Finite Group Databasebeta

← #30865306

Subgroups — #30865306

Order 17152.

914 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 235 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×#16864027C211yesmaximal, characteristic
2#22568844C212yesmaximal
3#22561200C214yesmaximal, complemented
4#22561196C214yesmaximal, complemented
5#22560290C212yesmaximal
6#22560289C211yesmaximal, characteristic
7C67×#1821C211yesmaximal, characteristic
8#16864027—28no—
9C2×#14524042C412yes—
10C2×#14524042C2212yes—
11C2×#14524034C412yes—
12C2×#14524034C2212yes—
13C2×#14524022C414yes—
14C2×#14524002C2211yescharacteristic
15C2×#14524002C2211yescharacteristic
16#16863454—28no—
17#16863419C2214yes—
18#16863360C2212yes—
19M16×Dic67C2214yes—
20#16863165C2214yes—
21#16863164C2214yes—
22#16863164C2214yes—
23C22×C4⋊C4×C67C2211yescharacteristic
24#407×C67C2212yes—
25#343×C67C2214yes—
26C22×Dic134C2×C414yes—
27C2×#11761612C2314yes—
28#14524042—28no—
29#14524042—24no—
30#14524042—24no—
31#14524034—28no—
32#14524034—24no—
33#14524034—24no—
34#14524022—216no—
35#14524002—24no—
36#14524002D418yes—
37C2×C4⋊Dic67C2×C414yes—
38C2×C4⋊Dic67C2×C411yescharacteristic
39C2×C4⋊Dic67C2×C411yescharacteristic
40C2×C4⋊Dic67C2311yescharacteristic
41C2×C4⋊Dic67C2311yescharacteristic
42C2×Dic67⋊C4C2×C414yes—
43C2×Dic67⋊C4C2×C414yes—
44C2×C22⋊Dic67C2×C412yes—
45C2×C22⋊Dic67C2×C412yes—
46C2×C4×Dic67C2×C412yes—
47C2×C4×Dic67C2312yes—
48Dic67⋊C8—28no—
49C22⋊(C67⋊C8)—24no—
50#14523953—24no—
51C8×Dic67—24no—
52C23×C4×C67C2×C412yes—
53C23×C4×C67C2311yescharacteristic
54C2×M16×C67C2314yes—
55C2×C4⋊C4×C67—24no—
56C2×C4⋊C4×C67C2×C412yes—
57C2×C4⋊C4×C67C2312yes—
58C22⋊C8×C67—24no—
59C2×Dic134—28no—
60C2×Dic134—84no—
61C22×Dic67C22×C412yes—
62C22×Dic67C22×C412yes—
63C22×Dic67C22×C412yes—
64C22×Dic67C22×C412yes—
65#11761612—28no—
66C2×C67⋊C8—24no—
67C4⋊Dic67—22no—
68C4⋊Dic67—22no—
69C4⋊Dic67—28no—
70C4⋊Dic67C22⋊C414yes—
71C4⋊Dic67C22⋊C414yes—
72Dic67⋊C4—28no—
73Dic67⋊C4—28no—
74C22⋊Dic67—24no—
75C22⋊Dic67C22⋊C418yes—
76C4×Dic67—24no—
77C4×Dic67C2×D414yes—
78C4×Dic67C2×D414yes—
79C24×C67C22×C411yescharacteristic
80C22×C4×C67—24no—
81C22×C4×C67—24no—
82C22×C4×C67—82no—
83C22×C4×C67C22×C411yescharacteristic
84C22×C4×C67C22×C414yes—
85C22×C4×C67C2×D414yes—
86C22×C4×C67C2411yescharacteristic
87M16×C67—28no—
88C2×C8×C67—24no—
89C4⋊C4×C67—24no—
90C4⋊C4×C67—22no—
91C4⋊C4×C67—22no—
92C23×C67—22no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67C2×C22⋊C412yes—
96C23×C67C23×C411yescharacteristic
97C2×C4×C67—22no—
98C2×C4×C67—28no—
99C2×C4×C67—24no—
100C2×C4×C67—44no—
101C2×C4×C67—84no—
102C2×C4×C67C2×C22⋊C412yes—
103C2×C4×C67C22×D412yes—
104C2×C4×C672−1+414yes—
105C2×Dic67—22no—
106C2×Dic67—22no—
107C2×Dic67—42no—
108C2×Dic67—42no—
109C2×Dic67—42no—
110C2×Dic67—42no—
111C2×Dic67C2×C22⋊C414yes—
112C2×Dic67C2×C22⋊C414yes—
113Dic134—88no—
114C8×C67—44no—
115C67⋊C8—44no—
116C22×C67—21no—
117C22×C67—21no—
118C22×C67—22no—
119C22×C67—22no—
120C22×C67—22no—
121C22×C67—44no—
122C22×C67C22×C22⋊C411yescharacteristic
123C22×C67#51912yes—
124C4×C67—22no—
125C4×C67—44no—
126C4×C67#53614yes—
127Dic67—24no—
128Dic67—24no—
129Dic67—42no—
130Dic67—42no—
131#21832—671nomaximal, Sylow
132C2×C67—22no—
