Finite Group Databasebeta

← #30744102

Subgroups — #30744102

Order 17152.

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

normal subgroups only

This group has 468 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
1C4×#14524043C211yesmaximal, characteristic
2#22559958C211yesmaximal, characteristic
3#22558033C214yesmaximal
4C67×#1745C211yesmaximal, characteristic
5C2×#14524043C411yescharacteristic
6C2×C4×Dic134—21no—
7C2×#14523954—21no—
8#16863302—21no—
9#16863294—21no—
10#16863294C2212yes—
11#16863266C411yescharacteristic
12C4×C4⋊Dic67—21no—
13C4×C4⋊Dic67C412yes—
14#16863151—21no—
15#16863151C2212yes—
16C2×#97×C67—21no—
17C43×C67C2211yescharacteristic
18#336×C67—21no—
19#336×C67C2212yes—
20C22×Dic134—21no—
21C22×C67⋊C8—22no—
22#14524043—24no—
23C4×Dic134—44no—
24C2×C4⋊Dic67—21no—
25C2×C4⋊Dic67—21no—
26C2×C4⋊Dic67—21no—
27C2×C4⋊Dic67C2×C412yes—
28C2×C4⋊Dic67C2×C412yes—
29C2×Dic67⋊C4—22no—
30C2×C4×Dic67—22no—
31#14523954—24no—
32C67⋊#95—22no—
33C67⋊#95—22no—
34C22×C8×C67—22no—
35C2×C42×C67C2×C411yescharacteristic
36C2×C42×C67C2×C411yescharacteristic
37C2×C42×C67D411yescharacteristic
38C2×C42×C67D411yescharacteristic
39C2×C42×C67D411yescharacteristic
40C2×C42×C67D411yescharacteristic
41C2×C42×C67C2311yescharacteristic
42#97×C67—24no—
43C2×Dic134—24no—
44C2×Dic134—42no—
45C2×Dic134—42no—
46C22×Dic67—22no—
47C22×Dic67—22no—
48C2×C67⋊C8—24no—
49C2×C67⋊C8—42no—
50C2×C67⋊C8—42no—
51C4⋊Dic67—24no—
52C4⋊Dic67—22no—
53C4⋊Dic67—42no—
54C4⋊Dic67—42no—
55C4⋊Dic67—42no—
56C4⋊Dic67M1614yes—
57Dic67⋊C4—44no—
58C4×Dic67—44no—
59C22×C4×C67C22⋊C411yescharacteristic
60C22×C4×C67C22⋊C411yescharacteristic
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22×C411yescharacteristic
64C22×C4×C67C2×D411yescharacteristic
65C22×C4×C67C2×D411yescharacteristic
66C2×C8×C67—24no—
67C2×C8×C67—42no—
68C2×C8×C67—42no—
69C42×C67—22no—
70C42×C67—22no—
71C42×C67—22no—
72C42×C67—22no—
73C42×C67—22no—
74C42×C67—22no—
75C42×C67—41no—
76C23×C67C2×C22⋊C411yescharacteristic
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67#10112yes—
93C2×C4×C67#10412yes—
94C2×C4×C67#10412yes—
95C2×C4×C67C2×M1612yes—
96C2×C4×C67AGL(1,ℤ/8)12yes—
97C2×C4×C67#13712yes—
98C2×Dic67—24no—
99C2×Dic67—42no—
100C2×Dic67—42no—
101C2×Dic67—42no—
102C2×Dic67—42no—
103C2×Dic67—42no—
104Dic134—48no—
105C8×C67—44no—
106C67⋊C8—44no—
107C22×C67#40711yescharacteristic
108C22×C67#41011yescharacteristic
109C22×C67C2×#10111yescharacteristic
110C22×C67#41811yescharacteristic
111C22×C67#41911yescharacteristic
112C22×C67C2×#10411yescharacteristic
113C22×C67C2×#10411yescharacteristic
114C4×C67—22no—
115C4×C67—22no—
116C4×C67—22no—
117C4×C67—22no—
118C4×C67—22no—
119C4×C67—22no—
120C4×C67—41no—
121Dic67—44no—
122Dic67—82no—
123#7638—671nomaximal, Sylow
124C2×C67#146711yescharacteristic
125C2×C67#149011yescharacteristic
126C2×C67#149111yescharacteristic
127C2×C67#150511yescharacteristic
128C2×C67#150611yescharacteristic
129C2×C67#152611yescharacteristic
130C2×C67#155611yescharacteristic
131C4×C4⋊Q8—671no—
