Finite Group Databasebeta

← #30748476

Subgroups — #30748476

Order 17152.

778 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 147 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#22560289C212yesmaximal
2#22558539C212yesmaximal
3#22558539C212yesmaximal
4C22×#125×C67C211yesmaximal, characteristic
5#16863360C2214yes—
6#16863165—24no—
7#16863122—24no—
8#16863122—24no—
9C2×#125×C67—24no—
10C2×#125×C67—24no—
11C2×#125×C67—24no—
12C22×C4⋊C4×C67C411yescharacteristic
13C22×C4⋊C4×C67C411yescharacteristic
14C22×C4⋊C4×C67C411yescharacteristic
15C22×C4⋊C4×C67C2211yescharacteristic
16C22×C4⋊C4×C67C2211yescharacteristic
17C22×C4⋊C4×C67C2211yescharacteristic
18C22×C42×C67C411yescharacteristic
19C2×#11761612—24no—
20C22⋊(C67⋊C8)—24no—
21C22⋊(C67⋊C8)—24no—
22C23×C4×C67C2×C411yescharacteristic
23C23×C4×C67C2×C411yescharacteristic
24C23×C4×C67C2×C411yescharacteristic
25C23×C4×C67C2×C411yescharacteristic
26C23×C4×C67C2×C411yescharacteristic
27C23×C4×C67C2×C411yescharacteristic
28C23×C4×C67C2311yescharacteristic
29#125×C67—232no—
30C2×C4⋊C4×C67—22no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67—22no—
38C2×C4⋊C4×C67—22no—
39C2×C4⋊C4×C67—22no—
40C2×C4⋊C4×C67—22no—
41C2×C4⋊C4×C67—22no—
42C2×C4⋊C4×C67—22no—
43C2×C4⋊C4×C67—22no—
44C2×C4⋊C4×C67—22no—
45C2×C4⋊C4×C67D414yes—
46C2×C4⋊C4×C67D414yes—
47C2×C4⋊C4×C67D414yes—
48C2×C42×C67—22no—
49C2×C42×C67—22no—
50C2×C42×C67—22no—
51#11761612—48no—
52C2×C67⋊C8—44no—
53C24×C67C22×C411yescharacteristic
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67—24no—
57C22×C4×C67—21no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—42no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67—42no—
71C22×C4×C67—42no—
72C22×C4×C67—42no—
73C22×C4×C67C22⋊C412yes—
74C22×C4×C67C22⋊C412yes—
75C22×C4×C67C22⋊C412yes—
76C22×C4×C67C22⋊C412yes—
77C22×C4×C67C22⋊C412yes—
78C22×C4×C67C22⋊C412yes—
79C22×C4×C67C2×D412yes—
80C22×C4×C67C2×D412yes—
81C22×C4×C67C2×D412yes—
82C4⋊C4×C67—28no—
83C4⋊C4×C67—28no—
84C4⋊C4×C67—28no—
85C4⋊C4×C67—28no—
86C4⋊C4×C67—28no—
87C4⋊C4×C67—28no—
88C42×C67—28no—
89C23×C67—22no—
90C23×C67—22no—
91C23×C67—22no—
92C23×C67C2×C22⋊C411yescharacteristic
93C23×C67C2×C22⋊C411yescharacteristic
94C23×C67C2×C22⋊C411yescharacteristic
95C2×C4×C67—24no—
96C2×C4×C67—24no—
97C2×C4×C67—22no—
98C2×C4×C67—24no—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—22no—
102C2×C4×C67—22no—
103C2×C4×C67—22no—
104C2×C4×C67—22no—
105C2×C4×C67—24no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—42no—
120C2×C4×C67—42no—
121C2×C4×C67—42no—
122C2×C4×C67#10114yes—
123C2×C4×C67#12014yes—
124C67⋊C8—84no—
125C22×C67—28no—
126C22×C67—21no—
127C22×C67—21no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67—21no—
131C22×C67—21no—
132C22×C67#37911yescharacteristic
133C22×C67#41012yes—
134C22×C67C2×#10112yes—
135C22×C67#41312yes—
136C4×C67—24no—
137C4×C67—24no—
138C4×C67—42no—
139C4×C67—42no—
140C4×C67—42no—
141C4×C67—42no—
142C4×C67—42no—
143#8554—671nomaximal, Sylow
144C2×C67—22no—
145C2×C67—22no—
146C2×C67—22no—
147C2×C67#178211yescharacteristic
148C2×C67#178411yescharacteristic
149C2×C67#178511yescharacteristic
150C22×#125D6711yescharacteristic
151#1821—672no—
152#1463—672no—
153#1463—672no—
154C67#855411yescharacteristic, complemented, Sylow
155C2×#125—24no—
156C2×#125—24no—
157C2×#125—24no—
158C22×C4⋊C4Dic6711yescharacteristic
159C22×C4⋊C4Dic6711yescharacteristic
160C22×C4⋊C4Dic6711yescharacteristic
161C22×C4⋊C4C2×D6711yescharacteristic
162C22×C4⋊C4C2×D6711yescharacteristic
163C22×C4⋊C4C2×D6711yescharacteristic
164C22×C42Dic6711yescharacteristic
165#407—674no—
166#343—1344no—
167#324—1344no—
168#324—1344no—
169C23×C4C2×Dic6711yescharacteristic
170C23×C4C2×Dic6711yescharacteristic
171C23×C4C2×Dic6711yescharacteristic
172C23×C4C2×Dic6711yescharacteristic
173C23×C4C2×Dic6711yescharacteristic
174C23×C4C2×Dic6711yescharacteristic
175C23×C4C22×D6711yescharacteristic
176C2×M16—1344no—
177#125—232no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C22⋊D6714yes—
194C2×C4⋊C4C22⋊D6714yes—
195C2×C4⋊C4C22⋊D6714yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C22⋊C8—1344no—
200C22⋊C8—1344no—
201C24C22×Dic6711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4C22⋊Dic6712yes—
222C22×C4C22⋊Dic6712yes—
223C22×C4C22⋊Dic6712yes—
224C22×C4C22⋊Dic6712yes—
225C22×C4C22⋊Dic6712yes—
226C22×C4C22⋊Dic6712yes—
227C22×C4C2×C22⋊D6712yes—
228C22×C4C2×C22⋊D6712yes—
229C22×C4C2×C22⋊D6712yes—
230M16—2688no—
231C2×C8—2684no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C42—28no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23C2×C22⋊Dic6711yescharacteristic
243C23C2×C22⋊Dic6711yescharacteristic
244C23C2×C22⋊Dic6711yescharacteristic
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#1452396214yes—
273C2×C4#1452401214yes—
274C8—5364no—
275C22—28no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1686325211yescharacteristic
283C22#1686337012yes—
284C22C2×#1452396212yes—
285C22#1686338012yes—
286C4—24no—
287C4—24no—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—42no—
292C4—42no—
293C2—22no—
294C2—22no—
295C2—22no—
296C2#2256012711yescharacteristic
297C2#2256013311yescharacteristic
298C2#2256013711yescharacteristic
subgroupG/H
centerC22#16863252
commutator, prime radicalC24×C67C22×C4
FrattiniC23×C4C22×D67
FittingC22×#125×C67C2
socleC22×C67#379
cosocleC23×C4×C67C23
hypercenterC22×#125D67
hypocenterC67#8554
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
154. C67#8554semidirect

Maximal subgroups

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

subgroupindex
#225602892
#225585392
C22×#125×C672
#855467