Finite Group Databasebeta

← #30764995

Subgroups — #30764995

Order 17152.

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

normal subgroups only

This group has 434 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#22562829C211yesmaximal, characteristic
2#22558523C211yesmaximal, characteristic
3#22558152C214yesmaximal, complemented
4C67×#1458C211yesmaximal, characteristic
5C2×#14524002C411yescharacteristic
6C22×C4⋊Dic67C411yescharacteristic
7#16863449—21no—
8#16863294—21no—
9#16863294—21no—
10#16863268—21no—
11#16863268—21no—
12#16863265C412yes—
13#16863261C2212yes—
14#16863122C2212yes—
15#16863120—21no—
16C2×#117×C67C2211yescharacteristic
17#431×C67—21no—
18#324×C67C2212yes—
19#323×C67—21no—
20C23×Dic67—21no—
21#14524002—22no—
22#14524002C814yescomplemented
23C2×C4⋊Dic67—21no—
24C2×C4⋊Dic67—21no—
25C2×C4⋊Dic67—41no—
26C2×C4⋊Dic67—41no—
27C2×C4⋊Dic67C2×C412yes—
28C2×Dic67⋊C4—22no—
29C2×Dic67⋊C4—22no—
30C2×C22⋊Dic67—21no—
31C2×C4×Dic67C2×C412yes—
32C4⋊(C67⋊C8)—41no—
33C22⋊(C67⋊C8)—22no—
34C4×C67⋊C8—41no—
35C67⋊#95—22no—
36C67⋊#95—22no—
37C23×C4×C67C2×C411yescharacteristic
38#117×C67—41no—
39#117×C67—41no—
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C22⋊C4×C67D411yescharacteristic
43C2×C22⋊C4×C67D411yescharacteristic
44C2×C42×C67D411yescharacteristic
45C2×C42×C67D411yescharacteristic
46C4⋊C8×C67—41no—
47C22⋊C8×C67—22no—
48C4×C8×C67—41no—
49C22×Dic67—22no—
50C22×Dic67—22no—
51C22×Dic67—42no—
52C22×Dic67—82no—
53C2×C67⋊C8—42no—
54C4⋊Dic67—24no—
55C4⋊Dic67—41no—
56C4⋊Dic67—41no—
57C4⋊Dic67—42no—
58Dic67⋊C4—82no—
59Dic67⋊C4—82no—
60C22⋊Dic67—24no—
61C4×Dic67—22no—
62C4×Dic67C2×C814yes—
63C24×C67C22⋊C411yescharacteristic
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—22no—
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C2×C812yes—
71C22×C4×C67M1612yes—
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C2×C8×C67—42no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C22⋊C4×C67—41no—
79C22⋊C4×C67—41no—
80C42×C67—21no—
81C42×C67—21no—
82C42×C67—21no—
83C42×C67—21no—
84C23×C67—21no—
85C23×C67—21no—
86C23×C67—41no—
87C23×C67—41no—
88C23×C67C22⋊C812yes—
89C23×C67C2×C22⋊C411yescharacteristic
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—22no—
93C2×C4×C67—22no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67C22⋊C812yes—
105C2×C4×C67#9911yescharacteristic
106C2×C4×C67#9911yescharacteristic
107C2×C4×C67#10112yes—
108C2×C4×C67C22×C811yescharacteristic
109C2×C4×C67C2×M1611yescharacteristic
110C2×Dic67—24no—
111C2×Dic67—42no—
112C2×Dic67—42no—
113C2×Dic67—42no—
114C2×Dic67—42no—
115C2×Dic67—42no—
116C2×Dic67—84no—
117C8×C67—82no—
118C67⋊C8—82no—
119C22×C67—21no—
120C22×C67—21no—
121C22×C67—22no—
122C22×C67—42no—
123C22×C67—42no—
124C22×C67—41no—
125C22×C67—41no—
126C22×C67C2×C22⋊C811yescharacteristic
127C22×C67C2×#9911yescharacteristic
128C22×C67C2×#10111yescharacteristic
129C4×C67—21no—
130C4×C67—21no—
131C4×C67—41no—
132C4×C67—41no—
133C4×C67—41no—
134C4×C67—41no—
135C4×C67#32412yes—
136C4×C67#32412yes—
137Dic67—44no—
138Dic67—82no—
139#11449—671nomaximal, Sylow
140C2×C67—21no—
141C2×C67—21no—
142C2×C67—42no—
143C2×C67C2×#32411yescharacteristic
144C2×C67#212911yescharacteristic
145C2×C67#213411yescharacteristic
