Finite Group Databasebeta

← #13121209

Subgroups — #13121209

Order 1664.

592 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#88178C212yesmaximal, complemented
2C2×#88132C212yesmaximal, complemented
3C13×#386C211yesmaximal, characteristic
4#11718882C211yesmaximal, characteristic
5#11718881C211yesmaximal, characteristic
6#11718609C212yesmaximal, complemented
7#11718609C212yesmaximal, complemented
8#11718317C212yesmaximal, complemented
9#11718308C212yesmaximal, complemented
10C22×D4×C13C2211yescharacteristic
11C23×C4×C13C2211yescharacteristic
12C23×Dic13C2211yescharacteristic
13C22×Dic26C2212yes—
14C2×C13×C22⋊C4C2212yes—
15C2×C13×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic13C2212yes—
17C2×C22⋊Dic13C2212yes—
18C2×C22⋊Dic13C2212yes—
19C2×C22⋊Dic13C2211yescharacteristic
20C2×C4⋊Dic13C2212yes—
21C2×C4⋊Dic13C2212yes—
22C2×C4⋊Dic13C2211yescharacteristic
23#88178—28no—
24C2×#6101C2212yes—
25C2×#6101C2212yes—
26C2×#6101C2212yes—
27C2×#6101C2212yes—
28C2×C4×Dic13C2211yescharacteristic
29#88132—28no—
30C13×#95C2212yes—
31#88071C2212yes—
32#88071C2212yes—
33C24×C13C2311yescharacteristic
34C24×C13C2311yescharacteristic
35C2×D4×C13—41no—
36C2×D4×C13—41no—
37C2×D4×C13Q814yescomplemented
38C22×C4×C13—24no—
39C22×C4×C13—41no—
40C22×C4×C13C2312yes—
41C22×C4×C13C2312yes—
42C22×C4×C13C2311yescharacteristic
43C22×Dic13—24no—
44C22×Dic13—41no—
45C22×Dic13C2312yes—
46C22×Dic13C2312yes—
47C22×Dic13C2312yes—
48C22×Dic13C2312yes—
49C2×Dic26—24no—
50C2×Dic26—24no—
51C2×Dic26—42no—
52C13×C22⋊C4—24no—
53C13×C22⋊C4—41no—
54C22⋊Dic13—24no—
55C22⋊Dic13—24no—
56C22⋊Dic13—24no—
57C22⋊Dic13—41no—
58C4⋊Dic13—24no—
59C4⋊Dic13—24no—
60C4⋊Dic13—41no—
61#6101—24no—
62#6101—24no—
63#6101—42no—
64#6101D418yescomplemented
65C4×Dic13—41no—
66#2683—131nomaximal, Sylow
67C23×C13—22no—
68C23×C13—22no—
69C23×C13—21no—
70C23×C13—21no—
71C23×C13—21no—
72C23×C13—21no—
73C23×C13—22no—
74C23×C13—22no—
75C23×C13C2×Q812yes—
76C23×C13C2×Q812yes—
77C23×C13C2411yescharacteristic
78D4×C13—44no—
79C2×C4×C13—22no—
80C2×C4×C13—21no—
81C2×C4×C13—21no—
82C2×C4×C13—22no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13—44no—
87C2×C4×C13C2×D414yes—
88C2×C4×C13C2×Q812yes—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—44no—
100C2×Dic13C2×D414yes—
101C2×Dic13C2×D414yes—
102Dic26—48no—
103C2×C22⋊Q8—132no—
104C2×C22⋊Q8—132no—
105#396—132no—
106#396—132no—
107#395—132no—
108#390—131no—
109#389—132no—
110#386—131no—
111#386D1311yescharacteristic
112C22×C13—22no—
113C22×C13—22no—
114C22×C13—24no—
115C22×C13—24no—
116C22×C13—42no—
117C22×C13—42no—
118C22×C13—41no—
119C22×C13—41no—
120C22×C13—41no—
121C22×C13—41no—
122C22×C13C22×D412yes—
123C22×C13C22×Q811yescharacteristic
124C22×C13H(5,𝔽2)11yescharacteristic
125C22×C13H(5,𝔽2)11yescharacteristic
126C22×C132−1+412yes—
127C4×C13—22no—
128C4×C13—24no—
129C4×C13—42no—
130Dic13—24no—
131Dic13—24no—
132Dic13—42no—
133Dic13—42no—
134C22×Q8—132no—
