Finite Group Databasebeta

← #16778559

Subgroups — #16778559

Order 3328.

624 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 118 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#13130752C212yesmaximal, complemented
2#13129652C214yesmaximal, complemented
3#127×Dic13C211yesmaximal, characteristic
4C2×C13×#127C411yescharacteristic
5C2×D4×Dic13—22no—
6C2×D4×Dic13C412yes—
7C2×#88112—22no—
8#11718882—22no—
9#11718882C2212yes—
10#11718596C411yescharacteristic
11C42×Dic13C2211yescharacteristic
12#11718290—22no—
13#11718290C2214yes—
14C22×D4×C13—22no—
15C22×D4×C13C2×C412yes—
16C23×Dic13—22no—
17C22×#1057—24no—
18C13×#127—24no—
19C2×C42×C13C2×C411yescharacteristic
20C2×C22⋊Dic13—22no—
21C2×C22⋊Dic13—24no—
22D4×Dic13—44no—
23D4×Dic13—44no—
24D4×Dic13—44no—
25C2×C4⋊Dic13—22no—
26C2×C4⋊Dic13C2×C412yes—
27C2×C4×Dic13C2×C411yescharacteristic
28C2×C4×Dic13D412yes—
29C2×C4×Dic13D412yes—
30C2×C4×Dic13C2311yescharacteristic
31#88112—44no—
32#7651—131nomaximal, Sylow
33C24×C13—22no—
34C2×D4×C13—28no—
35C2×D4×C13—22no—
36C2×D4×C13—42no—
37C2×D4×C13—42no—
38C2×D4×C13—44no—
39C2×D4×C13—44no—
40C2×D4×C13M1614yes—
41C22×C4×C13C22⋊C412yes—
42C22×C4×C13C22×C411yescharacteristic
43C22×Dic13—22no—
44C22×Dic13—44no—
45C22×Dic13—48no—
46C22×Dic13C22⋊C411yescharacteristic
47C22×Dic13C22⋊C411yescharacteristic
48C22×Dic13C2×D412yes—
49C2×#1057—44no—
50C2×#1057—44no—
51C2×#1057—44no—
52C42×C13—22no—
53C22⋊Dic13—44no—
54C22⋊Dic13—44no—
55C22⋊Dic13—44no—
56C4⋊Dic13—44no—
57C4⋊Dic13—42no—
58C4×Dic13—22no—
59C4×Dic13—42no—
60C4×Dic13—41no—
61C4×Dic13D814yes—
62C4×Dic13SD1614yes—
63C4×#127—131no—
64#1831—132no—
65#1270—134no—
66C23×C13—24no—
67C23×C13—42no—
68C23×C13—42no—
69C23×C13—44no—
70C23×C13—44no—
71C23×C13C2×C22⋊C411yescharacteristic
72D4×C13—44no—
73D4×C13—416no—
74D4×C13—44no—
75C2×C4×C13—22no—
76C2×C4×C13—21no—
77C2×C4×C13—21no—
78C2×C4×C13—22no—
79C2×C4×C13#10214yes—
80C2×C4×C13C2×M1612yes—
81C2×Dic13—22no—
82C2×Dic13—22no—
83C2×Dic13—21no—
84C2×Dic13—22no—
85C2×Dic13—42no—
86C2×Dic13—44no—
87C2×Dic13—88no—
88C2×Dic13#10012yes—
89C2×Dic13#10214yes—
90C2×Dic13C2×D812yes—
91C2×Dic13C2×SD1612yes—
92#1057—84no—
93C2×#127#26111yescharacteristic
94C2×C4×D4—132no—
95C2×C4×D4—262no—
96C2×C4⋊C8—262no—
97#390—132no—
98#390—262no—
99#384—131no—
100C43—131no—
101#336—134no—
102#336—262no—
103C22×C13—44no—
104C22×C13—48no—
105C22×C13—82no—
106C22×C13—82no—
107C22×C13—84no—
108C22×C13#40711yescharacteristic
109C22×C13#41011yescharacteristic
110C22×C13C2×#10011yescharacteristic
111C22×C13C2×#10212yes—
112C22×C13#42112yes—
113C4×C13—22no—
114C4×C13—24no—
115Dic13—22no—
116Dic13—22no—
117Dic13—41no—
118Dic13—82no—
119Dic13#33114yes—
120C22×D4—22no—
121C22×D4C2×#26112yes—
122C23×C4—262no—
123C22×C8—264no—
124#127—24no—
125C4×D4—524no—
126C4×D4—524no—
127C4×D4—524no—
128C2×C4⋊C4—132no—
129C2×C4⋊C4—262no—
130C2×C22⋊C4—262no—
131C2×C22⋊C4—264no—
132C2×C42—132no—
133C2×C42—131no—
134C2×C42—132no—
135C2×C42—131no—
136C2×C42C2×#26111yescharacteristic
137C4⋊C8—524no—
138C2×C13—84no—
139C2×C13#146611yescharacteristic
140C2×C13#148612yes—
141C2×C13#151712yes—
142C2×C13#153611yescharacteristic
143C2×C13C2×#33111yescharacteristic
144C24—22no—
145C2×D4—28no—
146C2×D4—22no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—44no—
150C2×D4—44no—
151C2×D4#612214yes—
152C22×C4—131no—
153C22×C4—132no—
154C22×C4—131no—
155C22×C4—262no—
156C22×C4—528no—
157C22×C4—524no—
158C22×C4C22⋊#26112yes—
159C22×C4C22×#26111yescharacteristic
160C2×C8—524no—
161C2×C8—524no—
162C2×C8—524no—
163C4⋊C4—522no—
164C4⋊C4—524no—
165C22⋊C4—524no—
166C22⋊C4—524no—
167C22⋊C4—524no—
168C42—22no—
169C42—134no—
170C42—134no—
171C42—262no—
172C42—522no—
173C42—521no—
174C13#765111yescharacteristic, complemented, Sylow
175C23—24no—
176C23—44no—
177C23—44no—
178C23—42no—
179C23—42no—
180C23C2×C22⋊#26111yescharacteristic
181D4—44no—
182D4—44no—
183D4—416no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—134no—
189C2×C4—132no—
190C2×C4—132no—
191C2×C4—132no—
192C2×C4—262no—
193C2×C4—262no—
194C2×C4—262no—
195C2×C4—261no—
196C2×C4—524no—
197C2×C4—522no—
198C2×C4—1048no—
199C2×C4#8811514yes—
200C2×C4C2×#612212yes—
201C8—1044no—
202C22—44no—
203C22—48no—
204C22—84no—
205C22—82no—
206C22—82no—
207C22C2×#8811512yes—
208C22#1171914012yes—
209C22#1171914211yescharacteristic
210C22D13⋊#10011yescharacteristic
211C22#1171916911yescharacteristic
212C4—22no—
213C4—24no—
214C4—134no—
215C4—262no—
216C4—262no—
217C4—521no—
218C4—1042no—
219C2—84no—
220C2#1313009611yescharacteristic
221C2#1313016112yes—
222C2#1313026512yes—
223C2#1313032511yescharacteristic
224C2D13⋊#33111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
Fitting, prime radicalC2×C13×#127C4
socleC23×C13C2×C22⋊C4
cosocleC2×C4×Dic13C23
hypercenterC2×#127#261
hypocenterC13#7651
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #13130752C2semidirect
2. #13129652C2semidirect
174. C13#7651semidirect

Maximal subgroups

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

subgroupindex
#131307522
#131296522
#127×Dic132
#765113