Finite Group Databasebeta

← #30765315

Subgroups — #30765315

Order 17152.

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

normal subgroups only

This group has 504 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#22558547C212yesmaximal, complemented
2#22558153C212yesmaximal, complemented
3#22558141C212yesmaximal, complemented
4C67×#2328C211yesmaximal, characteristic
5#16863360—22no—
6#16863122C2214yes—
7#16863120—22no—
8C2×#127×C67C411yescharacteristic
9C2×C4×D4×C67C411yescharacteristic
10C2×C4×D4×C67C2211yescharacteristic
11#390×C67C411yescharacteristic
12#390×C67C2211yescharacteristic
13#386×C67—22no—
14#386×C67—22no—
15#382×C67C2211yescharacteristic
16#376×C67C411yescharacteristic
17C2×#11761612—42no—
18C22⋊(C67⋊C8)—24no—
19C22⋊(C67⋊C8)—42no—
20C22×D4×C67—22no—
21C22×D4×C67C2×C411yescharacteristic
22C22×D4×C67C2×C411yescharacteristic
23C23×C4×C67—21no—
24C23×C4×C67D412yes—
25#127×C67—22no—
26#127×C67—22no—
27C4×D4×C67—42no—
28C4×D4×C67—42no—
29C2×C4⋊C4×C67C2×C411yescharacteristic
30C2×C4⋊C4×C67C2311yescharacteristic
31C2×C22⋊C4×C67—21no—
32C2×C22⋊C4×C67—22no—
33C2×C22⋊C4×C67—22no—
34C2×C22⋊C4×C67D412yes—
35C2×C42×C67C2×C411yescharacteristic
36C2×C42×C67C2×C411yescharacteristic
37C2×C42×C67C2×C411yescharacteristic
38#95×C67—21no—
39#95×C67—21no—
40#95×C67—22no—
41#11761612—84no—
42C2×C67⋊C8—44no—
43C24×C67—21no—
44C24×C67C22⋊C412yes—
45C2×D4×C67—28no—
46C2×D4×C67—21no—
47C2×D4×C67—21no—
48C2×D4×C67—21no—
49C2×D4×C67—41no—
50C2×D4×C67—41no—
51C2×D4×C67—41no—
52C2×D4×C67—41no—
53C2×D4×C67—82no—
54C2×D4×C67—82no—
55C2×D4×C67M1612yescomplemented
56C22×C4×C67—21no—
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—42no—
62C22×C4×C67—82no—
63C22×C4×C67—82no—
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22×C411yescharacteristic
67C22×C4×C67C2×D412yes—
68C4⋊C4×C67—41no—
69C4⋊C4×C67—41no—
70C22⋊C4×C67—42no—
71C22⋊C4×C67—42no—
72C22⋊C4×C67—82no—
73C22⋊C4×C67—82no—
74C42×C67—22no—
75C42×C67—21no—
76C42×C67—41no—
77C42×C67M1612yescomplemented
78C23×C67—22no—
79C23×C67—22no—
80C23×C67—22no—
81C23×C67—22no—
82C23×C67—42no—
83C23×C67—42no—
84C23×C67—42no—
85C23×C67—42no—
86C23×C67—41no—
87C23×C67—41no—
88C23×C67C2×C22⋊C411yescharacteristic
89D4×C67—42no—
90D4×C67—42no—
91D4×C67—42no—
92D4×C67—42no—
93D4×C67—88no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—22no—
97C2×C4×C67—21no—
98C2×C4×C67—22no—
99C2×C4×C67—41no—
100C2×C4×C67—44no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—42no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—42no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—82no—
111C2×C4×C67—84no—
112C2×C4×C67#9912yes—
113C2×C4×C67C2×M1611yescharacteristic
114C2×C4×C67C2×M1611yescharacteristic
115C67⋊C8—84no—
116C22×C67—21no—
117C22×C67—21no—
118C22×C67—44no—
119C22×C67—44no—
120C22×C67—42no—
121C22×C67—42no—
122C22×C67—42no—
123C22×C67—42no—
124C22×C67—82no—
125C22×C67—82no—
126C22×C67—81no—
127C22×C67—81no—
128C22×C67#40711yescharacteristic
129C22×C67C2×#9911yescharacteristic
130C22×C67#41311yescharacteristic
131C4×C67—22no—
