Finite Group Databasebeta

← #30771834

Subgroups — #30771834

Order 17152.

468 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#16863195C212yesmaximal
2#22560205C212yesmaximal
3#22558336C212yesmaximal
4C4×C4⋊Q8×C67C211yesmaximal, characteristic
5#16863420—22no—
6C2×#14523962C2214yes—
7#16863195—24no—
8#16863165—22no—
9C2×C4⋊Q8×C67C2211yescharacteristic
10C2×C4×Q8×C67C411yescharacteristic
11C2×C4×Q8×C67C2211yescharacteristic
12#391×C67C411yescharacteristic
13#391×C67C2211yescharacteristic
14#389×C67—22no—
15#384×C67C411yescharacteristic
16C4×C4⋊C4×C67—22no—
17C43×C67C411yescharacteristic
18C2×#11761612—24no—
19C4⋊(C67⋊C8)—42no—
20C4⋊(C67⋊C8)—42no—
21#14523962—24no—
22#14523962—24no—
23C22×Q8×C67C2×C411yescharacteristic
24C22×Q8×C67C2311yescharacteristic
25C4⋊Q8×C67—22no—
26C4⋊Q8×C67D412yes—
27C4⋊Q8×C67Q812yes—
28C4×Q8×C67—41no—
29C4×Q8×C67—41no—
30C4×Q8×C67—41no—
31C4×Q8×C67—41no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67C2×C411yescharacteristic
36C2×C4⋊C4×C67C2×C411yescharacteristic
37C2×C4⋊C4×C67D412yes—
38C2×C42×C67—21no—
39C2×C42×C67—21no—
40C2×C42×C67—21no—
41C2×C42×C67C2×C411yescharacteristic
42C2×C42×C67C2×C411yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67D412yes—
45#95×C67—22no—
46#95×C67—22no—
47#11761612—44no—
48#11761612—44no—
49C2×C67⋊C8—44no—
50C2×Q8×C67—21no—
51C2×Q8×C67—41no—
52C2×Q8×C67—41no—
53C2×Q8×C67—41no—
54C2×Q8×C67—41no—
55C2×Q8×C67C4⋊C412yes—
56C2×Q8×C67C2×D411yescharacteristic
57C2×Q8×C67C2×Q811yescharacteristic
58C2×Q8×C67#4112yes—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C412yes—
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22×C411yescharacteristic
67C22×C4×C67C2×D412yes—
68C4⋊C4×C67—24no—
69C4⋊C4×C67—24no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—42no—
72C4⋊C4×C67—41no—
73C4⋊C4×C67—44no—
74C4⋊C4×C67—42no—
75C42×C67—24no—
76C42×C67—21no—
77C42×C67—21no—
78C42×C67—22no—
79C42×C67—22no—
80C42×C67—21no—
81C42×C67—41no—
82C42×C67—42no—
83C42×C67—41no—
84C42×C67—41no—
85C42×C67C4⋊C412yes—
86C23×C67C2×C22⋊C411yescharacteristic
87Q8×C67—41no—
88Q8×C67—41no—
89Q8×C67—42no—
90Q8×C67—42no—
91Q8×C67—42no—
92C2×C4×C67—22no—
93C2×C4×C67—22no—
94C2×C4×C67—22no—
95C2×C4×C67—21no—
96C2×C4×C67—22no—
97C2×C4×C67—21no—
98C2×C4×C67—22no—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—22no—
102C2×C4×C67—21no—
103C2×C4×C67—22no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—41no—
112C2×C4×C67#9911yescharacteristic
113C2×C4×C67#9911yescharacteristic
114C2×C4×C67C2×C4⋊C411yescharacteristic
115C2×C4×C67#11711yescharacteristic
116C2×C4×C67#12112yes—
117C2×C4×C67C22⋊Q812yes—
118C67⋊C8—84no—
119C22×C67—21no—
120C22×C67—21no—
121C22×C67#38011yescharacteristic
122C22×C67C2×#9911yescharacteristic
123C22×C67#41011yescharacteristic
124C4×C67—24no—
125C4×C67—22no—
126C4×C67—21no—
127C4×C67—21no—
128C4×C67—22no—
129C4×C67—21no—
130C4×C67—42no—
131C4×C67—42no—
132C4×C67—41no—
133C4×C67—42no—
134C4×C67—81no—
135C4×C67#35612yescomplemented
136#12802—671nomaximal, Sylow
137C2×C67—21no—
138C2×C67—21no—
139C2×C67#179911yescharacteristic
