Finite Group Databasebeta

← #30774519

Subgroups — #30774519

Order 17152.

420 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 111 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#22560205C211yesmaximal, characteristic
2C4×#14523962C211yesmaximal, characteristic
3#22558336C214yesmaximal
4C67×#2567C211yesmaximal, characteristic
5#16863420—21no—
6C2×#14523962C2212yes—
7C2×#14523962C2212yes—
8C4×#11761612—21no—
9#16863165—21no—
10#16863165—21no—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×C4×Q8×C67C2211yescharacteristic
13#395×C67—21no—
14#395×C67—21no—
15#391×C67C2212yes—
16#389×C67—21no—
17#389×C67—21no—
18#384×C67C412yes—
19C4×C4⋊C4×C67C411yescharacteristic
20C2×#11761612—22no—
21C2×#11761612—22no—
22C4⋊(C67⋊C8)—41no—
23C4⋊(C67⋊C8)—41no—
24#14523962—22no—
25#14523962—24no—
26#14523962Q814yescomplemented
27#14523954—41no—
28C4×C67⋊C8—41no—
29C22×Q8×C67—21no—
30C22×Q8×C67C2311yescharacteristic
31C4⋊Q8×C67—41no—
32C4⋊Q8×C67—41no—
33C4×Q8×C67—41no—
34C4×Q8×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—22no—
38C2×C4⋊C4×C67—22no—
39C2×C4⋊C4×C67C2×C412yes—
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67C2×C412yes—
45C2×C42×C67D411yescharacteristic
46C2×C42×C67D411yescharacteristic
47#95×C67—21no—
48#95×C67—22no—
49#95×C67—22no—
50#11761612—24no—
51#11761612—42no—
52#11761612—42no—
53#11761612—42no—
54C2×C67⋊C8—42no—
55C2×C67⋊C8—42no—
56C2×Q8×C67—41no—
57C2×Q8×C67—41no—
58C2×Q8×C67—42no—
59C2×Q8×C67—82no—
60C2×Q8×C67C2×Q812yes—
61C2×Q8×C67#4112yes—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22×C411yescharacteristic
69C22×C4×C67C2×D411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C4⋊C4×C67—24no—
72C4⋊C4×C67—41no—
73C4⋊C4×C67—41no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—82no—
79C4⋊C4×C67—82no—
80C42×C67—21no—
81C42×C67—21no—
82C42×C67—21no—
83C42×C67—21no—
84C42×C67—21no—
85C42×C67—21no—
86C42×C67—42no—
87C23×C67C2×C22⋊C411yescharacteristic
88Q8×C67—42no—
89Q8×C67—42no—
90Q8×C67—84no—
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—22no—
98C2×C4×C67—24no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
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#9911yescharacteristic
111C2×C4×C67#9911yescharacteristic
112C2×C4×C67C4×Q812yes—
113C2×C4×C67C22⋊Q812yes—
114C2×C4×C67#12411yescharacteristic
115C2×C4×C67C4⋊Q811yescharacteristic
116C67⋊C8—44no—
117C67⋊C8—82no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67#39111yescharacteristic
121C22×C67C2×#9911yescharacteristic
122C22×C67#41011yescharacteristic
123C4×C67—21no—
124C4×C67—22no—
125C4×C67—21no—
126C4×C67—22no—
127C4×C67—21no—
128C4×C67—44no—
129C4×C67—41no—
130C4×C67—41no—
131C4×C67—41no—
132C4×C67—82no—
133C4×C67#35612yes—
134#13302—671nomaximal, Sylow
135C2×C67—21no—
136C2×C67—21no—
137C2×C67#199611yescharacteristic
138C2×C67C2×#35611yescharacteristic
139C2×C67#213411yescharacteristic
140#2567D6711yescharacteristic
141#1801—671no—
142C4×#101—671no—
143#1394—674no—
144C67#1330211yescharacteristic, complemented, Sylow
145C2×C4⋊Q8Dic6711yescharacteristic
146C2×C4×Q8C2×D6711yescharacteristic
147#423—1341no—
148C2×#101—672no—
149C2×#101—672no—
150C4×M16—1341no—
151#395—21no—
152#395—21no—
153#391C2×D6712yes—
154#389—21no—
155#389—21no—
156#384Dic6712yes—
157C4×C4⋊C4Dic6711yescharacteristic
158#343—1341no—
159#343—1341no—
160C22×Q8—21no—
161C22×Q8C22×D6711yescharacteristic
162C2×M16—1342no—
163C2×M16—1342no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C2×Dic6712yes—
173C2×C4⋊C4C2×Dic6711yescharacteristic
174C2×C4⋊C4C22⋊D6711yescharacteristic
175C2×C4⋊C4C22⋊D6711yescharacteristic
176C2×C42C2×Dic6711yescharacteristic
177C2×C42C2×Dic6712yes—
178C2×C42C22⋊D6711yescharacteristic
179C2×C42C22⋊D6711yescharacteristic
180C4⋊C8—2681no—
181C4⋊C8—2681no—
182#101—674no—
183#101—1344no—
184#101—1342no—
185#97—2681no—
186C4×C8—2681no—
187#95—21no—
188#95—22no—
189#95—22no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—82no—
194C2×Q8Q8×D6712yes—
195C2×Q8#1176162912yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4C22⋊Dic6711yescharacteristic
199C22×C4C22⋊Dic6711yescharacteristic
200C22×C4C22⋊Dic6711yescharacteristic
201C22×C4C22⋊Dic6711yescharacteristic
202C22×C4C22×Dic6711yescharacteristic
203C22×C4C2×C22⋊D6711yescharacteristic
204C22×C4C2×C22⋊D6711yescharacteristic
205M16—2682no—
206M16—2682no—
207M16—2682no—
208M16—1344no—
209C2×C8—2682no—
210C2×C8—2682no—
211C4⋊C4—24no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C23C2×C22⋊Dic6711yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4#1452395811yescharacteristic
251C2×C4#1452395811yescharacteristic
252C2×C4Q8×Dic6712yes—
253C2×C4#1452401912yes—
254C2×C4#1452403211yescharacteristic
255C2×C4#1452404111yescharacteristic
256C8—2684no—
257C8—5362no—
258C22—21no—
259C22—21no—
260C22#1686330311yescharacteristic
261C22C2×#1452395811yescharacteristic
262C22#1686337011yescharacteristic
263C4—21no—
264C4—22no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—44no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4#1686319512yes—
274C2—21no—
275C2—21no—
276C2#2256119511yescharacteristic
277C2C2×#1686319511yescharacteristic
278C2#2256190111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×Q8C22×D67
FittingC67×#2567C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC22×Q8×C67C23
hypercenter#2567D67
hypocenterC67#13302
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
26. #14523962Q8semidirect
144. C67#13302semidirect

Maximal subgroups

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

subgroupindex
#225602052
C4×#145239622
#225583362
C67×#25672
#1330267