132C4×C67—21no—
133C4×C67—21no—
134C4×C67—42no—
135C4×C67—42no—
136C4×C67—41no—
137C4×C67—81no—
138C4×C67—81no—
139#11533—671nomaximal, Sylow
140C2×C67—21no—
141C2×C67—21no—
142C2×C67—82no—
143C2×C67—82no—
144C2×C67#146511yescharacteristic
145C2×C67#212711yescharacteristic
146C2×C67#213411yescharacteristic
147#2328D6711yescharacteristic
148#1465—672no—
149#1324—672no—
150#1318—672no—
151C67#1153311yescharacteristic, complemented, Sylow
152C2×#127Dic6711yescharacteristic
153C2×C4×D4Dic6711yescharacteristic
154C2×C4×D4C2×D6711yescharacteristic
155#407—1342no—
156#390Dic6711yescharacteristic
157#390C2×D6711yescharacteristic
158#386—22no—
159#386—22no—
160#382C2×D6711yescharacteristic
161#376Dic6711yescharacteristic
162#324—674no—
163#323—1342no—
164C22×D4—22no—
165C22×D4C2×Dic6711yescharacteristic
166C22×D4C2×Dic6711yescharacteristic
167C23×C4—21no—
168C23×C4C22⋊D6712yes—
169C2×M16—2682no—
170#127—22no—
171#127—22no—
172C4×D4—42no—
173C4×D4—42no—
174C2×C4⋊C4C2×Dic6711yescharacteristic
175C2×C4⋊C4C22×D6711yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4C22⋊D6712yes—
180C2×C42C2×Dic6711yescharacteristic
181C2×C42C2×Dic6711yescharacteristic
182C2×C42C2×Dic6711yescharacteristic
183C22⋊C8—2682no—
184C22⋊C8—1344no—
185#95—21no—
186#95—21no—
187#95—22no—
188C24—21no—
189C24C22⋊Dic6712yes—
190C2×D4—28no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—82no—
199C2×D4—82no—
200C2×D4#1176161212yescomplemented
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4C22⋊Dic6711yescharacteristic
210C22×C4C22⋊Dic6711yescharacteristic
211C22×C4C22×Dic6711yescharacteristic
212C22×C4C2×C22⋊D6712yes—
213M16—5364no—
214C2×C8—2684no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C42—22no—
222C42—21no—
223C42—41no—
224C42#1176161212yescomplemented
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊Dic6711yescharacteristic
236D4—42no—
237D4—42no—
238D4—42no—
239D4—42no—
240D4—88no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4#1452395812yes—
260C2×C4C2×#1176161211yescharacteristic
261C2×C4C2×#1176161211yescharacteristic
262C8—5364no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—82no—
272C22—82no—
273C22—81no—
274C22—81no—
275C22#1686336011yescharacteristic
276C22C2×#1452395811yescharacteristic
277C22#1686338011yescharacteristic
278C4—21no—
279C4—21no—
280C4—22no—
281C4—42no—
282C4—42no—
283C4—41no—
284C4—81no—
285C4—81no—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—82no—
290C2#2255854711yescharacteristic
291C2#2256187811yescharacteristic
292C2#2256190111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2328C2
socleC22×C67C2×#99
prime radicalC2×#127×C67C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2328D67
hypocenterC67#11533
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. #22558547C2semidirect
2. #22558153C2semidirect
3. #22558141C2semidirect
55. C2×D4×C67M16semidirect
77. C42×C67M16semidirect
151. C67#11533semidirect
200. C2×D4#11761612semidirect
224. C42#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585472
#225581532
#225581412
C67×#23282
#1153367