Finite Group Databasebeta

← #30786885

Subgroups — #30786885

Order 17152.

1530 conjugacy classes of subgroups; only the 371 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 371 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
1C22×#14524037C211yesmaximal, characteristic
2C2×#16863264C211yesmaximal, characteristic, complemented
3#22558459C216yesmaximal, complemented
4#22558451C216yesmaximal
5C2×#384×C67C211yesmaximal, characteristic, complemented
6C2×#14524037—24no—
7C2×#14524037—24no—
8C2×#14524037C418yescomplemented
9C22×Dic67⋊C4C2213yes—
10C22×(C2×D67)⋊C4C2213yes—
11C22×(C2×D67)⋊C4C2214yes—
12#16863264—24no—
13C2×C67⋊#95C2216yes—
14C2×C67⋊#95C22112yes—
15C22×C4⋊C4×C67C2216yes—
16C22×C42×C67C2211yescharacteristic
17#384×C67—24no—
18C24×D67C2311yescharacteristic
19C23×Dic67C2314yes—
20C23×Dic67C2313yes—
21#14524037—232no—
22C2×Dic67⋊C4—26no—
23C2×Dic67⋊C4—26no—
24C2×Dic67⋊C4C2×C4112yes—
25C2×(C2×D67)⋊C4—26no—
26C2×(C2×D67)⋊C4—26no—
27C2×(C2×D67)⋊C4—412no—
28C2×(C2×D67)⋊C4C2×C4112yes—
29C67⋊#95—212no—
30C67⋊#95—412no—
31C23×C4×C67C2314yes—
32C23×C4×C67C2313yes—
33C2×C4⋊C4×C67—212no—
34C2×C4⋊C4×C67—412no—
35C2×C42×C67C2×C414yes—
36C2×C42×C67D414yes—
37C2×C42×C67D414yes—
38C23×D67—21no—
39C23×D67—21no—
40C23×D67—26no—
41C23×D67—26no—
42C23×D67C22×C412yes—
43C22×Dic67—23no—
44C22×Dic67—23no—
45C22×Dic67—26no—
46C22×Dic67—26no—
47C22×Dic67—24no—
48C22×Dic67—212no—
49C22×Dic67—212no—
50C22×Dic67C22×C416yes—
51Dic67⋊C4—224no—
52(C2×D67)⋊C4—224no—
53(C2×D67)⋊C4—416no—
54C24×C67C2411yescharacteristic
55C22×C4×C67—26no—
56C22×C4×C67—26no—
57C22×C4×C67—24no—
58C22×C4×C67—212no—
59C22×C4×C67—212no—
60C22×C4×C67C22×C416yes—
61C22×C4×C67C2×D416yes—
62C22×C4×C67C2×D416yes—
63C4⋊C4×C67—424no—
64C42×C67C22⋊C4116yescomplemented
65C23×C67C23×C411yescharacteristic
66C23×C67C22×D411yescharacteristic
67C23×C67C22×D411yescharacteristic
68C23×C67H(5,𝔽2)16yes—
69C23×C672−1+416yes—
70C22×D67—22no—
71C22×D67—212no—
72C22×D67—43no—
73C22×D67—43no—
74C22×D67—46no—
75C22×D67—44no—
76C22×D67—412no—
77C2×C4×C67—212no—
78C2×C4×C67—26no—
79C2×C4×C67—26no—
80C2×C4×C67—212no—
81C2×C4×C67—412no—
82C2×C4×C67—412no—
83C2×C4×C67—44no—
84C2×C4×C67C2×C22⋊C4112yes—
85C2×Dic67—26no—
86C2×Dic67—212no—
87C2×Dic67—43no—
88C2×Dic67—43no—
89C2×Dic67—46no—
90C2×Dic67—412no—
91C2×Dic67—44no—
92C2×Dic67—412no—
93C22×C67C22×C22⋊C411yescharacteristic
94C22×C67#52016yes—
95C22×C67#53513yes—
96C22×C67#53613yes—
97C22×C67#53716yes—
98C22×C67#56014yes—
99C22×C67#563112yes—
100C2×D67—46no—
101C2×D67—48no—
102C2×D67—81no—
103C2×D67—83no—
104C2×D67—84no—
105C4×C67—212no—
106C4×C67—84no—
107Dic67—46no—
108Dic67—84no—
109#14665—671nomaximal, Sylow
110C2×C67#231113yes—
111C2×C67#238314yes—
112C2×C67#281811yescharacteristic
113C2×C67#282113yes—
114C2×C67#282314yes—
115D67—82no—
116C22×#126—671no—
