Finite Group Databasebeta

← #16777888

Subgroups — #16777888

Order 3328.

228 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 36 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#13129779C211yesmaximal, characteristic, complemented
2#13129714C211yesmaximal, characteristic, complemented
3#13116057C211yesmaximal, characteristic
4C13×#537C411yescharacteristic
5C2×#88112—21no—
6C2×#88077—21no—
7#11718893—21no—
8#11718637C2211yescharacteristic
9#11718285—21no—
10#11718122C411yescharacteristic
11C2×C13×#41—21no—
12C22×#1057—41no—
13C13×C22⋊Q8—41no—
14C13×#121—41no—
15C13×#121—41no—
16C2×C13×C4⋊C4C2×C411yescharacteristic
17C2×C4⋊Dic13D411yescharacteristic
18C2×#6101—41no—
19C2×C4×Dic13D411yescharacteristic
20C2×#6099—21no—
21C22⋊#1057—41no—
22#88112—22no—
23#88112—41no—
24#88077—41no—
25#88077—41no—
26#88075—41no—
27#88072—41no—
28#7417—131nomaximal, Sylow
29C13×#41—41no—
30C13×#41—41no—
31C13×#41—41no—
32C13×#41—41no—
33C2×Q8×C13—21no—
34C2×D4×C13—21no—
35C2×D4×C13—41no—
36C2×D4×C13—41no—
37C22×C4×C13—21no—
38C22×C4×C13—41no—
39C22×C4×C13C22⋊C411yescharacteristic
40C22×Dic13—21no—
41C22×Dic13—21no—
42C2×#1057—42no—
43C2×#1057—81no—
44C2×#1057—81no—
45C2×#1057—81no—
46C13×C4⋊C4—41no—
47C13×C4⋊C4—41no—
48C13×C22⋊C4—42no—
49C4⋊Dic13—41no—
50#6101—81no—
51#6101—81no—
52C4×Dic13—21no—
53C4×Dic13D811yescharacteristic
54C4×Dic13SD1611yescharacteristic
55#6099—41no—
56#6099—41no—
57C2×#1055—41no—
58#2039—131no—
59#1325—131no—
60#1296—131no—
61C23×C13—41no—
62C23×C13#9911yescharacteristic
63Q8×C13—41no—
64Q8×C13—41no—
65D4×C13—41no—
66D4×C13—41no—
67D4×C13—81no—
68D4×C13—81no—
69D4×C13—161no—
70C2×C4×C13—21no—
71C2×C4×C13—21no—
72C2×C4×C13—41no—
73C2×C4×C13—41no—
74C2×C4×C13—81no—
75C2×C4×C13—81no—
76C2×C4×C13—81no—
77C2×C4×C13#10211yescharacteristic
78C2×C4×C13#10411yescharacteristic
79C2×Dic13—21no—
80C2×Dic13—21no—
81C2×Dic13—21no—
82C2×Dic13—21no—
83C2×Dic13—21no—
84C2×Dic13—21no—
85C2×Dic13—41no—
86C2×Dic13—81no—
87#1057—82no—
88#1057—161no—
89#1055—81no—
90#537#26111yescharacteristic
91C2×C4⋊C8—261no—
92C2×#104—261no—
93#417—261no—
94#384—131no—
95#343—131no—
96#324—261no—
97C22×C13—21no—
98C22×C13—21no—
99C22×C13—81no—
100C22×C13—81no—
101C22×C13—81no—
102C22×C13#32711yescharacteristic
103C22×C13AΣL(1,𝔽16)11yescharacteristic
104C22×C13#35511yescharacteristic
105C4×C13—21no—
106C4×C13—21no—
107C4×C13—81no—
108C4×C13—81no—
109Dic13—21no—
110Dic13—21no—
111Dic13—21no—
112Dic13—21no—
113Dic13—81no—
114C2×#41—21no—
115C2×M16—261no—
116C22×C8—521no—
117C22⋊Q8—41no—
118#121—41no—
119#121—41no—
120C2×C4⋊C4—131no—
121C2×C4⋊C4—521no—
122C2×C4⋊C4C2×#26111yescharacteristic
123C2×C42—131no—
124C4⋊C8—262no—
125C4⋊C8—521no—
126#104—521no—
127#104—521no—
128#103—521no—
129#102—521no—
130C22⋊C8—521no—
131C2×C13—21no—
132C2×C13—21no—
133C2×C13—161no—
134C2×C13#134511yescharacteristic
135C2×C13#140211yescharacteristic
136C2×C13#140511yescharacteristic
137#41—41no—
138#41—41no—
139#41—41no—
140#41—41no—
141C2×Q8—21no—
142C2×D4—21no—
143C2×D4—41no—
144C2×D4—41no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—261no—
148C22×C4—261no—
149C22×C4C22⋊#26111yescharacteristic
150M16—521no—
151M16—521no—
152C2×C8—521no—
153C2×C8—522no—
154C2×C8—1041no—
155C2×C8—1041no—
156C2×C8—1041no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—521no—
160C4⋊C4—1041no—
161C4⋊C4—1041no—
162C22⋊C4—42no—
163C42—131no—
164C42—131no—
165C42—261no—
166C13#741711yescharacteristic, complemented, Sylow
167C23—41no—
168C23#8811911yescharacteristic
169Q8—41no—
170Q8—41no—
171D4—41no—
172D4—41no—
173D4—81no—
174D4—81no—
175D4—161no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—261no—
184C2×C4—261no—
185C2×C4—261no—
186C2×C4—261no—
187C2×C4—261no—
188C2×C4—261no—
189C2×C4—521no—
190C2×C4—1041no—
191C2×C4#8811511yescharacteristic
192C2×C4#8811611yescharacteristic
193C8—1041no—
194C8—1042no—
195C8—2081no—
196C22—21no—
197C22—21no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22#1171829211yescharacteristic
202C22#1171829511yescharacteristic
203C22#1171830611yescharacteristic
204C4—21no—
205C4—21no—
206C4—81no—
207C4—81no—
208C4—261no—
209C4—261no—
210C4—261no—
211C4—261no—
212C4—1041no—
213C2—21no—
214C2—21no—
215C2—161no—
216C2#1312981811yescharacteristic
217C2#1312995511yescharacteristic
218C2#1312996111yescharacteristic
subgroupG/H
centerC22#11718306
commutatorC2×C13×C4⋊C4C2×C4
FrattiniC2×C4⋊C4C2×#261
Fitting, prime radicalC13×#537C4
socleC22×C13AΣL(1,𝔽16)
cosocle#11718637C22
hypercenter#537#261
hypocenterC13#7417
LCSG ▷ C2×C13×C4⋊C4 ▷ C22×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #13129779C2semidirect
2. #13129714C2semidirect
166. C13#7417semidirect

Maximal subgroups

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

subgroupindex
#131297792
#131297142
#131160572
#741713