146#2317—671no—
147#1458—671no—
148#1458D6711yescharacteristic
149#1324—674no—
150C67#1144911yescharacteristic, complemented, Sylow
151C2×#117—671no—
152C2×#117C2×D6711yescharacteristic
153C22×C4⋊C4—671no—
154#431—21no—
155#431—1341no—
156#389—1341no—
157#389—1341no—
158#385—1341no—
159#385—1341no—
160#384—672no—
161#382—672no—
162#324—672no—
163#324C2×D6712yes—
164#323—21no—
165#323—1341no—
166C23×C4—1341no—
167C23×C4C4×D6711yescharacteristic
168#117—41no—
169#117—41no—
170#117—674no—
171#117—1342no—
172C2×C4⋊C4—2681no—
173C2×C4⋊C4—2681no—
174C2×C4⋊C4—672no—
175C2×C4⋊C4—1341no—
176C2×C4⋊C4—1342no—
177C2×C4⋊C4—1342no—
178C2×C4⋊C4—1341no—
179C2×C4⋊C4C4×D6711yescharacteristic
180C2×C4⋊C4C22×D6711yescharacteristic
181C2×C22⋊C4—1341no—
182C2×C22⋊C4D26811yescharacteristic
183C2×C22⋊C4C22⋊D6711yescharacteristic
184C2×C42—672no—
185C2×C42D26811yescharacteristic
186C2×C42C22⋊D6711yescharacteristic
187C4⋊C8—41no—
188C4⋊C8—2681no—
189C22⋊C8—22no—
190C22⋊C8—1342no—
191C4×C8—41no—
192C4×C8—2681no—
193#95—1342no—
194#95—1342no—
195C24(C2×D67)⋊C411yescharacteristic
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—2682no—
200C22×C4—5362no—
201C22×C4—1342no—
202C22×C4—1342no—
203C22×C4(C2×D67)⋊C411yescharacteristic
204C22×C4(C2×D67)⋊C411yescharacteristic
205C22×C4(C2×D67)⋊C411yescharacteristic
206C22×C4C8×D6712yes—
207C22×C4#1176161112yes—
208C22×C4C2×C4×D6711yescharacteristic
209C22×C4C2×C22⋊D6711yescharacteristic
210C22×C4C2×D26811yescharacteristic
211C2×C8—42no—
212C2×C8—2682no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—2682no—
216C4⋊C4—2681no—
217C4⋊C4—2681no—
218C4⋊C4—5362no—
219C4⋊C4—5362no—
220C4⋊C4—1344no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—1344no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—674no—
229C42—1342no—
230C23—21no—
231C23—21no—
232C23—41no—
233C23—41no—
234C23(C2×D67)⋊C812yes—
235C23C2×(C2×D67)⋊C411yescharacteristic
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—2682no—
251C2×C4—2682no—
252C2×C4—2682no—
253C2×C4—2682no—
254C2×C4—2682no—
255C2×C4—5364no—
256C2×C4—1344no—
257C2×C4(C2×D67)⋊C812yes—
258C2×C4#1452395711yescharacteristic
259C2×C4#1452395711yescharacteristic
260C2×C4#1452396112yes—
261C2×C4C2×C8×D6711yescharacteristic
262C2×C4C2×#1176161111yescharacteristic
263C8—82no—
264C8—5362no—
265C22—21no—
266C22—21no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22C2×(C2×D67)⋊C811yescharacteristic
273C22C2×#1452395711yescharacteristic
274C22C2×#1452396111yescharacteristic
275C4—21no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—2684no—
282C4—5362no—
283C4#1686312112yes—
284C4#1686312112yes—
285C2—21no—
286C2—21no—
287C2—42no—
288C2C2×#1686312111yescharacteristic
289C2#2256188311yescharacteristic
290C2#2256190011yescharacteristic
subgroupG/H
centerC22C2×#14523957
commutatorC2×C4×C67C22×C8
FrattiniC2×C4⋊C4C22×D67
FittingC67×#1458C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#1458D67
hypocenterC67#11449
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #22558152C2semidirect
22. #14524002C8semidirect
150. C67#11449semidirect

Maximal subgroups

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

subgroupindex
#225628292
#225585232
#225581522
C67×#14582
#1144967