Finite Group Databasebeta

← #30765303

Subgroups — #30765303

Order 17152.

444 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 99 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#22558553C212yesmaximal, complemented
2#22558153C212yesmaximal, complemented
3#22558141C212yesmaximal
4C4×#124×C67C211yesmaximal, characteristic
5#16863453—22no—
6#16863122—22no—
7#16863122C2214yes—
8C2×#124×C67C411yescharacteristic
9C2×C4×Q8×C67C2211yescharacteristic
10C2×C4×D4×C67C411yescharacteristic
11#391×C67C411yescharacteristic
12#390×C67C2211yescharacteristic
13#388×C67—22no—
14#382×C67C2211yescharacteristic
15C4×C22⋊C4×C67—22no—
16C43×C67C411yescharacteristic
17C4⋊(C67⋊C8)—42no—
18C22⋊(C67⋊C8)—24no—
19#14523954—42no—
20C22×Q8×C67C2×C411yescharacteristic
21C22×D4×C67C2×C411yescharacteristic
22C23×C4×C67—21no—
23#124×C67—22no—
24#124×C67—22no—
25C4×Q8×C67—41no—
26C4×Q8×C67—41no—
27C4×D4×C67—41no—
28C4×D4×C67—41no—
29C2×C4⋊C4×C67C2×C411yescharacteristic
30C2×C4⋊C4×C67D412yes—
31C2×C4⋊C4×C67C2311yescharacteristic
32C2×C22⋊C4×C67—22no—
33C2×C22⋊C4×C67—21no—
34C2×C22⋊C4×C67—22no—
35C2×C42×C67—21no—
36C2×C42×C67—21no—
37C2×C42×C67C2×C411yescharacteristic
38C2×C42×C67C2×C411yescharacteristic
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67D412yes—
41#95×C67—22no—
42#95×C67—22no—
43C2×C67⋊C8—44no—
44C24×C67—21no—
45C2×Q8×C67—21no—
46C2×Q8×C67—21no—
47C2×Q8×C67—41no—
48C2×Q8×C67—41no—
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—44no—
57C22×C4×C67—41no—
58C22×C4×C67—41no—
59C22×C4×C67C22⋊C412yes—
60C22×C4×C67C22⋊C411yescharacteristic
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22×C411yescharacteristic
63C22×C4×C67C2×D412yes—
64C4⋊C4×C67—42no—
65C4⋊C4×C67—41no—
66C4⋊C4×C67—41no—
67C22⋊C4×C67—28no—
68C22⋊C4×C67—44no—
69C22⋊C4×C67—42no—
70C42×C67—21no—
71C42×C67—21no—
72C42×C67—22no—
73C42×C67—22no—
74C42×C67—21no—
75C42×C67—41no—
76C42×C67—42no—
77C42×C67—42no—
78C42×C67M1612yes—
79C23×C67—22no—
80C23×C67—42no—
81C23×C67—42no—
82C23×C67—41no—
83C23×C67—41no—
84C23×C67C2×C22⋊C411yescharacteristic
85Q8×C67—42no—
86Q8×C67—42no—
87D4×C67—42no—
88D4×C67—42no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—22no—
92C2×C4×C67—22no—
93C2×C4×C67—22no—
94C2×C4×C67—22no—
95C2×C4×C67—21no—
96C2×C4×C67—22no—
97C2×C4×C67—22no—
98C2×C4×C67—22no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—41no—
107C2×C4×C67—82no—
108C2×C4×C67—82no—
109C2×C4×C67#9911yescharacteristic
110C2×C4×C67#9911yescharacteristic
111C2×C4×C67C2×M1611yescharacteristic
112C2×C4×C67C2×M1611yescharacteristic
113C67⋊C8—84no—
114C22×C67—21no—
115C22×C67—21no—
116C22×C67—44no—
117C22×C67—42no—
118C22×C67—82no—
119C22×C67—81no—
120C22×C67—81no—
121C22×C67#40711yescharacteristic
122C22×C67C2×#9911yescharacteristic
123C22×C67#41311yescharacteristic
124C4×C67—21no—
125C4×C67—21no—
126C4×C67—21no—
127C4×C67—42no—
128C4×C67—42no—
129C4×C67—42no—
130C4×C67—41no—
131C4×C67—42no—
132C4×C67—81no—
133C4×C67#35512yes—
134#11530—671nomaximal, Sylow
135C2×C67—21no—
136C2×C67—21no—
137C2×C67—82no—
138C2×C67#146511yescharacteristic
139C2×C67#212611yescharacteristic
140C2×C67C2×#35511yescharacteristic
141C4×#124D6711yescharacteristic
142#1467—672no—
143#1324—672no—
144#1318—672no—
145C67#1153011yescharacteristic, complemented, Sylow
146C2×#124Dic6711yescharacteristic
147C2×C4×Q8C2×D6711yescharacteristic
148C2×C4×D4Dic6711yescharacteristic
149#432—1342no—
150#391Dic6711yescharacteristic
151#390C2×D6711yescharacteristic
152#388—22no—
153#382C2×D6711yescharacteristic
154C4×C22⋊C4—22no—
155C43Dic6711yescharacteristic
156#324—674no—
157#324—1342no—
158C22×Q8C2×Dic6711yescharacteristic
159C22×D4C2×Dic6711yescharacteristic
160C23×C4—21no—
161#124—22no—
162#124—22no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×D4—41no—
166C4×D4—41no—
167C2×C4⋊C4C2×Dic6711yescharacteristic
168C2×C4⋊C4C22⋊D6712yes—
169C2×C4⋊C4C22×D6711yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42C2×Dic6711yescharacteristic
176C2×C42C2×Dic6711yescharacteristic
177C2×C42C2×Dic6711yescharacteristic
178C2×C42C22⋊D6712yes—
179C4⋊C8—2682no—
180C22⋊C8—1344no—
181#97—2682no—
182#95—22no—
183#95—22no—
184C24—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4#1176161212yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C22⋊Dic6712yes—
200C22×C4C22⋊Dic6711yescharacteristic
201C22×C4C22⋊Dic6711yescharacteristic
202C22×C4C22×Dic6711yescharacteristic
203C22×C4C2×C22⋊D6712yes—
204C2×C8—2684no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C22⋊C4—28no—
209C22⋊C4—42no—
210C22⋊C4—44no—
211C42—22no—
212C42—22no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42#1176161212yes—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊Dic6711yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—42no—
229D4—42no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4#1452395811yescharacteristic
251C2×C4#1452395811yescharacteristic
252C2×C4C2×#1176161211yescharacteristic
253C2×C4C2×#1176161211yescharacteristic
254C8—5364no—
255C22—21no—
256C22—21no—
257C22—44no—
258C22—42no—
259C22—82no—
260C22—81no—
261C22—81no—
262C22#1686336011yescharacteristic
263C22C2×#1452395811yescharacteristic
264C22#1686338011yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—42no—
273C4—81no—
274C4#1686319312yes—
275C2—21no—
276C2—21no—
277C2—82no—
278C2#2255854711yescharacteristic
279C2#2256187411yescharacteristic
280C2C2×#1686319311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC4×#124×C67C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenterC4×#124D67
hypocenterC67#11530
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #22558553C2semidirect
2. #22558153C2semidirect
145. C67#11530semidirect

Maximal subgroups

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

subgroupindex
#225585532
#225581532
#225581412
C4×#124×C672
#1153067