Finite Group Databasebeta

← #30744146

Subgroups — #30744146

Order 17152.

468 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 119 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#22560352C212yesmaximal, complemented
2#22558035C212yesmaximal, complemented
3#22558030C212yesmaximal
4C4×#124×C67C211yesmaximal, characteristic
5#16863151—22no—
6#16863151—22no—
7#16863151C2214yes—
8C2×#124×C67C411yescharacteristic
9C2×C4×Q8×C67C411yescharacteristic
10C2×C4×D4×C67C411yescharacteristic
11#391×C67C2211yescharacteristic
12#390×C67C2211yescharacteristic
13#388×C67—22no—
14#382×C67C411yescharacteristic
15C4×C22⋊C4×C67—22no—
16C43×C67C2211yescharacteristic
17C22×C67⋊C8—24no—
18C22×Q8×C67C2×C411yescharacteristic
19C22×D4×C67C2×C411yescharacteristic
20C23×C4×C67—21no—
21#124×C67—22no—
22#124×C67—22no—
23C4×Q8×C67—42no—
24C4×D4×C67—42no—
25C2×C4⋊C4×C67—21no—
26C2×C4⋊C4×C67C2×C411yescharacteristic
27C2×C4⋊C4×C67C2×C411yescharacteristic
28C2×C22⋊C4×C67—22no—
29C2×C22⋊C4×C67—21no—
30C2×C22⋊C4×C67—22no—
31C2×C42×C67—21no—
32C2×C42×C67C2×C411yescharacteristic
33C2×C42×C67C2×C411yescharacteristic
34C2×C42×C67D412yes—
35C2×C42×C67D412yes—
36C2×C42×C67C2311yescharacteristic
37#95×C67—22no—
38#95×C67—22no—
39C2×C67⋊C8—44no—
40C2×C67⋊C8—44no—
41C2×C67⋊C8—44no—
42C24×C67—21no—
43C2×Q8×C67—21no—
44C2×Q8×C67—41no—
45C2×Q8×C67—41no—
46C2×Q8×C67M1612yes—
47C2×D4×C67—21no—
48C2×D4×C67—41no—
49C2×D4×C67—41no—
50C2×D4×C67M1612yes—
51C22×C4×C67—21no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67—44no—
55C22×C4×C67—42no—
56C22×C4×C67C22⋊C411yescharacteristic
57C22×C4×C67C22⋊C412yes—
58C22×C4×C67C22⋊C411yescharacteristic
59C22×C4×C67C22×C411yescharacteristic
60C22×C4×C67C2×D412yes—
61C4⋊C4×C67—41no—
62C4⋊C4×C67—41no—
63C4⋊C4×C67—42no—
64C22⋊C4×C67—28no—
65C22⋊C4×C67—42no—
66C22⋊C4×C67—44no—
67C42×C67—22no—
68C42×C67—21no—
69C42×C67—21no—
70C42×C67—24no—
71C42×C67—42no—
72C42×C67—41no—
73C42×C67—42no—
74C23×C67—22no—
75C23×C67—41no—
76C23×C67—41no—
77C23×C67—42no—
78C23×C67—42no—
79C23×C67C2×C22⋊C411yescharacteristic
80Q8×C67—42no—
81Q8×C67—42no—
82D4×C67—42no—
83D4×C67—42no—
84C2×C4×C67—21no—
85C2×C4×C67—22no—
86C2×C4×C67—21no—
87C2×C4×C67—22no—
88C2×C4×C67—22no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—21no—
92C2×C4×C67—22no—
93C2×C4×C67—22no—
94C2×C4×C67—22no—
95C2×C4×C67—41no—
96C2×C4×C67—41no—
97C2×C4×C67—41no—
98C2×C4×C67—41no—
99C2×C4×C67—42no—
100C2×C4×C67—42no—
101C2×C4×C67—84no—
102C2×C4×C67#10012yes—
103C2×C4×C67C2×M1611yescharacteristic
104C2×C4×C67C2×M1611yescharacteristic
105C2×C4×C67#13514yes—
106C67⋊C8—84no—
107C22×C67—42no—
108C22×C67—44no—
109C22×C67—81no—
110C22×C67—82no—
111C22×C67—81no—
112C22×C67#40711yescharacteristic
113C22×C67#41011yescharacteristic
114C22×C67C2×#10011yescharacteristic
115C22×C67#41612yes—
116C22×C67#42112yes—
117C4×C67—21no—
118C4×C67—21no—
119C4×C67—24no—
120C4×C67—22no—
121C4×C67—41no—
122C4×C67—42no—
123C4×C67—42no—
124C4×C67—81no—
