Finite Group Databasebeta

← #30765258

Subgroups — #30765258

Order 17152.

418 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 99 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#22558543C211yesmaximal, characteristic
2#22558543C211yesmaximal, characteristic
3#22558157C214yesmaximal
4C67×#2317C211yesmaximal, characteristic
5#16863453—21no—
6#16863453—21no—
7#16863122—21no—
8#16863122—21no—
9#16863120C2212yes—
10#16863120C2212yes—
11C2×#117×C67C411yescharacteristic
12C2×#117×C67C2211yescharacteristic
13C22×C4⋊C4×C67C411yescharacteristic
14#389×C67—21no—
15#389×C67—21no—
16#385×C67C412yes—
17#385×C67C2212yes—
18#384×C67—21no—
19#382×C67—21no—
20C4⋊(C67⋊C8)—41no—
21C4⋊(C67⋊C8)—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—22no—
24#14523954—41no—
25#14523954—41no—
26C23×C4×C67C2×C411yescharacteristic
27C23×C4×C67C2×C412yes—
28#117×C67—24no—
29#117×C67—41no—
30#117×C67—41no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—41no—
37C2×C4⋊C4×C67—41no—
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C4⋊C4×C67D411yescharacteristic
40C2×C22⋊C4×C67C2×C412yes—
41C2×C22⋊C4×C67C2×C411yescharacteristic
42C2×C22⋊C4×C67C2311yescharacteristic
43C2×C42×C67—21no—
44C2×C42×C67D411yescharacteristic
45C2×C42×C67D411yescharacteristic
46#95×C67—22no—
47#95×C67—22no—
48C2×C67⋊C8—42no—
49C2×C67⋊C8—42no—
50C24×C67C22×C411yescharacteristic
51C22×C4×C67—22no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67—22no—
55C22×C4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—42no—
58C22×C4×C67—82no—
59C22×C4×C67C22⋊C411yescharacteristic
60C22×C4×C67C22⋊C411yescharacteristic
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C2×D411yescharacteristic
64C22×C4×C67C2×D411yescharacteristic
65C4⋊C4×C67—24no—
66C4⋊C4×C67—41no—
67C4⋊C4×C67—41no—
68C4⋊C4×C67—42no—
69C4⋊C4×C67—41no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—82no—
72C4⋊C4×C67—82no—
73C22⋊C4×C67—22no—
74C22⋊C4×C67—41no—
75C22⋊C4×C67—41no—
76C22⋊C4×C67M1614yes—
77C42×C67—24no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—21no—
81C42×C67—21no—
82C23×C67—21no—
83C23×C67—21no—
84C23×C67—41no—
85C23×C67—41no—
86C23×C67C2×C22⋊C411yescharacteristic
87C23×C67C2×M1612yes—
88C2×C4×C67—24no—
89C2×C4×C67—24no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—24no—
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—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—84no—
109C2×C4×C67#9911yescharacteristic
110C2×C4×C67#9911yescharacteristic
111C2×C4×C67#10112yes—
112C67⋊C8—82no—
113C67⋊C8—82no—
114C22×C67—21no—
115C22×C67—21no—
116C22×C67—22no—
117C22×C67—42no—
118C22×C67—42no—
119C22×C67—41no—
120C22×C67—41no—
121C22×C67#40711yescharacteristic
122C22×C67C2×#9911yescharacteristic
123C22×C67C2×#10111yescharacteristic
124C4×C67—21no—
125C4×C67—21no—
126C4×C67—22no—
127C4×C67—44no—
128C4×C67—41no—
129C4×C67—41no—
130C4×C67—41no—
131C4×C67—41no—
132C4×C67—82no—
133#11520—671nomaximal, Sylow
134C2×C67—21no—
135C2×C67—21no—
136C2×C67—42no—
137C2×C67#146311yescharacteristic
138C2×C67#212211yescharacteristic
139C2×C67#212311yescharacteristic
140#2317D6711yescharacteristic
141#1464—671no—
142#1464—671no—
143#1326—674no—
144C67#1152011yescharacteristic, complemented, Sylow
145C2×#117Dic6711yescharacteristic
146C2×#117C2×D6711yescharacteristic
147C22×C4⋊C4Dic6711yescharacteristic
148#432—1341no—
149#432—1341no—
150#389—21no—
151#389—21no—
152#385Dic6712yes—
153#385C2×D6712yes—
154#384—21no—
155#382—21no—
156#324—1341no—
157#324—1341no—
158#323—672no—
159#323—672no—
160C23×C4C2×Dic6711yescharacteristic
161C23×C4C2×Dic6712yes—
162#117—24no—
163#117—41no—
164#117—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4C22⋊D6711yescharacteristic
173C2×C4⋊C4C22⋊D6711yescharacteristic
174C2×C22⋊C4C2×Dic6712yes—
175C2×C22⋊C4C2×Dic6711yescharacteristic
176C2×C22⋊C4C22×D6711yescharacteristic
177C2×C42—21no—
178C2×C42C22⋊D6711yescharacteristic
179C2×C42C22⋊D6711yescharacteristic
180C4⋊C8—2681no—
181C4⋊C8—2681no—
182C22⋊C8—1342no—
183C22⋊C8—1342no—
184#97—2681no—
185#97—2681no—
186#95—22no—
187#95—22no—
188C24C22×Dic6711yescharacteristic
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—82no—
197C22×C4C22⋊Dic6711yescharacteristic
198C22×C4C22⋊Dic6711yescharacteristic
199C22×C4C22⋊Dic6711yescharacteristic
200C22×C4C22⋊Dic6711yescharacteristic
201C22×C4C2×C22⋊D6711yescharacteristic
202C22×C4C2×C22⋊D6711yescharacteristic
203C2×C8—2682no—
204C2×C8—2682no—
205C4⋊C4—24no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C22⋊C4—22no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4#1176161214yes—
217C42—24no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23C2×C22⋊Dic6711yescharacteristic
227C23C2×#1176161212yes—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—84no—
249C2×C4#1452395811yescharacteristic
250C2×C4#1452395811yescharacteristic
251C2×C4#1452396212yes—
252C8—5362no—
253C8—5362no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22#1686336011yescharacteristic
262C22C2×#1452395811yescharacteristic
263C22C2×#1452396211yescharacteristic
264C4—21no—
265C4—21no—
266C4—22no—
267C4—44no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C2—21no—
274C2—21no—
275C2—42no—
276C2#2255853911yescharacteristic
277C2#2256185911yescharacteristic
278C2#2256186311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutator, prime radicalC24×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2317C2
socleC22×C67C2×#99
cosocleC2×C22⋊C4×C67C23
hypercenter#2317D67
hypocenterC67#11520
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
144. C67#11520semidirect

Maximal subgroups

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

subgroupindex
#225585432
#225581572
C67×#23172
#1152067