135C22×D4C2×D1311yescharacteristic
136C23×C4—131no—
137C23×C4C2×D1311yescharacteristic
138C22⋊Q8—268no—
139C22⋊Q8—268no—
140C2×C4⋊C4—132no—
141C2×C4⋊C4—132no—
142C2×C4⋊C4—132no—
143C2×C4⋊C4—132no—
144C2×C4⋊C4—132no—
145C2×C4⋊C4—132no—
146C2×C4⋊C4—131no—
147C2×C22⋊C4—132no—
148C2×C22⋊C4—132no—
149C2×C22⋊C4—132no—
150C2×C22⋊C4—131no—
151C2×C22⋊C4C2×D1312yes—
152C2×C22⋊C4C2×D1311yescharacteristic
153C2×C42—131no—
154#95—132no—
155#95—132no—
156#95C2×D1312yes—
157C2×C13—42no—
158C2×C13—42no—
159C2×C13#53712yes—
160C2×C13#54311yescharacteristic
161C2×C13D4211yescharacteristic
162C2×C13D4×Q812yes—
163C2×C13#56311yescharacteristic
164C24C22×D1311yescharacteristic
165C24C22×D1311yescharacteristic
166C2×Q8—264no—
167C2×Q8—264no—
168C2×Q8—522no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4Dic2614yescomplemented
172C22×C4—24no—
173C22×C4—41no—
174C22×C4—132no—
175C22×C4—132no—
176C22×C4—132no—
177C22×C4—132no—
178C22×C4—264no—
179C22×C4—521no—
180C22×C4C22×D1312yes—
181C22×C4C22×D1311yescharacteristic
182C22×C4C22×D1312yes—
183C4⋊C4—138no—
184C4⋊C4—264no—
185C4⋊C4—264no—
186C4⋊C4—264no—
187C4⋊C4—264no—
188C4⋊C4—522no—
189C4⋊C4—521no—
190C22⋊C4—24no—
191C22⋊C4—41no—
192C22⋊C4—264no—
193C22⋊C4—264no—
194C22⋊C4—264no—
195C22⋊C4—521no—
196C42—521no—
197C13#268311yescharacteristic, complemented, Sylow
198C23—22no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—22no—
205C23—22no—
206C23C2×Dic2612yes—
207C23C2×Dic2612yes—
208C23C23×D1311yescharacteristic
209Q8—528no—
210D4—44no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—134no—
220C2×C4—134no—
221C2×C4—262no—
222C2×C4—262no—
223C2×C4—262no—
224C2×C4—262no—
225C2×C4—262no—
226C2×C4—262no—
227C2×C4—262no—
228C2×C4—262no—
229C2×C4—262no—
230C2×C4—262no—
231C2×C4—524no—
232C2×C4C2×Dic2612yes—
233C2×C4D4×D1314yes—
234C22—22no—
235C22—22no—
236C22—24no—
237C22—24no—
238C22—41no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22C22×Dic2611yescharacteristic
245C22C2×D4×D1312yes—
246C22#8825111yescharacteristic
247C22#8825111yescharacteristic
248C22#8825712yes—
249C4—22no—
250C4—24no—
251C4—42no—
252C4—264no—
253C4—264no—
254C4—522no—
255C4—522no—
256C2—42no—
257C2—42no—
258C2#1171918811yescharacteristic
259C2D4×Dic2612yes—
260C2#1171929511yescharacteristic
261C2#1171931312yes—
262C2#1171935611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#386C2
prime radicalC22×D4×C13C22
hypercenter#386D13
hypocenterC13#2683
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#88178C2semidirect
2. C2×#88132C2semidirect
6. #11718609C2semidirect
7. #11718609C2semidirect
8. #11718317C2semidirect
9. #11718308C2semidirect
37. C2×D4×C13Q8semidirect
64. #6101D4semidirect
171. C2×D4Dic26semidirect
197. C13#2683semidirect

Maximal subgroups

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

subgroupindex
C2×#881782
C2×#881322
C13×#3862
#117188822
#117188812
#117186092
#117183172
#117183082
#268313