Finite Group Databasebeta

← #30766046

Subgroups — #30766046

Order 17152.

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

normal subgroups only

This group has 452 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#22559948C211yesmaximal, characteristic
2#22558523C211yesmaximal, characteristic
3#22558215C214yesmaximal, complemented
4C67×#2621C211yesmaximal, characteristic
5C22×#11761612—21no—
6#16863449—21no—
7#16863165—21no—
8#16863165C2212yes—
9#16863122C2212yes—
10#16863120—21no—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×C22⋊Q8×C67—21no—
13C2×#117×C67C2211yescharacteristic
14#398×C67—21no—
15#398×C67C2212yes—
16#397×C67—21no—
17#397×C67C412yes—
18#391×C67—21no—
19#382×C67C411yescharacteristic
20C2×#11761612—24no—
21C2×#11761612—22no—
22C2×#11761612—41no—
23C2×#11761612—42no—
24C22×C67⋊C8—41no—
25C4⋊(C67⋊C8)—41no—
26C22⋊(C67⋊C8)—22no—
27C4×C67⋊C8—41no—
28C22×Q8×C67—21no—
29C23×C4×C67D411yescharacteristic
30C4⋊Q8×C67—41no—
31C4⋊Q8×C67—41no—
32C22⋊Q8×C67—42no—
33C22⋊Q8×C67—42no—
34#117×C67—41no—
35#117×C67—41no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—22no—
39C2×C4⋊C4×C67C2×C412yes—
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67C2×C411yescharacteristic
44C2×C22⋊C4×C67D411yescharacteristic
45C2×C42×C67C2×C411yescharacteristic
46C2×C42×C67D411yescharacteristic
47#95×C67—21no—
48#95×C67—22no—
49#95×C67—22no—
50#95×C67C2×C412yes—
51#11761612—24no—
52#11761612—44no—
53#11761612—42no—
54C2×C67⋊C8—42no—
55C2×C67⋊C8—44no—
56C2×C67⋊C8—42no—
57C24×C67C22⋊C411yescharacteristic
58C2×Q8×C67—41no—
59C2×Q8×C67—41no—
60C2×Q8×C67—42no—
61C2×Q8×C67—42no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—22no—
65C22×C4×C67—22no—
66C22×C4×C67—22no—
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67D812yes—
71C22×C4×C67SD1612yes—
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C4⋊C4×C67—41no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—42no—
78C4⋊C4×C67—42no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—42no—
82C22⋊C4×C67—41no—
83C22⋊C4×C67—41no—
84C22⋊C4×C67—44no—
85C42×C67—21no—
86C42×C67—21no—
87C42×C67—21no—
88C42×C67—21no—
89C23×C67—21no—
90C23×C67—21no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67#10212yes—
94C23×C67C2×C22⋊C411yescharacteristic
95Q8×C67—82no—
96Q8×C67—82no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—22no—
102C2×C4×C67—22no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67#9911yescharacteristic
118C2×C4×C67#9911yescharacteristic
119C2×C4×C67#10212yes—
120C2×C4×C67#10412yes—
121C2×C4×C67C2×D811yescharacteristic
122C2×C4×C67C2×SD1611yescharacteristic
123C67⋊C8—44no—
124C67⋊C8—82no—
125C22×C67—21no—
126C22×C67—21no—
127C22×C67—22no—
128C22×C67—42no—
129C22×C67—41no—
130C22×C67—41no—
131C22×C67—81no—
132C22×C67C2×#9911yescharacteristic
133C22×C67C2×#10211yescharacteristic
134C22×C67C2×#10411yescharacteristic
135C4×C67—21no—
136C4×C67—21no—
137C4×C67—41no—
138C4×C67—41no—
139C4×C67—41no—
140C4×C67—41no—
141C4×C67—82no—
142C4×C67—82no—
143C4×C67#32712yes—
144C4×C67#32712yes—
145#11700—671nomaximal, Sylow
146C2×C67—21no—
147C2×C67—21no—
148C2×C67—42no—
149C2×C67C2×#32711yescharacteristic
150C2×C67#212311yescharacteristic
151C2×C67#213511yescharacteristic
152#2621D6711yescharacteristic
153#1742—671no—
154#1458—671no—
155#1346—674no—
156C67#1170011yescharacteristic, complemented, Sylow
157C22×M16—1341no—
158C2×C4⋊Q8Dic6711yescharacteristic
159C2×C22⋊Q8—21no—
160C2×#117C2×D6711yescharacteristic
161#431—1341no—
162#398—21no—
163#398C2×D6712yes—
164#397—21no—
165#397Dic6712yes—
166#391—21no—
167#382Dic6711yescharacteristic
168#343—672no—
169#343—1341no—
170#324—672no—
171#323—1341no—
172C22×Q8—21no—
173C23×C4C22⋊D6711yescharacteristic
174C2×M16—2681no—
175C2×M16—2682no—
176C2×M16—1344no—
177C2×M16—1342no—
178C22×C8—2681no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183#117—41no—
184#117—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C2×Dic6712yes—
189C2×C4⋊C4C22⋊D6711yescharacteristic
190C2×C4⋊C4C22×D6711yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4C2×Dic6711yescharacteristic
193C2×C22⋊C4C22⋊D6711yescharacteristic
194C2×C42C2×Dic6711yescharacteristic
195C2×C42C22⋊D6711yescharacteristic
196C4⋊C8—2681no—
197C22⋊C8—1342no—
198C4×C8—2681no—
199#95—21no—
200#95—22no—
201#95—22no—
202#95C2×Dic6712yes—
203C24C22⋊Dic6711yescharacteristic
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—42no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4C22⋊Dic6711yescharacteristic
214C22×C4C22⋊Dic6711yescharacteristic
215C22×C4C22⋊Dic6711yescharacteristic
216C22×C4#1176161312yes—
217C22×C4#1176161612yes—
218C22×C4C22×Dic6711yescharacteristic
219C22×C4C2×C22⋊D6711yescharacteristic
220C22×C4C2×C22⋊D6711yescharacteristic
221M16—2684no—
222M16—2682no—
223M16—1344no—
224C2×C8—2682no—
225C2×C8—2684no—
226C2×C8—2682no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—44no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23#1452396412yes—
246C23C2×C22⋊Dic6711yescharacteristic
247Q8—82no—
248Q8—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4#1452395811yescharacteristic
270C2×C4#1452395811yescharacteristic
271C2×C4#1452396412yes—
272C2×C4#1452397012yes—
273C2×C4C2×#1176161311yescharacteristic
274C2×C4C2×#1176161611yescharacteristic
275C8—2684no—
276C8—5362no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22C2×#1452395811yescharacteristic
285C22C2×#1452396411yescharacteristic
286C22C2×#1452397011yescharacteristic
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4#1686313012yes—
296C4#1686313012yes—
297C2—21no—
298C2—21no—
299C2—42no—
300C2C2×#1686313011yescharacteristic
301C2#2256186311yescharacteristic
302C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2621C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2621D67
hypocenterC67#11700
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
3. #22558215C2semidirect
156. C67#11700semidirect

Maximal subgroups

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

subgroupindex
#225599482
#225585232
#225582152
C67×#26212
#1170067