Finite Group Databasebeta

← #30751556

Subgroups — #30751556

Order 17152.

694 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 139 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#22568548C211yesmaximal, characteristic
2#22560277C211yesmaximal, characteristic
3C2×#16863132C212yesmaximal
4#22558088C212yesmaximal
5C2×#376×C67C211yesmaximal, characteristic
6C2×#14524022C2211yescharacteristic
7C2×C22⋊(C67⋊C8)C2212yes—
8#16863302C412yescomplemented
9#16863294C412yescomplemented
10#16863280—22no—
11#16863269—21no—
12#16863259—21no—
13#16863256C2211yescharacteristic
14#16863151—22no—
15#16863132—24no—
16C22×C42×C67—21no—
17C22×C42×C67C2211yescharacteristic
18#376×C67—24no—
19C2×#95×C67—21no—
20C2×#95×C67C2212yes—
21C22×Dic134C2×C411yescharacteristic
22C22×C67⋊C8—22no—
23#14524022—22no—
24#14524022—22no—
25C2×C4⋊Dic67C2×C411yescharacteristic
26C2×C4⋊Dic67C2×C412yes—
27C2×Dic67⋊C4—22no—
28C2×Dic67⋊C4—21no—
29C2×C22⋊Dic67—21no—
30C2×C22⋊Dic67—21no—
31C22⋊(C67⋊C8)—22no—
32C22⋊(C67⋊C8)—22no—
33C67⋊#95—22no—
34C67⋊#95—22no—
35C67⋊#95—22no—
36C67⋊#95—21no—
37C23×C4×C67—22no—
38C23×C4×C67D411yescharacteristic
39C23×C4×C67D411yescharacteristic
40C23×C4×C67C2311yescharacteristic
41C2×C42×C67—28no—
42C2×C42×C67—24no—
43C2×C42×C67—22no—
44C2×C42×C67—22no—
45C2×C42×C67C2×C412yes—
46C2×C42×C67D412yes—
47#95×C67—24no—
48#95×C67—22no—
49#95×C67D414yes—
50C2×Dic134—22no—
51C2×Dic134—42no—
52C22×Dic67—22no—
53C22×Dic67—21no—
54C22×Dic67—21no—
55C2×C67⋊C8—44no—
56C2×C67⋊C8—42no—
57C4⋊Dic67—22no—
58C4⋊Dic67—42no—
59Dic67⋊C4—44no—
60Dic67⋊C4—42no—
61C22⋊Dic67—41no—
62C22⋊Dic67—41no—
63C22⋊Dic67—42no—
64C24×C67C2×D411yescharacteristic
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—24no—
68C22×C4×C67—28no—
69C22×C4×C67—24no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—22no—
73C22×C4×C67—22no—
74C22×C4×C67—42no—
75C22×C4×C67—44no—
76C22×C4×C67—42no—
77C22×C4×C67—42no—
78C22×C4×C67C22⋊C412yes—
79C22×C4×C67C22⋊C412yes—
80C22×C4×C67C22×C411yescharacteristic
81C22×C4×C67C2×D412yes—
82C22×C4×C67C2×D411yescharacteristic
83C22×C4×C67#4112yes—
84C42×C67—216no—
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—
94C2×C4×C67—24no—
95C2×C4×C67—24no—
96C2×C4×C67—22no—
97C2×C4×C67—22no—
98C2×C4×C67—22no—
99C2×C4×C67—22no—
100C2×C4×C67—28no—
101C2×C4×C67—22no—
102C2×C4×C67—22no—
103C2×C4×C67—22no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—44no—
107C2×C4×C67—48no—
108C2×C4×C67—48no—
109C2×C4×C67—42no—
110C2×C4×C67—44no—
111C2×C4×C67—42no—
112C2×C4×C67#10414yes—
113C2×C4×C67C4×D412yes—
114C2×C4×C67#12112yes—
115C2×Dic67—41no—
116C2×Dic67—41no—
117C2×Dic67—42no—
118C2×Dic67—42no—
119C2×Dic67—42no—
120C2×Dic67—42no—
121C2×Dic67—42no—
122Dic134—82no—
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—48no—
141C4×C67—42no—
142Dic67—81no—
143Dic67—81no—
144Dic67—82no—
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#2855—671no—
154C2×#376D6711yescharacteristic
155#1817—671no—
156C2×#328—672no—
157#1294—672no—
158C67#914511yescharacteristic, complemented, Sylow
