Finite Group Databasebeta

← #30765146

Subgroups — #30765146

Order 17152.

430 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 109 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×(C22⋊D67)⋊C8C211yesmaximal, characteristic
2#22562005C212yesmaximal, complemented
3#22558133C211yesmaximal, characteristic, complemented
4#22558132C212yesmaximal, complemented
5C67×#1314C211yesmaximal, characteristic, complemented
6C23×C67⋊C8C411yescharacteristic
7C22×C8×D67C411yescharacteristic
8C2×C4×C22⋊D67C411yescharacteristic
9(C22⋊D67)⋊C8—22no—
10(C22⋊D67)⋊C8—22no—
11C2×Dic67⋊C8C411yescharacteristic
12C2×(C2×D67)⋊C8C2211yescharacteristic
13C2×C8×Dic67C2211yescharacteristic
14Dic67⋊C16—22no—
15C22⋊(C67⋊C16)C2212yes—
16C16×Dic67—22no—
17C2×C22⋊C8×C67C2211yescharacteristic
18C22⋊C16×C67C2212yes—
19C22×C22⋊D67C811yescharacteristic
20C22×C4×D67C2×C411yescharacteristic
21C22×C67⋊C8—21no—
22C22×C67⋊C8—21no—
23C22×C67⋊C8—42no—
24C22×C67⋊C8C2×C411yescharacteristic
25C22×C67⋊C8C2×C411yescharacteristic
26C2×C8×D67—21no—
27C2×C8×D67—21no—
28C2×C8×D67—42no—
29C4×C22⋊D67—22no—
30C4×C22⋊D67—22no—
31C2×Dic67⋊C4C811yescharacteristic
32C2×C22⋊Dic67C811yescharacteristic
33C2×(C2×D67)⋊C4C811yescharacteristic
34C2×C4×Dic67C2×C411yescharacteristic
35C2×C67⋊C16—22no—
36Dic67⋊C8—22no—
37(C2×D67)⋊C8—22no—
38C8×Dic67D412yes—
39C8×Dic67D412yes—
40C23×C4×C67C2×C411yescharacteristic
41C22×C8×C67C2×C411yescharacteristic
42C22×C8×C67C2311yescharacteristic
43C2×C16×C67—22no—
44C22⋊C8×C67—22no—
45C23×D67C2×C811yescharacteristic
46C2×C22⋊D67—22no—
47C2×C22⋊D67—22no—
48C2×C22⋊D67C1614yescomplemented
49C22×Dic67C2×C811yescharacteristic
50C22×Dic67C2×C811yescharacteristic
51C22×Dic67C2×C811yescharacteristic
52C2×C4×D67—21no—
53C2×C4×D67—21no—
54C2×C4×D67—41no—
55C2×C4×D67—41no—
56C2×C67⋊C8—22no—
57C2×C67⋊C8—21no—
58C2×C67⋊C8—42no—
59C2×C67⋊C8—42no—
60C2×C67⋊C8—41no—
61C2×C67⋊C8C22⋊C411yescharacteristic
62C2×C67⋊C8C22⋊C411yescharacteristic
63C8×D67—42no—
64C8×D67—42no—
65Dic67⋊C4—22no—
66C22⋊Dic67—22no—
67(C2×D67)⋊C4—22no—
68C4×Dic67C22⋊C412yes—
69C4×Dic67M1612yes—
70C67⋊C16—42no—
71C24×C67C2×C811yescharacteristic
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—41no—
75C22×C4×C67—41no—
76C22×C4×C67C2×C811yescharacteristic
77C22×C4×C67C22×C411yescharacteristic
78C2×C8×C67—21no—
79C2×C8×C67—41no—
80C2×C8×C67—41no—
81C2×C8×C67C2×D412yes—
82C16×C67—42no—
83C23×C67—21no—
84C23×C67—21no—
85C23×C67—41no—
86C23×C67—41no—
87C23×C67C2×C1612yes—
88C23×C67C22×C811yescharacteristic
89C22×D67—21no—
90C22×D67—21no—
91C22×D67—41no—
92C22×D67—41no—
93C22×D67C2×C1612yes—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—42no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67C2×C22⋊C411yescharacteristic
102C2×C4×C67C2×M1611yescharacteristic
103C22⋊D67—28no—
104C2×Dic67—21no—
105C2×Dic67—41no—
106C2×Dic67—41no—
107C2×Dic67C22⋊C812yes—
108C2×Dic67C22⋊C812yes—
109C2×Dic67C2×C1612yes—
110C2×Dic67M3212yes—
111C4×D67—42no—
112C4×D67—41no—
113C4×D67—41no—
114C8×C67—22no—
115C8×C67—41no—
116C67⋊C8—21no—
117C67⋊C8—41no—
118C67⋊C8#10012yes—
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×C67C22×C1611yescharacteristic
128C22×C67C2×M3211yescharacteristic
129C2×D67—22no—
130C2×D67—42no—
131C2×D67—41no—
132C2×D67—41no—
133C2×D67—42no—
134C4×C67—21no—
135C4×C67—41no—
136C4×C67#41011yescharacteristic
137C4×C67C2×#10011yescharacteristic
138Dic67—22no—
139Dic67—41no—
140Dic67C22⋊C1614yes—
