Finite Group Databasebeta

← #30744092

Subgroups — #30744092

Order 17152.

520 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 135 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#22560062C212yesmaximal, complemented
2#22558035C212yesmaximal, complemented
3#22558030C212yesmaximal
4C4×#124×C67C211yesmaximal, characteristic
5C2×C4⋊(C67⋊C8)—22no—
6C2×#14523954—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—
18C4⋊(C67⋊C8)—28no—
19#14523954—44no—
20C22×Q8×C67C2×C411yescharacteristic
21C22×D4×C67C2×C411yescharacteristic
22C23×C4×C67—21no—
23#124×C67—22no—
24#124×C67—22no—
25C4×Q8×C67—42no—
26C4×D4×C67—42no—
27C2×C4⋊C4×C67—21no—
28C2×C4⋊C4×C67C2×C411yescharacteristic
29C2×C4⋊C4×C67C2×C411yescharacteristic
30C2×C22⋊C4×C67—22no—
31C2×C22⋊C4×C67—21no—
32C2×C22⋊C4×C67—22no—
33C2×C42×C67—21no—
34C2×C42×C67C2×C411yescharacteristic
35C2×C42×C67C2×C411yescharacteristic
36C2×C42×C67D412yes—
37C2×C42×C67D412yes—
38C2×C42×C67C2311yescharacteristic
39#95×C67—22no—
40#95×C67—22no—
41C2×C67⋊C8—28no—
42C2×C67⋊C8—44no—
43C2×C67⋊C8—44no—
44C24×C67—21no—
45C2×Q8×C67—21no—
46C2×Q8×C67—41no—
47C2×Q8×C67—41no—
48C2×Q8×C67M1612yes—
49C2×D4×C67—21no—
50C2×D4×C67—41no—
51C2×D4×C67—41no—
52C2×D4×C67M1612yes—
53C22×C4×C67—21no—
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67—42no—
57C22×C4×C67—44no—
58C22×C4×C67C22⋊C411yescharacteristic
59C22×C4×C67C22⋊C412yes—
60C22×C4×C67C22⋊C411yescharacteristic
61C22×C4×C67C22×C411yescharacteristic
62C22×C4×C67C2×D412yes—
63C4⋊C4×C67—41no—
64C4⋊C4×C67—41no—
65C4⋊C4×C67—42no—
66C22⋊C4×C67—28no—
67C22⋊C4×C67—42no—
68C22⋊C4×C67—42no—
69C22⋊C4×C67—42no—
70C42×C67—22no—
71C42×C67—24no—
72C42×C67—21no—
73C42×C67—21no—
74C42×C67—22no—
75C42×C67—22no—
76C42×C67—41no—
77C42×C67—42no—
78C23×C67—22no—
79C23×C67—41no—
80C23×C67—41no—
81C23×C67—42no—
82C23×C67—42no—
83C23×C67C2×C22⋊C411yescharacteristic
84Q8×C67—42no—
85Q8×C67—42no—
86D4×C67—42no—
87D4×C67—42no—
88C2×C4×C67—21no—
89C2×C4×C67—22no—
90C2×C4×C67—21no—
91C2×C4×C67—22no—
92C2×C4×C67—22no—
93C2×C4×C67—22no—
94C2×C4×C67—21no—
95C2×C4×C67—22no—
96C2×C4×C67—22no—
97C2×C4×C67—22no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—42no—
103C2×C4×C67—42no—
104C2×C4×C67—84no—
105C2×C4×C67#10012yes—
106C2×C4×C67#10414yes—
107C2×C4×C67C2×M1611yescharacteristic
108C2×C4×C67C2×M1611yescharacteristic
109C2×C4×C67AGL(1,ℤ/8)12yes—
110C2×C4×C67#13712yes—
111C67⋊C8—48no—
112C22×C67—42no—
113C22×C67—44no—
114C22×C67—81no—
115C22×C67—82no—
116C22×C67—81no—
117C22×C67#40711yescharacteristic
118C22×C67#41011yescharacteristic
119C22×C67C2×#10011yescharacteristic
120C22×C67#41712yes—
121C22×C67C2×#10412yes—
122C4×C67—21no—
123C4×C67—21no—
124C4×C67—22no—
125C4×C67—22no—
126C4×C67—41no—
127C4×C67—42no—
128C4×C67—42no—
129C4×C67—81no—
130C4×C67—81no—
131C4×C67#32914yes—
132#7635—671nomaximal, Sylow
133C2×C67—82no—
134C2×C67#146611yescharacteristic
135C2×C67#148912yes—
