Finite Group Databasebeta

← #30766881

Subgroups — #30766881

Order 17152.

460 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 107 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#22560201C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558215C214yesmaximal, complemented
4C67×#2486C211yesmaximal, characteristic
5#16863420—21no—
6C2×#14523960—21no—
7C2×C22⋊(C67⋊C8)—21no—
8#16863165C2212yes—
9#16863122C2212yes—
10#16863120—21no—
11C2×#123×C67—21no—
12C2×C4×D4×C67C2211yescharacteristic
13C22×C4⋊C4×C67C411yescharacteristic
14#398×C67C2212yes—
15#397×C67C412yes—
16#389×C67—21no—
17#381×C67—21no—
18#380×C67—21no—
19#380×C67C411yescharacteristic
20C2×#11761612—22no—
21C22×C67⋊C8—41no—
22C4⋊(C67⋊C8)—41no—
23C4⋊(C67⋊C8)—41no—
24#14523960—42no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—41no—
27C22×D4×C67D411yescharacteristic
28C23×C4×C67—21no—
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31#123×C67—42no—
32#123×C67—42no—
33C4×D4×C67—41no—
34C4×D4×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—22no—
38C2×C4⋊C4×C67—41no—
39C2×C4⋊C4×C67—41no—
40C2×C4⋊C4×C67C2×C412yes—
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67C2×C411yescharacteristic
45C2×C22⋊C4×C67D411yescharacteristic
46C2×C42×C67D411yescharacteristic
47#95×C67—22no—
48#95×C67—22no—
49#95×C67C2×C412yes—
50#11761612—42no—
51#11761612—42no—
52C2×C67⋊C8—42no—
53C2×C67⋊C8—42no—
54C2×C67⋊C8—82no—
55C24×C67C22⋊C411yescharacteristic
56C24×C67C22⋊C411yescharacteristic
57C2×D4×C67—41no—
58C2×D4×C67—41no—
59C2×D4×C67D812yes—
60C2×D4×C67SD1612yes—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—22no—
66C22×C4×C67—22no—
67C22×C4×C67—42no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67—42no—
71C22×C4×C67—42no—
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22×C411yescharacteristic
75C22×C4×C67C2×D411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C4⋊C4×C67—41no—
78C4⋊C4×C67—42no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—42no—
83C4⋊C4×C67—42no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—44no—
88C42×C67—21no—
89C42×C67—21no—
90C23×C67—21no—
91C23×C67—21no—
92C23×C67—21no—
93C23×C67—21no—
94C23×C67—41no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67#10212yes—
99C23×C67#10212yes—
100C23×C67C2×C22⋊C411yescharacteristic
101D4×C67—42no—
102D4×C67—42no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—22no—
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—41no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
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—81no—
122C2×C4×C67—84no—
123C2×C4×C67#9911yescharacteristic
124C2×C4×C67#9911yescharacteristic
125C2×C4×C67C2×D811yescharacteristic
126C2×C4×C67C2×SD1611yescharacteristic
127C67⋊C8—82no—
128C67⋊C8—82no—
129C22×C67—21no—
130C22×C67—21no—
131C22×C67—22no—
132C22×C67—22no—
133C22×C67—42no—
134C22×C67—42no—
135C22×C67—42no—
136C22×C67—41no—
137C22×C67—41no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—81no—
141C22×C67C2×#9911yescharacteristic
142C22×C67C2×#10211yescharacteristic
143C22×C67#42111yescharacteristic
144C4×C67—21no—
145C4×C67—21no—
146C4×C67—21no—
147C4×C67—41no—
148C4×C67—41no—
149C4×C67—41no—
150C4×C67—82no—
151C4×C67—82no—
152C4×C67#35212yes—
153#11881—671nomaximal, Sylow
154C2×C67—21no—
155C2×C67—21no—
156C2×C67—42no—
157C2×C67—42no—
158C2×C67#151911yescharacteristic
159C2×C67C2×#35211yescharacteristic
160C2×C67#213511yescharacteristic
161#2486D6711yescharacteristic
162#1800—671no—
163#1459—671no—
164#1346—674no—
165C67#1188111yescharacteristic, complemented, Sylow
166C2×#123—21no—
167C2×C4×D4C2×D6711yescharacteristic
168C22×C4⋊C4Dic6711yescharacteristic
169#423—1341no—
170C2×#100—1341no—
171C2×C22⋊C8—1341no—
172#398C2×D6712yes—
173#397Dic6712yes—
174#389—21no—
175#381—21no—
176#380—21no—
177#380Dic6711yescharacteristic
178#343—672no—
179#324—672no—
180#323—1341no—
181C22×D4C22⋊D6711yescharacteristic
182C23×C4—21no—
183C23×C4C2×Dic6711yescharacteristic
184C23×C4C22⋊D6711yescharacteristic
185C2×M16—1342no—
186C22×C8—2681no—
187#123—42no—
188#123—42no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C2×Dic6712yes—
197C2×C4⋊C4C22×D6711yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4C2×Dic6711yescharacteristic
201C2×C22⋊C4C22⋊D6711yescharacteristic
202C2×C42C22⋊D6711yescharacteristic
203C4⋊C8—2681no—
204C4⋊C8—2681no—
205#100—2682no—
206C22⋊C8—2681no—
207C22⋊C8—1342no—
208#95—22no—
209#95—22no—
210#95C2×Dic6712yes—
211C24C22⋊Dic6711yescharacteristic
212C24C22⋊Dic6711yescharacteristic
213C2×D4—41no—
214C2×D4—41no—
215C2×D4#1176161312yes—
216C2×D4#1176161612yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊Dic6711yescharacteristic
229C22×C4C22⋊Dic6711yescharacteristic
230C22×C4C22×Dic6711yescharacteristic
231C22×C4C2×C22⋊D6711yescharacteristic
232C22×C4C2×C22⋊D6711yescharacteristic
233M16—2682no—
234M16—2682no—
235C2×C8—2682no—
236C2×C8—2682no—
237C2×C8—5362no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#1452396412yes—
260C23#1452396412yes—
261C23C2×C22⋊Dic6711yescharacteristic
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—84no—
284C2×C4#1452395811yescharacteristic
285C2×C4#1452395811yescharacteristic
286C2×C4C2×#1176161311yescharacteristic
287C2×C4C2×#1176161611yescharacteristic
288C8—5362no—
289C8—5362no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#1452395811yescharacteristic
303C22C2×#1452396411yescharacteristic
304C22#1686341311yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4#1686318612yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2255877711yescharacteristic
319C2C2×#1686318611yescharacteristic
320C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2486C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C4⋊C4×C67C23
hypercenter#2486D67
hypocenterC67#11881
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #22558215C2semidirect
165. C67#11881semidirect

Maximal subgroups

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

subgroupindex
#225602012
#225585272
#225582152
C67×#24862
#1188167