Finite Group Databasebeta

← #37571774

Subgroups — #37571774

Order 24448.

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

normal subgroups only

This group has 200 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#27133383C211yesmaximal, characteristic
2#27133066C211yesmaximal, characteristic, complemented
3#27132945C211yesmaximal, characteristic, complemented
4#27132410C211yesmaximal, characteristic, complemented
5#27132393C211yesmaximal, characteristic, complemented
6D4⋊(C191⋊C8)C211yesmaximal, characteristic
7#461×C191C211yesmaximal, characteristic
8#20259322C2211yescharacteristic
9#20259315—22no—
10#20259309—21no—
11#20259301—21no—
12C4×Dic382C2211yescharacteristic
13#20259287—21no—
14#20259281—21no—
15#20259255—22no—
16#20259253—21no—
17C4⋊(C191⋊C8)C2211yescharacteristic
18#20259245—21no—
19C4×C191⋊C8C2211yescharacteristic
20C2×SD16×C191—21no—
21C4⋊Q8×C191C2211yescharacteristic
22C4×D4×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24#103×C191—21no—
25C2×Dic382—21no—
26C2×Dic382D411yescharacteristic
27C2×C191⋊C8—21no—
28C2×C191⋊C8—21no—
29C4⋊Dic191—21no—
30C4⋊Dic191D411yescharacteristic
31C4⋊Dic191D412yes—
32Dic191⋊C4—22no—
33Dic191⋊C4—21no—
34C22⋊Dic191—22no—
35C22⋊Dic191—22no—
36C4×Dic191—22no—
37C4×Dic191—21no—
38C2×Q8×C191—21no—
39C2×D4×C191D411yescharacteristic
40C22×C4×C191—21no—
41SD16×C191—42no—
42C2×C8×C191—21no—
43C4⋊C4×C191—21no—
44C4⋊C4×C191—21no—
45C4⋊C4×C191D411yescharacteristic
46C22⋊C4×C191—21no—
47C42×C191C2311yescharacteristic
48Dic382—21no—
49Dic382—21no—
50Dic382—41no—
51Dic382—41no—
52C2×Dic191—21no—
53C2×Dic191—21no—
54C2×Dic191—22no—
55C191⋊C8—22no—
56C191⋊C8—41no—
57C23×C191—21no—
58Q8×C191—41no—
59Q8×C191—41no—
60D4×C191—21no—
61D4×C191—21no—
62C2×C4×C191—21no—
63C2×C4×C191—41no—
64C2×C4×C191—41no—
65C2×C4×C191—41no—
66C2×C4×C191C2×D411yescharacteristic
67C2×C4×C191C2×D411yescharacteristic
68C2×C4×C191C2×D411yescharacteristic
69C8×C191—41no—
70C22×C191—41no—
71C22×C191—41no—
72C22×C191—41no—
73C22×C191#12011yescharacteristic
74C4×C191—21no—
75C4×C191—41no—
76C4×C191—81no—
77C4×C191#13511yescharacteristic
78C4×C191#13511yescharacteristic
79C4×C191AGL(1,ℤ/8)11yescharacteristic
80C4×C191#13711yescharacteristic
81Dic191—42no—
82Dic191—41no—
83Dic191—41no—
84C2×C191—81no—
85C2×C191#44911yescharacteristic
86C2×C191#45211yescharacteristic
87C2×C191#45611yescharacteristic
88C191#165011yescharacteristic, complemented, Sylow
89#1650—1911nomaximal, Sylow
90#556—1911no—
91#500—1911no—
92#475—1911no—
93#461D19111yescharacteristic
94#332—1911no—
95Q8⋊C8—1911no—
96D4⋊C8—1911no—
97C2×SD16—21no—
98C4⋊Q8—1911no—
99C4⋊Q8C2×D19111yescharacteristic
100#126—3822no—
101#124—3821no—
102C22⋊Q8—3821no—
103C4×Q8—3821no—
104C4×Q8—1911no—
105C4×D4C2×D19111yescharacteristic
106#117—3821no—
107#107—3822no—
108#106—3821no—
109C4⋊C8—1911no—
110C4⋊C8C2×D19111yescharacteristic
111#103—21no—
112#103—3821no—
113C4×C8—1911no—
114C2×Q8—21no—
115C2×Q8—3821no—
116C2×Q8—1911no—
117C2×D4C22⋊D19111yescharacteristic
118C22×C4—21no—
119SD16—42no—
120C2×C8—21no—
121C2×C8—3821no—
122C2×C8—3821no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—3822no—
126C4⋊C4—3821no—
127C4⋊C4—3821no—
128C4⋊C4—1911no—
129C4⋊C4—1912no—
130C4⋊C4C22⋊D19111yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—3822no—
133C22⋊C4—3822no—
134C42—3822no—
135C42—3821no—
136C42C22×D19111yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—3821no—
141Q8—3821no—
142Q8—7641no—
143Q8—7641no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—3821no—
151C2×C4—3821no—
152C2×C4—3822no—
153C2×C4D4×D19111yescharacteristic
154C2×C4D4×D19111yescharacteristic
155C2×C4C2×C22⋊D19111yescharacteristic
156C8—41no—
157C8—3822no—
158C8—7641no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2025929011yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—7642no—
167C4—7641no—
168C4—7641no—
169C4#2025934211yescharacteristic
170C4#2025934311yescharacteristic
171C4#2025934811yescharacteristic
172C4#2025935111yescharacteristic
173C2—81no—
174C2#2713280611yescharacteristic
175C2#2713282211yescharacteristic
176C2#2713284411yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#461×C191C2
socleC22×C191#120
prime radicalC2×D4×C191D4
hypercenter#461D191
hypocenterC191#1650
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #27133066C2semidirect
3. #27132945C2semidirect
4. #27132410C2semidirect
5. #27132393C2semidirect
88. C191#1650semidirect

Maximal subgroups

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

subgroupindex
#271333832
#271330662
#271329452
#271324102
#271323932
D4⋊(C191⋊C8)2
#461×C1912
#1650191