Finite Group Databasebeta

← #30766565

Subgroups — #30766565

Order 17152.

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

normal subgroups only

This group has 434 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#22560289C211yesmaximal, characteristic
2#22558539C211yesmaximal, characteristic
3#22558236C214yesmaximal, complemented
4C2×#400×C67C211yesmaximal, characteristic
5#16863360—21no—
6#16863360—21no—
7#16863165—21no—
8#16863165C2212yes—
9#16863122—21no—
10#16863122C2212yes—
11C22×C4⋊C4×C67C411yescharacteristic
12C22×C4⋊C4×C67C2211yescharacteristic
13#400×C67—21no—
14#400×C67—21no—
15#400×C67C412yes—
16#400×C67C2212yes—
17C2×#95×C67—21no—
18C2×#95×C67—21no—
19C2×#95×C67C411yescharacteristic
20C2×#11761612—22no—
21C2×#11761612—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—41no—
24C22⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—41no—
26C23×C4×C67—21no—
27C23×C4×C67—21no—
28C23×C4×C67C2×C411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67—41no—
33C2×C4⋊C4×C67—41no—
34C2×C4⋊C4×C67—41no—
35C2×C4⋊C4×C67—41no—
36C2×C4⋊C4×C67C2×C412yes—
37C2×C4⋊C4×C67C2×C411yescharacteristic
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C4⋊C4×C67D411yescharacteristic
40C2×C4⋊C4×C67C2311yescharacteristic
41#95×C67—22no—
42#95×C67—22no—
43#95×C67—21no—
44#95×C67—22no—
45#95×C67—22no—
46#95×C67C2×C412yes—
47#11761612—42no—
48#11761612—42no—
49#11761612—82no—
50C2×C67⋊C8—42no—
51C2×C67⋊C8—42no—
52C24×C67C22⋊C411yescharacteristic
53C22×C4×C67—21no—
54C22×C4×C67—21no—
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—22no—
62C22×C4×C67—41no—
63C22×C4×C67—42no—
64C22×C4×C67—42no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67—44no—
68C22×C4×C67—42no—
69C22×C4×C67—41no—
70C22×C4×C67—41no—
71C22×C4×C67C22⋊C411yescharacteristic
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—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—42no—
82C4⋊C4×C67—42no—
83C23×C67—22no—
84C23×C67—22no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67#9911yescharacteristic
88C23×C67#9911yescharacteristic
89C23×C67C2×C22⋊C411yescharacteristic
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—42no—
100C2×C4×C67—42no—
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—41no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—44no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—81no—
114C2×C4×C67—84no—
115C2×C4×C67—82no—
116C2×C4×C67—82no—
117C2×C4×C67#13712yes—
118C67⋊C8—82no—
119C67⋊C8—82no—
120C22×C67—24no—
121C22×C67—24no—
122C22×C67—21no—
123C22×C67—21no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67—21no—
127C22×C67—21no—
128C22×C67#35212yes—
129C22×C67#35512yes—
130C22×C67C2×#9911yescharacteristic
131C22×C67#41911yescharacteristic
132C22×C67#42111yescharacteristic
133C4×C67—22no—
134C4×C67—41no—
135C4×C67—41no—
136C4×C67—41no—
137C4×C67—41no—
138C4×C67—41no—
139C4×C67—82no—
140C4×C67—82no—
141#11814—671nomaximal, Sylow
142C2×C67—21no—
143C2×C67—22no—
144C2×C67—22no—
145C2×C67—21no—
146C2×C67#151111yescharacteristic
147C2×C67C2×#35211yescharacteristic
148C2×C67C2×#35511yescharacteristic
149C2×#400D6711yescharacteristic
150#1821—671no—
151#1463—671no—
152#1353—674no—
153C67#1181411yescharacteristic, complemented, Sylow
154C22×C4⋊C4Dic6711yescharacteristic
155C22×C4⋊C4C2×D6711yescharacteristic
156#407—1341no—
157#407—1341no—
158#400—21no—
159#400—21no—
160#400Dic6712yes—
161#400C2×D6712yes—
162C2×#95—21no—
163C2×#95—21no—
164C2×#95Dic6711yescharacteristic
165#343—672no—
166#343—1341no—
167#324—672no—
168#324—1341no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4C2×Dic6711yescharacteristic
172C23×C4C22⋊D6711yescharacteristic
173C23×C4C22⋊D6711yescharacteristic
174C2×M16—2681no—
175C2×M16—1342no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4C2×Dic6712yes—
182C2×C4⋊C4C2×Dic6711yescharacteristic
183C2×C4⋊C4C22⋊D6711yescharacteristic
184C2×C4⋊C4C22⋊D6711yescharacteristic
185C2×C4⋊C4C22×D6711yescharacteristic
186C22⋊C8—2681no—
187C22⋊C8—2681no—
188C22⋊C8—2681no—
189C22⋊C8—1342no—
190#95—22no—
191#95—22no—
192#95—21no—
193#95—22no—
194#95—22no—
195#95C2×Dic6712yes—
196C24C22⋊Dic6711yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4C22⋊Dic6711yescharacteristic
216C22×C4C22⋊Dic6711yescharacteristic
217C22×C4C22⋊Dic6711yescharacteristic
218C22×C4C22×Dic6711yescharacteristic
219C22×C4C2×C22⋊D6711yescharacteristic
220C22×C4C2×C22⋊D6711yescharacteristic
221M16—2682no—
222M16—2682no—
223M16—5362no—
224C2×C8—2682no—
225C2×C8—2682no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23#1452395811yescharacteristic
237C23#1452395811yescharacteristic
238C23C2×C22⋊Dic6711yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—84no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#1452407412yes—
267C8—5362no—
268C8—5362no—
269C22—24no—
270C22—24no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22#1686318612yes—
278C22#1686319312yes—
279C22C2×#1452395811yescharacteristic
280C22#1686340511yescharacteristic
281C22#1686341311yescharacteristic
282C4—22no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—82no—
289C4—82no—
290C2—21no—
291C2—21no—
292C2—22no—
293C2—22no—
294C2#2255874211yescharacteristic
295C2C2×#1686318611yescharacteristic
296C2C2×#1686319311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC2×#400×C67C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenterC2×#400D67
hypocenterC67#11814
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
3. #22558236C2semidirect
153. C67#11814semidirect

Maximal subgroups

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

subgroupindex
#225602892
#225585392
#225582362
C2×#400×C672
#1181467