Finite Group Databasebeta

← #37572919

Subgroups — #37572919

Order 24448.

444 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 97 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#27133138C211yesmaximal, characteristic, complemented
2#27132668C212yesmaximal, complemented
3C2×#20259239C211yesmaximal, characteristic, complemented
4#27132423C212yesmaximal, complemented
5C2×#127×C191C211yesmaximal, characteristic
6C2×#15690271C2212yes—
7D4×Dic191—24no—
8#20259281C2212yes—
9C2×C22⋊Dic191—22no—
10#20259243—24no—
11#20259239—22no—
12C22×D4×C191—22no—
13C22×D4×C191C2211yescharacteristic
14C22×D4×C191C2211yescharacteristic
15#127×C191—22no—
16#127×C191C414yescomplemented
17C2×C42×C191C2211yescharacteristic
18C22×Dic191—24no—
19#15690271—22no—
20#15690271—22no—
21C2×C191⋊C8—22no—
22C4⋊Dic191D414yescomplemented
23C22⋊Dic191—24no—
24C22⋊Dic191—22no—
25C4×Dic191—22no—
26C24×C191—21no—
27C24×C191D412yes—
28C2×D4×C191—28no—
29C2×D4×C191—22no—
30C2×D4×C191—22no—
31C2×D4×C191—21no—
32C2×D4×C191—22no—
33C2×D4×C191—22no—
34C2×D4×C191—48no—
35C2×D4×C191C2×C412yes—
36C2×D4×C191C2×C412yescomplemented
37C2×D4×C191D412yes—
38C22×C4×C191—21no—
39C22×C4×C191C2311yescharacteristic
40C42×C191—21no—
41C42×C191C2×C412yescomplemented
42C2×Dic191—24no—
43C2×Dic191—84no—
44C191⋊C8—42no—
45C23×C191—22no—
46C23×C191—22no—
47C23×C191—22no—
48C23×C191—44no—
49C23×C191—44no—
50C23×C191—41no—
51C23×C191—41no—
52C23×C191C22⋊C414yes—
53C23×C191C2×D411yescharacteristic
54D4×C191—24no—
55D4×C191—24no—
56D4×C191—24no—
57D4×C191—416no—
58D4×C191—42no—
59C2×C4×C191—24no—
60C2×C4×C191—22no—
61C2×C4×C191C22×C411yescharacteristic
62C2×C4×C191C2×D412yes—
63C2×C4×C191C2×D411yescharacteristic
64C2×C4×C191#4112yes—
65C22×C191—22no—
66C22×C191—24no—
67C22×C191—44no—
68C22×C191—44no—
69C22×C191—44no—
70C22×C191—42no—
71C22×C191—82no—
72C22×C191—82no—
73C22×C191—82no—
74C22×C191C2×C22⋊C411yescharacteristic
75C22×C191#12011yescharacteristic
76C22×C191#12311yescharacteristic
77C4×C191—24no—
78C4×C191C4×D412yes—
79C4×C191#12112yes—
80Dic191—44no—
81C2×C191—21no—
82C2×C191—21no—
83C2×C191—44no—
84C2×C191—82no—
85C2×C191#38611yescharacteristic
86C2×C191#45312yes—
87C191#188711yescharacteristic, complemented, Sylow
88#1887—1911nomaximal, Sylow
89C2×#127D19111yescharacteristic
90#518—1911no—
91#418—1912no—
92C2×#100—1911no—
93#339—1912no—
94C22×D4—22no—
95C22×D4C2×D19111yescharacteristic
96C22×D4C2×D19111yescharacteristic
97C2×M16—1912no—
98#127—22no—
99#127Dic19114yescomplemented
100C4×D4—3824no—
101#117—1912no—
102C2×C22⋊C4—3822no—
103C2×C42C2×D19111yescharacteristic
104#102—3824no—
105#100—3822no—
106C24—21no—
107C24C22⋊D19112yes—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—48no—
115C2×D4C2×Dic19112yes—
116C2×D4C2×Dic19112yescomplemented
117C2×D4C22⋊D19112yes—
118C22×C4—21no—
119C22×C4—3824no—
120C22×C4C22×D19111yescharacteristic
121M16—3822no—
122M16—3822no—
123C2×C8—3822no—
124C4⋊C4—1914no—
125C22⋊C4—3824no—
126C22⋊C4—3822no—
127C42—21no—
128C42—3822no—
129C42C2×Dic19112yescomplemented
130C23—22no—
131C23—22no—
132C23—22no—
133C23—44no—
134C23—44no—
135C23—41no—
136C23—41no—
137C23C22⋊Dic19114yes—
138C23C2×C22⋊D19111yescharacteristic
139D4—24no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—3824no—
147C2×C4—15284no—
148C2×C4C22×Dic19111yescharacteristic
149C2×C4D4×D19112yes—
150C2×C4C2×C22⋊D19111yescharacteristic
151C2×C4#1569028612yes—
152C8—7642no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—44no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22C2×C22⋊Dic19111yescharacteristic
163C22#2025929011yescharacteristic
164C22#2025930311yescharacteristic
165C4—24no—
166C4—7644no—
167C4D4×Dic19112yes—
168C4#2025929412yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—82no—
173C2#2713254311yescharacteristic
174C2#2713283012yes—
subgroupG/H
centerC22#20259290
commutatorC2×C4×C191C22×C4
FrattiniC22×C4C22×D191
Fitting, prime radicalC2×#127×C191C2
socleC22×C191#120
cosocleC22×C4×C191C23
hypercenterC2×#127D191
hypocenterC191#1887
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #27133138C2semidirect
2. #27132668C2semidirect
3. C2×#20259239C2semidirect
4. #27132423C2semidirect
16. #127×C191C4semidirect
22. C4⋊Dic191D4semidirect
36. C2×D4×C191C2×C4semidirect
41. C42×C191C2×C4semidirect
87. C191#1887semidirect
99. #127Dic191semidirect
116. C2×D4C2×Dic191semidirect
129. C42C2×Dic191semidirect

Maximal subgroups

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

subgroupindex
#271331382
#271326682
C2×#202592392
#271324232
C2×#127×C1912
#1887191