Finite Group Databasebeta

← #37571491

Subgroups — #37571491

Order 24448.

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

normal subgroups only

This group has 282 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#27133244C211yesmaximal, characteristic
2#27132804C212yesmaximal, complemented
3#27132583C211yesmaximal, characteristic, complemented
4#27132398C212yesmaximal, complemented
5#324×C191C211yesmaximal, characteristic, complemented
6C2×#15690287—21no—
7C22×Dic382C2211yescharacteristic
8C2×#15690276—22no—
9#20259302—22no—
10#20259297—22no—
11#20259296—21no—
12#20259296C2212yes—
13C2×Dic191⋊C4—21no—
14#20259247—22no—
15C191⋊#95—21no—
16C191⋊#95C2211yescharacteristic
17C2×C4⋊C4×C191C2211yescharacteristic
18C22⋊C8×C191C2212yes—
19#15690287—42no—
20#15690287—42no—
21C2×Dic382—21no—
22C2×Dic382—22no—
23C2×Dic382—42no—
24C2×D764—21no—
25C2×D764—21no—
26C2×D764D412yes—
27C2×C22⋊D191—22no—
28C22×Dic191—21no—
29C22×Dic191—21no—
30C2×C4×D191—22no—
31#15690276—44no—
32C4⋊Dic191—21no—
33C4⋊Dic191D412yes—
34Dic191⋊C4—42no—
35(C2×D191)⋊C4—22no—
36(C2×D191)⋊C4—22no—
37C22×C4×C191D411yescharacteristic
38C22×C4×C191D411yescharacteristic
39C22×C4×C191C2311yescharacteristic
40C2×C8×C191—22no—
41C4⋊C4×C191—22no—
42C22×D191—22no—
43Dic382—24no—
44Dic382—42no—
45Dic382—42no—
46D764—42no—
47D764—42no—
48D764—81no—
49C22⋊D191—42no—
50C22⋊D191—42no—
51C2×Dic191—22no—
52C2×Dic191—22no—
53C2×Dic191—22no—
54C2×Dic191—41no—
55C2×Dic191—41no—
56C2×Dic191—41no—
57C2×Dic191—41no—
58C4×D191—44no—
59C23×C191C2×D411yescharacteristic
60C2×C4×C191—21no—
61C2×C4×C191—21no—
62C2×C4×C191—21no—
63C2×C4×C191—41no—
64C2×C4×C191—41no—
65C2×C4×C191SD1612yescomplemented
66C2×C4×C191C2×D412yes—
67C8×C191—42no—
68C22×C191—21no—
69C22×C191—21no—
70C22×C191#12011yescharacteristic
71C22×C191C2×SD1611yescharacteristic
72C22×C191AGL(1,ℤ/8)11yescharacteristic
73C2×D191—42no—
74C2×D191—42no—
75C2×D191—42no—
76C4×C191—22no—
77C4×C191—22no—
78C4×C191—41no—
79Dic191—42no—
80Dic191—42no—
81Dic191—42no—
82C2×C191—21no—
83C2×C191—21no—
84C2×C191#45011yescharacteristic
85C2×C191#45611yescharacteristic
86C2×C191B(4,𝔽2)11yescharacteristic
87D191—82no—
88C191#160411yescharacteristic, complemented, Sylow
89#1604—1911nomaximal, Sylow
90#537—1911no—
91#448—1912no—
92#395—1911no—
93#327—1912no—
94#324D19111yescharacteristic, complemented
95C2×#41—3821no—
96C22×Q8—1911no—
97C2×SD16—3822no—
98C22⋊Q8—3822no—
99#121—3822no—
100#121—3821no—
101#121—1912no—
102C2×C4⋊C4—3821no—
103C2×C4⋊C4C2×D19111yescharacteristic
104#103—3822no—
105C22⋊C8C2×D19112yes—
106#95—3821no—
107#95—1911no—
108#41—7642no—
109#41—7642no—
110C2×Q8—3821no—
111C2×Q8—3822no—
112C2×Q8—7642no—
113C2×D4—3821no—
114C2×D4—3821no—
115C2×D4—3822no—
116C2×D4—1912no—
117C22×C4—3822no—
118C22×C4—3821no—
119C22×C4—3821no—
120C22×C4D76411yescharacteristic
121C22×C4D76411yescharacteristic
122C22×C4C22×D19111yescharacteristic
123SD16—7644no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—3821no—
127C4⋊C4—1912no—
128C4⋊C4—7642no—
129C22⋊C4—3822no—
130C22⋊C4—3822no—
131C23—3822no—
132C23C2×D76411yescharacteristic
133Q8—3824no—
134Q8—7642no—
135Q8—7642no—
136D4—15281no—
137D4—7642no—
138D4—7642no—
139D4—7642no—
140D4—7642no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—3822no—
147C2×C4—3822no—
148C2×C4—3822no—
149C2×C4—7644no—
150C2×C4—7641no—
151C2×C4—7641no—
152C2×C4—7641no—
153C2×C4—7641no—
154C2×C4#1569027612yescomplemented
155C2×C4D4×D19112yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—7642no—
160C22—7642no—
161C22—7642no—
162C22#2025929211yescharacteristic
163C22C2×#1569027611yescharacteristic
164C22#2025935011yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—7642no—
169C4—7642no—
170C4—7642no—
171C2—21no—
172C2—21no—
173C2—15282no—
174C2#2713281611yescharacteristic
175C2#2713284811yescharacteristic
176C2#2713285111yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#324×C191C2
socleC22×C191#120
prime radical#27133244C2
hypercenter#324D191
hypocenterC191#1604
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27132804C2semidirect
3. #27132583C2semidirect
4. #27132398C2semidirect
5. #324×C191C2semidirect
65. C2×C4×C191SD16semidirect
88. C191#1604semidirect
94. #324D191semidirect
154. C2×C4#15690276semidirect

Maximal subgroups

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

subgroupindex
#271332442
#271328042
#271325832
#271323982
#324×C1912
#1604191