Finite Group Databasebeta

← #30864751

Subgroups — #30864751

Order 17152.

1470 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1470 conjugacy classes of subgroups — too many to draw. Only its 251 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×#16863908C211yesmaximal, characteristic
2C2×#16863358C212yesmaximal, complemented
3#22560505C218yesmaximal, complemented
4#22560280C211yesmaximal, characteristic, complemented
5#22560120C212yesmaximal, complemented
6C67×#1818C211yesmaximal, characteristic, complemented
7C22×#11761628C2211yescharacteristic
8C23×C4×D67C2211yescharacteristic
9C22×#11761611C2214yes—
10#16863908—28no—
11C2×#14524027C412yes—
12C2×#14524023C412yes—
13C2×#14524018C412yes—
14C2×#14524013C412yes—
15C2×C4×D268C2214yes—
16#16863358—28no—
17C2×C22⋊(C67⋊C8)C2212yes—
18C2×(C2×D67)⋊C8C2214yes—
19C2×(C2×D67)⋊C8C2218yescomplemented
20#16863151C2214yescomplemented
21C2×#117×C67C2211yescharacteristic
22C2×C22⋊C8×C67C2212yes—
23#336×C67C2214yescomplemented
24C24×D67C2×C411yescharacteristic
25C2×#11761628—24no—
26C2×#11761628—44no—
27C22×Dic134C2×C411yescharacteristic
28C22×D268C2×C414yes—
29C22×D268C2×C411yescharacteristic
30C22×C22⋊D67C2×C412yes—
31C23×Dic67C2×C411yescharacteristic
32C22×C4×D67—44no—
33C22×C4×D67C2×C412yes—
34C22×C4×D67D418yes—
35C22×C4×D67C2312yes—
36C22×C4×D67C2312yes—
37C2×#11761611—28no—
38C2×#11761611—48no—
39C22×C67⋊C8C2314yes—
40#14524027—28no—
41#14524023—28no—
42#14524018—28no—
43#14524013—28no—
44C4×D268—28no—
45C4×D268—28no—
46C2×C4⋊Dic67C2×C414yes—
47C2×(C2×D67)⋊C4C2×C414yes—
48C2×(C2×D67)⋊C4C2×C414yes—
49C22⋊(C67⋊C8)—24no—
50(C2×D67)⋊C8—28no—
51(C2×D67)⋊C8—48no—
52C23×C4×C67C2311yescharacteristic
53C22×C8×C67C2314yes—
54#117×C67—24no—
55C2×C4⋊C4×C67C2×C412yes—
56C2×C22⋊C4×C67C2×C412yes—
57C2×C42×C67C2312yes—
58C22⋊C8×C67—24no—
59C23×D67—42no—
60C23×D67—44no—
61C23×D67C22⋊C414yes—
62C23×D67C22×C412yes—
63C23×D67C22×C412yes—
64#11761628—416no—
65C2×Dic134—22no—
66C2×Dic134—42no—
67C2×D268—28no—
68C2×D268—22no—
69C2×D268—48no—
70C2×D268—42no—
71C2×C22⋊D67—24no—
72C2×C22⋊D67—44no—
73C22×Dic67—42no—
74C22×Dic67C22⋊C414yes—
75C22×Dic67C22×C412yes—
76C22×Dic67C22×C412yes—
77C2×C4×D67—28no—
78C2×C4×D67—24no—
79C2×C4×D67—28no—
80C2×C4×D67—416no—
81C2×C4×D67—44no—
82C2×C4×D67C22⋊C418yes—
83C2×C4×D67C2×D414yes—
84C2×C4×D67C2×D414yes—
85#11761611—416no—
86C2×C67⋊C8—24no—
87C2×C67⋊C8—28no—
88C4⋊Dic67—28no—
89(C2×D67)⋊C4—28no—
90(C2×D67)⋊C4—28no—
91C24×C67C22×C411yescharacteristic
92C22×C4×C67—24no—
93C22×C4×C67C22×C414yes—
