Finite Group Databasebeta

← #30765508

Subgroups — #30765508

Order 17152.

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

normal subgroups only

This group has 426 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#22558550C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558157C214yesmaximal, complemented
4C67×#2487C211yesmaximal, characteristic
5#16863453—21no—
6C2×C22⋊(C67⋊C8)—21no—
7#16863122—21no—
8#16863120—21no—
9#16863120C2212yes—
10#16863120C2212yes—
11C2×C4×D4×C67C2211yescharacteristic
12C22×C4⋊C4×C67C411yescharacteristic
13#400×C67—21no—
14#399×C67—21no—
15#389×C67—21no—
16#386×C67C412yes—
17#385×C67C2212yes—
18#380×C67—21no—
19#380×C67C411yescharacteristic
20C22×C67⋊C8—41no—
21C4⋊(C67⋊C8)—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—22no—
24C22⋊(C67⋊C8)—41no—
25#14523954—41no—
26C22×D4×C67C2×C411yescharacteristic
27C23×C4×C67C2×C411yescharacteristic
28C23×C4×C67C2×C412yes—
29C23×C4×C67D411yescharacteristic
30C4×D4×C67—41no—
31C4×D4×C67—41no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—41no—
37C2×C4⋊C4×C67—41no—
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67C2×C412yes—
41C2×C22⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67C2311yescharacteristic
43C2×C42×C67D411yescharacteristic
44#95×C67—21no—
45#95×C67—22no—
46#95×C67—22no—
47C2×C67⋊C8—42no—
48C2×C67⋊C8—42no—
49C2×C67⋊C8—82no—
50C24×C67C22⋊C411yescharacteristic
51C24×C67C22×C411yescharacteristic
52C2×D4×C67—22no—
53C2×D4×C67—22no—
54C2×D4×C67—41no—
55C22×C4×C67—21no—
56C22×C4×C67—21no—
57C22×C4×C67—21no—
58C22×C4×C67—21no—
59C22×C4×C67—22no—
60C22×C4×C67—22no—
61C22×C4×C67—42no—
62C22×C4×C67—42no—
63C22×C4×C67—42no—
64C22×C4×C67—82no—
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C2×D411yescharacteristic
69C22×C4×C67C2×D411yescharacteristic
70C4⋊C4×C67—41no—
71C4⋊C4×C67—42no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—41no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—82no—
76C22⋊C4×C67—24no—
77C22⋊C4×C67—41no—
78C22⋊C4×C67—41no—
79C22⋊C4×C67—82no—
80C42×C67—21no—
81C42×C67—21no—
82C23×C67—21no—
83C23×C67—21no—
84C23×C67—21no—
85C23×C67—21no—
86C23×C67—41no—
87C23×C67—41no—
88C23×C67—41no—
89C23×C67—41no—
90C23×C67#10112yes—
91C23×C67C2×C22⋊C411yescharacteristic
92C23×C67#13112yes—
93D4×C67—44no—
94C2×C4×C67—22no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—42no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
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—81no—
114C2×C4×C67—84no—
115C2×C4×C67#9911yescharacteristic
116C2×C4×C67#9911yescharacteristic
117C67⋊C8—82no—
118C67⋊C8—82no—
119C22×C67—21no—
120C22×C67—21no—
121C22×C67—22no—
122C22×C67—22no—
123C22×C67—42no—
124C22×C67—42no—
125C22×C67—42no—
126C22×C67—41no—
127C22×C67—41no—
128C22×C67—41no—
129C22×C67—41no—
130C22×C67—81no—
131C22×C67#40811yescharacteristic
132C22×C67C2×#9911yescharacteristic
133C22×C67C2×#10111yescharacteristic
134C4×C67—21no—
135C4×C67—41no—
136C4×C67—41no—
137C4×C67—41no—
138C4×C67—41no—
139C4×C67—82no—
140C4×C67—82no—
141C4×C67#35212yes—
142#11576—671nomaximal, Sylow
143C2×C67—21no—
144C2×C67—21no—
145C2×C67—42no—
146C2×C67—42no—
147C2×C67#147211yescharacteristic
148C2×C67C2×#35211yescharacteristic
149C2×C67#212211yescharacteristic
150#2487D6711yescharacteristic
151#1466—671no—
152#1459—671no—
153#1326—674no—
154C67#1157611yescharacteristic, complemented, Sylow
155C2×C4×D4C2×D6711yescharacteristic
156C22×C4⋊C4Dic6711yescharacteristic
157#432—1341no—
158C2×C22⋊C8—1341no—
159#400—21no—
160#399—21no—
161#389—21no—
162#386Dic6712yes—
163#385C2×D6712yes—
164#380—21no—
165#380Dic6711yescharacteristic
166#324—1341no—
167#323—672no—
168#323—672no—
169#323—1341no—
170C22×D4C2×Dic6711yescharacteristic
171C23×C4C2×Dic6711yescharacteristic
172C23×C4C2×Dic6712yes—
173C23×C4C22⋊D6711yescharacteristic
174C22×C8—2681no—
175C4×D4—41no—
176C4×D4—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4C22⋊D6711yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4C2×Dic6712yes—
186C2×C22⋊C4C22⋊D6711yescharacteristic
187C2×C22⋊C4C22×D6711yescharacteristic
188C2×C42C22⋊D6711yescharacteristic
189C4⋊C8—2681no—
190C22⋊C8—2681no—
191C22⋊C8—1342no—
192C22⋊C8—1342no—
193#97—2681no—
194#95—21no—
195#95—22no—
196#95—22no—
197C24C22⋊Dic6711yescharacteristic
198C24C22×Dic6711yescharacteristic
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—82no—
212C22×C4C22⋊Dic6711yescharacteristic
213C22×C4C22⋊Dic6711yescharacteristic
214C22×C4C22⋊Dic6711yescharacteristic
215C22×C4C2×C22⋊D6711yescharacteristic
216C22×C4C2×C22⋊D6711yescharacteristic
217C2×C8—2682no—
218C2×C8—2682no—
219C2×C8—5362no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—82no—
226C22⋊C4—24no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23#1452396212yes—
241C23C2×C22⋊Dic6711yescharacteristic
242C23#1452405012yes—
243D4—44no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—84no—
265C2×C4#1452395811yescharacteristic
266C2×C4#1452395811yescharacteristic
267C8—5362no—
268C8—5362no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22#1686336311yescharacteristic
282C22C2×#1452395811yescharacteristic
283C22C2×#1452396211yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—82no—
290C4—82no—
291C4#1686318612yes—
292C2—21no—
293C2—21no—
294C2—42no—
295C2—42no—
296C2#2255857311yescharacteristic
297C2C2×#1686318611yescharacteristic
298C2#2256185911yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC24×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2487C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C22⋊C4×C67C23
hypercenter#2487D67
hypocenterC67#11576
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #22558157C2semidirect
154. C67#11576semidirect

Maximal subgroups

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

subgroupindex
#225585502
#225585272
#225581572
C67×#24872
#1157667