Finite Group Databasebeta

← #30749402

Subgroups — #30749402

Order 17152.

498 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 131 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#22560201C211yesmaximal, characteristic
2#22558530C211yesmaximal, characteristic
3#22558112C212yesmaximal, complemented
4#22558067C212yesmaximal, complemented
5C67×#2349C211yesmaximal, characteristic
6#16863449C2212yes—
7#16863420C2212yes—
8C2×#14523960—21no—
9#16863165—21no—
10#16863120—22no—
11C2×C4×D4×C67—21no—
12C2×C4×D4×C67C2211yescharacteristic
13#389×C67—21no—
14#385×C67—22no—
15#382×C67C2211yescharacteristic
16#380×C67C412yes—
17C4×C4⋊C4×C67C2211yescharacteristic
18C2×#95×C67C412yes—
19C2×#11761612—22no—
20C4⋊(C67⋊C8)—22no—
21C4⋊(C67⋊C8)—22no—
22C4⋊(C67⋊C8)—22no—
23#14523960—42no—
24C22⋊(C67⋊C8)—22no—
25C4×C67⋊C8—22no—
26C22×D4×C67D411yescharacteristic
27C23×C4×C67—22no—
28C23×C4×C67C2×C412yes—
29C4×D4×C67—22no—
30C4×D4×C67—22no—
31C4×D4×C67—44no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67D411yescharacteristic
36C2×C22⋊C4×C67—22no—
37C2×C22⋊C4×C67D412yes—
38C2×C42×C67—21no—
39C2×C42×C67C2311yescharacteristic
40#95×C67—22no—
41#95×C67—21no—
42#95×C67C2×C411yescharacteristic
43#95×C67C2×C412yes—
44#95×C67C2×C411yescharacteristic
45#11761612—42no—
46#11761612—42no—
47C2×C67⋊C8—42no—
48C2×C67⋊C8—42no—
49C24×C67C22⋊C412yes—
50C2×D4×C67—22no—
51C2×D4×C67—22no—
52C2×D4×C67SD1612yes—
53C2×D4×C67Q1612yes—
54C22×C4×C67—21no—
55C22×C4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—21no—
58C22×C4×C67—22no—
59C22×C4×C67—44no—
60C22×C4×C67—44no—
61C22×C4×C67—82no—
62C22×C4×C67—82no—
63C22×C4×C67C22⋊C412yes—
64C22×C4×C67C22×C411yescharacteristic
65C22×C4×C67C2×D411yescharacteristic
66C22×C4×C67C2×D411yescharacteristic
67C4⋊C4×C67—22no—
68C4⋊C4×C67—42no—
69C4⋊C4×C67—42no—
70C4⋊C4×C67—82no—
71C22⋊C4×C67—24no—
72C22⋊C4×C67—44no—
73C42×C67—42no—
74C42×C67#4112yes—
75C42×C67#4112yes—
76C23×C67—22no—
77C23×C67—22no—
78C23×C67—42no—
79C23×C67—42no—
80C23×C67#10314yes—
81C23×C67C2×C22⋊C411yescharacteristic
82D4×C67—24no—
83D4×C67—24no—
84C2×C4×C67—22no—
85C2×C4×C67—22no—
86C2×C4×C67—41no—
87C2×C4×C67—41no—
88C2×C4×C67—41no—
89C2×C4×C67—41no—
90C2×C4×C67—42no—
91C2×C4×C67—41no—
92C2×C4×C67—41no—
93C2×C4×C67—42no—
94C2×C4×C67—42no—
95C2×C4×C67—41no—
96C2×C4×C67—42no—
97C2×C4×C67—82no—
98C2×C4×C67—82no—
99C2×C4×C67—88no—
100C2×C4×C67#11711yescharacteristic
101C2×C4×C67#11711yescharacteristic
102C2×C4×C67#12312yes—
103C2×C4×C67#12312yes—
104C2×C4×C67C2×SD1611yescharacteristic
105C2×C4×C67C2×Q1611yescharacteristic
106C2×C4×C67AGL(1,ℤ/8)12yes—
107C67⋊C8—82no—
108C67⋊C8—82no—
109C22×C67—21no—
110C22×C67—21no—
111C22×C67—24no—
112C22×C67—44no—
113C22×C67—44no—
114C22×C67—42no—
115C22×C67—42no—
116C22×C67#38111yescharacteristic
117C22×C67C2×#10311yescharacteristic
118C22×C67#41811yescharacteristic
119C4×C67—22no—
120C4×C67—21no—
121C4×C67—21no—
122C4×C67—42no—
123C4×C67—42no—
124C4×C67—81no—
125C4×C67—82no—