125C4×C67—81no—
126#7649—671nomaximal, Sylow
127C2×C67—82no—
128C2×C67#146611yescharacteristic
129C2×C67#148812yes—
130C2×C67#152112yes—
131C2×C67#153311yescharacteristic
132C2×C67#154311yescharacteristic
133C4×#124D6711yescharacteristic
134#1839—672no—
135#1272—672no—
136#1270—672no—
137C67#764911yescharacteristic, complemented, Sylow
138C2×#124Dic6711yescharacteristic
139C2×C4×Q8Dic6711yescharacteristic
140C2×C4×D4Dic6711yescharacteristic
141#391C2×D6711yescharacteristic
142#390C2×D6711yescharacteristic
143#388—22no—
144#382Dic6711yescharacteristic
145C4×C22⋊C4—22no—
146C43C2×D6711yescharacteristic
147#336—674no—
148#336—1342no—
149#336—1342no—
150C22×Q8C2×Dic6711yescharacteristic
151C22×D4C2×Dic6711yescharacteristic
152C23×C4—21no—
153C22×C8—1344no—
154#124—22no—
155#124—22no—
156C4×Q8—42no—
157C4×D4—42no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×Dic6711yescharacteristic
160C2×C4⋊C4C2×Dic6711yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42—21no—
165C2×C42C2×Dic6711yescharacteristic
166C2×C42C2×Dic6711yescharacteristic
167C2×C42C22⋊D6712yes—
168C2×C42C22⋊D6712yes—
169C2×C42C22×D6711yescharacteristic
170#95—22no—
171#95—22no—
172C24—21no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8#1176161212yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4#1176161212yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4C22⋊Dic6712yes—
187C22×C4C22⋊Dic6711yescharacteristic
188C22×C4C22⋊Dic6711yescharacteristic
189C22×C4C22×Dic6711yescharacteristic
190C22×C4C2×C22⋊D6712yes—
191C2×C8—2684no—
192C2×C8—2684no—
193C2×C8—2684no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C22⋊C4—28no—
198C22⋊C4—42no—
199C22⋊C4—44no—
200C42—22no—
201C42—21no—
202C42—21no—
203C42—24no—
204C42—42no—
205C42—41no—
206C42—42no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23C2×C22⋊Dic6711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4#1452396012yes—
236C2×C4C2×#1176161211yescharacteristic
237C2×C4C2×#1176161211yescharacteristic
238C2×C4#1452406314yes—
239C8—5364no—
240C22—42no—
241C22—44no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22#1686336011yescharacteristic
246C22#1686337011yescharacteristic
247C22C2×#1452396011yescharacteristic
248C22#1686339212yes—
249C22#1686341312yes—
250C4—24no—
251C4—21no—
252C4—21no—
253C4—22no—
254C4—42no—
255C4—41no—
256C4—42no—
257C4—81no—
258C4—81no—
259C2—82no—
260C2#2255855011yescharacteristic
261C2#2255864312yes—
262C2#2255878612yes—
263C2#2255883311yescharacteristic
264C2#2255887111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC4×#124×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC22×D4×C67C2×C4
cosocleC2×C42×C67C23
hypercenterC4×#124D67
hypocenterC67#7649
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #22560352C2semidirect
2. #22558035C2semidirect
137. C67#7649semidirect

Maximal subgroups

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

subgroupindex
#225603522
#225580352
#225580302
C4×#124×C672
#764967