Finite Group Databasebeta

← #30766448

Subgroups — #30766448

Order 17152.

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

normal subgroups only

This group has 460 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
1C4×#14523958C211yesmaximal, characteristic
2#22558547C211yesmaximal, characteristic
3#22558230C214yesmaximal, complemented
4C67×#2563C211yesmaximal, characteristic
5C2×#14523958—21no—
6#16863360—21no—
7C4×C22⋊Dic67—21no—
8#16863163C2212yes—
9#16863122—21no—
10#16863120C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic, complemented
12C2×C4×D4×C67C2211yescharacteristic
13#393×C67C412yescomplemented
14#393×C67C2212yes—
15#390×C67—21no—
16#382×C67—21no—
17#380×C67—21no—
18#380×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic, complemented
20C2×#11761612—41no—
21C2×C22⋊Dic67—22no—
22C2×C4×Dic67—41no—
23#14523958—24no—
24C22⋊(C67⋊C8)—22no—
25C22⋊(C67⋊C8)—41no—
26C67⋊#95—41no—
27C22×Q8×C67C2×C412yes—
28C22×D4×C67D411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4⋊Q8×C67—41no—
32C4⋊Q8×C67—41no—
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×C67D411yescharacteristic
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67C2×C411yescharacteristic
42C2×C22⋊C4×C67C2311yescharacteristic
43C2×C42×C67—21no—
44C2×C42×C67C2×C411yescharacteristic
45#95×C67—22no—
46#95×C67—22no—
47#95×C67C2×C412yes—
48C22×Dic67—42no—
49#11761612—82no—
50C2×C67⋊C8—42no—
51C22⋊Dic67—24no—
52C22⋊Dic67—42no—
53C4×Dic67—82no—
54C24×C67C22⋊C411yescharacteristic
55C24×C67C2×D411yescharacteristic
56C2×Q8×C67—42no—
57C2×Q8×C67—82no—
58C2×D4×C67—22no—
59C2×D4×C67—22no—
60C2×D4×C67—41no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—22no—
66C22×C4×C67—22no—
67C22×C4×C67—42no—
68C22×C4×C67—42no—
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C4⋊C4×C67—41no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C22⋊C4×C67—41no—
80C22⋊C4×C67—41no—
81C22⋊C4×C67—82no—
82C22⋊C4×C67—82no—
83C42×C67—21no—
84C42×C67—21no—
85C42×C67—42no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67—21no—
90C23×C67—21no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67C2×C22⋊C411yescharacteristic
96C23×C67#13712yes—
97Q8×C67—84no—
98D4×C67—44no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—22no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—81no—
117C2×C4×C67—81no—
118C2×C4×C67#9911yescharacteristic
119C2×C4×C67#9911yescharacteristic
120C2×C4×C67#10412yes—
121C2×Dic67—44no—
122C2×Dic67—82no—
123C2×Dic67—82no—
124C67⋊C8—82no—
125C22×C67—21no—
126C22×C67—21no—
127C22×C67—22no—
128C22×C67—42no—
129C22×C67—42no—
130C22×C67—41no—
131C22×C67—41no—
132C22×C67—41no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—81no—
136C22×C67—81no—
137C22×C67C2×#9911yescharacteristic
138C22×C67#41911yescharacteristic
139C22×C67C2×#10411yescharacteristic
140C4×C67—21no—
141C4×C67—22no—
142C4×C67—41no—
143C4×C67—41no—
144C4×C67—41no—
145C4×C67—82no—
146C4×C67—82no—
147C4×C67#35412yes—
148Dic67—84no—
149#11787—671nomaximal, Sylow
150C2×C67—21no—
151C2×C67—21no—
152C2×C67—42no—
153C2×C67—81no—
154C2×C67#150811yescharacteristic
155C2×C67#212211yescharacteristic
156C2×C67C2×#35411yescharacteristic
157#2563D6711yescharacteristic
158C4×#99—671no—
159#1465—671no—
160#1351—674no—
161C67#1178711yescharacteristic, complemented, Sylow
162C2×C4⋊Q8Dic6711yescharacteristic, complemented
163C2×C4×D4C2×D6711yescharacteristic
164C2×#99—1341no—
165#407—1341no—
166#393Dic6712yescomplemented
167#393C2×D6712yes—
168#390—21no—
169#382—21no—
170#380—21no—
171#380—21no—
172C4×C4⋊C4Dic6711yescharacteristic, complemented
173C4×C22⋊C4—1341no—
174#342—672no—
175#324—1341no—
176#323—672no—
177C22×Q8C2×Dic6712yes—
178C22×D4C22⋊D6711yescharacteristic
179C23×C4C22⋊D6711yescharacteristic
180C23×C4C22⋊D6711yescharacteristic
181C2×M16—2681no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C22⋊D6711yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—1342no—
193C2×C22⋊C4C2×Dic6711yescharacteristic
194C2×C22⋊C4C22×D6711yescharacteristic
195C2×C42—21no—
196C2×C42—2681no—
197C2×C42C2×Dic6711yescharacteristic
198#99—1344no—
199C22⋊C8—2681no—
200C22⋊C8—1342no—
201#95—22no—
202#95—22no—
203#95—2681no—
204#95C2×Dic6712yes—
205C24C22⋊Dic6711yescharacteristic
206C24C2×C22⋊D6711yescharacteristic
207C2×Q8—42no—
208C2×Q8—82no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—2682no—
221C22×C4C22⋊Dic6711yescharacteristic
222C22×C4C22⋊Dic6711yescharacteristic
223C22×C4C22⋊Dic6711yescharacteristic
224C22×C4C22×Dic6711yescharacteristic
225C22×C4C2×C22⋊D6711yescharacteristic
226M16—5362no—
227C2×C8—2682no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—2682no—
238C22⋊C4—1344no—
239C42—21no—
240C42—21no—
241C42—42no—
242C42—5362no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C2×C22⋊Dic6711yescharacteristic
253C23#1452407412yes—
254Q8—84no—
255D4—44no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—2684no—
276C2×C4—5362no—
277C2×C4—5362no—
278C2×C4#1452395811yescharacteristic
279C2×C4#1452395811yescharacteristic
280C2×C4#1452397012yes—
281C8—5362no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22C2×#1452395811yescharacteristic
295C22#1686340511yescharacteristic
296C22C2×#1452397011yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—82no—
303C4—82no—
304C4—5364no—
305C4#1686319112yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2255872911yescharacteristic
311C2#2256185911yescharacteristic
312C2C2×#1686319111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2563C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2563D67
hypocenterC67#11787
LCSG ▷ C22×C4×C67 ▷ C23×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. #22558230C2semidirect
11. C2×C4⋊Q8×C67C4semidirect
13. #393×C67C4semidirect
19. C4×C4⋊C4×C67C4semidirect
161. C67#11787semidirect
162. C2×C4⋊Q8Dic67semidirect
166. #393Dic67semidirect
172. C4×C4⋊C4Dic67semidirect

Maximal subgroups

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

subgroupindex
C4×#145239582
#225585472
#225582302
C67×#25632
#1178767