Finite Group Databasebeta

← #30765487

Subgroups — #30765487

Order 17152.

350 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 91 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#22558553C211yesmaximal, characteristic
2#22558533C211yesmaximal, characteristic
3#22558155C212yesmaximal
4#22558153C212yesmaximal
5C67×#2565C211yesmaximal, characteristic
6#16863453—21no—
7#16863449—21no—
8#16863122—21no—
9#16863122—21no—
10#16863122C2212yes—
11#16863122C2212yes—
12C2×#125×C67C411yescharacteristic
13C2×C4×Q8×C67C2211yescharacteristic
14#398×C67—21no—
15#398×C67—21no—
16#391×C67C411yescharacteristic
17#391×C67C411yescharacteristic
18#387×C67—21no—
19#387×C67—21no—
20#384×C67C2211yescharacteristic
21#382×C67C2211yescharacteristic
22C4×C4⋊C4×C67C411yescharacteristic
23C4⋊(C67⋊C8)—41no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—22no—
27#14523954—41no—
28C4×C67⋊C8—41no—
29C22×Q8×C67C2×C411yescharacteristic
30#125×C67—41no—
31#125×C67—41no—
32C4×Q8×C67—41no—
33C4×Q8×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—21no—
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C4⋊C4×C67C2311yescharacteristic
44C2×C42×C67C2×C411yescharacteristic
45C2×C42×C67C2×C411yescharacteristic
46C2×C42×C67C2×C411yescharacteristic
47C2×C42×C67D411yescharacteristic
48C2×C42×C67D411yescharacteristic
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—
58C2×Q8×C67—22no—
59C2×Q8×C67—22no—
60C2×Q8×C67—41no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22×C411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C4⋊C4×C67—41no—
73C4⋊C4×C67—41no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—42no—
78C4⋊C4×C67—42no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—41no—
83C42×C67—21no—
84C42×C67—21no—
85C42×C67—21no—
86C42×C67—21no—
87C42×C67—21no—
88C42×C67—21no—
89C42×C67—22no—
90C42×C67—22no—
91C23×C67C2×C22⋊C411yescharacteristic
92Q8×C67—44no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—22no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—41no—
113C2×C4×C67—41no—
114C2×C4×C67—42no—
115C2×C4×C67—41no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67#9911yescharacteristic
119C2×C4×C67#9911yescharacteristic
120C2×C4×C67#10112yes—
121C2×C4×C67#13112yes—
122C67⋊C8—82no—
123C67⋊C8—82no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67#40811yescharacteristic
127C22×C67C2×#9911yescharacteristic
128C22×C67C2×#10111yescharacteristic
129C4×C67—21no—
130C4×C67—21no—
131C4×C67—22no—
132C4×C67—22no—
133C4×C67—42no—
134C4×C67—41no—
135C4×C67—41no—
136C4×C67—41no—
137C4×C67—41no—
138C4×C67—81no—
139C4×C67—81no—
140C4×C67—81no—
141#11571—671nomaximal, Sylow
142C2×C67—21no—
143C2×C67—21no—
144C2×C67#147211yescharacteristic
145C2×C67#212611yescharacteristic
146C2×C67#213311yescharacteristic
147#2565D6711yescharacteristic
148#1467—671no—
149#1461—671no—
150#1325—672no—
151#1324—672no—
152C67#1157111yescharacteristic, complemented, Sylow
153C2×#125Dic6711yescharacteristic
154C2×C4×Q8C2×D6711yescharacteristic
155#432—1341no—
156#431—1341no—
157#398—21no—
158#398—21no—
159#391Dic6711yescharacteristic
160#391Dic6711yescharacteristic
161#387—21no—
162#387—21no—
163#384C2×D6711yescharacteristic
164#382C2×D6711yescharacteristic
165C4×C4⋊C4Dic6711yescharacteristic
166#324—672no—
167#324—672no—
168#324—1341no—
169#324—1341no—
170C22×Q8C2×Dic6711yescharacteristic
171#125—41no—
172#125—41no—
173C4×Q8—41no—
174C4×Q8—41no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C2×Dic6711yescharacteristic
181C2×C4⋊C4C2×Dic6711yescharacteristic
182C2×C4⋊C4C22⋊D6711yescharacteristic
183C2×C4⋊C4C22⋊D6711yescharacteristic
184C2×C4⋊C4C22×D6711yescharacteristic
185C2×C42C2×Dic6711yescharacteristic
186C2×C42C2×Dic6711yescharacteristic
187C2×C42C2×Dic6711yescharacteristic
188C2×C42C22⋊D6711yescharacteristic
189C2×C42C22⋊D6711yescharacteristic
190C4⋊C8—2681no—
191C4⋊C8—2681no—
192C22⋊C8—1342no—
193C22⋊C8—1342no—
194#97—2681no—
195C4×C8—2681no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4C22⋊Dic6711yescharacteristic
211C22×C4C22⋊Dic6711yescharacteristic
212C22×C4C22⋊Dic6711yescharacteristic
213C22×C4C22⋊Dic6711yescharacteristic
214C22×C4C22×Dic6711yescharacteristic
215C22×C4C2×C22⋊D6711yescharacteristic
216C22×C4C2×C22⋊D6711yescharacteristic
217C2×C8—2682no—
218C2×C8—2682no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—22no—
237C42—22no—
238C23C2×C22⋊Dic6711yescharacteristic
239Q8—44no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4#1452395811yescharacteristic
266C2×C4#1452395811yescharacteristic
267C2×C4#1452396212yes—
268C2×C4#1452405012yes—
269C8—5362no—
270C8—5362no—
271C22—21no—
272C22—21no—
273C22#1686336311yescharacteristic
274C22C2×#1452395811yescharacteristic
275C22C2×#1452396211yescharacteristic
276C4—21no—
277C4—21no—
278C4—22no—
279C4—22no—
280C4—42no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—81no—
286C4—81no—
287C4—81no—
288C2—21no—
289C2—21no—
290C2#2255857311yescharacteristic
291C2#2256187411yescharacteristic
292C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2565C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2565D67
hypocenterC67#11571
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2565

Complements

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

NH ≅ G/Ntype
152. C67#11571semidirect

Maximal subgroups

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

subgroupindex
#225585532
#225585332
#225581552
#225581532
C67×#25652
#1157167