Finite Group Databasebeta

← #30765219

Subgroups — #30765219

Order 17152.

574 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 99 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#22558539C212yesmaximal, complemented
2#22558143C212yesmaximal, complemented
3#22558141C212yesmaximal, complemented
4C67×#2311C211yesmaximal, characteristic
5#16863360—22no—
6#16863122—22no—
7#16863122C2214yes—
8C22×#41×C67C411yescharacteristic
9C2×#117×C67C411yescharacteristic
10C22×C4⋊C4×C67C2211yescharacteristic
11#391×C67C411yescharacteristic
12#391×C67C2211yescharacteristic
13#390×C67C411yescharacteristic
14#390×C67C2211yescharacteristic
15#386×C67—22no—
16#380×C67—22no—
17C2×#11761612—42no—
18C22⋊(C67⋊C8)—24no—
19C22⋊(C67⋊C8)—42no—
20C2×#41×C67—22no—
21C2×#41×C67—22no—
22C2×#41×C67—42no—
23C2×#41×C67—42no—
24C22×Q8×C67C2×C411yescharacteristic
25C22×D4×C67—21no—
26C22×D4×C67C2×C411yescharacteristic
27C23×C4×C67—21no—
28C23×C4×C67C2×C411yescharacteristic
29C23×C4×C67D412yes—
30#117×C67—42no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67—42no—
33C2×C4⋊C4×C67C2×C411yescharacteristic
34C2×C4⋊C4×C67D412yes—
35C2×C4⋊C4×C67C2311yescharacteristic
36C2×C22⋊C4×C67—21no—
37C2×C22⋊C4×C67—22no—
38C2×C22⋊C4×C67—22no—
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67C2×C411yescharacteristic
41#95×C67—22no—
42#95×C67—22no—
43#11761612—84no—
44C2×C67⋊C8—44no—
45C24×C67—21no—
46C24×C67C22⋊C411yescharacteristic
47#41×C67—48no—
48#41×C67—48no—
49C2×Q8×C67—21no—
50C2×Q8×C67—41no—
51C2×Q8×C67—41no—
52C2×Q8×C67M1612yescomplemented
53C2×D4×C67—24no—
54C2×D4×C67—21no—
55C2×D4×C67—41no—
56C2×D4×C67—41no—
57C2×D4×C67—42no—
58C2×D4×C67—42no—
59C2×D4×C67M1612yescomplemented
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—24no—
65C22×C4×C67—22no—
66C22×C4×C67—21no—
67C22×C4×C67—22no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—42no—
71C22×C4×C67—42no—
72C22×C4×C67—82no—
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C412yes—
75C22×C4×C67C22×C411yescharacteristic
76C22×C4×C67C2×D412yes—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—42no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—42no—
81C22⋊C4×C67—42no—
82C22⋊C4×C67—82no—
83C22⋊C4×C67—82no—
84C42×C67—41no—
85C42×C67—41no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—22no—
89C23×C67—41no—
90C23×C67—41no—
91C23×C67—42no—
92C23×C67—42no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67#9912yes—
96C23×C67C2×C22⋊C411yescharacteristic
97Q8×C67—42no—
98Q8×C67—42no—
99D4×C67—42no—
100D4×C67—42no—
101D4×C67—48no—
102C2×C4×C67—24no—
103C2×C4×C67—24no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—22no—
109C2×C4×C67—22no—
110C2×C4×C67—22no—
111C2×C4×C67—48no—
112C2×C4×C67—44no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—82no—
124C2×C4×C67#10012yes—
125C2×C4×C67C2×M1611yescharacteristic
126C2×C4×C67C2×M1611yescharacteristic
127C67⋊C8—84no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67—22no—
131C22×C67—22no—
132C22×C67—22no—
133C22×C67—42no—
134C22×C67—42no—
135C22×C67—44no—
136C22×C67—42no—
137C22×C67—82no—
138C22×C67—81no—
139C22×C67—81no—
140C22×C67#40711yescharacteristic
141C22×C67C2×#9911yescharacteristic
142C22×C67C2×#10011yescharacteristic
143C4×C67—22no—
144C4×C67—21no—
145C4×C67—21no—
146C4×C67—42no—
147C4×C67—42no—
148C4×C67—42no—
149C4×C67—81no—
150C4×C67—81no—
151#11508—671nomaximal, Sylow
152C2×C67—21no—
153C2×C67—21no—
154C2×C67—42no—
155C2×C67—82no—
156C2×C67#146211yescharacteristic
157C2×C67#212711yescharacteristic
158C2×C67#213311yescharacteristic
159#2311D6711yescharacteristic
160#1463—672no—
161#1319—672no—
162#1318—672no—
163C67#1150811yescharacteristic, complemented, Sylow
164C22×#41Dic6711yescharacteristic
165C2×#117Dic6711yescharacteristic
166C22×C4⋊C4C2×D6711yescharacteristic
167#407—1342no—
168#391Dic6711yescharacteristic
169#391C2×D6711yescharacteristic
170#390Dic6711yescharacteristic
171#390C2×D6711yescharacteristic
172#386—22no—
173#380—22no—
174#324—674no—
175#324—1342no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—42no—
179C2×#41—42no—
180C22×Q8C2×Dic6711yescharacteristic
181C22×D4—21no—
182C22×D4C2×Dic6711yescharacteristic
183C23×C4—21no—
184C23×C4C2×Dic6711yescharacteristic
185C23×C4C22⋊D6712yes—
186C2×M16—2682no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C2×Dic6711yescharacteristic
191C2×C4⋊C4C22⋊D6712yes—
192C2×C4⋊C4C22×D6711yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C42C2×Dic6711yescharacteristic
197C2×C42C2×Dic6711yescharacteristic
198C22⋊C8—2682no—
199C22⋊C8—1344no—
200#95—22no—
201#95—22no—
202C24—21no—
203C24C22⋊Dic6711yescharacteristic
204#41—48no—
205#41—48no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8#1176161212yescomplemented
210C2×D4—24no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4#1176161212yescomplemented
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4C22⋊Dic6711yescharacteristic
231C22×C4C22⋊Dic6712yes—
232C22×C4C22×Dic6711yescharacteristic
233C22×C4C2×C22⋊D6712yes—
234M16—5364no—
235C2×C8—2684no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C42—41no—
244C42—41no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23#1452395812yes—
255C23C2×C22⋊Dic6711yescharacteristic
256Q8—42no—
257Q8—42no—
258D4—42no—
259D4—42no—
260D4—48no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—82no—
283C2×C4#1452396012yes—
284C2×C4C2×#1176161211yescharacteristic
285C2×C4C2×#1176161211yescharacteristic
286C8—5364no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—44no—
295C22—42no—
296C22—82no—
297C22—81no—
298C22—81no—
299C22#1686336011yescharacteristic
300C22C2×#1452395811yescharacteristic
301C22C2×#1452396011yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—82no—
314C2#2255853611yescharacteristic
315C2#2256187811yescharacteristic
316C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2311C2
socleC22×C67C2×#99
prime radicalC22×#41×C67C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2311D67
hypocenterC67#11508
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2311

Complements

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

NH ≅ G/Ntype
1. #22558539C2semidirect
2. #22558143C2semidirect
3. #22558141C2semidirect
52. C2×Q8×C67M16semidirect
59. C2×D4×C67M16semidirect
163. C67#11508semidirect
209. C2×Q8#11761612semidirect
216. C2×D4#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585392
#225581432
#225581412
C67×#23112
#1150867