Finite Group Databasebeta

← #30765456

Subgroups — #30765456

Order 17152.

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

normal subgroups only

This group has 444 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#22558143C214yesmaximal, complemented
4C67×#2547C211yesmaximal, characteristic
5#16863360—21no—
6C2×C22⋊(C67⋊C8)—21no—
7#16863122C2212yes—
8#16863122C2212yes—
9#16863120—21no—
10#16863120—21no—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×#124×C67C412yes—
13C2×C4×D4×C67C2211yescharacteristic
14#397×C67—21no—
15#397×C67—21no—
16#391×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×C67C2×C412yes—
27C22×D4×C67C2×C411yescharacteristic
28C23×C4×C67D411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C4⋊Q8×C67—41no—
31C4⋊Q8×C67—41no—
32#124×C67—44no—
33C4×D4×C67—41no—
34C4×D4×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67C2311yescharacteristic
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67D411yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67C2×C412yes—
45#95×C67—21no—
46#95×C67—22no—
47#95×C67—22no—
48#11761612—82no—
49C2×C67⋊C8—42no—
50C2×C67⋊C8—42no—
51C2×C67⋊C8—82no—
52C24×C67C22⋊C411yescharacteristic
53C24×C67C22⋊C411yescharacteristic
54C2×Q8×C67—24no—
55C2×Q8×C67—42no—
56C2×Q8×C67—42no—
57C2×D4×C67—22no—
58C2×D4×C67—22no—
59C2×D4×C67—41no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—22no—
65C22×C4×C67—22no—
66C22×C4×C67—42no—
67C22×C4×C67—42no—
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22×C411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C22×C4×C67C2×D411yescharacteristic
73C4⋊C4×C67—41no—
74C4⋊C4×C67—42no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C22⋊C4×C67—41no—
79C22⋊C4×C67—41no—
80C22⋊C4×C67—44no—
81C22⋊C4×C67—44no—
82C42×C67—21no—
83C42×C67—21no—
84C42×C67—42no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67—21no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67—41no—
94C23×C67#10012yes—
95C23×C67C2×C22⋊C411yescharacteristic
96Q8×C67—84no—
97D4×C67—44no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—22no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—81no—
116C2×C4×C67—81no—
117C2×C4×C67#9911yescharacteristic
118C2×C4×C67#9911yescharacteristic
119C2×C4×C67#13112yes—
120C67⋊C8—82no—
121C67⋊C8—82no—
122C22×C67—21no—
123C22×C67—21no—
124C22×C67—22no—
125C22×C67—42no—
126C22×C67—42no—
127C22×C67—41no—
128C22×C67—41no—
129C22×C67—41no—
130C22×C67—41no—
131C22×C67—41no—
132C22×C67—81no—
133C22×C67—81no—
134C22×C67#40811yescharacteristic
135C22×C67C2×#9911yescharacteristic
136C22×C67C2×#10011yescharacteristic
137C4×C67—21no—
138C4×C67—22no—
139C4×C67—41no—
140C4×C67—41no—
141C4×C67—41no—
142C4×C67—82no—
143C4×C67—82no—
144C4×C67#35412yes—
145#11564—671nomaximal, Sylow
146C2×C67—21no—
147C2×C67—21no—
148C2×C67—42no—
149C2×C67—81no—
150C2×C67#147111yescharacteristic
151C2×C67C2×#35411yescharacteristic
152C2×C67#213511yescharacteristic
153#2547D6711yescharacteristic
154#1465—671no—
155#1459—671no—
156#1319—674no—
157C67#1156411yescharacteristic, complemented, Sylow
158C2×C4⋊Q8Dic6711yescharacteristic
159C2×#124Dic6712yes—
160C2×C4×D4C2×D6711yescharacteristic
161#407—1341no—
162C2×C22⋊C8—1341no—
163#397—21no—
164#397—21no—
165#391C2×D6712yes—
166#380—21no—
167#380—21no—
168C4×C4⋊C4Dic6711yescharacteristic
169#324—672no—
170#324—672no—
171#323—1341no—
172#323—1341no—
173C22×Q8C2×Dic6712yes—
174C22×D4C2×Dic6711yescharacteristic
175C23×C4C22⋊D6711yescharacteristic
176C23×C4C22⋊D6711yescharacteristic
177C2×M16—2681no—
178C22×C8—2681no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181#124—44no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4C22×D6711yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊D6711yescharacteristic
191C2×C22⋊C4C22⋊D6711yescharacteristic
192C2×C42C2×Dic6711yescharacteristic
193C2×C42C2×Dic6712yes—
194C22⋊C8—2681no—
195C22⋊C8—2681no—
196C22⋊C8—1342no—
197C22⋊C8—1342no—
198#95—21no—
199#95—22no—
200#95—22no—
201C24C22⋊Dic6711yescharacteristic
202C24C22⋊Dic6711yescharacteristic
203C2×Q8—24no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C22⋊Dic6711yescharacteristic
218C22×C4C22⋊Dic6711yescharacteristic
219C22×C4C22×Dic6711yescharacteristic
220C22×C4C2×C22⋊D6711yescharacteristic
221C22×C4C2×C22⋊D6711yescharacteristic
222M16—5362no—
223C2×C8—2682no—
224C2×C8—2682no—
225C2×C8—5362no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C42—21no—
236C42—21no—
237C42—42no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#1452396012yes—
248C23C2×C22⋊Dic6711yescharacteristic
249Q8—84no—
250D4—44no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4#1452395811yescharacteristic
271C2×C4#1452395811yescharacteristic
272C2×C4#1452405012yes—
273C8—5362no—
274C8—5362no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22#1686336311yescharacteristic
288C22C2×#1452395811yescharacteristic
289C22C2×#1452396011yescharacteristic
290C4—21no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4#1686319112yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#2255856911yescharacteristic
303C2C2×#1686319111yescharacteristic
304C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2547C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2547D67
hypocenterC67#11564
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #22558143C2semidirect
157. C67#11564semidirect

Maximal subgroups

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

subgroupindex
#225585472
#225585272
#225581432
C67×#25472
#1156467