Finite Group Databasebeta

← #30744388

Subgroups — #30744388

Order 17152.

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

normal subgroups only

This group has 394 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#22560352C211yesmaximal, characteristic
2#22559958C211yesmaximal, characteristic
3#22558032C214yesmaximal
4C67×#2859C211yesmaximal, characteristic
5C2×#14523954—21no—
6#16863151—21no—
7#16863151—21no—
8#16863151—21no—
9#16863151C2212yes—
10#16863151C2212yes—
11C2×#125×C67—21no—
12#389×C67—21no—
13#389×C67—21no—
14#389×C67C412yes—
15#389×C67C2212yes—
16#384×C67C411yescharacteristic
17#384×C67C411yescharacteristic
18#382×C67—21no—
19C43×C67C2211yescharacteristic
20C22×C67⋊C8—22no—
21C22×C67⋊C8—22no—
22#14523954—24no—
23#125×C67—44no—
24C2×C4⋊C4×C67—21no—
25C2×C4⋊C4×C67—21no—
26C2×C4⋊C4×C67—22no—
27C2×C4⋊C4×C67—21no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67—22no—
30C2×C4⋊C4×C67C2×C412yes—
31C2×C4⋊C4×C67C2×C412yes—
32C2×C42×C67C2×C411yescharacteristic
33C2×C42×C67C2×C411yescharacteristic
34C2×C42×C67D411yescharacteristic
35C2×C42×C67D411yescharacteristic
36C2×C42×C67D411yescharacteristic
37C2×C42×C67D411yescharacteristic
38C2×C42×C67C2311yescharacteristic
39#95×C67—22no—
40#95×C67—22no—
41C2×C67⋊C8—24no—
42C2×C67⋊C8—42no—
43C2×C67⋊C8—42no—
44C2×C67⋊C8—42no—
45C2×C67⋊C8—42no—
46C2×C67⋊C8—42no—
47C22×C4×C67—22no—
48C22×C4×C67—22no—
49C22×C4×C67C22⋊C411yescharacteristic
50C22×C4×C67C22⋊C411yescharacteristic
51C22×C4×C67C22⋊C411yescharacteristic
52C22×C4×C67C22⋊C411yescharacteristic
53C22×C4×C67C22×C411yescharacteristic
54C22×C4×C67C2×D411yescharacteristic
55C22×C4×C67C2×D411yescharacteristic
56C4⋊C4×C67—24no—
57C4⋊C4×C67—42no—
58C4⋊C4×C67—42no—
59C4⋊C4×C67—42no—
60C4⋊C4×C67—42no—
61C4⋊C4×C67—44no—
62C4⋊C4×C67—42no—
63C4⋊C4×C67—44no—
64C42×C67—22no—
65C42×C67—22no—
66C42×C67—22no—
67C42×C67—22no—
68C42×C67—22no—
69C42×C67—41no—
70C42×C67—41no—
71C23×C67C2×C22⋊C411yescharacteristic
72C2×C4×C67—21no—
73C2×C4×C67—21no—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
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—42no—
89C2×C4×C67—42no—
90C2×C4×C67—42no—
91C2×C4×C67—42no—
92C2×C4×C67—42no—
93C2×C4×C67—42no—
94C2×C4×C67#10112yes—
95C2×C4×C67#10412yes—
96C2×C4×C67#13112yes—
97C2×C4×C67AGL(1,ℤ/8)12yes—
98C2×C4×C67#13712yes—
99C67⋊C8—44no—
100C67⋊C8—82no—
101C22×C67#40811yescharacteristic
102C22×C67#41011yescharacteristic
103C22×C67C2×#10111yescharacteristic
104C22×C67#41811yescharacteristic
105C22×C67#41911yescharacteristic
106C22×C67C2×#10411yescharacteristic
107C22×C67#42111yescharacteristic
108C4×C67—22no—
109C4×C67—22no—
110C4×C67—22no—
111C4×C67—22no—
112C4×C67—22no—
113C4×C67—41no—
114C4×C67—41no—
115C4×C67—82no—
116C4×C67—82no—
117#7703—671nomaximal, Sylow
118C2×C67#147011yescharacteristic
119C2×C67#148411yescharacteristic
120C2×C67#149711yescharacteristic
121C2×C67#150611yescharacteristic
122C2×C67#151411yescharacteristic
123C2×C67#152811yescharacteristic
124C2×C67#155611yescharacteristic
125#2859D6711yescharacteristic
126#1839—671no—
127#1745—671no—
128#1271—674no—
129C67#770311yescharacteristic, complemented, Sylow
130C2×#125—21no—
131C2×#97—1341no—
132#389—21no—
133#389—21no—
134#389Dic6712yes—
135#389C2×D6712yes—
136#384Dic6711yescharacteristic
137#384Dic6711yescharacteristic
138#382—21no—
139C43C2×D6711yescharacteristic
140#336—672no—
141#336—672no—
142#336—1341no—
143#336—1341no—
144#336—1341no—
145C22×C8—1342no—
146C22×C8—1342no—
147#125—44no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4C2×Dic6712yes—
155C2×C4⋊C4C2×Dic6712yes—
156C2×C42C2×Dic6711yescharacteristic
157C2×C42C2×Dic6711yescharacteristic
158C2×C42C22⋊D6711yescharacteristic
159C2×C42C22⋊D6711yescharacteristic
160C2×C42C22⋊D6711yescharacteristic
161C2×C42C22⋊D6711yescharacteristic
162C2×C42C22×D6711yescharacteristic
163#97—1344no—
164#95—22no—
165#95—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4C22⋊Dic6711yescharacteristic
169C22×C4C22⋊Dic6711yescharacteristic
170C22×C4C22⋊Dic6711yescharacteristic
171C22×C4C22⋊Dic6711yescharacteristic
172C22×C4C22×Dic6711yescharacteristic
173C22×C4C2×C22⋊D6711yescharacteristic
174C22×C4C2×C22⋊D6711yescharacteristic
175C2×C8—2682no—
176C2×C8—2682no—
177C2×C8—2682no—
178C2×C8—2682no—
179C2×C8—2682no—
180C2×C8—1344no—
181C4⋊C4—24no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42—41no—
195C42—41no—
196C23C2×C22⋊Dic6711yescharacteristic
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4#1452396212yes—
220C2×C4#1452397012yes—
221C2×C4#1452405012yes—
222C2×C4#1452406912yes—
223C2×C4#1452407412yes—
224C8—2684no—
225C8—5362no—
226C22#1686336311yescharacteristic
227C22#1686337011yescharacteristic
228C22C2×#1452396211yescharacteristic
229C22#1686340111yescharacteristic
230C22#1686340511yescharacteristic
231C22C2×#1452397011yescharacteristic
232C22#1686341311yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—41no—
239C4—41no—
240C4—82no—
241C4—82no—
242C2#2255856511yescharacteristic
243C2#2255862611yescharacteristic
244C2#2255868211yescharacteristic
245C2#2255872111yescharacteristic
246C2#2255875611yescharacteristic
247C2#2255881111yescharacteristic
248C2#2255892611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2859C2
socle, prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2859D67
hypocenterC67#7703
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C43 ◁ #2859

Complements

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

NH ≅ G/Ntype
129. C67#7703semidirect

Maximal subgroups

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

subgroupindex
#225603522
#225599582
#225580322
C67×#28592
#770367