Finite Group Databasebeta

← #30748484

Subgroups — #30748484

Order 17152.

842 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 139 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, complemented
2#22558539C212yesmaximal, complemented
3#22558536C212yesmaximal
4C22×#126×C67C211yesmaximal, characteristic
5#16863360C2214yes—
6#16863165—24no—
7#16863122—24no—
8#16863120—24no—
9C2×#126×C67—24no—
10C2×#126×C67—24no—
11C2×#126×C67—24no—
12C22×C4⋊C4×C67C411yescharacteristic
13C22×C4⋊C4×C67C2211yescharacteristic
14C22×C4⋊C4×C67C2211yescharacteristic
15C22×C22⋊C4×C67C411yescharacteristic
16C22×C22⋊C4×C67C411yescharacteristic
17C22×C22⋊C4×C67C2211yescharacteristic
18C22×C42×C67C411yescharacteristic
19C2×#11761612—24no—
20C22⋊(C67⋊C8)—24no—
21C22⋊(C67⋊C8)—24no—
22C25×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#126×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×C67D414yes—
38C2×C4⋊C4×C67D414yes—
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67—22no—
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67—22no—
44C2×C22⋊C4×C67—22no—
45C2×C22⋊C4×C67—22no—
46C2×C22⋊C4×C67—22no—
47C2×C22⋊C4×C67D414yes—
48C2×C42×C67—22no—
49C2×C42×C67—22no—
50C2×C42×C67—22no—
51#11761612—48no—
52C2×C67⋊C8—44no—
53C24×C67—21no—
54C24×C67—21no—
55C24×C67—42no—
56C24×C67—42no—
57C24×C67—42no—
58C24×C67C22⋊C412yes—
59C24×C67C22×C411yescharacteristic
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—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—42no—
71C22×C4×C67—42no—
72C22×C4×C67—42no—
73C22×C4×C67—42no—
74C22×C4×C67—42no—
75C22×C4×C67—42no—
76C22×C4×C67C22⋊C412yes—
77C22×C4×C67C22⋊C412yes—
78C22×C4×C67C22⋊C412yes—
79C22×C4×C67C22⋊C412yes—
80C22×C4×C67C22⋊C412yes—
81C22×C4×C67C2×D412yes—
82C22×C4×C67C2×D412yes—
83C22×C4×C67C2×D412yes—
84C4⋊C4×C67—28no—
85C4⋊C4×C67—28no—
86C4⋊C4×C67—28no—
87C22⋊C4×C67—28no—
88C22⋊C4×C67—28no—
89C22⋊C4×C67—28no—
90C42×C67—28no—
91C23×C67—22no—
92C23×C67—22no—
93C23×C67—22no—
94C23×C67—22no—
95C23×C67—24no—
96C23×C67—42no—
97C23×C67—42no—
98C23×C67—42no—
99C23×C67—42no—
100C23×C67—42no—
101C23×C67—42no—
102C23×C67—42no—
103C23×C67—42no—
104C23×C67—88no—
105C23×C67C2×C22⋊C411yescharacteristic
106C23×C67C2×C22⋊C411yescharacteristic
107C23×C67C2×C22⋊C411yescharacteristic
108C2×C4×C67—24no—
109C2×C4×C67—22no—
110C2×C4×C67—22no—
111C2×C4×C67—22no—
112C2×C4×C67—22no—
113C2×C4×C67—22no—
114C2×C4×C67—24no—
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—42no—
123C2×C4×C67—42no—
124C2×C4×C67—42no—
125C2×C4×C67—42no—
126C2×C4×C67—42no—
127C2×C4×C67—42no—
128C2×C4×C67—42no—
129C2×C4×C67—42no—
130C2×C4×C67—42no—
131C2×C4×C67#12014yes—
132C67⋊C8—84no—
133C22×C67—28no—
134C22×C67—21no—
135C22×C67—21no—
136C22×C67—21no—
137C22×C67—21no—
138C22×C67—21no—
139C22×C67—21no—
140C22×C67—42no—
141C22×C67—42no—
142C22×C67—42no—
143C22×C67—84no—
144C22×C67—84no—
145C22×C67—84no—
146C22×C67#37911yescharacteristic
147C22×C67#41012yes—
148C22×C67#41312yes—
149C22×C67#41312yes—
150C4×C67—24no—
151C4×C67—42no—
152C4×C67—42no—
