×Dic191—22no—
47C2×Dic191—41no—
48C2×Dic191—41no—
49C2×Dic191—41no—
50C2×Dic191—41no—
51C2×Dic191—41no—
52C2×Dic191—81no—
53C2×Dic191—81no—
54C23×C191—21no—
55C23×C191—21no—
56C23×C191—41no—
57C23×C191—41no—
58C23×C191Q1612yescomplemented
59C23×C191C2×D411yescharacteristic
60C2×C4×C191—21no—
61C2×C4×C191—41no—
62C2×C4×C191—41no—
63C2×C4×C191C2×D412yes—
64C8×C191—42no—
65C22×C191—21no—
66C22×C191—21no—
67C22×C191—22no—
68C22×C191—42no—
69C22×C191—41no—
70C22×C191—41no—
71C22×C191—81no—
72C22×C191#12011yescharacteristic
73C22×C191C2×Q1611yescharacteristic
74C22×C191#13711yescharacteristic
75C4×C191—22no—
76C4×C191—41no—
77Dic191—42no—
78Dic191—42no—
79Dic191—42no—
80Dic191—81no—
81C2×C191—21no—
82C2×C191—21no—
83C2×C191—42no—
84C2×C191#45111yescharacteristic
85C2×C191#45511yescharacteristic
86C2×C191B(4,𝔽2)11yescharacteristic
87C191#160211yescharacteristic, complemented, Sylow
88#1602—1911nomaximal, Sylow
89#543—1911no—
90#451—1912no—
91#393—1911no—
92#328—1912no—
93#323D19111yescharacteristic
94C22×Q8—1911no—
95C2×Q16—3822no—
96C22⋊Q8—3822no—
97C22⋊Q8—3822no—
98C22⋊Q8—3821no—
99C22⋊Q8—1912no—
100C2×C22⋊C4—3821no—
101C2×C22⋊C4—3821no—
102C2×C22⋊C4C2×D19111yescharacteristic
103#103—3822no—
104C22⋊C8C2×D19112yes—
105#95—3821no—
106#95—1911no—
107C24D76411yescharacteristic
108C2×Q8—3821no—
109C2×Q8—3822no—
110C2×Q8—1912no—
111C2×Q8—7642no—
112C22×C4—3821no—
113C22×C4—3821no—
114C22×C4—3821no—
115C22×C4—3821no—
116C22×C4D76411yescharacteristic
117C22×C4C22×D19111yescharacteristic
118Q16—7644no—
119C2×C8—22no—
120C4⋊C4—3822no—
121C4⋊C4—3821no—
122C4⋊C4—3822no—
123C4⋊C4—1912no—
124C22⋊C4—22no—
125C22⋊C4—3822no—
126C22⋊C4—3822no—
127C22⋊C4—7642no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic76412yescomplemented
133C23C2×D76411yescharacteristic
134Q8—3824no—
135Q8—15281no—
136Q8—7642no—
137Q8—7642no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—3822no—
142C2×C4—3822no—
143C2×C4—3822no—
144C2×C4—15281no—
145C2×C4—15281no—
146C2×C4—7641no—
147C2×C4—7641no—
148C2×C4—7641no—
149C2×C4—7641no—
150C2×C4—7641no—
151C2×C4D4×D19112yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#2025929211yescharacteristic
161C22C2×Dic76411yescharacteristic
162C22#2025935511yescharacteristic
163C4—22no—
164C4—41no—
165C4—15281no—
166C4—7642no—
167C4—7642no—
168C4—7642no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2713282111yescharacteristic
173C2#2713284311yescharacteristic
174C2#2713285111yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#323×C191C2
socleC22×C191#120
prime radicalC24×C191D4
hypercenter#323D191
hypocenterC191#1602
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27132821C2semidirect
4. #27132401C2semidirect
58. C23×C191Q16semidirect
87. C191#1602semidirect
132. C23Dic764semidirect

Maximal subgroups

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

subgroupindex
#271332892
#271328212
#271325752
#271324012
#323×C1912
#1602191