141#11487—671nomaximal, Sylow
142C2×C67—21no—
143C2×C67—21no—
144C2×C67—42no—
145C2×C67#146011yescharacteristic
146C2×C67C2×C22⋊C1611yescharacteristic
147C2×C67#211511yescharacteristic
148D67—42no—
149C2×C8×D4—671no—
150#2162—672no—
151#1314—671no—
152#1314—672no—
153#1314D6711yescharacteristic, complemented
154C67#1148711yescharacteristic, complemented, Sylow
155C23×C8—671no—
156C23×C8—671no—
157C2×C4×D4—671no—
158C8×D4—1342no—
159C8×D4—1342no—
160C2×C4⋊C8—671no—
161C2×C22⋊C8—671no—
162C2×C22⋊C8C2×D6711yescharacteristic
163C2×C4×C8—671no—
164C4⋊C16—1342no—
165C22⋊C16—672no—
166C22⋊C16C2×D6712yes—
167C4×C16—1342no—
168C22×D4—671no—
169C23×C4—671no—
170C23×C4C4×D6711yescharacteristic
171C22×C8—2682no—
172C22×C8—2682no—
173C22×C8—671no—
174C22×C8—671no—
175C22×C8—1341no—
176C22×C8—1341no—
177C22×C8—1341no—
178C22×C8—1341no—
179C22×C8C4×D6711yescharacteristic
180C22×C8C22×D6711yescharacteristic
181C4×D4—1342no—
182C4×D4—1342no—
183C2×C4⋊C4—671no—
184C2×C22⋊C4—671no—
185C2×C22⋊C4—671no—
186C2×C42—671no—
187C2×C16—22no—
188C2×C16—1342no—
189C4⋊C8—1342no—
190C22⋊C8—22no—
191C22⋊C8—1342no—
192C4×C8—672no—
193C4×C8—672no—
194C24—671no—
195C24C8×D6711yescharacteristic
196C2×D4—674no—
197C2×D4—1342no—
198C2×D4—1342no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—2681no—
204C22×C4—2681no—
205C22×C4—671no—
206C22×C4—671no—
207C22×C4—671no—
208C22×C4—1341no—
209C22×C4—1341no—
210C22×C4C8×D6711yescharacteristic
211C22×C4C2×C4×D6711yescharacteristic
212C2×C8—21no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—2682no—
216C2×C8—2682no—
217C2×C8—2682no—
218C2×C8—2682no—
219C2×C8—2681no—
220C2×C8—671no—
221C2×C8—671no—
222C2×C8—1341no—
223C2×C8—1342no—
224C2×C8D4×D6712yes—
225C4⋊C4—1342no—
226C22⋊C4—1342no—
227C22⋊C4—1342no—
228C42—672no—
229C42—672no—
230C16—42no—
231C16—2682no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23—2681no—
237C23—2681no—
238C23—672no—
239C23—1341no—
240C23—1341no—
241C23C16×D6712yes—
242C23C2×C8×D6711yescharacteristic
243D4—1348no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—2682no—
252C2×C4—2681no—
253C2×C4—2681no—
254C2×C4—2681no—
255C2×C4—2681no—
256C2×C4—672no—
257C2×C4—672no—
258C2×C4—672no—
259C2×C4—672no—
260C2×C4—1341no—
261C2×C4C22⋊C4×D6711yescharacteristic
262C2×C4M16×D6711yescharacteristic
263C8—22no—
264C8—41no—
265C8—2681no—
266C8—672no—
267C8—1341no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—2682no—
276C22—2682no—
277C22—2681no—
278C22—2681no—
279C22—1342no—
280C22C22⋊C8×D6711yescharacteristic
281C22C2×C16×D6711yescharacteristic
282C22M32×D6711yescharacteristic
283C4—21no—
284C4—41no—
285C4—2681no—
286C4—674no—
287C4—1342no—
288C4#1686336811yescharacteristic
289C4#100×D6711yescharacteristic
290C2—21no—
291C2—21no—
292C2—42no—
293C2—2682no—
294C2#2255852811yescharacteristic
295C2C22⋊C16×D6711yescharacteristic
296C2#2256182811yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D67
commutatorC22×C67C22×C16
FrattiniC22×C8C22×D67
FittingC67×#1314C2
socleC22×C67C2×C22⋊C8
prime radicalC22×C22⋊D67C8
cosocleC22×C8×C67C23
hypercenter#1314D67
hypocenterC67#11487
LCSG ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #22562005C2semidirect
3. #22558133C2semidirect
4. #22558132C2semidirect
5. C67×#1314C2semidirect
48. C2×C22⋊D67C16semidirect
153. #1314D67semidirect
154. C67#11487semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D67)⋊C82
#225620052
#225581332
#225581322
C67×#13142
#1148767