133C2×C67—42no—
134C2×C67#231011yescharacteristic
135C2×C67#288811yescharacteristic
136C2×C67#288911yescharacteristic
137C2×#544—671no—
138#2921—672no—
139#1998—674no—
140#1997—674no—
141#1822—672no—
142#1821—671no—
143#1821D6711yescharacteristic
144C67#2183211yescharacteristic, complemented, Sylow
145#544—1348no—
146C2×C4⋊Q8—672no—
147C2×C4⋊Q8—672no—
148C2×#125—672no—
149C2×#125—672no—
150C2×C22⋊Q8—674no—
151C2×#117—671no—
152C2×#117—671no—
153C22×C4⋊C4C2×D6711yescharacteristic
154#433—1348no—
155#423—674no—
156#407—672no—
157#407C2×D6712yes—
158C4×M16—674no—
159#343—674no—
160#343—674no—
161#343—674no—
162#343C2×D6714yes—
163C22×Q8—674no—
164C23×C4C4×D6712yes—
165C23×C4C22×D6711yescharacteristic
166C2×M16—674no—
167C2×M16C22×D6714yes—
168C4⋊Q8—1344no—
169C4⋊Q8—1344no—
170C4⋊Q8—1348no—
171#125—1344no—
172#125—1344no—
173#125—1348no—
174C22⋊Q8—13416no—
175#117—678no—
176#117—1344no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—671no—
179C2×C4⋊C4—671no—
180C2×C4⋊C4—674no—
181C2×C4⋊C4—674no—
182C2×C4⋊C4—674no—
183C2×C4⋊C4—671no—
184C2×C4⋊C4—671no—
185C2×C4⋊C4C4×D6712yes—
186C2×C4⋊C4C22×D6712yes—
187C2×C22⋊C4—672no—
188C2×C22⋊C4—672no—
189C2×C42—672no—
190C2×C42—672no—
191C4⋊C8—1348no—
192C22⋊C8—24no—
193C22⋊C8—1344no—
194#97—1344no—
195C4×C8—1344no—
196C24C2×C4×D6711yescharacteristic
197C2×Q8—5364no—
198C2×Q8—1348no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—82no—
202C22×C4—672no—
203C22×C4—672no—
204C22×C4—672no—
205C22×C4—672no—
206C22×C4C2×C4×D6711yescharacteristic
207C22×C4C2×C4×D6714yes—
208C22×C4D4×D6714yes—
209C22×C4C23×D6711yescharacteristic
210M16—28no—
211M16—1348no—
212C2×C8—24no—
213C2×C8—1344no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—674no—
218C4⋊C4—674no—
219C4⋊C4—1348no—
220C4⋊C4—1348no—
221C4⋊C4—1348no—
222C4⋊C4—1342no—
223C4⋊C4—1342no—
224C22⋊C4—678no—
225C22⋊C4—1344no—
226C42—674no—
227C42—674no—
228C42—1344no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23C22⋊C4×D6712yes—
233C23C22×C4×D6711yescharacteristic
234Q8—5368no—
235C2×C4—22no—
236C2×C4—28no—
237C2×C4—24no—
238C2×C4—44no—
239C2×C4—84no—
240C2×C4—2682no—
241C2×C4—2682no—
242C2×C4—2682no—
243C2×C4—2682no—
244C2×C4—674no—
245C2×C4—674no—
246C2×C4—1342no—
247C2×C4—1342no—
248C2×C4C22⋊C4×D6712yes—
249C2×C4C2×D4×D6712yes—
250C2×C4#1452409014yes—
251C8—44no—
252C8—2684no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—44no—
259C22C2×C22⋊C4×D6711yescharacteristic
260C22#1686387312yes—
261C4—22no—
262C4—44no—
263C4—2682no—
264C4—2682no—
265C4—1344no—
266C4—1344no—
267C4#1686396814yes—
268C2—22no—
269C2—42no—
270C2#2256279411yescharacteristic
271C2#2256868511yescharacteristic
272C2#2256868911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D67
commutatorC23×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#1821C2
socleC22×C67C22×C22⋊C4
prime radicalC24×C67C22×C4
cosocleC22×C4×C67C24
hypercenter#1821D67
hypocenterC67#21832
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #22561200C2semidirect
4. #22561196C2semidirect
144. C67#21832semidirect

Maximal subgroups

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

subgroupindex
C2×#168640272
#225688442
#225612002
#225611962
#225602902
#225602892
C67×#18212
#2183267