Finite Group Databasebeta

← #30744396

Subgroups — #30744396

Order 17152.

500 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 119 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#22560277C211yesmaximal, characteristic
2#22559971C211yesmaximal, characteristic
3#22558032C214yesmaximal, complemented
4C67×#2857C211yesmaximal, characteristic
5C2×C22⋊(C67⋊C8)—21no—
6C2×C22⋊(C67⋊C8)—21no—
7C2×#14523954—21no—
8#16863151—21no—
9#16863151C2212yes—
10#16863151C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×C22⋊Q8×C67—21no—
13C22×C42×C67C2211yescharacteristic
14#391×C67—21no—
15#389×C67—21no—
16#389×C67C2212yes—
17#384×C67C411yescharacteristic
18#380×C67—21no—
19#380×C67C412yes—
20C22×C67⋊C8—22no—
21C22×C67⋊C8—22no—
22C22⋊(C67⋊C8)—24no—
23C22⋊(C67⋊C8)—42no—
24#14523954—42no—
25C22×Q8×C67—21no—
26C23×C4×C67C2×C411yescharacteristic
27C23×C4×C67D411yescharacteristic
28C23×C4×C67D411yescharacteristic
29C4⋊Q8×C67—42no—
30C22⋊Q8×C67—44no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67C2×C412yes—
36C2×C4⋊C4×C67C2×C412yes—
37C2×C22⋊C4×C67—22no—
38C2×C42×C67—42no—
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67D411yescharacteristic
41C2×C42×C67D411yescharacteristic
42C2×C42×C67C2311yescharacteristic
43#95×C67—22no—
44#95×C67—22no—
45C2×C67⋊C8—24no—
46C2×C67⋊C8—42no—
47C2×C67⋊C8—42no—
48C2×C67⋊C8—42no—
49C2×C67⋊C8—42no—
50C2×C67⋊C8—42no—
51C24×C67C22⋊C411yescharacteristic
52C2×Q8×C67—42no—
53C2×Q8×C67—42no—
54C2×Q8×C67—42no—
55C22×C4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—22no—
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—22no—
63C22×C4×C67—41no—
64C22×C4×C67—41no—
65C22×C4×C67—41no—
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22×C411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C4⋊C4×C67—24no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—42no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—44no—
78C22⋊C4×C67—44no—
79C42×C67—22no—
80C42×C67—22no—
81C42×C67—41no—
82C42×C67—41no—
83C23×C67—21no—
84C23×C67—21no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67#10112yes—
90C23×C67C2×C22⋊C411yescharacteristic
91Q8×C67—84no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—44no—
106C2×C4×C67—44no—
107C2×C4×C67—44no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67#9912yes—
115C2×C4×C67#10412yes—
116C2×C4×C67#13112yes—
117C2×C4×C67AGL(1,ℤ/8)12yes—
118C2×C4×C67#13712yes—
119C67⋊C8—44no—
120C67⋊C8—82no—
121C22×C67—22no—
122C22×C67—22no—
123C22×C67—22no—
124C22×C67—41no—
125C22×C67—41no—
126C22×C67—41no—
127C22×C67—41no—
128C22×C67#40811yescharacteristic
129C22×C67C2×#9911yescharacteristic
130C22×C67C2×#10111yescharacteristic
131C22×C67#41811yescharacteristic
132C22×C67#41911yescharacteristic
133C22×C67C2×#10411yescharacteristic
134C22×C67#42111yescharacteristic
135C4×C67—22no—
136C4×C67—22no—
137C4×C67—22no—
138C4×C67—22no—
139C4×C67—22no—
140C4×C67—41no—
141C4×C67—82no—
142C4×C67—82no—
143#7705—671nomaximal, Sylow
144C2×C67—42no—
145C2×C67#147211yescharacteristic
146C2×C67#148511yescharacteristic
147C2×C67#149611yescharacteristic
148C2×C67#150711yescharacteristic
149C2×C67#151111yescharacteristic
150C2×C67#153111yescharacteristic
151C2×C67#155611yescharacteristic
152#2857D6711yescharacteristic
153#1817—671no—
154#1749—671no—
155#1271—674no—
156C67#770511yescharacteristic, complemented, Sylow
157C2×C4⋊Q8Dic6711yescharacteristic
158C2×C22⋊Q8—21no—
159C22×C42C2×D6711yescharacteristic
160C2×C22⋊C8—1341no—
161C2×C22⋊C8—1341no—
162C2×#97—1341no—
163#391—21no—
164#389—21no—
165#389C2×D6712yes—
166#384Dic6711yescharacteristic
167#380—21no—
168#380Dic6712yes—
169#336—672no—
170#336—672no—
171#336—1341no—
172C22×Q8—21no—
173C23×C4C2×Dic6711yescharacteristic
174C23×C4C22⋊D6711yescharacteristic
175C23×C4C22⋊D6711yescharacteristic
176C22×C8—1342no—
177C22×C8—1342no—
178C4⋊Q8—42no—
179C22⋊Q8—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C2×Dic6712yes—
185C2×C4⋊C4C2×Dic6712yes—
186C2×C22⋊C4—22no—
187C2×C42—42no—
188C2×C42C2×Dic6711yescharacteristic
189C2×C42C22⋊D6711yescharacteristic
190C2×C42C22⋊D6711yescharacteristic
191C2×C42C22×D6711yescharacteristic
192C22⋊C8—2682no—
193C22⋊C8—1344no—
194#97—2682no—
195#95—22no—
196#95—22no—
197C24C22⋊Dic6711yescharacteristic
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4C22⋊Dic6711yescharacteristic
213C22×C4C22⋊Dic6711yescharacteristic
214C22×C4C22⋊Dic6711yescharacteristic
215C22×C4C22×Dic6711yescharacteristic
216C22×C4C2×C22⋊D6711yescharacteristic
217C22×C4C2×C22⋊D6711yescharacteristic
218C2×C8—2682no—
219C2×C8—2682no—
220C2×C8—2682no—
221C2×C8—2682no—
222C2×C8—2682no—
223C2×C8—1344no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C22⋊C4—44no—
231C42—22no—
232C42—22no—
233C42—41no—
234C42—41no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23#1452396212yes—
242C23C2×C22⋊Dic6711yescharacteristic
243Q8—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4#1452395812yes—
267C2×C4#1452397012yes—
268C2×C4#1452405012yes—
269C2×C4#1452406912yes—
270C2×C4#1452407412yes—
271C8—2684no—
272C8—5362no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22#1686336311yescharacteristic
281C22C2×#1452395811yescharacteristic
282C22C2×#1452396211yescharacteristic
283C22#1686340111yescharacteristic
284C22#1686340511yescharacteristic
285C22C2×#1452397011yescharacteristic
286C22#1686341311yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—82no—
294C4—82no—
295C2—42no—
296C2#2255857311yescharacteristic
297C2#2255863011yescharacteristic
298C2#2255867811yescharacteristic
299C2#2255872511yescharacteristic
300C2#2255874211yescharacteristic
301C2#2255882411yescharacteristic
302C2#2255892611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2857C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2857D67
hypocenterC67#7705
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
3. #22558032C2semidirect
156. C67#7705semidirect

Maximal subgroups

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

subgroupindex
#225602772
#225599712
#225580322
C67×#28572
#770567