Finite Group Databasebeta

← #30748333

Subgroups — #30748333

Order 17152.

818 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 131 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#22560292C211yesmaximal, characteristic, complemented
2#22560286C211yesmaximal, characteristic, complemented
3#22558738C214yesmaximal, complemented
4C2×#540×C67C211yesmaximal, characteristic
5#16863360C2212yes—
6#16863259C2212yes—
7#16863165—22no—
8#16863163—22no—
9#16863130—28no—
10#540×C67—28no—
11C2×#127×C67C411yescharacteristic, complemented
12C2×#125×C67C411yescharacteristic, complemented
13C2×#123×C67—22no—
14C2×#121×C67—22no—
15C2×#121×C67C2212yes—
16C2×C4×D4×C67C412yescomplemented
17C2×#117×C67C2211yescharacteristic
18C2×#11761612—22no—
19C2×C22⋊Dic67—22no—
20C22⋊(C67⋊C8)—24no—
21C67⋊#95—24no—
22C22×D4×C67—22no—
23C22×D4×C67—22no—
24C22×D4×C67C2×C412yes—
25C23×C4×C67—22no—
26C23×C4×C67C2311yescharacteristic
27#127×C67—24no—
28#125×C67—24no—
29#123×C67—216no—
30#121×C67—216no—
31#121×C67—24no—
32#121×C67D418yes—
33C4×D4×C67—24no—
34C4×D4×C67—24no—
35#117×C67—24no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67C2×C412yes—
38C2×C4⋊C4×C67D412yes—
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67D412yes—
42C2×C42×C67C2×C411yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C22×Dic67—42no—
45#11761612—44no—
46C2×C67⋊C8—42no—
47C22⋊Dic67—44no—
48C24×C67—22no—
49C24×C67C2×D411yescharacteristic
50C2×D4×C67—28no—
51C2×D4×C67—28no—
52C2×D4×C67—22no—
53C2×D4×C67—44no—
54C2×D4×C67—84no—
55C2×D4×C67—84no—
56C2×D4×C67C22⋊C414yes—
57C22×C4×C67—22no—
58C22×C4×C67—28no—
59C22×C4×C67—42no—
60C22×C4×C67—84no—
61C22×C4×C67C22⋊C412yes—
62C22×C4×C67C22⋊C412yes—
63C22×C4×C67C22×C411yescharacteristic
64C22×C4×C67C2×D411yescharacteristic
65C22×C4×C67C2×D414yes—
66C4⋊C4×C67—28no—
67C4⋊C4×C67—22no—
68C4⋊C4×C67—24no—
69C4⋊C4×C67C22⋊C414yes—
70C22⋊C4×C67—28no—
71C22⋊C4×C67—24no—
72C22⋊C4×C67—28no—
73C42×C67—22no—
74C42×C67—22no—
75C23×C67—22no—
76C23×C67—24no—
77C23×C67—42no—
78C23×C67—44no—
79C23×C67—44no—
80C23×C67—44no—
81C23×C67C2×C22⋊C411yescharacteristic
82C23×C67#12012yes—
83D4×C67—44no—
84D4×C67—44no—
85D4×C67—88no—
86D4×C67—88no—
87C2×C4×C67—22no—
88C2×C4×C67—22no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—24no—
92C2×C4×C67—44no—
93C2×C4×C67—44no—
94C2×C4×C67—42no—
95C2×C4×C67—42no—
96C2×C4×C67—88no—
97C2×C4×C67C2×C22⋊C412yes—
98C2×C4×C67#12012yes—
99C2×Dic67—82no—
100C2×Dic67—84no—
101C67⋊C8—82no—
102C22×C67—22no—
103C22×C67—24no—
104C22×C67—22no—
105C22×C67—44no—
106C22×C67—44no—
107C22×C67—44no—
108C22×C67—82no—
109C22×C67—88no—
110C22×C67#37911yescharacteristic
111C22×C67#41012yes—
112C4×C67—22no—
113C4×C67—22no—
114C4×C67—44no—
115C4×C67—42no—
116C4×C67—42no—
117Dic67—162no—
118#8518—671nomaximal, Sylow
119C2×C67—22no—
120C2×C67—42no—
121C2×C67—84no—
122C2×C67#178211yescharacteristic
123C2×C67#179512yes—
