Finite Group Databasebeta

← #30766124

Subgroups — #30766124

Order 17152.

448 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 115 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#22559942C211yesmaximal, characteristic
2#22558523C211yesmaximal, characteristic
3#22558233C214yesmaximal
4C67×#2628C211yesmaximal, characteristic
5C2×#14524002—21no—
6#16863449—21no—
7#16863163—21no—
8#16863163C2212yes—
9#16863122—21no—
10#16863120C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic, complemented
12C2×#117×C67C2211yescharacteristic
13#400×C67C412yescomplemented
14#399×C67C2212yes—
15#395×C67—21no—
16#393×C67—21no—
17#389×C67—21no—
18#382×C67C411yescharacteristic, complemented
19#380×C67—21no—
20#14524002—24no—
21#14524002—42no—
22C2×C4⋊Dic67—41no—
23C2×C22⋊Dic67—22no—
24C2×C4×Dic67—41no—
25C4⋊(C67⋊C8)—41no—
26C22⋊(C67⋊C8)—22no—
27C4×C67⋊C8—41no—
28C22×Q8×C67—21no—
29C23×C4×C67D411yescharacteristic
30C4⋊Q8×C67—41no—
31C4⋊Q8×C67—41no—
32#117×C67—41no—
33#117×C67—41no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67C2×C412yes—
38C2×C4⋊C4×C67C2×C411yescharacteristic
39C2×C4⋊C4×C67D411yescharacteristic
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67C2311yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67D411yescharacteristic
45#95×C67—22no—
46#95×C67—21no—
47#95×C67—22no—
48#95×C67C2×C412yes—
49C22×Dic67—42no—
50C2×C67⋊C8—42no—
51C4⋊Dic67—44no—
52C22⋊Dic67—24no—
53C22⋊Dic67—42no—
54C4×Dic67—44no—
55C24×C67C2×D411yescharacteristic
56C2×Q8×C67—42no—
57C2×Q8×C67—82no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—22no—
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67SD1612yes—
68C22×C4×C67Q1612yes—
69C22×C4×C67C22×C411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C4⋊C4×C67—41no—
72C4⋊C4×C67—41no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—82no—
78C22⋊C4×C67—41no—
79C22⋊C4×C67—41no—
80C22⋊C4×C67—82no—
81C42×C67—21no—
82C42×C67—21no—
83C42×C67—21no—
84C42×C67—21no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67—41no—
88C23×C67—41no—
89C23×C67C2×C22⋊C411yescharacteristic
90C23×C67C2×SD1611yescharacteristic
91C23×C67C2×Q1611yescharacteristic
92Q8×C67—84no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—22no—
96C2×C4×C67—22no—
97C2×C4×C67—22no—
98C2×C4×C67—22no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67#9911yescharacteristic
114C2×C4×C67#9911yescharacteristic
115C2×C4×C67#10312yes—
116C2×C4×C67#10312yes—
117C2×C4×C67#10412yes—
118C2×Dic67—44no—
119C2×Dic67—82no—
120C2×Dic67—82no—
121C67⋊C8—82no—
122C22×C67—21no—
123C22×C67—21no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67—41no—
127C22×C67—41no—
128C22×C67—41no—
129C22×C67—41no—
130C22×C67—81no—
131C22×C67C2×#9911yescharacteristic
132C22×C67C2×#10311yescharacteristic
133C22×C67C2×#10411yescharacteristic
134C4×C67—22no—
135C4×C67—41no—
136C4×C67—41no—
137C4×C67—41no—
138C4×C67—41no—
139C4×C67—82no—
140C4×C67—82no—
141C4×C67#32812yes—
142C4×C67#32812yes—
143Dic67—84no—
144#11718—671nomaximal, Sylow
145C2×C67—21no—
146C2×C67—21no—
147C2×C67—41no—
148C2×C67—41no—
149C2×C67C2×#32811yescharacteristic
150C2×C67#212311yescharacteristic
151C2×C67#212811yescharacteristic
152#2628D6711yescharacteristic
153#1740—671no—
154#1458—671no—
155#1352—674no—
156C67#1171811yescharacteristic, complemented, Sylow
157C2×C4⋊Q8Dic6711yescharacteristic, complemented
158C2×#117—1341no—
159C2×#117C2×D6711yescharacteristic
160#431—1341no—
161#400Dic6712yescomplemented
162#399C2×D6712yes—
163#395—21no—
164#393—21no—
165#389—21no—
166#382Dic6711yescharacteristic, complemented
167#380—21no—
168#342—672no—
169#342—1341no—
170#324—1341no—
171#323—672no—
172C22×Q8—21no—
173C23×C4C22⋊D6711yescharacteristic
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—41no—
177#117—41no—
178#117—2682no—
179#117—1344no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—2681no—
184C2×C4⋊C4C2×Dic6712yes—
185C2×C4⋊C4C2×Dic6711yescharacteristic
186C2×C4⋊C4C22⋊D6711yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1342no—
189C2×C22⋊C4C22⋊D6711yescharacteristic
190C2×C22⋊C4C22×D6711yescharacteristic
191C2×C42—2681no—
192C2×C42C2×Dic6711yescharacteristic
193C2×C42C22⋊D6711yescharacteristic
194C4⋊C8—2681no—
195C22⋊C8—1342no—
196C4×C8—2681no—
197#95—22no—
198#95—21no—
199#95—22no—
200#95C2×Dic6712yes—
201C24C2×C22⋊D6711yescharacteristic
202C2×Q8—42no—
203C2×Q8—82no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—2682no—
210C22×C4C22⋊Dic6711yescharacteristic
211C22×C4C22⋊Dic6711yescharacteristic
212C22×C4C22⋊Dic6711yescharacteristic
213C22×C4C22⋊Dic6711yescharacteristic
214C22×C4#1176161512yes—
215C22×C4#1176161812yes—
216C22×C4C22×Dic6711yescharacteristic
217C22×C4C2×C22⋊D6711yescharacteristic
218C2×C8—2682no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C4⋊C4—2684no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C22⋊C4—2682no—
231C22⋊C4—1344no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—2684no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×C22⋊Dic6711yescharacteristic
242C23C2×#1176161511yescharacteristic
243C23C2×#1176161811yescharacteristic
244Q8—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—2684no—
266C2×C4—5362no—
267C2×C4—5362no—
268C2×C4#1452395811yescharacteristic
269C2×C4#1452395811yescharacteristic
270C2×C4#1452396712yes—
271C2×C4#1452396712yes—
272C2×C4#1452397012yes—
273C8—5362no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#1452395811yescharacteristic
284C22C2×#1452396711yescharacteristic
285C22C2×#1452397011yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4—5364no—
294C4#1686313312yes—
295C4#1686313312yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#1686313311yescharacteristic
301C2#2256186311yescharacteristic
302C2#2256188111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2628C2
socleC22×C67C2×#99
prime radicalC24×C67C2×D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2628D67
hypocenterC67#11718
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C67C4semidirect
13. #400×C67C4semidirect
18. #382×C67C4semidirect
156. C67#11718semidirect
157. C2×C4⋊Q8Dic67semidirect
161. #400Dic67semidirect
166. #382Dic67semidirect

Maximal subgroups

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

subgroupindex
#225599422
#225585232
#225582332
C67×#26282
#1171867