126C4×C67#48112yes—
127C4×C67#48412yes—
128#8765—671nomaximal, Sylow
129C2×C67—21no—
130C2×C67—21no—
131C2×C67—44no—
132C2×C67#182411yescharacteristic
133C2×C67#182711yescharacteristic
134C2×C67#182911yescharacteristic
135#2349D6711yescharacteristic
136#1800—671no—
137#1460—671no—
138#1303—672no—
139#1286—672no—
140C67#876511yescharacteristic, complemented, Sylow
141C2×C4×D4—21no—
142C2×C4×D4C2×D6711yescharacteristic
143#431—672no—
144#423—672no—
145C2×#100—1341no—
146#389—21no—
147#385—22no—
148#382C2×D6711yescharacteristic
149#380Dic6712yes—
150C4×C4⋊C4C2×D6711yescharacteristic
151C2×#95Dic6712yes—
152#343—1341no—
153#323—1342no—
154C22×D4C22⋊D6711yescharacteristic
155C23×C4—22no—
156C23×C4C2×Dic6712yes—
157C2×M16—1342no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22⋊D6711yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊D6712yes—
167C2×C42—21no—
168C2×C42C22×D6711yescharacteristic
169C4⋊C8—1342no—
170C4⋊C8—1342no—
171C4⋊C8—1342no—
172#100—2682no—
173C22⋊C8—1342no—
174C4×C8—1342no—
175#95—22no—
176#95—21no—
177#95C2×Dic6711yescharacteristic
178#95C2×Dic6712yes—
179#95C2×Dic6711yescharacteristic
180C24C22⋊Dic6712yes—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4#1176161512yes—
184C2×D4#1176161812yes—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4C22⋊Dic6712yes—
195C22×C4C22×Dic6711yescharacteristic
196C22×C4C2×C22⋊D6711yescharacteristic
197C22×C4C2×C22⋊D6711yescharacteristic
198M16—2682no—
199M16—2682no—
200C2×C8—2682no—
201C2×C8—2682no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—82no—
206C22⋊C4—24no—
207C22⋊C4—44no—
208C42—42no—
209C42#1176162812yes—
210C42#1176162812yes—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23#1452396714yes—
216C23C2×C22⋊Dic6711yescharacteristic
217D4—24no—
218D4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—88no—
235C2×C4#1452400211yescharacteristic
236C2×C4#1452400211yescharacteristic
237C2×C4#1452402612yes—
238C2×C4#1452402612yes—
239C2×C4C2×#1176161511yescharacteristic
240C2×C4C2×#1176161811yescharacteristic
241C2×C4#1452406912yes—
242C8—5362no—
243C8—5362no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22#1686325911yescharacteristic
252C22C2×#1452396711yescharacteristic
253C22#1686340111yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—82no—
260C4—81no—
261C4#1686371412yes—
262C4#1686373112yes—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#2256029911yescharacteristic
267C2#2256031311yescharacteristic
268C2#2256032011yescharacteristic
subgroupG/H
centerC22#16863259
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2349C2
socleC22×C67#381
prime radicalC22×D4×C67D4
cosocleC2×C42×C67C23
hypercenter#2349D67
hypocenterC67#8765
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #22558112C2semidirect
4. #22558067C2semidirect
140. C67#8765semidirect

Maximal subgroups

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

subgroupindex
#225602012
#225585302
#225581122
#225580672
C67×#23492
#876567