Finite Group Databasebeta

← #30751557

Subgroups — #30751557

Order 17152.

694 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#16863242C211yesmaximal, characteristic
2#22560277C211yesmaximal, characteristic
3C2×#16863133C212yesmaximal
4#22558087C212yesmaximal
5C67×#2855C211yesmaximal, characteristic
6C22×C4×Dic67—21no—
7C2×C22⋊(C67⋊C8)C2212yes—
8#16863242—24no—
9C2×C67⋊#95—21no—
10C2×C67⋊#95C2212yes—
11#16863151—22no—
12#16863133—24no—
13C2×C22⋊Q8×C67C2211yescharacteristic
14C22×C42×C67C2211yescharacteristic
15#391×C67C412yescomplemented
16#389×C67C412yescomplemented
17#387×C67—22no—
18#385×C67—21no—
19#381×C67—21no—
20#380×C67C2211yescharacteristic
21C23×Dic67—22no—
22C22×C67⋊C8—22no—
23C2×C4×Dic67—24no—
24C2×C4×Dic67—28no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—22no—
27C67⋊#95—24no—
28C67⋊#95—22no—
29C67⋊#95D414yes—
30C22×Q8×C67C2×C411yescharacteristic
31C23×C4×C67D411yescharacteristic
32C23×C4×C67D411yescharacteristic
33C23×C4×C67C2311yescharacteristic
34C22⋊Q8×C67—22no—
35C22⋊Q8×C67—22no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67C2×C411yescharacteristic
39C2×C4⋊C4×C67C2×C412yes—
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67—21no—
42C2×C42×C67—22no—
43C2×C42×C67—22no—
44C2×C42×C67C2×C412yes—
45C2×C42×C67D412yes—
46#95×C67—22no—
47#95×C67—22no—
48#95×C67—22no—
49#95×C67—21no—
50C22×Dic67—28no—
51C22×Dic67—24no—
52C22×Dic67—44no—
53C22×Dic67—42no—
54C22×Dic67—42no—
55C2×C67⋊C8—44no—
56C2×C67⋊C8—42no—
57C4×Dic67—216no—
58C24×C67C2×D411yescharacteristic
59C2×Q8×C67—22no—
60C2×Q8×C67—42no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—24no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—22no—
67C22×C4×C67—22no—
68C22×C4×C67—21no—
69C22×C4×C67—22no—
70C22×C4×C67—21no—
71C22×C4×C67—42no—
72C22×C4×C67C22⋊C412yes—
73C22×C4×C67C22⋊C412yes—
74C22×C4×C67C22×C411yescharacteristic
75C22×C4×C67C2×D412yes—
76C22×C4×C67C2×D411yescharacteristic
77C22×C4×C67#4112yes—
78C4⋊C4×C67—22no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—44no—
81C4⋊C4×C67—42no—
82C22⋊C4×C67—41no—
83C22⋊C4×C67—41no—
84C22⋊C4×C67—42no—
85C42×C67—24no—
86C42×C67—24no—
87C23×C67—22no—
88C23×C67—22no—
89C23×C67C2×C22⋊C411yescharacteristic
90C23×C67#12011yescharacteristic
91C23×C67#12311yescharacteristic
92C23×C67#13512yes—
93C23×C67#13712yes—
94Q8×C67—82no—
95C2×C4×C67—24no—
96C2×C4×C67—24no—
97C2×C4×C67—22no—
98C2×C4×C67—22no—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—22no—
102C2×C4×C67—22no—
103C2×C4×C67—22no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
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#10414yes—
116C2×C4×C67C4×D412yes—
117C2×C4×C67#12112yes—
118C2×Dic67—28no—
119C2×Dic67—44no—
120C2×Dic67—48no—
121C2×Dic67—48no—
122C2×Dic67—44no—
123C67⋊C8—82no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67—21no—
127C22×C67—21no—
128C22×C67—24no—
129C22×C67—24no—
130C22×C67#38611yescharacteristic
131C22×C67#41012yes—
132C22×C67#41611yescharacteristic
133C22×C67#41911yescharacteristic
134C22×C67C2×#10412yes—
135C22×C67#45212yes—
136C22×C67#48512yes—
137C4×C67—24no—
138C4×C67—22no—
139C4×C67—22no—
140C4×C67—42no—
