Finite Group Databasebeta

← #37571496

Subgroups — #37571496

Order 24448.

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

normal subgroups only

This group has 238 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#27133294C211yesmaximal, characteristic
2#27132821C212yesmaximal
3#27132583C211yesmaximal, characteristic
4#27132401C212yesmaximal
5#324×C191C211yesmaximal, characteristic
6C22×Dic382C2211yescharacteristic
7C2×Dic764—22no—
8#20259321—22no—
9#20259313—22no—
10#20259301—21no—
11#20259301C2212yes—
12#20259281—21no—
13C2×Dic191⋊C4—21no—
14#20259247—22no—
15C191⋊#95—21no—
16C191⋊#95C2211yescharacteristic
17C2×C4⋊C4×C191C2211yescharacteristic
18C22⋊C8×C191C2212yes—
19C2×Dic382—21no—
20C2×Dic382—22no—
21C2×Dic382—42no—
22C2×Dic382D412yes—
23C22×Dic191—21no—
24C22×Dic191—21no—
25Dic764—44no—
26C4⋊Dic191—21no—
27C4⋊Dic191—21no—
28C4⋊Dic191—21no—
29C4⋊Dic191D412yes—
30Dic191⋊C4—22no—
31Dic191⋊C4—22no—
32Dic191⋊C4—42no—
33C22⋊Dic191—22no—
34C4×Dic191—22no—
35C22×C4×C191D411yescharacteristic
36C22×C4×C191D411yescharacteristic
37C22×C4×C191C2311yescharacteristic
38C2×C8×C191—22no—
39C4⋊C4×C191—22no—
40Dic382—24no—
41Dic382—42no—
42Dic382—42no—
43Dic382—81no—
44C2×Dic191—22no—
45C2×Dic191—22no—
46C2×Dic191—22no—
47C2×Dic191—22no—
48C2×Dic191—41no—
49C2×Dic191—41no—
50C2×Dic191—41no—
51C2×Dic191—41no—
52C23×C191C2×D411yescharacteristic
53C2×C4×C191—21no—
54C2×C4×C191—21no—
55C2×C4×C191—21no—
56C2×C4×C191—41no—
57C2×C4×C191—41no—
58C2×C4×C191Q1612yescomplemented
59C2×C4×C191C2×D412yes—
60C8×C191—42no—
61C22×C191—21no—
62C22×C191—21no—
63C22×C191#12011yescharacteristic
64C22×C191C2×Q1611yescharacteristic
65C22×C191#13711yescharacteristic
66C4×C191—22no—
67C4×C191—22no—
68C4×C191—41no—
69Dic191—42no—
70Dic191—42no—
71Dic191—42no—
72Dic191—42no—
73C2×C191—21no—
74C2×C191—21no—
75C2×C191#45111yescharacteristic
76C2×C191#45611yescharacteristic
77C2×C191B(4,𝔽2)11yescharacteristic
78C191#160511yescharacteristic, complemented, Sylow
79#1605—1911nomaximal, Sylow
80#544—1911no—
81#451—1912no—
82#395—1911no—
83#328—1912no—
84#324D19111yescharacteristic
85C22×Q8—1911no—
86C2×Q16—3822no—
87C4⋊Q8—3822no—
88#125—3822no—
89C22⋊Q8—3821no—
90C22⋊Q8—1912no—
91#117—3821no—
92C2×C4⋊C4—3821no—
93C2×C4⋊C4C2×D19111yescharacteristic
94#103—3822no—
95C22⋊C8C2×D19112yes—
96#95—3821no—
97#95—1911no—
98C2×Q8—3821no—
99C2×Q8—3822no—
100C2×Q8—1912no—
101C2×Q8—7642no—
102C22×C4—3821no—
103C22×C4—3821no—
104C22×C4D76411yescharacteristic
105C22×C4D76411yescharacteristic
106C22×C4C22×D19111yescharacteristic
107Q16—7644no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—3822no—
111C4⋊C4—3821no—
112C4⋊C4—3821no—
113C4⋊C4—3821no—
114C4⋊C4—3822no—
115C4⋊C4—1912no—
116C4⋊C4—7642no—
117C22⋊C4—3822no—
118C42—3822no—
119C23C2×D76411yescharacteristic
120Q8—3824no—
121Q8—15281no—
122Q8—7642no—
123Q8—7642no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—3822no—
130C2×C4—3822no—
131C2×C4—3822no—
132C2×C4—3822no—
133C2×C4—7641no—
134C2×C4—7641no—
135C2×C4—7641no—
136C2×C4—7641no—
137C2×C4Dic76412yescomplemented
138C2×C4D4×D19112yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#2025929211yescharacteristic
143C22C2×Dic76411yescharacteristic
144C22#2025935511yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—7642no—
149C4—7642no—
150C4—7642no—
151C4—7642no—
152C2—21no—
153C2—21no—
154C2#2713282111yescharacteristic
155C2#2713284811yescharacteristic
156C2#2713285111yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#324×C191C2
socleC22×C191#120
prime radicalC23×C191C2×D4
hypercenter#324D191
hypocenterC191#1605
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
58. C2×C4×C191Q16semidirect
78. C191#1605semidirect
137. C2×C4Dic764semidirect

Maximal subgroups

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

subgroupindex
#271332942
#271328212
#271325832
#271324012
#324×C1912
#1605191