124C2×#540D6711yescharacteristic
125#1822—671no—
126#1820—671no—
127#1510—674no—
128C67#851811yescharacteristic, complemented, Sylow
129#540—28no—
130C2×#127Dic6711yescharacteristic, complemented
131C2×#125Dic6711yescharacteristic, complemented
132C2×#123—22no—
133C2×#121—22no—
134C2×#121C2×D6712yes—
135C2×C4×D4Dic6712yescomplemented
136C2×#117C2×D6711yescharacteristic
137#407—672no—
138#381—672no—
139#343—1342no—
140#342—1342no—
141#327—1348no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4C2×Dic6712yes—
145C23×C4—22no—
146C23×C4C22×D6711yescharacteristic
147C2×M16—1342no—
148#127—24no—
149#125—24no—
150#123—216no—
151#121—216no—
152#121—24no—
153#121C22⋊D6718yes—
154C4×D4—24no—
155C4×D4—24no—
156#117—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×Dic6712yes—
159C2×C4⋊C4C22⋊D6712yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1342no—
163C2×C22⋊C4C22⋊D6712yes—
164C2×C42C2×Dic6711yescharacteristic
165C2×C42C2×Dic6711yescharacteristic
166C22⋊C8—1344no—
167#95—1344no—
168C24—22no—
169C24C2×C22⋊D6711yescharacteristic
170C2×D4—28no—
171C2×D4—28no—
172C2×D4—22no—
173C2×D4—44no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4C22⋊Dic6714yes—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—42no—
180C22×C4—84no—
181C22×C4—2682no—
182C22×C4C22⋊Dic6712yes—
183C22×C4C22⋊Dic6712yes—
184C22×C4C22×Dic6711yescharacteristic
185C22×C4C2×C22⋊D6711yescharacteristic
186C22×C4C2×C22⋊D6714yes—
187M16—2684no—
188C2×C8—2682no—
189C4⋊C4—28no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4C22⋊Dic6714yes—
193C22⋊C4—28no—
194C22⋊C4—24no—
195C22⋊C4—28no—
196C22⋊C4—2684no—
197C42—22no—
198C42—22no—
199C23—22no—
200C23—24no—
201C23—42no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23C2×C22⋊Dic6711yescharacteristic
206C23#1452401212yes—
207D4—44no—
208D4—44no—
209D4—88no—
210D4—88no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—88no—
221C2×C4—5362no—
222C2×C4—5364no—
223C2×C4C2×C22⋊Dic6712yes—
224C2×C4#1452401212yes—
225C8—5362no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—82no—
233C22—88no—
234C22#1686325211yescharacteristic
235C22#1686337012yes—
236C4—22no—
237C4—22no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—10722no—
242C2—22no—
243C2—42no—
244C2—84no—
245C2#2256012711yescharacteristic
246C2#2256017912yes—
subgroupG/H
centerC22#16863252
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
Fitting, prime radicalC2×#540×C67C2
socleC22×C67#379
cosocleC23×C4×C67C23
hypercenterC2×#540D67
hypocenterC67#8518
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. #22560292C2semidirect
2. #22560286C2semidirect
3. #22558738C2semidirect
11. C2×#127×C67C4semidirect
12. C2×#125×C67C4semidirect
16. C2×C4×D4×C67C4semidirect
128. C67#8518semidirect
130. C2×#127Dic67semidirect
131. C2×#125Dic67semidirect
135. C2×C4×D4Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602922
#225602862
#225587382
C2×#540×C672
#851867