136C2×C67#150912yes—
137C2×C67C2×#32911yescharacteristic
138C2×C67#156011yescharacteristic
139C4×#124D6711yescharacteristic
140#1767—672no—
141#1272—672no—
142#1270—672no—
143C67#763511yescharacteristic, complemented, Sylow
144C2×#124Dic6711yescharacteristic
145C2×C4×Q8Dic6711yescharacteristic
146C2×C4×D4Dic6711yescharacteristic
147C2×C4⋊C8—1342no—
148C2×#97—1342no—
149#391C2×D6711yescharacteristic
150#390C2×D6711yescharacteristic
151#388—22no—
152#382Dic6711yescharacteristic
153C4×C22⋊C4—22no—
154C43C2×D6711yescharacteristic
155#336—674no—
156C22×Q8C2×Dic6711yescharacteristic
157C22×D4C2×Dic6711yescharacteristic
158C23×C4—21no—
159C22×C8—1344no—
160#124—22no—
161#124—22no—
162C4×Q8—42no—
163C4×D4—42no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C2×Dic6711yescharacteristic
166C2×C4⋊C4C2×Dic6711yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C42—21no—
171C2×C42C2×Dic6711yescharacteristic
172C2×C42C2×Dic6711yescharacteristic
173C2×C42C22⋊D6712yes—
174C2×C42C22⋊D6712yes—
175C2×C42C22×D6711yescharacteristic
176C4⋊C8—1348no—
177#97—2684no—
178#95—22no—
179#95—22no—
180C24—21no—
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8#1176161212yes—
185C2×D4—21no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4#1176161212yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4C22⋊Dic6712yes—
195C22×C4C22⋊Dic6711yescharacteristic
196C22×C4C22⋊Dic6711yescharacteristic
197C22×C4C22×Dic6711yescharacteristic
198C22×C4C2×C22⋊D6712yes—
199C2×C8—2684no—
200C2×C8—2684no—
201C2×C8—1348no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C22⋊C4—28no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—22no—
210C42—24no—
211C42—21no—
212C42—21no—
213C42—22no—
214C42—22no—
215C42—41no—
216C42—42no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23C2×C22⋊Dic6711yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—84no—
244C2×C4#1452396012yes—
245C2×C4#1452397014yes—
246C2×C4C2×#1176161211yescharacteristic
247C2×C4C2×#1176161211yescharacteristic
248C2×C4#1452406812yes—
249C2×C4#1452407312yes—
250C8—2688no—
251C22—42no—
252C22—44no—
253C22—82no—
254C22—81no—
255C22—81no—
256C22#1686336011yescharacteristic
257C22#1686337011yescharacteristic
258C22C2×#1452396011yescharacteristic
259C22#1686339712yes—
260C22C2×#1452397012yes—
261C4—22no—
262C4—22no—
263C4—21no—
264C4—21no—
265C4—42no—
266C4—42no—
267C4—41no—
268C4—81no—
269C4—81no—
270C4#1686313614yes—
271C2—82no—
272C2#2255855011yescharacteristic
273C2#2255864812yes—
274C2#2255873412yes—
275C2C2×#1686313611yescharacteristic
276C2#2255894411yescharacteristic
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#7635
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. #22560062C2semidirect
2. #22558035C2semidirect
143. C67#7635semidirect

Maximal subgroups

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

subgroupindex
#225600622
#225580352
#225580302
C4×#124×C672
#763567