Finite Group Databasebeta

← #2853×C241 #44504520

Subgroups — #2853×C241 #44504520

Order 30848.

582 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 216 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
1C2×#123×C241C211yesmaximal, characteristic
2C22×C42×C241C211yesmaximal, characteristic, complemented
3#388×C241C214yesmaximal, complemented
4#387×C241C214yesmaximal, complemented
5#386×C241C212yesmaximal
6#385×C241C212yesmaximal, complemented
7#381×C241C211yesmaximal, characteristic, complemented
8C22×D4×C241C2211yescharacteristic
9C23×C4×C241C2212yes—
10C23×C4×C241C2211yescharacteristic
11#123×C241—24no—
12C2×C4⋊C4×C241C2212yes—
13C2×C4⋊C4×C241C2214yes—
14C2×C22⋊C4×C241C2211yescharacteristic
15C2×C22⋊C4×C241C2212yes—
16C2×C22⋊C4×C241C2212yes—
17C2×C22⋊C4×C241C2214yes—
18C2×C42×C241—24no—
19C2×C42×C241C2214yes—
20#95×C241C2214yes—
21#95×C241C2214yes—
22#95×C241C2214yes—
23C24×C241C2311yescharacteristic
24C24×C241C2311yescharacteristic
25C2×D4×C241—22no—
26C2×D4×C241—24no—
27C22×C4×C241—24no—
28C22×C4×C241—24no—
29C22×C4×C241—28no—
30C22×C4×C241D414yes—
31C22×C4×C241C2312yes—
32C22×C4×C241C2314yes—
33C22×C4×C241C2311yescharacteristic
34C22×C4×C241C2312yes—
35C22×C4×C241C2314yes—
36C4⋊C4×C241—24no—
37C4⋊C4×C241—28no—
38C22⋊C4×C241—22no—
39C22⋊C4×C241—24no—
40C22⋊C4×C241—24no—
41C22⋊C4×C241—28no—
42C42×C241—28no—
43C23×C241—24no—
44C23×C241—21no—
45C23×C241—24no—
46C23×C241—22no—
47C23×C241C2×D412yes—
48C23×C241#4114yes—
49C23×C241C2411yescharacteristic
50D4×C241—44no—
51C2×C4×C241—216no—
52C2×C4×C241—28no—
53C2×C4×C241—24no—
54C2×C4×C241—24no—
55C2×C4×C241—24no—
56C2×C4×C241—22no—
57C2×C4×C241—21no—
58C2×C4×C241—24no—
59C2×C4×C241—24no—
60C2×C4×C241—24no—
61C2×C4×C241—24no—
62C2×C4×C241—22no—
63C2×C4×C241C2×D414yes—
64C2×C4×C241#4118yes—
65C22×C241—24no—
66C22×C241—28no—
67C22×C241—41no—
68C22×C241—42no—
69C22×C241—44no—
70C22×C241#12314yes—
71C22×C241C22×D411yescharacteristic
72C22×C241C2×#4112yes—
73C22×C241C2×#4114yes—
74C4×C241—24no—
75C4×C241—28no—
76C4×C241—41no—
77C4×C241—42no—
78C4×C241—44no—
79C2×C241—24no—
80C2×C241—81no—
81C2×C241C2×#12311yescharacteristic
82C2×C241#52512yes—
83C2×C241#52914yes—
84C241#285311yescharacteristic, direct product factor, Sylow
85#2853C24111yesmaximal, characteristic, direct product factor, Sylow
86C2×#123C2×C24111yescharacteristic
87C22×C42C2×C24111yescharacteristic, complemented
88#388C2×C24114yescomplemented
89#387C2×C24114yescomplemented
90#386C2×C24112yes—
91#385C2×C24112yescomplemented
92#381C2×C24111yescharacteristic, complemented
93C22×D4C22×C24111yescharacteristic
94C23×C4C22×C24112yes—
95C23×C4C22×C24111yescharacteristic
96#123—24no—
97C2×C4⋊C4C22×C24112yes—
98C2×C4⋊C4C22×C24114yes—
99C2×C22⋊C4C22×C24111yescharacteristic
100C2×C22⋊C4C22×C24112yes—
101C2×C22⋊C4C22×C24112yes—
102C2×C22⋊C4C22×C24114yes—
103C2×C42—24no—
104C2×C42C22×C24114yes—
105#95C22×C24114yes—
106#95C22×C24114yes—
107#95C22×C24114yes—
108C24C23×C24111yescharacteristic
109C24C23×C24111yescharacteristic
110C2×D4—22no—
111C2×D4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—28no—
115C22×C4D4×C24114yes—
116C22×C4C23×C24114yes—
117C22×C4C23×C24111yescharacteristic
118C22×C4C23×C24112yes—
119C22×C4C23×C24112yes—
120C22×C4C23×C24114yes—
121C4⋊C4—24no—
122C4⋊C4—28no—
123C22⋊C4—22no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—28no—
127C42—28no—
128C23—24no—
129C23—21no—
130C23—24no—
131C23—22no—
132C23C2×D4×C24112yes—
133C23#41×C24114yes—
134C23C24×C24111yescharacteristic
135D4—44no—
136C2×C4—216no—
137C2×C4—28no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4C2×D4×C24114yes—
149C2×C4#41×C24118yes—
150C22—24no—
151C22—28no—
152C22—41no—
153C22—42no—
154C22—44no—
155C22#123×C24114yes—
156C22C22×D4×C24111yescharacteristic
157C22C2×#41×C24112yes—
158C22C2×#41×C24114yes—
159C4—24no—
160C4—28no—
161C4—41no—
162C4—42no—
163C4—44no—
164C2—24no—
165C2—81no—
166C2C2×#123×C24111yescharacteristic
167C2#525×C24112yes—
168C2#529×C24114yes—
subgroupG/H
center, socleC23×C241C24
commutator, Frattini, cosocleC23C24×C241
prime radicalC22×D4×C241C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C241 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C42×C241C2semidirect
3. #388×C241C2semidirect
4. #387×C241C2semidirect
6. #385×C241C2semidirect
7. #381×C241C2semidirect
84. C241#2853direct (N × H)
85. #2853C241direct (N × H)
87. C22×C42C2×C241semidirect
88. #388C2×C241semidirect
89. #387C2×C241semidirect
91. #385C2×C241semidirect
92. #381C2×C241semidirect

Maximal subgroups

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

subgroupindex
C2×#123×C2412
C22×C42×C2412
#388×C2412
#387×C2412
#386×C2412
#385×C2412
#381×C2412
#2853241