Finite Group Databasebeta

← #30766405

Subgroups — #30766405

Order 17152.

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

normal subgroups only

This group has 480 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#22558209C214yesmaximal, complemented
4C67×#2547C211yesmaximal, characteristic
5C2×#14523958—21no—
6#16863360—21no—
7C4×C22⋊Dic67—21no—
8#16863163C2212yes—
9#16863122—21no—
10#16863120C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic, complemented
12C2×#124×C67—21no—
13C2×C4×D4×C67C2211yescharacteristic
14#397×C67C412yescomplemented
15#397×C67C2212yes—
16#391×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×C67—21no—
28C22×D4×C67D411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4⋊Q8×C67—41no—
32C4⋊Q8×C67—41no—
33#124×C67—44no—
34C4×D4×C67—41no—
35C4×D4×C67—41no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67C2×C412yes—
39C2×C4⋊C4×C67D411yescharacteristic
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67—22no—
42C2×C22⋊C4×C67C2×C411yescharacteristic
43C2×C22⋊C4×C67C2311yescharacteristic
44C2×C42×C67—21no—
45C2×C42×C67C2×C411yescharacteristic
46#95×C67—22no—
47#95×C67—22no—
48#95×C67C2×C412yes—
49C22×Dic67—42no—
50#11761612—82no—
51C2×C67⋊C8—42no—
52C22⋊Dic67—24no—
53C22⋊Dic67—42no—
54C4×Dic67—82no—
55C24×C67C22⋊C411yescharacteristic
56C24×C67C2×D411yescharacteristic
57C2×Q8×C67—42no—
58C2×Q8×C67—42no—
59C2×Q8×C67—42no—
60C2×D4×C67—22no—
61C2×D4×C67—22no—
62C2×D4×C67—41no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—22no—
68C22×C4×C67—22no—
69C22×C4×C67—42no—
70C22×C4×C67—42no—
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22×C411yescharacteristic
75C22×C4×C67C2×D411yescharacteristic
76C4⋊C4×C67—41no—
77C4⋊C4×C67—42no—
78C4⋊C4×C67—42no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C22⋊C4×C67—41no—
82C22⋊C4×C67—41no—
83C22⋊C4×C67—44no—
84C22⋊C4×C67—44no—
85C42×C67—21no—
86C42×C67—21no—
87C42×C67—42no—
88C23×C67—21no—
89C23×C67—21no—
90C23×C67—21no—
91C23×C67—21no—
92C23×C67—21no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67C2×C22⋊C411yescharacteristic
98C23×C67AGL(1,ℤ/8)12yes—
99Q8×C67—84no—
100D4×C67—44no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—22no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—81no—
119C2×C4×C67—81no—
120C2×C4×C67#9911yescharacteristic
121C2×C4×C67#9911yescharacteristic
122C2×C4×C67#10412yes—
123C2×Dic67—44no—
124C2×Dic67—82no—
125C2×Dic67—82no—
126C67⋊C8—82no—
127C22×C67—21no—
128C22×C67—21no—
129C22×C67—22no—
130C22×C67—42no—
131C22×C67—42no—
132C22×C67—41no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—41no—
136C22×C67—41no—
137C22×C67—81no—
138C22×C67—81no—
139C22×C67C2×#9911yescharacteristic
140C22×C67#41811yescharacteristic
141C22×C67C2×#10411yescharacteristic
142C4×C67—21no—
143C4×C67—22no—
144C4×C67—41no—
145C4×C67—41no—
146C4×C67—41no—
147C4×C67—82no—
148C4×C67—82no—
149C4×C67#35412yes—
150Dic67—84no—
151#11777—671nomaximal, Sylow
152C2×C67—21no—
153C2×C67—21no—
154C2×C67—42no—
155C2×C67—81no—
156C2×C67#150711yescharacteristic
157C2×C67#212311yescharacteristic
158C2×C67C2×#35411yescharacteristic
159#2547D6711yescharacteristic
160C4×#99—671no—
161#1465—671no—
162#1344—674no—
163C67#1177711yescharacteristic, complemented, Sylow
164C2×C4⋊Q8Dic6711yescharacteristic, complemented
165C2×#124—21no—
166C2×C4×D4C2×D6711yescharacteristic
167C2×#99—1341no—
168#407—1341no—
169#397Dic6712yescomplemented
170#397C2×D6712yes—
171#391—21no—
172#380—21no—
173#380—21no—
174C4×C4⋊C4Dic6711yescharacteristic, complemented
175C4×C22⋊C4—1341no—
176#342—672no—
177#324—1341no—
178#323—672no—
179C22×Q8—21no—
180C22×D4C22⋊D6711yescharacteristic
181C23×C4C22⋊D6711yescharacteristic
182C23×C4C22⋊D6711yescharacteristic
183C2×M16—2681no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186#124—44no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×Dic6712yes—
192C2×C4⋊C4C22⋊D6711yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1342no—
196C2×C22⋊C4C2×Dic6711yescharacteristic
197C2×C22⋊C4C22×D6711yescharacteristic
198C2×C42—21no—
199C2×C42—2681no—
200C2×C42C2×Dic6711yescharacteristic
201#99—1344no—
202C22⋊C8—2681no—
203C22⋊C8—1342no—
204#95—22no—
205#95—22no—
206#95—2681no—
207#95C2×Dic6712yes—
208C24C22⋊Dic6711yescharacteristic
209C24C2×C22⋊D6711yescharacteristic
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—2682no—
225C22×C4C22⋊Dic6711yescharacteristic
226C22×C4C22⋊Dic6711yescharacteristic
227C22×C4C22⋊Dic6711yescharacteristic
228C22×C4C22×Dic6711yescharacteristic
229C22×C4C2×C22⋊D6711yescharacteristic
230M16—5362no—
231C2×C8—2682no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—2682no—
242C22⋊C4—1344no—
243C42—21no—
244C42—21no—
245C42—42no—
246C42—5362no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C2×C22⋊Dic6711yescharacteristic
257C23#1452406912yes—
258Q8—84no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—2684no—
280C2×C4—5362no—
281C2×C4—5362no—
282C2×C4#1452395811yescharacteristic
283C2×C4#1452395811yescharacteristic
284C2×C4#1452397012yes—
285C8—5362no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22C2×#1452395811yescharacteristic
299C22#1686340111yescharacteristic
300C22C2×#1452397011yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—82no—
307C4—82no—
308C4—5364no—
309C4#1686319112yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2255872511yescharacteristic
315C2#2256186311yescharacteristic
316C2C2×#1686319111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2547C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2547D67
hypocenterC67#11777
LCSG ▷ C22×C4×C67 ▷ C23×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. #22558209C2semidirect
11. C2×C4⋊Q8×C67C4semidirect
14. #397×C67C4semidirect
19. C4×C4⋊C4×C67C4semidirect
163. C67#11777semidirect
164. C2×C4⋊Q8Dic67semidirect
169. #397Dic67semidirect
174. 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
#225582092
C67×#25472
#1177767