Finite Group Databasebeta

← #30766889

Subgroups — #30766889

Order 17152.

408 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 107 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#22560205C211yesmaximal, characteristic
2#22558533C211yesmaximal, characteristic
3#22558215C214yesmaximal
4C67×#2546C211yesmaximal, characteristic
5#16863449—21no—
6#16863420—21no—
7C2×#14523962—21no—
8#16863165C2212yes—
9#16863122—21no—
10#16863122C2212yes—
11C2×C4⋊Q8×C67—21no—
12C2×C4⋊Q8×C67C411yescharacteristic
13C2×C4×Q8×C67C2211yescharacteristic
14#398×C67C412yes—
15#398×C67C2212yes—
16#391×C67—21no—
17#389×C67—21no—
18#389×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic
20C2×#11761612—22no—
21C4⋊(C67⋊C8)—41no—
22C4⋊(C67⋊C8)—41no—
23C4⋊(C67⋊C8)—41no—
24#14523962—42no—
25C22⋊(C67⋊C8)—22no—
26C4×C67⋊C8—41no—
27C22×Q8×C67—21no—
28C22×Q8×C67D411yescharacteristic
29C4⋊Q8×C67—41no—
30C4⋊Q8×C67—41no—
31C4⋊Q8×C67—42no—
32C4⋊Q8×C67—42no—
33C4×Q8×C67—41no—
34C4×Q8×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—22no—
39C2×C4⋊C4×C67C2×C412yes—
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C4⋊C4×C67C2311yescharacteristic
43C2×C42×C67—21no—
44C2×C42×C67C2×C411yescharacteristic
45C2×C42×C67D411yescharacteristic
46C2×C42×C67D411yescharacteristic
47#95×C67—22no—
48#95×C67—22no—
49#95×C67C2×C412yes—
50#11761612—42no—
51#11761612—42no—
52C2×C67⋊C8—42no—
53C2×C67⋊C8—42no—
54C2×Q8×C67—41no—
55C2×Q8×C67—41no—
56C2×Q8×C67—42no—
57C2×Q8×C67—42no—
58C2×Q8×C67—42no—
59C2×Q8×C67D812yes—
60C2×Q8×C67SD1612yes—
61C22×C4×C67—22no—
62C22×C4×C67—22no—
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22×C411yescharacteristic
68C22×C4×C67C2×D411yescharacteristic
69C22×C4×C67C2×D411yescharacteristic
70C4⋊C4×C67—24no—
71C4⋊C4×C67—41no—
72C4⋊C4×C67—41no—
73C4⋊C4×C67—41no—
74C4⋊C4×C67—42no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—44no—
78C4⋊C4×C67—42no—
79C4⋊C4×C67—42no—
80C42×C67—21no—
81C42×C67—21no—
82C42×C67—21no—
83C42×C67—21no—
84C42×C67—21no—
85C42×C67—21no—
86C42×C67—42no—
87C23×C67C2×C22⋊C411yescharacteristic
88Q8×C67—42no—
89Q8×C67—42no—
90Q8×C67—84no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—22no—
98C2×C4×C67—24no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—42no—
103C2×C4×C67—42no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67#9911yescharacteristic
111C2×C4×C67#9911yescharacteristic
112C2×C4×C67#10212yes—
113C2×C4×C67#10212yes—
114C2×C4×C67C2×D811yescharacteristic
115C2×C4×C67C2×SD1611yescharacteristic
116C67⋊C8—82no—
117C67⋊C8—82no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67C2×#9911yescharacteristic
121C22×C67C2×#10211yescharacteristic
122C22×C67#42111yescharacteristic
123C4×C67—21no—
124C4×C67—21no—
125C4×C67—22no—
126C4×C67—22no—
127C4×C67—21no—
128C4×C67—44no—
129C4×C67—41no—
130C4×C67—41no—
131C4×C67—41no—
132C4×C67—82no—
133C4×C67#35612yes—
134#11883—671nomaximal, Sylow
135C2×C67—21no—
136C2×C67—21no—
137C2×C67#151911yescharacteristic
138C2×C67#212311yescharacteristic
139C2×C67C2×#35611yescharacteristic
140#2546D6711yescharacteristic
141#1801—671no—
142#1461—671no—
143#1346—674no—
144C67#1188311yescharacteristic, complemented, Sylow
145C2×C4⋊Q8—21no—
146C2×C4⋊Q8Dic6711yescharacteristic
147C2×C4×Q8C2×D6711yescharacteristic
148#431—1341no—
149#423—1341no—
150C2×#101—1341no—
151#398Dic6712yes—
152#398C2×D6712yes—
153#391—21no—
154#389—21no—
155#389—21no—
156C4×C4⋊C4Dic6711yescharacteristic
157#343—672no—
158#324—672no—
159#324—1341no—
160C22×Q8—21no—
161C22×Q8C22⋊D6711yescharacteristic
162C2×M16—1342no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4×Q8—41no—
168C4×Q8—41no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C2×Dic6712yes—
174C2×C4⋊C4C2×Dic6711yescharacteristic
175C2×C4⋊C4C22⋊D6711yescharacteristic
176C2×C4⋊C4C22×D6711yescharacteristic
177C2×C42—21no—
178C2×C42C2×Dic6711yescharacteristic
179C2×C42C22⋊D6711yescharacteristic
180C2×C42C22⋊D6711yescharacteristic
181C4⋊C8—2681no—
182C4⋊C8—2681no—
183C4⋊C8—2681no—
184#101—2682no—
185C22⋊C8—1342no—
186C4×C8—2681no—
187#95—22no—
188#95—22no—
189#95C2×Dic6712yes—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8#1176161312yes—
196C2×Q8#1176161612yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4C22⋊Dic6711yescharacteristic
200C22×C4C22⋊Dic6711yescharacteristic
201C22×C4C22⋊Dic6711yescharacteristic
202C22×C4C22⋊Dic6711yescharacteristic
203C22×C4C22×Dic6711yescharacteristic
204C22×C4C2×C22⋊D6711yescharacteristic
205C22×C4C2×C22⋊D6711yescharacteristic
206M16—2682no—
207M16—2682no—
208C2×C8—2682no—
209C2×C8—2682no—
210C4⋊C4—24no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C23C2×C22⋊Dic6711yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4#1452395811yescharacteristic
251C2×C4#1452395811yescharacteristic
252C2×C4#1452396412yes—
253C2×C4#1452396412yes—
254C2×C4C2×#1176161311yescharacteristic
255C2×C4C2×#1176161611yescharacteristic
256C8—5362no—
257C8—5362no—
258C22—21no—
259C22—21no—
260C22C2×#1452395811yescharacteristic
261C22C2×#1452396411yescharacteristic
262C22#1686341311yescharacteristic
263C4—21no—
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—44no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4#1686319512yes—
274C2—21no—
275C2—21no—
276C2#2255877711yescharacteristic
277C2#2256186311yescharacteristic
278C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2546C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2546D67
hypocenterC67#11883
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
144. C67#11883semidirect

Maximal subgroups

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

subgroupindex
#225602052
#225585332
#225582152
C67×#25462
#1188367