Finite Group Databasebeta

← #30772097

Subgroups — #30772097

Order 17152.

638 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 115 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#22560133C212yesmaximal
2#22558336C214yesmaximal, complemented
3C2×#391×C67C211yesmaximal, characteristic
4C2×#14523962—22no—
5C2×#14523962C2214yes—
6#16863360—22no—
7C23×Q8×C67C2211yescharacteristic
8C22×C4⋊C4×C67C411yescharacteristic
9C22×C42×C67C411yescharacteristic
10#391×C67—22no—
11#391×C67C412yes—
12#391×C67C2212yes—
13C2×#95×C67—22no—
14C2×#11761612—24no—
15#14523962—28no—
16#14523962—44no—
17C22⋊(C67⋊C8)—42no—
18C22⋊(C67⋊C8)—42no—
19C22×Q8×C67—24no—
20C22×Q8×C67—41no—
21C22×Q8×C67—41no—
22C22×Q8×C67—42no—
23C22×Q8×C67C2×C411yescharacteristic
24C22×Q8×C67D412yes—
25C22×Q8×C67C2311yescharacteristic
26C23×C4×C67—21no—
27C23×C4×C67—21no—
28C23×C4×C67C2×C411yescharacteristic
29C23×C4×C67D412yes—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67—41no—
32C2×C4⋊C4×C67—41no—
33C2×C4⋊C4×C67C2×C412yes—
34C2×C42×C67—21no—
35C2×C42×C67—42no—
36C2×C42×C67C2×C412yes—
37#95×C67—24no—
38#95×C67—24no—
39#11761612—44no—
40#11761612—44no—
41C2×C67⋊C8—44no—
42C24×C67C22⋊C411yescharacteristic
43C2×Q8×C67—24no—
44C2×Q8×C67—22no—
45C2×Q8×C67—24no—
46C2×Q8×C67—41no—
47C2×Q8×C67—41no—
48C2×Q8×C67—42no—
49C2×Q8×C67—41no—
50C2×Q8×C67—41no—
51C2×Q8×C67—48no—
52C2×Q8×C67—48no—
53C2×Q8×C67#4114yes—
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—24no—
59C22×C4×C67—22no—
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—24no—
63C22×C4×C67—22no—
64C22×C4×C67—42no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67—44no—
68C22×C4×C67—41no—
69C22×C4×C67—41no—
70C22×C4×C67C22⋊C412yes—
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C2×D412yes—
74C4⋊C4×C67—42no—
75C4⋊C4×C67—42no—
76C42×C67—42no—
77C42×C67—42no—
78C23×C67—22no—
79C23×C67—22no—
80C23×C67—21no—
81C23×C67—21no—
82C23×C67#9911yescharacteristic
83C23×C67#9911yescharacteristic
84C23×C67C2×C22⋊C411yescharacteristic
85Q8×C67—44no—
86Q8×C67—48no—
87Q8×C67—44no—
88C2×C4×C67—24no—
89C2×C4×C67—24no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—24no—
97C2×C4×C67—22no—
98C2×C4×C67—22no—
99C2×C4×C67—22no—
100C2×C4×C67—44no—
101C2×C4×C67—44no—
102C2×C4×C67—48no—
103C2×C4×C67—42no—
104C2×C4×C67—42no—
105C2×C4×C67—44no—
106C2×C4×C67—42no—
107C2×C4×C67—82no—
108C2×C4×C67—82no—
109C2×C4×C67#9912yes—
110C2×C4×C67#11712yes—
111C2×C4×C67#12314yes—
112C67⋊C8—84no—
113C22×C67—24no—
114C22×C67—24no—
115C22×C67—21no—
116C22×C67—21no—
117C22×C67—21no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67—21no—
121C22×C67#35512yes—
122C22×C67#35612yes—
123C22×C67#38111yescharacteristic
124C22×C67C2×#9911yescharacteristic
125C22×C67C2×#9911yescharacteristic
126C4×C67—24no—
127C4×C67—22no—
128C4×C67—22no—
129C4×C67—44no—
130C4×C67—42no—
131C4×C67—82no—
132#12860—671nomaximal, Sylow
133C2×C67—21no—
134C2×C67—22no—
135C2×C67—22no—
136C2×C67—21no—
137C2×C67#181911yescharacteristic
138C2×C67C2×#35511yescharacteristic
139C2×C67C2×#35611yescharacteristic
140C2×#391D6711yescharacteristic
141#1784—672no—
142#1394—674no—
143C67#1286011yescharacteristic, complemented, Sylow
144C23×Q8C2×D6711yescharacteristic
145C22×C4⋊C4Dic6711yescharacteristic
146C22×C42Dic6711yescharacteristic
147C2×#101—674no—
148C2×#101—1342no—
149#407—1342no—
150#391—22no—
151#391Dic6712yes—
152#391C2×D6712yes—
153C2×#95—22no—
154C22×Q8—24no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—42no—
158C22×Q8C2×Dic6711yescharacteristic
159C22×Q8C22⋊D6712yes—
160C22×Q8C22×D6711yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×Dic6711yescharacteristic
164C23×C4C22⋊D6712yes—
165C2×M16—1344no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4C2×Dic6712yes—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42C2×Dic6712yes—
173#101—2684no—
174#101—1348no—
175C22⋊C8—2682no—
176C22⋊C8—2682no—
177#95—24no—
178#95—24no—
179C24C22⋊Dic6711yescharacteristic
180C2×Q8—24no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—48no—
189C2×Q8—48no—
190C2×Q8#1176162814yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C22⋊Dic6712yes—
208C22×C4C22⋊Dic6711yescharacteristic
209C22×C4C22×Dic6711yescharacteristic
210C22×C4C2×C22⋊D6712yes—
211M16—2684no—
212M16—2684no—
213C2×C8—2684no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C42—42no—
217C42—42no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23#1452395811yescharacteristic
223C23#1452395811yescharacteristic
224C23C2×C22⋊Dic6711yescharacteristic
225Q8—44no—
226Q8—48no—
227Q8—44no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4#1452395812yes—
250C2×C4#1452400212yes—
251C2×C4#1452402614yes—
252C8—5364no—
253C22—24no—
254C22—24no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22#1686319312yes—
262C22#1686319512yes—
263C22#1686325911yescharacteristic
264C22C2×#1452395811yescharacteristic
265C22C2×#1452395811yescharacteristic
266C4—24no—
267C4—22no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C2—21no—
273C2—21no—
274C2—22no—
275C2—22no—
276C2#2256028311yescharacteristic
277C2C2×#1686319311yescharacteristic
278C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×Q8C22×D67
FittingC2×#391×C67C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC22×Q8×C67C23
hypercenterC2×#391D67
hypocenterC67#12860
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. #22558336C2semidirect
143. C67#12860semidirect

Maximal subgroups

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

subgroupindex
#225601332
#225583362
C2×#391×C672
#1286067