Finite Group Databasebeta

← C67×#34375 #30713626

Subgroups — C67×#34375 #30713626

Order 17152.

1898 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#533×C67C211yesmaximal, characteristic
2C67×#2381C214yesmaximal, complemented
3C67×#2371C214yesmaximal, complemented
4C67×#2355C218yesmaximal, complemented
5C67×#2332C218yesmaximal
6C67×#2314C214yesmaximal
7C67×#2283C212yesmaximal
8#533×C67C4116yescomplemented
9C2×C4⋊Q8×C67C2212yes—
10C2×#125×C67C2212yes—
11C2×C22⋊Q8×C67C2214yes—
12C2×C4×Q8×C67C2214yes—
13C2×#117×C67C2212yes—
14C22×C42×C67C2211yescharacteristic
15#391×C67C22116yes—
16#388×C67C22116yes—
17#387×C67C22116yes—
18#387×C67C22116yes—
19#387×C67C22116yes—
20#384×C67C2214yes—
21#383×C67C2214yes—
22#382×C67C2214yes—
23#382×C67C2214yes—
24#381×C67C2214yes—
25#380×C67C2214yes—
26C4×C4⋊C4×C67C2214yes—
27C4×C4⋊C4×C67C2218yes—
28C4×C4⋊C4×C67C22116yes—
29C4×C22⋊C4×C67C2214yes—
30#376×C67C2214yes—
31C22×Q8×C67C2314yes—
32C23×C4×C67C2311yescharacteristic
33C23×C4×C67C2312yes—
34C4⋊Q8×C67C2×C4116yes—
35#125×C67C2×C4116yes—
36C22⋊Q8×C67C2×C4132yes—
37C4×Q8×C67C2×C4132yes—
38#117×C67C2×C4116yes—
39C2×C4⋊C4×C67C2318yes—
40C2×C4⋊C4×C67C23116yes—
41C2×C4⋊C4×C67C2314yes—
42C2×C4⋊C4×C67C2318yes—
43C2×C4⋊C4×C67C2314yes—
44C2×C22⋊C4×C67C2318yes—
45C2×C22⋊C4×C67C2314yes—
46C2×C42×C67C2×C418yes—
47C2×C42×C67C2318yes—
48C2×C42×C67C23116yes—
49C2×C42×C67C2312yes—
50C2×C42×C67C2311yescharacteristic
51C2×C42×C67C2311yescharacteristic
52C2×C42×C67C2314yes—
53#95×C67C2318yes—
54#95×C67C2318yes—
55#95×C67C2318yes—
56#95×C67C2318yes—
57#95×C67C23116yes—
58#95×C67C23116yes—
59C24×C67C2411yescharacteristic
60C2×Q8×C67—44no—
61C2×Q8×C67—44no—
62C2×Q8×C67C22×C4116yes—
63C22×C4×C67—24no—
64C22×C4×C67—44no—
65C22×C4×C67C22×C414yes—
66C22×C4×C67C22×C418yes—
67C22×C4×C67C24116yes—
68C22×C4×C67C2412yes—
69C22×C4×C67C2418yes—
70C22×C4×C67C2411yescharacteristic
71C22×C4×C67C2412yes—
72C22×C4×C67C2411yescharacteristic
73C4⋊C4×C67—216no—
74C4⋊C4×C67—416no—
75C4⋊C4×C67C22×C4116yes—
76C4⋊C4×C67C22×C4132yes—
77C4⋊C4×C67C22×C4116yes—
78C22⋊C4×C67—48no—
79C22⋊C4×C67C22×C4116yes—
80C42×C67—48no—
81C42×C67—416no—
82C42×C67C22×C418yes—
83C42×C67C22×C414yes—
84C42×C67C22×C414yes—
85C42×C67C22×C4116yes—
86C23×C67—22no—
87C23×C67—24no—
88C23×C67C23×C412yes—
89C23×C67C2511yescharacteristic
90Q8×C67—416no—
91C2×C4×C67—22no—
92C2×C4×C67—24no—
93C2×C4×C67—28no—
94C2×C4×C67—28no—
95C2×C4×C67—24no—
96C2×C4×C67—28no—
97C2×C4×C67—216no—
98C2×C4×C67—216no—
99C2×C4×C67—216no—
100C2×C4×C67—48no—
101C2×C4×C67C23×C412yes—
102C2×C4×C67C23×C414yes—
103C2×C4×C67C23×C414yes—
104C2×C4×C67C23×C4116yes—
105C2×C4×C67C23×C412yes—
106C2×C4×C672−1+414yes—
