Finite Group Databasebeta

← #16777712

Subgroups — #16777712

Order 3328.

356 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 38 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#13129762C212yesmaximal, complemented
2#13114754C211yesmaximal, characteristic
3C22×C13×#41C411yescharacteristic
4C2×#88074—22no—
5#11718872C411yescharacteristic
6#11718872C2211yescharacteristic
7#11718305—22no—
8C2×C13×#41—41no—
9C2×C13×#41—41no—
10C2×C13×#41—44no—
11C22×Q8×C13C811yescharacteristic, complemented
12C22×D4×C13—21no—
13C22×D4×C13C811yescharacteristic, complemented
14C23×C4×C13—21no—
15C23×C4×C13C2×C411yescharacteristic
16C2×C22⋊Dic13—21no—
17C2×C22⋊Dic13D412yes—
18C22⋊#1057—42no—
19#88074—44no—
20#88071—41no—
21#88071—41no—
22#7339—131nomaximal, Sylow
23C24×C13—21no—
24C24×C13C22⋊C411yescharacteristic
25C13×#41—44no—
26C13×#41—44no—
27C13×#41—44no—
28C13×#41—44no—
29C2×Q8×C13—42no—
30C2×Q8×C13—41no—
31C2×D4×C13—24no—
32C2×D4×C13—41no—
33C2×D4×C13—42no—
34C2×D4×C13—41no—
35C2×D4×C13—41no—
36C2×D4×C13—42no—
37C22×C4×C13—21no—
38C22×C4×C13—21no—
39C22×C4×C13—21no—
40C22×C4×C13—42no—
41C22×C4×C13—42no—
42C22×C4×C13—42no—
43C22×C4×C13—42no—
44C22×C4×C13C2×C811yescharacteristic
45C22×C4×C13M1611yescharacteristic
46C22×Dic13—22no—
47C22×Dic13—41no—
48C2×#1057—82no—
49C22⋊Dic13—42no—
50C22⋊Dic13—42no—
51#1782—131no—
52#1317—132no—
53C23×C13—22no—
54C23×C13—22no—
55C23×C13—41no—
56C23×C13—41no—
57C23×C13—81no—
58C23×C13—82no—
59C23×C13C22⋊C811yescharacteristic
60C23×C13#9911yescharacteristic
61C23×C13#10011yescharacteristic
62Q8×C13—42no—
63Q8×C13—42no—
64D4×C13—44no—
65D4×C13—42no—
66D4×C13—44no—
67D4×C13—42no—
68C2×C4×C13—22no—
69C2×C4×C13—21no—
70C2×C4×C13—21no—
71C2×C4×C13—41no—
72C2×C4×C13—42no—
73C2×C4×C13—48no—
74C2×C4×C13—41no—
75C2×C4×C13—42no—
76C2×C4×C13—41no—
77C2×C4×C13—41no—
78C2×Dic13—42no—
79C2×Dic13—82no—
80C2×Dic13—81no—
81C2×Dic13—82no—
82#1057—162no—
83C22×#41#26111yescharacteristic
84C2×#99—262no—
85#381—131no—
86#381—131no—
87#323—262no—
88C22×C13—21no—
89C22×C13—21no—
90C22×C13—41no—
91C22×C13—41no—
92C22×C13—41no—
93C22×C13—42no—
94C22×C13—82no—
95C22×C13—82no—
96C22×C13—82no—
97C22×C13—82no—
98C22×C13#32311yescharacteristic
99C22×C13AΣL(1,𝔽16)12yes—
100C4×C13—41no—
101C4×C13—42no—
102C4×C13—41no—
103Dic13—82no—
104Dic13—161no—
105C2×#41—41no—
106C2×#41—41no—
107C2×#41—44no—
108C22×Q8#105711yescharacteristic, complemented
109C22×D4—21no—
110C22×D4#105711yescharacteristic, complemented
111C23×C4—21no—
112C23×C4C2×#26111yescharacteristic
113C2×C22⋊C4—132no—
114C2×C22⋊C4—261no—
115#99—524no—
116C22⋊C8—522no—
117#95—521no—
118#95—521no—
119C2×C13—22no—
120C2×C13—21no—
121C2×C13—41no—
122C2×C13—41no—
123C2×C13—82no—
124C2×C13#131611yescharacteristic
125C24—21no—
126C24C22⋊#26111yescharacteristic
127#41—44no—
128#41—44no—
129#41—44no—
130#41—44no—
131C2×Q8—42no—
132C2×Q8—41no—
133C2×D4—24no—
134C2×D4—41no—
135C2×D4—42no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—42no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—262no—
147C22×C4—521no—
148C22×C4C2×#105711yescharacteristic
149C22×C4#612211yescharacteristic
150C2×C8—1042no—
151C22⋊C4—522no—
152C22⋊C4—522no—
153C13#733911yescharacteristic, complemented, Sylow
154C23—22no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—81no—
159C23—82no—
160C23#8811911yescharacteristic
161C23C22⋊#105711yescharacteristic
162C23#8812111yescharacteristic
163Q8—42no—
164Q8—42no—
165D4—44no—
166D4—42no—
167D4—42no—
168D4—44no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—48no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—522no—
180C2×C4—1042no—
181C2×C4—1041no—
182C2×C4—1042no—
183C8—2082no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22#1171830511yescharacteristic
195C22#1171830612yes—
196C4—41no—
197C4—42no—
198C4—41no—
199C4—1042no—
200C4—2081no—
201C2—22no—
202C2—21no—
203C2—41no—
204C2—41no—
205C2—82no—
206C2#1312976011yescharacteristic
subgroupG/H
centerC2#13129760
commutatorC22×C4×C13C2×C8
FrattiniC23×C4C2×#261
Fitting, prime radicalC22×C13×#41C4
socleC2×C13#1316
cosocle#11718872C22
hypercenterC22×#41#261
hypocenterC13#7339
LCSG ▷ C22×C4×C13 ▷ C23×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #13129762C2semidirect
11. C22×Q8×C13C8semidirect
13. C22×D4×C13C8semidirect
108. C22×Q8#1057semidirect
110. C22×D4#1057semidirect
153. C13#7339semidirect

Maximal subgroups

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

subgroupindex
#131297622
#131147542
#733913