159C2×C22⋊Q8—671no—
160C22×C42—21no—
161C22×C42C2×D6711yescharacteristic
162C2×C22⋊C8—672no—
163#391—672no—
164#389—672no—
165#387—1342no—
166#385—1341no—
167#381—1341no—
168#380—671no—
169#376—24no—
170C2×#95—21no—
171C2×#95C2×D6712yes—
172#336—1342no—
173#328—1344no—
174C22×Q8—671no—
175C23×C4—22no—
176C23×C4C22⋊D6711yescharacteristic
177C23×C4C22⋊D6711yescharacteristic
178C23×C4C22×D6711yescharacteristic
179C22×C8—1342no—
180C22⋊Q8—1342no—
181C22⋊Q8—1342no—
182C2×C4⋊C4—672no—
183C2×C4⋊C4—671no—
184C2×C4⋊C4—1342no—
185C2×C4⋊C4—1341no—
186C2×C22⋊C4—1341no—
187C2×C22⋊C4—1341no—
188C2×C42—24no—
189C2×C42—28no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C4×D6712yes—
193C2×C42D26812yes—
194C22⋊C8—1342no—
195C22⋊C8—1342no—
196#95—24no—
197#95—22no—
198#95—1342no—
199#95—1342no—
200#95—1342no—
201#95—1341no—
202#95C22⋊D6714yes—
203C24C2×C22⋊D6711yescharacteristic
204C2×Q8—2682no—
205C2×Q8—1342no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—1342no—
220C22×C4—1341no—
221C22×C4—1341no—
222C22×C4(C2×D67)⋊C412yes—
223C22×C4(C2×D67)⋊C412yes—
224C22×C4C2×C4×D6711yescharacteristic
225C22×C4C2×C22⋊D6712yes—
226C22×C4C2×D26811yescharacteristic
227C22×C4#1176162812yes—
228C2×C8—2684no—
229C2×C8—2682no—
230C4⋊C4—2682no—
231C4⋊C4—2684no—
232C4⋊C4—2682no—
233C4⋊C4—1342no—
234C22⋊C4—2681no—
235C22⋊C4—2681no—
236C22⋊C4—2682no—
237C42—216no—
238C42—24no—
239C42—24no—
240C23—22no—
241C23—22no—
242C23C2×(C2×D67)⋊C411yescharacteristic
243C23#1452401211yescharacteristic
244C23#1452402611yescharacteristic
245C23#1452406312yes—
246C23#1452407312yes—
247Q8—5362no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—28no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—48no—
261C2×C4—44no—
262C2×C4—48no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—2681no—
267C2×C4—2681no—
268C2×C4—2682no—
269C2×C4—2682no—
270C2×C4—2682no—
271C2×C4—2682no—
272C2×C4—2682no—
273C2×C4#1452397114yes—
274C2×C4C4×C22⋊D6712yes—
275C2×C4#1452401712yes—
276C8—5362no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—24no—
283C22#1686327511yescharacteristic
284C22#1686336912yes—
285C22#1686339011yescharacteristic
286C22#1686340411yescharacteristic
287C22C2×#1452397112yes—
288C22#1686355812yes—
289C22#1686373712yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—48no—
294C4—42no—
295C4—5361no—
296C4—5361no—
297C4—5362no—
298C2—22no—
299C2—22no—
300C2#2255869312yes—
301C2#2255872012yes—
302C2#2256052111yescharacteristic
303C2#2256065211yescharacteristic
304C2#2256066711yescharacteristic
subgroupG/H
centerC23C2×(C2×D67)⋊C4
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC2×#376×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenterC2×#376D67
hypocenterC67#9145
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
8. #16863302C4semidirect
9. #16863294C4semidirect
158. C67#9145semidirect

Maximal subgroups

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

subgroupindex
#225685482
#225602772
C2×#168631322
#225580882
C2×#376×C672
#914567