141C4×C67—81no—
142C4×C67—81no—
143C4×C67—82no—
144Dic67—48no—
145#9145—671nomaximal, Sylow
146C2×C67—22no—
147C2×C67—22no—
148C2×C67#150012yes—
149C2×C67#150612yes—
150C2×C67#187311yescharacteristic
151C2×C67#189611yescharacteristic
152C2×C67#189911yescharacteristic
153#2855D6711yescharacteristic
154C2×#376—671no—
155#1817—671no—
156C2×#328—672no—
157#1294—672no—
158C67#914511yescharacteristic, complemented, Sylow
159C2×C22⋊Q8C2×D6711yescharacteristic
160C22×C42—1341no—
161C22×C42C2×D6711yescharacteristic
162C2×C22⋊C8—672no—
163#391Dic6712yescomplemented
164#389Dic6712yescomplemented
165#387—22no—
166#385—21no—
167#381—21no—
168#380C2×D6711yescharacteristic
169#376—1344no—
170C2×#95—672no—
171C2×#95—1341no—
172#336—1342no—
173#328—1344no—
174C22×Q8C2×Dic6711yescharacteristic
175C23×C4—1342no—
176C23×C4C22⋊D6711yescharacteristic
177C23×C4C22⋊D6711yescharacteristic
178C23×C4C22×D6711yescharacteristic
179C22×C8—1342no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C2×Dic6711yescharacteristic
185C2×C4⋊C4C2×Dic6712yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—1344no—
191C2×C42—1348no—
192C2×C42C2×Dic6712yes—
193C2×C42C22⋊D6712yes—
194C22⋊C8—1342no—
195C22⋊C8—1342no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—674no—
201#95—1344no—
202#95—1342no—
203C24C2×C22⋊D6711yescharacteristic
204C2×Q8—22no—
205C2×Q8—42no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—2684no—
218C22×C4—2682no—
219C22×C4—2682no—
220C22×C4—1348no—
221C22×C4—1344no—
222C22×C4C22⋊Dic6712yes—
223C22×C4C22⋊Dic6712yes—
224C22×C4C22×Dic6711yescharacteristic
225C22×C4D4×D6712yes—
226C22×C4C2×C22⋊D6711yescharacteristic
227C22×C4#1176162712yes—
228C2×C8—2684no—
229C2×C8—2682no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—42no—
237C42—24no—
238C42—24no—
239C42—13416no—
240C23—22no—
241C23—22no—
242C23C2×C22⋊Dic6711yescharacteristic
243C23#1452401111yescharacteristic
244C23#1452402411yescharacteristic
245C23#1452406312yes—
246C23#1452407412yes—
247Q8—82no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—2688no—
269C2×C4—2684no—
270C2×C4—2688no—
271C2×C4—2684no—
272C2×C4—1348no—
273C2×C4#1452397014yes—
274C2×C4D4×Dic6712yes—
275C2×C4#1452401512yes—
276C8—5362no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—24no—
283C22#1686327611yescharacteristic
284C22#1686337012yes—
285C22#1686339211yescharacteristic
286C22#1686340511yescharacteristic
287C22C2×#1452397012yes—
288C22#1686355612yes—
289C22#1686373412yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—81no—
295C4—81no—
296C4—82no—
297C4—2688no—
298C2—22no—
299C2—22no—
300C2#2255869512yes—
301C2#2255872112yes—
302C2#2256052211yescharacteristic
303C2#2256065311yescharacteristic
304C2#2256066811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2855C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenter#2855D67
hypocenterC67#9145
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
15. #391×C67C4semidirect
16. #389×C67C4semidirect
158. C67#9145semidirect
163. #391Dic67semidirect
164. #389Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168632422
#225602772
C2×#168631332
#225580872
C67×#28552
#914567