94C22×C4×C67C22×C411yescharacteristic
95C22×C4×C67C2×D414yes—
96C22×C4×C67C2411yescharacteristic
97C2×C8×C67—24no—
98C2×C8×C67—28no—
99C4⋊C4×C67—24no—
100C22⋊C4×C67—24no—
101C42×C67—24no—
102C23×C67—22no—
103C23×C67—24no—
104C23×C67C2×C22⋊C412yes—
105C23×C67C23×C411yescharacteristic
106C22×D67—24no—
107C22×D67—24no—
108C22×D67—24no—
109C22×D67—22no—
110C22×D67—24no—
111C22×D67—28no—
112C22×D67—44no—
113C22×D67—48no—
114C22×D67—416no—
115C22×D67C2×C22⋊C414yes—
116C2×C4×C67—22no—
117C2×C4×C67—28no—
118C2×C4×C67—24no—
119C2×C4×C67—28no—
120C2×C4×C67C2×C22⋊C412yes—
121C2×C4×C67C22×D412yes—
122C2×C4×C67H(5,𝔽2)14yes—
123C22⋊D67—48no—
124C2×Dic67—24no—
125C2×Dic67—22no—
126C2×Dic67—24no—
127C2×Dic67—44no—
128C2×Dic67C2×C22⋊C414yes—
129D268—44no—
130D268—416no—
131C4×D67—28no—
132C4×D67—28no—
133C4×D67—48no—
134C4×D67—48no—
135Dic134—44no—
136C8×C67—44no—
137C67⋊C8—44no—
138C22×C67—22no—
139C22×C67—24no—
140C22×C67—44no—
141C22×C67C22×C22⋊C411yescharacteristic
142C22×C67#51812yes—
143C22×C67#56814yes—
144C2×D67—24no—
145C2×D67—24no—
146C2×D67—24no—
147C2×D67—416no—
148C2×D67—48no—
149C2×D67—42no—
150C2×D67—44no—
151C2×D67—48no—
152C4×C67—22no—
153C4×C67—44no—
154C4×C67#53414yes—
155Dic67—24no—
156Dic67—42no—
157#21743—671nomaximal, Sylow
158C2×C67—42no—
159C2×C67#230811yescharacteristic
160C2×C67#287912yes—
161C2×C67#297814yes—
162D67—44no—
163D67—82no—
164C2×#525—671no—
165C2×#407—672no—
166#1870—678no—
167#1818—671no—
168#1818D6711yescharacteristic, complemented
169#1780—672no—
170C67#2174311yescharacteristic, complemented, Sylow
171C22×#41—671no—
172C24×C4—671no—
173C22×M16—674no—
174#525—1348no—
175C2×#123—672no—
176C2×C22⋊Q8—672no—
177C2×#121—672no—
178C2×#120—672no—
179C2×C4×D4—674no—
180C2×#117C2×D6711yescharacteristic
181#407—1348no—
182C2×C22⋊C8—678no—
183C2×C22⋊C8—672no—
184C2×C22⋊C8—674no—
185C2×C22⋊C8C2×D6712yes—
186#336—674no—
187#336C2×D6714yescomplemented
188C25—671no—
189C2×#41—2684no—
190C2×#41—1344no—
191C22×Q8—671no—
192C22×D4—672no—
193C22×D4—674no—
194C22×D4—671no—
195C23×C4—2684no—
196C23×C4—671no—
197C23×C4—672no—
198C23×C4—678no—
199C23×C4—672no—
200C23×C4—672no—
201C23×C4C22×D6711yescharacteristic
202C2×M16—2688no—
203C2×M16—1348no—
204C22×C8—674no—
205C22×C8C22×D6714yes—
206#123—1348no—
207C22⋊Q8—1348no—
208#121—1348no—
209#120—1348no—
210C4×D4—1348no—
211C4×D4—1348no—
212#117—24no—
213C2×C4⋊C4—674no—
214C2×C4⋊C4C4×D6712yes—
215C2×C22⋊C4—674no—
216C2×C22⋊C4—674no—
217C2×C22⋊C4C4×D6712yes—
218C2×C42C22×D6712yes—
219C22⋊C8—24no—
220C22⋊C8—2688no—
