Finite Group Databasebeta

← #30765482

Subgroups — #30765482

Order 17152.

480 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 95 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#22558547C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558155C214yesmaximal, complemented
4C67×#2542C211yesmaximal, characteristic
5#16863360—21no—
6C2×C22⋊(C67⋊C8)—21no—
7#16863122C2212yes—
8#16863122C2212yes—
9#16863120—21no—
10#16863120—21no—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×C22⋊Q8×C67—21no—
13C2×C22⋊Q8×C67—21no—
14C2×C4×D4×C67C2211yescharacteristic
15#399×C67—21no—
16#399×C67—21no—
17#390×C67C412yes—
18#384×C67C2212yes—
19C4×C4⋊C4×C67C411yescharacteristic
20C2×#11761612—41no—
21C22×C67⋊C8—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—22no—
24C22⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—41no—
26C22×Q8×C67—21no—
27C22×D4×C67C2×C411yescharacteristic
28C23×C4×C67D411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C4⋊Q8×C67—41no—
31C4⋊Q8×C67—41no—
32C22⋊Q8×C67—44no—
33C22⋊Q8×C67—44no—
34C4×D4×C67—41no—
35C4×D4×C67—41no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—22no—
39C2×C4⋊C4×C67—22no—
40C2×C4⋊C4×C67C2×C412yes—
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67—22no—
44C2×C22⋊C4×C67D411yescharacteristic
45C2×C22⋊C4×C67D411yescharacteristic
46C2×C42×C67C2×C412yes—
47C2×C42×C67C2×C411yescharacteristic
48#95×C67—21no—
49#11761612—82no—
50C2×C67⋊C8—42no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—82no—
53C24×C67C22⋊C411yescharacteristic
54C24×C67C22⋊C411yescharacteristic
55C2×Q8×C67—42no—
56C2×Q8×C67—42no—
57C2×Q8×C67—41no—
58C2×Q8×C67—41no—
59C2×D4×C67—22no—
60C2×D4×C67—22no—
61C2×D4×C67—41no—
62C22×C4×C67—22no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—22no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C4⋊C4×C67—24no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—42no—
78C4⋊C4×C67—44no—
79C4⋊C4×C67—44no—
80C4⋊C4×C67—42no—
81C22⋊C4×C67—41no—
82C22⋊C4×C67—41no—
83C22⋊C4×C67—44no—
84C22⋊C4×C67—44no—
85C42×C67—21no—
86C42×C67—21no—
87C42×C67—24no—
88C23×C67—21no—
89C23×C67—21no—
90C23×C67—21no—
91C23×C67—21no—
92C23×C67—21no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67#10112yes—
98C23×C67C2×C22⋊C411yescharacteristic
99Q8×C67—82no—
100Q8×C67—82no—
101D4×C67—44no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—22no—
105C2×C4×C67—24no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—81no—
120C2×C4×C67—81no—
121C2×C4×C67#9911yescharacteristic
122C2×C4×C67#9911yescharacteristic
123C2×C4×C67#13112yes—
124C67⋊C8—82no—
125C67⋊C8—82no—
126C22×C67—21no—
127C22×C67—21no—
128C22×C67—22no—
129C22×C67—42no—
130C22×C67—42no—
131C22×C67—41no—
132C22×C67—41no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—41no—
136C22×C67—81no—
137C22×C67—81no—
138C22×C67#40811yescharacteristic
139C22×C67C2×#9911yescharacteristic
140C22×C67C2×#10111yescharacteristic
141C4×C67—21no—
142C4×C67—22no—
143C4×C67—44no—
144C4×C67—41no—
145C4×C67—41no—
146C4×C67—41no—
147C4×C67—82no—
148C4×C67#35312yes—
149#11570—671nomaximal, Sylow
150C2×C67—21no—
151C2×C67—21no—
152C2×C67—42no—
153C2×C67—81no—
154C2×C67#147211yescharacteristic
155C2×C67C2×#35311yescharacteristic
156C2×C67#213511yescharacteristic
157#2542D6711yescharacteristic
158#1465—671no—
159#1459—671no—
160#1325—674no—
161C67#1157011yescharacteristic, complemented, Sylow
162C2×C4⋊Q8Dic6711yescharacteristic
163C2×C22⋊Q8—21no—
164C2×C22⋊Q8—21no—
165C2×C4×D4C2×D6711yescharacteristic
166#407—1341no—
167C2×C22⋊C8—1341no—
168#399—21no—
169#399—21no—
170#390Dic6712yes—
171#384C2×D6712yes—
172C4×C4⋊C4Dic6711yescharacteristic
173#324—672no—
174#324—672no—
175#323—1341no—
176#323—1341no—
177C22×Q8—21no—
178C22×D4C2×Dic6711yescharacteristic
179C23×C4C22⋊D6711yescharacteristic
180C23×C4C22⋊D6711yescharacteristic
181C2×M16—2681no—
182C22×C8—2681no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C2×Dic6712yes—
194C2×C4⋊C4C22×D6711yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C22⋊D6711yescharacteristic
198C2×C22⋊C4C22⋊D6711yescharacteristic
199C2×C42C2×Dic6712yes—
200C2×C42C2×Dic6711yescharacteristic
201C22⋊C8—2681no—
202C22⋊C8—2681no—
203C22⋊C8—1342no—
204C22⋊C8—1342no—
205#95—21no—
206C24C22⋊Dic6711yescharacteristic
207C24C22⋊Dic6711yescharacteristic
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C22⋊Dic6711yescharacteristic
224C22×C4C22⋊Dic6711yescharacteristic
225C22×C4C22×Dic6711yescharacteristic
226C22×C4C2×C22⋊D6711yescharacteristic
227C22×C4C2×C22⋊D6711yescharacteristic
228M16—5362no—
229C2×C8—2682no—
230C2×C8—2682no—
231C2×C8—5362no—
232C4⋊C4—24no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C42—21no—
243C42—21no—
244C42—24no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1452396212yes—
255C23C2×C22⋊Dic6711yescharacteristic
256Q8—82no—
257Q8—82no—
258D4—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4#1452395811yescharacteristic
279C2×C4#1452395811yescharacteristic
280C2×C4#1452405012yes—
281C8—5362no—
282C8—5362no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22#1686336311yescharacteristic
296C22C2×#1452395811yescharacteristic
297C22C2×#1452396211yescharacteristic
298C4—21no—
299C4—22no—
300C4—44no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4#1686318912yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2255857311yescharacteristic
311C2C2×#1686318911yescharacteristic
312C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2542C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2542D67
hypocenterC67#11570
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #22558155C2semidirect
161. C67#11570semidirect

Maximal subgroups

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

subgroupindex
#225585472
#225585272
#225581552
C67×#25422
#1157067