117C2×#384D6711yescharacteristic, complemented
118C2×#383—671no—
119#1441—676no—
120#1439—676no—
121C67#1466511yescharacteristic, complemented, Sylow
122C2×#126—678no—
123C2×#126—1344no—
124C2×#126—1344no—
125C22×C4⋊C4—673no—
126C22×C4⋊C4C2×D6716yes—
127C22×C22⋊C4—674no—
128C22×C22⋊C4—673no—
129C22×C42C2×D6711yescharacteristic
130#384—24no—
131#383—1344no—
132C2×#95—6712no—
133C2×#95—676no—
134C25—671no—
135C23×C4—673no—
136C23×C4—674no—
137C23×C4C22×D6714yes—
138C23×C4C22×D6713yes—
139#126—13432no—
140C2×C4⋊C4—212no—
141C2×C4⋊C4—412no—
142C2×C4⋊C4—6712no—
143C2×C4⋊C4—1346no—
144C2×C4⋊C4—1346no—
145C2×C22⋊C4—26812no—
146C2×C22⋊C4—6712no—
147C2×C22⋊C4—1346no—
148C2×C22⋊C4—1346no—
149C2×C42C4×D6714yes—
150C2×C42D26814yes—
151C2×C42C22⋊D6714yes—
152#95—26812no—
153#95—13412no—
154C24—672no—
155C24—1341no—
156C24—1341no—
157C24—1346no—
158C24—1346no—
159C24C23×D6711yescharacteristic
160C22×C4—26no—
161C22×C4—26no—
162C22×C4—24no—
163C22×C4—212no—
164C22×C4—212no—
165C22×C4—676no—
166C22×C4—1344no—
167C22×C4—13412no—
168C22×C4—13412no—
169C22×C4—1343no—
170C22×C4—1343no—
171C22×C4—1346no—
172C22×C4—1346no—
173C22×C4C2×C4×D6716yes—
174C22×C4C2×C22⋊D6716yes—
175C22×C4C2×D26816yes—
176C4⋊C4—424no—
177C4⋊C4—13424no—
178C22⋊C4—26816no—
179C22⋊C4—13424no—
180C42(C2×D67)⋊C4116yescomplemented
181C23—2684no—
182C23—26812no—
183C23—2683no—
184C23—2683no—
185C23—2686no—
186C23—1342no—
187C23—13412no—
188C23C22×C4×D6711yescharacteristic
189C23C22×C22⋊D6711yescharacteristic
190C23C22×D26811yescharacteristic
191C23#1452408816yes—
192C23#1452409116yes—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—26no—
196C2×C4—212no—
197C2×C4—44no—
198C2×C4—412no—
199C2×C4—412no—
200C2×C4—2683no—
201C2×C4—2683no—
202C2×C4—2686no—
203C2×C4—2684no—
204C2×C4—26812no—
205C2×C4—26812no—
206C2×C4—1346no—
207C2×C4—13412no—
208C2×C4C2×(C2×D67)⋊C4112yes—
209C22—2688no—
210C22—2686no—
211C22—5361no—
212C22—5363no—
213C22—5364no—
214C22C22×(C2×D67)⋊C411yescharacteristic
215C22#1686387716yes—
216C22#1686396313yes—
217C22#1686397013yes—
218C22#1686397716yes—
219C22#1686413314yes—
220C22#16864145112yes—
221C4—212no—
222C4—84no—
223C4—2686no—
224C4—5364no—
225C2—5362no—
226C2#2256280213yes—
227C2#2256340214yes—
228C2#2256825011yescharacteristic
229C2#2256826813yes—
230C2#2256828314yes—
subgroupG/H
center, FrattiniC24C23×D67
commutatorC23×C67C23×C4
FittingC2×#384×C67C2
socle, cosocleC24×C67C24
prime radicalC24×D67C23
hypercenterC2×#384D67
hypocenterC67#14665
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. C2×#16863264C2semidirect
3. #22558459C2semidirect
5. C2×#384×C67C2semidirect
8. C2×#14524037C4semidirect
64. C42×C67C22⋊C4semidirect
117. C2×#384D67semidirect
121. C67#14665semidirect
180. C42(C2×D67)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145240372
C2×#168632642
#225584592
#225584512
C2×#384×C672
#1466567