221C22⋊C8—1348no—
222C22⋊C8—1344no—
223C24—2682no—
224C24—2684no—
225C24—674no—
226C24—672no—
227C24—672no—
228C24C2×C4×D6711yescharacteristic
229#41—26816no—
230C2×Q8—2682no—
231C2×Q8—1342no—
232C2×D4—2682no—
233C2×D4—2688no—
234C2×D4—2684no—
235C2×D4—1348no—
236C2×D4—1344no—
237C2×D4—1342no—
238C22×C4—24no—
239C22×C4—2684no—
240C22×C4—26816no—
241C22×C4—2682no—
242C22×C4—674no—
243C22×C4—678no—
244C22×C4—672no—
245C22×C4—674no—
246C22×C4—674no—
247C22×C4—672no—
248C22×C4—1344no—
249C22×C4—1348no—
250C22×C4—1348no—
251C22×C4C2×C4×D6714yes—
252C22×C4C2×C4×D6711yescharacteristic
253C22×C4D4×D6714yes—
254C22×C4C23×D6711yescharacteristic
255M16—26816no—
256C2×C8—24no—
257C2×C8—28no—
258C2×C8—1344no—
259C2×C8—1348no—
260C4⋊C4—24no—
261C4⋊C4—1348no—
262C22⋊C4—24no—
263C22⋊C4—1348no—
264C22⋊C4—1348no—
265C42—24no—
266C23—22no—
267C23—24no—
268C23—2684no—
269C23—2688no—
270C23—26816no—
271C23—674no—
272C23—1342no—
273C23—1344no—
274C23—1348no—
275C23—1344no—
276C23—1344no—
277C23—1344no—
278C23C22⋊C4×D6712yes—
279C23C22×C4×D6711yescharacteristic
280Q8—2684no—
281D4—2684no—
282D4—26816no—
283D4—2688no—
284C2×C4—22no—
285C2×C4—28no—
286C2×C4—24no—
287C2×C4—28no—
288C2×C4—2688no—
289C2×C4—2688no—
290C2×C4—2684no—
291C2×C4—674no—
292C2×C4—1342no—
293C2×C4—1344no—
294C2×C4—1344no—
295C2×C4—1348no—
296C2×C4—1348no—
297C2×C4C22⋊C4×D6712yes—
298C2×C4C2×D4×D6712yes—
299C2×C4#1452408914yes—
300C8—44no—
301C8—2684no—
302C22—24no—
303C22—22no—
304C22—44no—
305C22—26816no—
306C22—2688no—
307C22—2688no—
308C22—2684no—
309C22—2682no—
310C22—1344no—
311C22—1344no—
312C22—1344no—
313C22C2×C22⋊C4×D6711yescharacteristic
314C22#1686387012yes—
315C22#1686416114yes—
316C4—22no—
317C4—44no—
318C4—2682no—
319C4—1344no—
320C4#1686395714yes—
321C2—42no—
322C2—2684no—
323C2—5362no—
324C2#2256278611yescharacteristic
325C2#2256865112yes—
326C2#2256927014yes—
subgroupG/H
center, FrattiniC22×C4C23×D67
commutator, socleC23×C67C23×C4
FittingC67×#1818C2
prime radicalC2×#16863908C2
cosocleC22×C4×C67C24
hypercenter#1818D67
hypocenterC67#21743
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. C2×#16863358C2semidirect
3. #22560505C2semidirect
4. #22560280C2semidirect
5. #22560120C2semidirect
6. C67×#1818C2semidirect
19. C2×(C2×D67)⋊C8C22semidirect
20. #16863151C22semidirect
23. #336×C67C22semidirect
168. #1818D67semidirect
170. C67#21743semidirect
187. #336C2×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168639082
C2×#168633582
#225605052
#225602802
#225601202
C67×#18182
#2174367