Finite Group Databasebeta

← #28362006

Subgroups — #28362006

Order 13952.

238 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 57 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#20295181C212yesmaximal
2#20294946C211yesmaximal, characteristic
3#20294763C212yesmaximal
4#20294752C211yesmaximal, characteristic
5#544×C109C211yesmaximal, characteristic
6C22×Dic218C2211yescharacteristic
7C2×#13182019—22no—
8C2×#13182008—21no—
9#16812622—22no—
10C22⋊#11724784C2212yes—
11#16812606—21no—
12#16812606C2211yescharacteristic
13C4⋊Q8×C109—22no—
14#125×C109—22no—
15C22⋊Q8×C109—21no—
16C22⋊Q8×C109C2212yes—
17#117×C109—21no—
18C2×C4⋊C4×C109C2211yescharacteristic
19C2×Dic218—22no—
20C2×Dic218—42no—
21C22×Dic109—21no—
22C22×Dic109—21no—
23#13182019—44no—
24C2×#11724784—22no—
25#13182008—42no—
26C2×Q8×C109—21no—
27C2×Q8×C109D412yes—
28C22×C4×C109D411yescharacteristic
29C22×C4×C109D411yescharacteristic
30C22×C4×C109C2311yescharacteristic
31C4⋊C4×C109—21no—
32C4⋊C4×C109—21no—
33C4⋊C4×C109—21no—
34C4⋊C4×C109—22no—
35C4⋊C4×C109—22no—
36C4⋊C4×C109—22no—
37C4⋊C4×C109D412yes—
38C22⋊C4×C109—22no—
39C42×C109—22no—
40C23×C109C2×D411yescharacteristic
41Q8×C109—42no—
42Q8×C109—81no—
43C2×C4×C109—21no—
44C2×C4×C109—21no—
45C2×C4×C109—22no—
46C2×C4×C109—22no—
47C2×C4×C109—21no—
48C2×C4×C109—41no—
49C2×C4×C109—41no—
50C2×C4×C109Q1612yescomplemented
51C2×C4×C109C2×D412yes—
52C2×Dic109—22no—
53C2×Dic109—22no—
54C2×Dic109—41no—
55C2×Dic109—41no—
56C2×Dic109—41no—
57C2×Dic109—41no—
58Dic218—24no—
59Dic218—42no—
60#11724784—42no—
61C22×C109—21no—
62C22×C109—21no—
63C22×C109#12011yescharacteristic
64C22×C109C2×Q1611yescharacteristic
65C22×C109#13711yescharacteristic
66C4×C109—22no—
67C4×C109—22no—
68C4×C109—42no—
69C4×C109—41no—
70C4×C109—42no—
71Dic109—42no—
72Dic109—42no—
73C2×C109—21no—
74C2×C109—21no—
75C2×C109#45111yescharacteristic
76C2×C109#45611yescharacteristic
77C2×C109B(4,𝔽2)11yescharacteristic
78#1605—1091nomaximal, Sylow
79C109#160511yescharacteristic, complemented, Sylow
80#544D10911yescharacteristic
81#451—1092no—
82#395—1091no—
83#328—1092no—
84#324—1091no—
85C22×Q8—1091no—
86C2×Q16—2182no—
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—21no—
90C22⋊Q8C2×D10912yes—
91#117—21no—
92C2×C4⋊C4—2181no—
93C2×C4⋊C4C2×D10911yescharacteristic
94#103—2182no—
95C22⋊C8—1092no—
96#95—1091no—
97#95—2181no—
98C2×Q8—21no—
99C2×Q8—4362no—
100C2×Q8—2182no—
101C2×Q8C22⋊D10912yes—
102C22×C4—2181no—
103C22×C4—2181no—
104C22×C4C22⋊D10911yescharacteristic
105C22×C4C22⋊D10911yescharacteristic
106C22×C4C22×D10911yescharacteristic
107Q16—4364no—
108C2×C8—2182no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—4362no—
116C4⋊C4C22⋊D10912yes—
117C22⋊C4—22no—
118C42—22no—
119C23C2×C22⋊D10911yescharacteristic
120Q8—42no—
121Q8—81no—
122Q8—4362no—
123Q8—2184no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—4361no—
132C2×C4—4361no—
133C2×C4—4361no—
134C2×C4—4361no—
135C2×C4—2182no—
136C2×C4—2182no—
137C2×C4#1318201912yescomplemented
138C2×C4C2×C22⋊D10912yes—
139C8—4362no—
140C22—21no—
141C22—21no—
142C22#1681266811yescharacteristic
143C22C2×#1318201911yescharacteristic
144C22#1681273011yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—4362no—
151C4—4362no—
152C2—21no—
153C2—21no—
154C2#2029518211yescharacteristic
155C2#2029520911yescharacteristic
156C2#2029521311yescharacteristic
subgroupG/H
centerC22#16812668
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting#544×C109C2
socleC22×C109#120
prime radicalC23×C109C2×D4
hypercenter#544D109
hypocenterC109#1605
LCSG ▷ C22×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
50. C2×C4×C109Q16semidirect
79. C109#1605semidirect
137. C2×C4#13182019semidirect

Maximal subgroups

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

subgroupindex
#202951812
#202949462
#202947632
#202947522
#544×C1092
#1605109