140C2×C67C2×#35611yescharacteristic
141C2×C67#213211yescharacteristic
142C4×C4⋊Q8D6711yescharacteristic
143C2×#356—672no—
144#1801—672no—
145#1394—672no—
146C67#1280211yescharacteristic, complemented, Sylow
147C2×C4⋊Q8C2×D6711yescharacteristic
148C2×C4×Q8Dic6711yescharacteristic
149C2×C4×Q8C2×D6711yescharacteristic
150#423—1342no—
151C2×#101—674no—
152#391Dic6711yescharacteristic
153#391C2×D6711yescharacteristic
154#389—22no—
155#384Dic6711yescharacteristic
156C4×C4⋊C4—22no—
157C43Dic6711yescharacteristic
158#356—1344no—
159#343—1342no—
160C22×Q8C2×Dic6711yescharacteristic
161C22×Q8C22×D6711yescharacteristic
162C2×M16—1344no—
163C4⋊Q8—22no—
164C4⋊Q8Dic13412yes—
165C4⋊Q8D26812yes—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×Q8—41no—
169C4×Q8—41no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C2×Dic6711yescharacteristic
174C2×C4⋊C4C2×Dic6711yescharacteristic
175C2×C4⋊C4C22⋊D6712yes—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C2×Dic6711yescharacteristic
180C2×C42C2×Dic6711yescharacteristic
181C2×C42C2×Dic6711yescharacteristic
182C2×C42C22⋊D6712yes—
183C4⋊C8—2682no—
184C4⋊C8—2682no—
185#101—1344no—
186#101—1344no—
187#95—22no—
188#95—22no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8C4⋊Dic6712yes—
195C2×Q8C2×D26811yescharacteristic
196C2×Q8C2×Dic13411yescharacteristic
197C2×Q8#1176162812yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4C22⋊Dic6712yes—
203C22×C4C22⋊Dic6711yescharacteristic
204C22×C4C22⋊Dic6711yescharacteristic
205C22×C4C22×Dic6711yescharacteristic
206C22×C4C2×C22⋊D6712yes—
207M16—2684no—
208M16—2684no—
209C2×C8—2684no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C42—24no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—41no—
224C42—42no—
225C42—41no—
226C42—41no—
227C42C4⋊Dic6712yes—
228C23C2×C22⋊Dic6711yescharacteristic
229Q8—41no—
230Q8—41no—
231Q8—42no—
232Q8—42no—
233Q8—42no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4#1452395811yescharacteristic
255C2×C4#1452395811yescharacteristic
256C2×C4C2×C4⋊Dic6711yescharacteristic
257C2×C4#1452400211yescharacteristic
258C2×C4#1452401712yes—
259C2×C4#1452402212yes—
260C8—5364no—
261C22—21no—
262C22—21no—
263C22#1686325611yescharacteristic
264C22C2×#1452395811yescharacteristic
265C22#1686337011yescharacteristic
266C4—24no—
267C4—21no—
268C4—22no—
269C4—21no—
270C4—21no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—42no—
275C4—42no—
276C4—81no—
277C4#1686319512yescomplemented
278C2—21no—
279C2—21no—
280C2#2256019711yescharacteristic
281C2C2×#1686319511yescharacteristic
282C2#2256189411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×Q8C22×D67
FittingC4×C4⋊Q8×C67C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC22×Q8×C67C23
hypercenterC4×C4⋊Q8D67
hypocenterC67#12802
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
135. C4×C67#356semidirect
146. C67#12802semidirect
277. C4#16863195semidirect

Maximal subgroups

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

subgroupindex
C2×#168631952
#225602052
#225583362
C4×C4⋊Q8×C672
#1280267