Finite Group Databasebeta

← #28389613

Subgroups — #28389613

Order 14464.

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

normal subgroups only

This group has 246 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#20299179C211yesmaximal, characteristic
2#20298917C212yesmaximal, complemented
3#20298537C211yesmaximal, characteristic, complemented
4#20298333C212yesmaximal, complemented
5#324×C113C211yesmaximal, characteristic, complemented
6C2×#14268647—21no—
7#16813602—22no—
8#16813597—22no—
9#16813596—21no—
10#16813596C2212yes—
11C2×C4⋊Dic113C2211yescharacteristic
12#16813554—22no—
13#16813544—22no—
14#16813529—21no—
15#16813529—21no—
16#16813529C2211yescharacteristic
17C2×C4⋊C4×C113C2211yescharacteristic
18C22⋊C8×C113C2212yes—
19#14268647—42no—
20#14268647—42no—
21C2×Dic226—21no—
22C2×D452—21no—
23C2×D452—21no—
24C2×D452D412yes—
25C2×C22⋊D113—22no—
26C22×Dic113—21no—
27C22×Dic113—21no—
28C2×C4×D113—22no—
29C4⋊Dic113—21no—
30C4⋊Dic113—22no—
31C4⋊Dic113D412yes—
32#14268624—22no—
33#14268624—22no—
34C22×C4×C113D411yescharacteristic
35C22×C4×C113D411yescharacteristic
36C22×C4×C113C2311yescharacteristic
37C2×C8×C113—22no—
38C4⋊C4×C113—22no—
39C23×C113C2×D411yescharacteristic
40C22×D113—22no—
41C2×C4×C113—21no—
42C2×C4×C113—21no—
43C2×C4×C113—21no—
44C2×C4×C113—41no—
45C2×C4×C113—41no—
46C2×C4×C113SD1612yes—
47C2×C4×C113C2×D412yes—
48C22⋊D113—42no—
49C22⋊D113—42no—
50C2×Dic113—22no—
51C2×Dic113—41no—
52C2×Dic113—41no—
53C2×Dic113—42no—
54C2×Dic113—41no—
55C2×Dic113—41no—
56D452—42no—
57D452—42no—
58D452—81no—
59C4×D113—44no—
60Dic226—42no—
61C8×C113—42no—
62C22×C113—21no—
63C22×C113—21no—
64C22×C113#12011yescharacteristic
65C22×C113C2×SD1611yescharacteristic
66C22×C113AGL(1,ℤ/8)11yescharacteristic
67C2×D113—42no—
68C2×D113—42no—
69C2×D113—42no—
70C4×C113—22no—
71C4×C113—22no—
72C4×C113—41no—
73Dic113—42no—
74Dic113—81no—
75Dic113—81no—
76C2×C113—21no—
77C2×C113—21no—
78C2×C113#45011yescharacteristic
79C2×C113#45411yescharacteristic
80C2×C113#45811yescharacteristic
81D113—82no—
82#1616—1131nomaximal, Sylow
83C113#161611yescharacteristic, complemented, Sylow
84#537—1131no—
85#481—1132no—
86#400—1131no—
87#327—1132no—
88#324D11311yescharacteristic, complemented
89C2×#41—2261no—
90C22⋊Q8—2262no—
91#121—1132no—
92#121—2262no—
93#121—2261no—
94C2×C4⋊C4—1131no—
95C2×C4⋊C4C2×D11311yescharacteristic
96#106—2262no—
97#102—2262no—
98C22⋊C8C2×D11312yes—
99#95—1131no—
100#95—2261no—
101#95—2261no—
102#41—4522no—
103#41—4522no—
104C2×Q8—2261no—
105C2×D4—1132no—
106C2×D4—2261no—
107C2×D4—2261no—
108C2×D4—2262no—
109C22×C4—2262no—
110C22×C4—2261no—
111C22×C4—2261no—
112C22×C4D45211yescharacteristic
113C22×C4D45211yescharacteristic
114C22×C4C22×D11311yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—1132no—
118C4⋊C4—2261no—
119C4⋊C4—2262no—
120C22⋊C4—2262no—
121C22⋊C4—2262no—
122C23—2262no—
123C23C2×D45211yescharacteristic
124Q8—4522no—
125D4—9041no—
126D4—4522no—
127D4—4522no—
128D4—4522no—
129D4—4522no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—4524no—
136C2×C4—4521no—
137C2×C4—4521no—
138C2×C4—4522no—
139C2×C4—4521no—
140C2×C4—4521no—
141C2×C4—2262no—
142C2×C4#1426863612yes—
143C2×C4D4×D11312yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—4522no—
148C22—4522no—
149C22—4522no—
150C22#1681359211yescharacteristic
151C22C2×#1426863611yescharacteristic
152C22#1681365011yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—9041no—
157C4—9041no—
158C4—4522no—
159C2—21no—
160C2—21no—
161C2—9042no—
162C2#2029875111yescharacteristic
163C2#2029877311yescharacteristic
164C2#2029878911yescharacteristic
subgroupG/H
centerC22#16813592
commutator, cosocleC22×C4×C113C23
FrattiniC22×C4C22×D113
Fitting#324×C113C2
socleC22×C113#120
prime radical#20299179C2
hypercenter#324D113
hypocenterC113#1616
LCSG ▷ C22×C4×C113 ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20298917C2semidirect
3. #20298537C2semidirect
4. #20298333C2semidirect
5. #324×C113C2semidirect
83. C113#1616semidirect
88. #324D113semidirect

Maximal subgroups

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

subgroupindex
#202991792
#202989172
#202985372
#202983332
#324×C1132
#1616113