Finite Group Databasebeta

← #44509441

Subgroups — #44509441

Order 30848.

246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 45 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#29507906C212yesmaximal, complemented
2#29507884C211yesmaximal, characteristic
3#29507510C211yesmaximal, characteristic, complemented
4#29507507C212yesmaximal, complemented
5#559×C241C211yesmaximal, characteristic, complemented
6#21440687C2211yescharacteristic
7#21440668—21no—
8#21440625—22no—
9#21440570—21no—
10#21440559C2211yescharacteristic
11#21440557C2212yes—
12#21440556—22no—
13#127×C241C2211yescharacteristic
14#124×C241—22no—
15#123×C241—21no—
16#123×C241C2212yes—
17C4×Q8×C241—21no—
18C4×D4×C241—22no—
19#16838770—21no—
20#16838769—41no—
21C2×C22⋊D241—22no—
22D4×D241—41no—
23#16838758—41no—
24#16838755—41no—
25#16838754—21no—
26#16838749—42no—
27C22⋊Dic241—22no—
28#16838747—42no—
29C2×Q8×C241—21no—
30C2×D4×C241—22no—
31C2×D4×C241D411yescharacteristic
32C2×D4×C241C2311yescharacteristic
33C22×C4×C241—21no—
34C22×C4×C241D412yes—
35C4⋊C4×C241—21no—
36C4⋊C4×C241—22no—
37C22⋊C4×C241—22no—
38C22⋊C4×C241—21no—
39C22⋊C4×C241—22no—
40C22⋊C4×C241D412yes—
41C42×C241—22no—
42C42×C241D411yescharacteristic
43C22×D241—22no—
44Dic482—21no—
45D964—21no—
46C22⋊D241—42no—
47C22⋊D241—41no—
48C2×Dic241—22no—
49C2×Dic241—42no—
50C4×D241—41no—
51#14515349—41no—
52C23×C241—21no—
53C23×C241C2×D412yes—
54Q8×C241—42no—
55D4×C241—41no—
56D4×C241—41no—
57D4×C241—44no—
58C2×C4×C241—21no—
59C2×C4×C241—22no—
60C2×C4×C241—22no—
61C2×C4×C241—42no—
62C2×C4×C241—81no—
63C2×C4×C241C2×D411yescharacteristic
64C22×C241—22no—
65C22×C241—42no—
66C22×C241—42no—
67C22×C241—41no—
68C22×C241#12011yescharacteristic
69C2×D241—41no—
70C2×D241—82no—
71C4×C241—22no—
72C4×C241—21no—
73C4×C241—42no—
74C4×C241—42no—
75Dic241—41no—
76Dic241—82no—
77C2×C241—21no—
78C2×C241—42no—
79C2×C241—81no—
80C2×C241B(4,𝔽2)11yescharacteristic
81D241—81no—
82C241#219811yescharacteristic, complemented, Sylow
83#2198—2411nomaximal, Sylow
84#559D24111yescharacteristic, complemented
85#458—2412no—
86#453—2411no—
87#353—2411no—
88#352—2412no—
89H(5,𝔽2)—2411no—
90AGL(1,ℤ/8)—4821no—
91#127C2×D24111yescharacteristic
92#124—22no—
93#123—21no—
94#123—4822no—
95#123C2×D24112yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—4821no—
99#100—2411no—
100#99—4822no—
101#99—2412no—
102#41—9641no—
103#41—4821no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—9641no—
107C2×D4—4822no—
108C2×D4C22⋊D24111yescharacteristic
109C2×D4C22×D24111yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D24112yes—
112SD16—9641no—
113D8—9641no—
114M16—4821no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—9642no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—9642no—
122C22⋊C4—4822no—
123C22⋊C4C22⋊D24112yes—
124C42—22no—
125C42C22⋊D24111yescharacteristic
126C23—21no—
127C23—4822no—
128C23C2×C22⋊D24112yes—
129Q8—42no—
130Q8—4821no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—9642no—
135D4—9641no—
136D4—4821no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—9641no—
143C2×C4—9642no—
144C2×C4—4822no—
145C2×C4C2×C22⋊D24111yescharacteristic
146C8—9641no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—19282no—
152C22—9641no—
153C22#2144061111yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—19282no—
159C4—9641no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—19281no—
164C2#2950790411yescharacteristic
subgroupG/H
centerC2#29507904
commutator, cosocleC2×D4×C241C23
FrattiniC2×D4C22×D241
Fitting#559×C241C2
socleC2×C241B(4,𝔽2)
prime radical#29507884C2
hypercenter#559D241
hypocenterC241#2198
LCSG ▷ C2×D4×C241 ▷ C22×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #29507906C2semidirect
3. #29507510C2semidirect
4. #29507507C2semidirect
5. #559×C241C2semidirect
82. C241#2198semidirect
84. #559D241semidirect

Maximal subgroups

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

subgroupindex
#295079062
#295078842
#295075102
#295075072
#559×C2412
#2198241