132#1745—671no—
133#1745D6711yescharacteristic
134#1271—674no—
135C67#763811yescharacteristic, complemented, Sylow
136C2×C4⋊Q8—671no—
137C2×C4×Q8—1341no—
138C2×#97—21no—
139C2×#97—1341no—
140#391—1341no—
141#389—672no—
142#389—1341no—
143#384—671no—
144C4×C4⋊C4—672no—
145C4×C4⋊C4—1341no—
146C43C2×D6711yescharacteristic
147#336—21no—
148#336—672no—
149#336—1341no—
150#336C2×D6712yes—
151C22×Q8—1341no—
152C22×C8—22no—
153C22×C8—1342no—
154C4⋊Q8—1344no—
155C4×Q8—2684no—
156C2×C4⋊C4—672no—
157C2×C4⋊C4—672no—
158C2×C4⋊C4—1341no—
159C2×C4⋊C4—1341no—
160C2×C4⋊C4—1342no—
161C2×C4⋊C4—1341no—
162C2×C42—1342no—
163C2×C42C4×D6711yescharacteristic
164C2×C42C4×D6711yescharacteristic
165C2×C42D26811yescharacteristic
166C2×C42D26811yescharacteristic
167C2×C42C22⋊D6711yescharacteristic
168C2×C42C22⋊D6711yescharacteristic
169C2×C42C22×D6711yescharacteristic
170#97—24no—
171#97—1344no—
172#95—1342no—
173#95—1342no—
174C2×Q8—2682no—
175C2×Q8—2682no—
176C2×Q8—1344no—
177C22×C4—1342no—
178C22×C4—1342no—
179C22×C4(C2×D67)⋊C411yescharacteristic
180C22×C4(C2×D67)⋊C411yescharacteristic
181C22×C4(C2×D67)⋊C411yescharacteristic
182C22×C4(C2×D67)⋊C411yescharacteristic
183C22×C4C2×C4×D6711yescharacteristic
184C22×C4C2×C22⋊D6711yescharacteristic
185C22×C4C2×D26811yescharacteristic
186C2×C8—24no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—2682no—
190C2×C8—2682no—
191C2×C8—1344no—
192C4⋊C4—2682no—
193C4⋊C4—2682no—
194C4⋊C4—2682no—
195C4⋊C4—2684no—
196C4⋊C4—674no—
197C4⋊C4—1344no—
198C4⋊C4—1342no—
199C42—22no—
200C42—22no—
201C42—22no—
202C42—22no—
203C42—22no—
204C42—22no—
205C42—41no—
206C42—2684no—
207C23C2×(C2×D67)⋊C411yescharacteristic
208Q8—2688no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—2682no—
225C2×C4—2682no—
226C2×C4—2682no—
227C2×C4—2682no—
228C2×C4—2682no—
229C2×C4—1344no—
230C2×C4#1452396112yes—
231C2×C4#1452396912yes—
232C2×C4#1452397112yes—
233C2×C4M16×D6712yes—
234C2×C4#1452406812yes—
235C2×C4#1452407612yes—
236C8—44no—
237C8—2684no—
238C22#1686335911yescharacteristic
239C22#1686336911yescharacteristic
240C22C2×#1452396111yescharacteristic
241C22#1686340011yescharacteristic
242C22#1686340611yescharacteristic
243C22C2×#1452396911yescharacteristic
244C22C2×#1452397111yescharacteristic
245C4—22no—
246C4—22no—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—22no—
251C4—41no—
252C4—2684no—
253C4—5362no—
254C2#2255855211yescharacteristic
255C2#2255865111yescharacteristic
256C2#2255865711yescharacteristic
257C2#2255871611yescharacteristic
258C2#2255872211yescharacteristic
259C2#2255880311yescharacteristic
260C2#2255892711yescharacteristic
subgroupG/H
centerC23C2×(C2×D67)⋊C4
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1745C2
socle, prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#1745D67
hypocenterC67#7638
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
135. C67#7638semidirect

Maximal subgroups

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

subgroupindex
C4×#145240432
#225599582
#225580332
C67×#17452
#763867