153C4×C67—42no—
154C4×C67—42no—
155C4×C67—42no—
156#8556—671nomaximal, Sylow
157C2×C67—22no—
158C2×C67—22no—
159C2×C67—22no—
160C2×C67—82no—
161C2×C67#178211yescharacteristic
162C2×C67#178511yescharacteristic
163C2×C67#178511yescharacteristic
164C22×#126D6711yescharacteristic
165#1821—672no—
166#1463—672no—
167#1462—672no—
168C67#855611yescharacteristic, complemented, Sylow
169C2×#126—24no—
170C2×#126—24no—
171C2×#126—24no—
172C22×C4⋊C4Dic6711yescharacteristic
173C22×C4⋊C4C2×D6711yescharacteristic
174C22×C4⋊C4C2×D6711yescharacteristic
175C22×C22⋊C4Dic6711yescharacteristic
176C22×C22⋊C4Dic6711yescharacteristic
177C22×C22⋊C4C2×D6711yescharacteristic
178C22×C42Dic6711yescharacteristic
179#407—674no—
180#343—1344no—
181#324—1344no—
182#323—1344no—
183C25C2×Dic6711yescharacteristic
184C23×C4C2×Dic6711yescharacteristic
185C23×C4C2×Dic6711yescharacteristic
186C23×C4C2×Dic6711yescharacteristic
187C23×C4C2×Dic6711yescharacteristic
188C23×C4C2×Dic6711yescharacteristic
189C23×C4C22×D6711yescharacteristic
190C2×M16—1344no—
191#126—232no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4C22⋊D6714yes—
200C2×C4⋊C4C22⋊D6714yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C22⋊D6714yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C22⋊C8—1344no—
214C22⋊C8—1344no—
215C24—21no—
216C24—21no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24C22⋊Dic6712yes—
221C24C22×Dic6711yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C22⋊Dic6712yes—
239C22×C4C22⋊Dic6712yes—
240C22×C4C22⋊Dic6712yes—
241C22×C4C22⋊Dic6712yes—
242C22×C4C22⋊Dic6712yes—
243C22×C4C2×C22⋊D6712yes—
244C22×C4C2×C22⋊D6712yes—
245C22×C4C2×C22⋊D6712yes—
246M16—2688no—
247C2×C8—2684no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C22⋊C4—28no—
252C22⋊C4—28no—
253C22⋊C4—28no—
254C42—28no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—24no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—88no—
269C23C2×C22⋊Dic6711yescharacteristic
270C23C2×C22⋊Dic6711yescharacteristic
271C23C2×C22⋊Dic6711yescharacteristic
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4#1452401214yes—
296C8—5364no—
297C22—28no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—84no—
308C22—84no—
309C22—84no—
310C22#1686325211yescharacteristic
311C22#1686337012yes—
312C22#1686338012yes—
313C22#1686338012yes—
314C4—24no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4—42no—
319C4—42no—
320C2—22no—
321C2—22no—
322C2—22no—
323C2—82no—
324C2#2256012711yescharacteristic
325C2#2256013711yescharacteristic
326C2#2256013711yescharacteristic
subgroupG/H
centerC22#16863252
commutatorC24×C67C22×C4
FrattiniC23×C4C22×D67
FittingC22×#126×C67C2
socleC22×C67#379
prime radicalC25×C67C2×C4
cosocleC23×C4×C67C23
hypercenterC22×#126D67
hypocenterC67#8556
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. #22560289C2semidirect
2. #22558539C2semidirect
168. C67#8556semidirect

Maximal subgroups

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

subgroupindex
#225602892
#225585392
#225585362
C22×#126×C672
#855667