Finite Group Databasebeta

← #44509458

Subgroups — #44509458

Order 30848.

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

normal subgroups only

This group has 226 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
2#29507894C211yesmaximal, characteristic
3#29507512C211yesmaximal, characteristic, complemented
4#29507507C212yesmaximal, complemented
5#563×C241C211yesmaximal, characteristic, complemented
6#21440687C2211yescharacteristic
7#21440673—21no—
8#21440625—22no—
9#21440570—21no—
10#21440559C2211yescharacteristic
11#21440557C2212yes—
12#21440556—22no—
13C4⋊Q8×C241—21no—
14#126×C241—22no—
15#125×C241—21no—
16#124×C241C2211yescharacteristic
17#123×C241C2212yes—
18C22⋊Q8×C241—22no—
19#16838770—21no—
20#16838769—41no—
21C2×C22⋊D241—22no—
22D4×D241—41no—
23#16838760—41no—
24#16838757—41no—
25#16838754—21no—
26#16838749—42no—
27C22⋊Dic241—22no—
28#16838747—42no—
29C2×Q8×C241—21no—
30C2×Q8×C241D411yescharacteristic
31C2×D4×C241C2311yescharacteristic
32C22×C4×C241D412yes—
33C4⋊C4×C241—21no—
34C4⋊C4×C241—22no—
35C4⋊C4×C241—22no—
36C4⋊C4×C241—21no—
37C4⋊C4×C241—22no—
38C22⋊C4×C241—22no—
39C22⋊C4×C241—22no—
40C22⋊C4×C241D412yes—
41C42×C241—21no—
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×C241C2×D412yes—
53Q8×C241—41no—
54Q8×C241—81no—
55D4×C241—41no—
56C2×C4×C241—21no—
57C2×C4×C241—22no—
58C2×C4×C241—21no—
59C2×C4×C241—22no—
60C2×C4×C241—42no—
61C2×C4×C241C2×D411yescharacteristic
62C22×C241—22no—
63C22×C241—42no—
64C22×C241#12011yescharacteristic
65C2×D241—41no—
66C2×D241—82no—
67C4×C241—21no—
68C4×C241—42no—
69C4×C241—41no—
70C4×C241—42no—
71C4×C241—41no—
72Dic241—41no—
73Dic241—82no—
74C2×C241—21no—
75C2×C241—42no—
76C2×C241B(4,𝔽2)11yescharacteristic
77D241—81no—
78C241#220111yescharacteristic, complemented, Sylow
79#2201—2411nomaximal, Sylow
80#563D24111yescharacteristic, complemented
81#458—2412no—
82#455—2411no—
83#354—2411no—
84#352—2412no—
85H(5,𝔽2)—2411no—
86#137—4821no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D24111yescharacteristic
91#123—4822no—
92#123C2×D24112yes—
93C22⋊Q8—22no—
94#104—4821no—
95#100—2411no—
96#99—4822no—
97#99—2412no—
98#41—9641no—
99#41—4821no—
100C2×Q8—21no—
101C2×Q8C22⋊D24111yescharacteristic
102C2×D4—9641no—
103C2×D4—4822no—
104C2×D4C22×D24111yescharacteristic
105C22×C4C22⋊D24112yes—
106Q16—9641no—
107SD16—9641no—
108M16—4821no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—9642no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—9642no—
118C22⋊C4—4822no—
119C22⋊C4C22⋊D24112yes—
120C42—21no—
121C42C22⋊D24111yescharacteristic
122C23—4822no—
123C23C2×C22⋊D24112yes—
124Q8—41no—
125Q8—81no—
126Q8—4821no—
127D4—41no—
128D4—9642no—
129D4—9641no—
130D4—4821no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—9641no—
137C2×C4—9642no—
138C2×C4—4822no—
139C2×C4C2×C22⋊D24111yescharacteristic
140C8—9641no—
141C22—22no—
142C22—42no—
143C22—19282no—
144C22—9641no—
145C22#2144061111yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—19282no—
152C4—9641no—
153C2—21no—
154C2—42no—
155C2—19281no—
156C2#2950790411yescharacteristic
subgroupG/H
centerC2#29507904
commutator, cosocleC2×D4×C241C23
FrattiniC2×D4C22×D241
Fitting#563×C241C2
socleC2×C241B(4,𝔽2)
prime radical#21440687C22
hypercenter#563D241
hypocenterC241#2201
LCSG ▷ C2×D4×C241 ▷ C22×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #29507512C2semidirect
4. #29507507C2semidirect
5. #563×C241C2semidirect
78. C241#2201semidirect
80. #563D241semidirect

Maximal subgroups

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

subgroupindex
#295079062
#295078942
#295075122
#295075072
#563×C2412
#2201241