Finite Group Databasebeta

← #30765362

Subgroups — #30765362

Order 17152.

380 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 87 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#22558543C211yesmaximal, characteristic
2#22558523C211yesmaximal, characteristic
3#22558157C212yesmaximal
4#22558139C212yesmaximal
5C67×#2484C211yesmaximal, characteristic
6#16863453—21no—
7#16863449—21no—
8#16863122—21no—
9#16863122—21no—
10#16863120C2212yes—
11#16863120C2212yes—
12C2×#117×C67C2211yescharacteristic
13C22×C4⋊C4×C67C411yescharacteristic
14#400×C67—21no—
15#398×C67—21no—
16#387×C67—21no—
17#387×C67—21no—
18#385×C67C411yescharacteristic
19#385×C67C2211yescharacteristic
20#381×C67C411yescharacteristic
21#381×C67C411yescharacteristic
22#381×C67C2211yescharacteristic
23C4⋊(C67⋊C8)—41no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—22no—
27#14523954—41no—
28C4×C67⋊C8—41no—
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67C2×C411yescharacteristic
31C23×C4×C67C2×C411yescharacteristic
32#117×C67—41no—
33#117×C67—41no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—41no—
39C2×C4⋊C4×C67—41no—
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67C2×C411yescharacteristic
43C2×C22⋊C4×C67C2×C411yescharacteristic
44C2×C22⋊C4×C67C2×C411yescharacteristic
45C2×C22⋊C4×C67C2311yescharacteristic
46C2×C42×C67D411yescharacteristic
47C2×C42×C67D411yescharacteristic
48#95×C67—21no—
49#95×C67—21no—
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56C2×C67⋊C8—42no—
57C2×C67⋊C8—42no—
58C24×C67C22×C411yescharacteristic
59C22×C4×C67—21no—
60C22×C4×C67—22no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—22no—
67C22×C4×C67—41no—
68C22×C4×C67—42no—
69C22×C4×C67—41no—
70C22×C4×C67—41no—
71C22×C4×C67—41no—
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—42no—
82C4⋊C4×C67—42no—
83C4⋊C4×C67—41no—
84C4⋊C4×C67—41no—
85C22⋊C4×C67—22no—
86C22⋊C4×C67—41no—
87C22⋊C4×C67—41no—
88C22⋊C4×C67—41no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—21no—
92C42×C67—21no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67C2×C22⋊C411yescharacteristic
98C23×C67#13112yes—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—22no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—41no—
112C2×C4×C67—42no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—42no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—82no—
129C2×C4×C67#9911yescharacteristic
130C2×C4×C67#9911yescharacteristic
131C67⋊C8—82no—
132C67⋊C8—82no—
133C22×C67—21no—
134C22×C67—21no—
135C22×C67—22no—
136C22×C67—42no—
137C22×C67—42no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67#40811yescharacteristic
141C22×C67C2×#9911yescharacteristic
142C22×C67#41311yescharacteristic
143C4×C67—21no—
144C4×C67—21no—
145C4×C67—41no—
146C4×C67—41no—
147C4×C67—42no—
148C4×C67—41no—
149C4×C67—41no—
150C4×C67—41no—
151C4×C67—81no—
152C4×C67—81no—
153C4×C67—81no—
154#11544—671nomaximal, Sylow
155C2×C67—21no—
156C2×C67—21no—
157C2×C67—42no—
158C2×C67#146811yescharacteristic
159C2×C67#212011yescharacteristic
160C2×C67#212311yescharacteristic
161#2484D6711yescharacteristic
162#1464—671no—
163#1458—671no—
164#1326—672no—
165#1317—672no—
166C67#1154411yescharacteristic, complemented, Sylow
167C2×#117C2×D6711yescharacteristic
168C22×C4⋊C4Dic6711yescharacteristic
169#432—1341no—
170#431—1341no—
171#400—21no—
172#398—21no—
173#387—21no—
174#387—21no—
175#385Dic6711yescharacteristic
176#385C2×D6711yescharacteristic
177#381Dic6711yescharacteristic
178#381Dic6711yescharacteristic
179#381C2×D6711yescharacteristic
180#324—1341no—
181#324—1341no—
182#323—672no—
183#323—672no—
184C23×C4C2×Dic6711yescharacteristic
185C23×C4C2×Dic6711yescharacteristic
186C23×C4C2×Dic6711yescharacteristic
187#117—41no—
188#117—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C22⋊D6711yescharacteristic
196C2×C4⋊C4C22⋊D6711yescharacteristic
197C2×C22⋊C4C2×Dic6711yescharacteristic
198C2×C22⋊C4C2×Dic6711yescharacteristic
199C2×C22⋊C4C2×Dic6711yescharacteristic
200C2×C22⋊C4C22×D6711yescharacteristic
201C2×C42C22⋊D6711yescharacteristic
202C2×C42C22⋊D6711yescharacteristic
203C4⋊C8—2681no—
204C4⋊C8—2681no—
205C22⋊C8—1342no—
206C22⋊C8—1342no—
207#97—2681no—
208C4×C8—2681no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217C24C22×Dic6711yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4C22⋊Dic6711yescharacteristic
232C22×C4C22⋊Dic6711yescharacteristic
233C22×C4C22⋊Dic6711yescharacteristic
234C22×C4C22⋊Dic6711yescharacteristic
235C22×C4C2×C22⋊D6711yescharacteristic
236C22×C4C2×C22⋊D6711yescharacteristic
237C2×C8—2682no—
238C2×C8—2682no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C22⋊C4—22no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊Dic6711yescharacteristic
259C23#1452405012yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—82no—
290C2×C4#1452395811yescharacteristic
291C2×C4#1452395811yescharacteristic
292C8—5362no—
293C8—5362no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22#1686336311yescharacteristic
302C22C2×#1452395811yescharacteristic
303C22#1686338011yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—81no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2255855711yescharacteristic
319C2#2256185011yescharacteristic
320C2#2256186311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutator, prime radicalC24×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2484C2
socleC22×C67C2×#99
cosocleC2×C22⋊C4×C67C23
hypercenter#2484D67
hypocenterC67#11544
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2484

Complements

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

NH ≅ G/Ntype
166. C67#11544semidirect

Maximal subgroups

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

subgroupindex
#225585432
#225585232
#225581572
#225581392
C67×#24842
#1154467