Finite Group Databasebeta

← #30765478

Subgroups — #30765478

Order 17152.

424 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 95 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#22558547C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558153C214yesmaximal, complemented
4C67×#2563C211yesmaximal, characteristic
5#16863360—21no—
6C2×C22⋊(C67⋊C8)—21no—
7#16863122C2212yes—
8#16863122C2212yes—
9#16863120—21no—
10#16863120—21no—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×C4×D4×C67C2211yescharacteristic
13#393×C67—21no—
14#393×C67—21no—
15#390×C67C412yes—
16#382×C67C2212yes—
17#380×C67—21no—
18#380×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic
20C2×#11761612—41no—
21C22×C67⋊C8—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—22no—
24C22⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—41no—
26C22×Q8×C67—21no—
27C22×D4×C67C2×C411yescharacteristic
28C23×C4×C67D411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C4⋊Q8×C67—41no—
31C4⋊Q8×C67—41no—
32C4×D4×C67—41no—
33C4×D4×C67—41no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67C2×C412yes—
37C2×C4⋊C4×C67C2311yescharacteristic
38C2×C22⋊C4×C67—22no—
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67D411yescharacteristic
41C2×C22⋊C4×C67D411yescharacteristic
42C2×C42×C67C2×C412yes—
43C2×C42×C67C2×C411yescharacteristic
44#95×C67—21no—
45#95×C67—22no—
46#95×C67—22no—
47#11761612—82no—
48C2×C67⋊C8—42no—
49C2×C67⋊C8—42no—
50C2×C67⋊C8—82no—
51C24×C67C22⋊C411yescharacteristic
52C24×C67C22⋊C411yescharacteristic
53C2×Q8×C67—42no—
54C2×Q8×C67—82no—
55C2×D4×C67—22no—
56C2×D4×C67—22no—
57C2×D4×C67—41no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67—42no—
65C22×C4×C67—42no—
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22×C411yescharacteristic
69C22×C4×C67C2×D411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C4⋊C4×C67—41no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—41no—
76C22⋊C4×C67—41no—
77C22⋊C4×C67—41no—
78C22⋊C4×C67—82no—
79C22⋊C4×C67—82no—
80C42×C67—21no—
81C42×C67—21no—
82C42×C67—24no—
83C23×C67—21no—
84C23×C67—21no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—41no—
89C23×C67—41no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67#10112yes—
93C23×C67C2×C22⋊C411yescharacteristic
94Q8×C67—84no—
95D4×C67—44no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—22no—
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—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—81no—
114C2×C4×C67—81no—
115C2×C4×C67#9911yescharacteristic
116C2×C4×C67#9911yescharacteristic
117C2×C4×C67#13112yes—
118C67⋊C8—82no—
119C67⋊C8—82no—
120C22×C67—21no—
121C22×C67—21no—
122C22×C67—22no—
123C22×C67—42no—
124C22×C67—42no—
125C22×C67—41no—
126C22×C67—41no—
127C22×C67—41no—
128C22×C67—41no—
129C22×C67—41no—
130C22×C67—81no—
131C22×C67—81no—
132C22×C67#40811yescharacteristic
133C22×C67C2×#9911yescharacteristic
134C22×C67C2×#10111yescharacteristic
135C4×C67—21no—
136C4×C67—22no—
137C4×C67—41no—
138C4×C67—41no—
139C4×C67—41no—
140C4×C67—82no—
141C4×C67—82no—
142C4×C67#35412yes—
143#11569—671nomaximal, Sylow
144C2×C67—21no—
145C2×C67—21no—
146C2×C67—42no—
147C2×C67—81no—
148C2×C67#147211yescharacteristic
149C2×C67C2×#35411yescharacteristic
150C2×C67#213411yescharacteristic
151#2563D6711yescharacteristic
152#1465—671no—
153#1459—671no—
154#1324—674no—
155C67#1156911yescharacteristic, complemented, Sylow
156C2×C4⋊Q8Dic6711yescharacteristic
157C2×C4×D4C2×D6711yescharacteristic
158#407—1341no—
159C2×C22⋊C8—1341no—
160#393—21no—
161#393—21no—
162#390Dic6712yes—
163#382C2×D6712yes—
164#380—21no—
165#380—21no—
166C4×C4⋊C4Dic6711yescharacteristic
167#324—672no—
168#324—672no—
169#323—1341no—
170#323—1341no—
171C22×Q8—21no—
172C22×D4C2×Dic6711yescharacteristic
173C23×C4C22⋊D6711yescharacteristic
174C23×C4C22⋊D6711yescharacteristic
175C2×M16—2681no—
176C22×C8—2681no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C2×Dic6712yes—
184C2×C4⋊C4C22×D6711yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4C22⋊D6711yescharacteristic
188C2×C22⋊C4C22⋊D6711yescharacteristic
189C2×C42C2×Dic6712yes—
190C2×C42C2×Dic6711yescharacteristic
191C22⋊C8—2681no—
192C22⋊C8—2681no—
193C22⋊C8—1342no—
194C22⋊C8—1342no—
195#95—21no—
196#95—22no—
197#95—22no—
198C24C22⋊Dic6711yescharacteristic
199C24C22⋊Dic6711yescharacteristic
200C2×Q8—42no—
201C2×Q8—82no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4C22⋊Dic6711yescharacteristic
214C22×C4C22⋊Dic6711yescharacteristic
215C22×C4C22×Dic6711yescharacteristic
216C22×C4C2×C22⋊D6711yescharacteristic
217C22×C4C2×C22⋊D6711yescharacteristic
218M16—5362no—
219C2×C8—2682no—
220C2×C8—2682no—
221C2×C8—5362no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C42—21no—
232C42—21no—
233C42—24no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#1452396212yes—
244C23C2×C22⋊Dic6711yescharacteristic
245Q8—84no—
246D4—44no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4#1452395811yescharacteristic
267C2×C4#1452395811yescharacteristic
268C2×C4#1452405012yes—
269C8—5362no—
270C8—5362no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22#1686336311yescharacteristic
284C22C2×#1452395811yescharacteristic
285C22C2×#1452396211yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4#1686319112yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2255857311yescharacteristic
299C2C2×#1686319111yescharacteristic
300C2#2256190111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2563C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2563D67
hypocenterC67#11569
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #22558153C2semidirect
155. C67#11569semidirect

Maximal subgroups

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

subgroupindex
#225585472
#225585272
#225581532
C67×#25632
#1156967