107C22×C67—22no—
108C22×C67—24no—
109C22×C67—44no—
110C22×C67C24×C411yescharacteristic
111C22×C67C2×2−1+412yes—
112C22×C67#58514yes—
113C4×C67—22no—
114C4×C67—24no—
115C4×C67—24no—
116C4×C67—216no—
117C4×C67—416no—
118C4×C67#51914yes—
119#34375C6711yesmaximal, characteristic, direct product factor, Sylow
120C2×C67—42no—
121C2×C67C2×#51911yescharacteristic
122C2×C67#342812yes—
123C2×C67#355714yes—
124C2×#533C2×C6711yescharacteristic
125#2381C2×C6714yescomplemented
126#2371C2×C6714yescomplemented
127#2355C2×C6718yescomplemented
128#2332C2×C6718yes—
129#2314C2×C6714yes—
130#2283C2×C6712yes—
131C67#3437511yescharacteristic, direct product factor, Sylow
132#533C4×C67116yescomplemented
133C2×C4⋊Q8C22×C6712yes—
134C2×#125C22×C6712yes—
135C2×C22⋊Q8C22×C6714yes—
136C2×C4×Q8C22×C6714yes—
137C2×#117C22×C6712yes—
138C22×C42C22×C6711yescharacteristic
139#391C22×C67116yes—
140#388C22×C67116yes—
141#387C22×C67116yes—
142#387C22×C67116yes—
143#387C22×C67116yes—
144#384C22×C6714yes—
145#383C22×C6714yes—
146#382C22×C6714yes—
147#382C22×C6714yes—
148#381C22×C6714yes—
149#380C22×C6714yes—
150C4×C4⋊C4C22×C6718yes—
151C4×C4⋊C4C22×C67116yes—
152C4×C4⋊C4C22×C6714yes—
153C4×C22⋊C4C22×C6714yes—
154#376C22×C6714yes—
155C22×Q8C23×C6714yes—
156C23×C4C23×C6711yescharacteristic
157C23×C4C23×C6712yes—
158C4⋊Q8C2×C4×C67116yes—
159#125C2×C4×C67116yes—
160C22⋊Q8C2×C4×C67132yes—
161C4×Q8C2×C4×C67132yes—
162#117C2×C4×C67116yes—
163C2×C4⋊C4C23×C6718yes—
164C2×C4⋊C4C23×C67116yes—
165C2×C4⋊C4C23×C6714yes—
166C2×C4⋊C4C23×C6718yes—
167C2×C4⋊C4C23×C6714yes—
168C2×C22⋊C4C23×C6718yes—
169C2×C22⋊C4C23×C6714yes—
170C2×C42C2×C4×C6718yes—
171C2×C42C23×C6718yes—
172C2×C42C23×C67116yes—
173C2×C42C23×C6714yes—
174C2×C42C23×C6712yes—
175C2×C42C23×C6711yescharacteristic
176C2×C42C23×C6711yescharacteristic
177#95C23×C6718yes—
178#95C23×C6718yes—
179#95C23×C6718yes—
180#95C23×C6718yes—
181#95C23×C67116yes—
182#95C23×C67116yes—
183C24C24×C6711yescharacteristic
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8C22×C4×C67116yes—
187C22×C4—24no—
188C22×C4—44no—
189C22×C4C22×C4×C6714yes—
190C22×C4C22×C4×C6718yes—
191C22×C4C24×C67116yes—
192C22×C4C24×C6711yescharacteristic
193C22×C4C24×C6712yes—
194C22×C4C24×C6712yes—
195C22×C4C24×C6718yes—
196C22×C4C24×C6711yescharacteristic
197C4⋊C4—216no—
198C4⋊C4—416no—
199C4⋊C4C22×C4×C67116yes—
200C4⋊C4C22×C4×C67132yes—
201C4⋊C4C22×C4×C67116yes—
202C22⋊C4—48no—
203C22⋊C4C22×C4×C67116yes—
204C42—48no—
205C42—416no—
206C42C22×C4×C6718yes—
207C42C22×C4×C6714yes—
208C42C22×C4×C6714yes—
209C42C22×C4×C67116yes—
210C23—22no—
211C23—24no—
212C23C23×C4×C6712yes—
213C23C25×C6711yescharacteristic
214Q8—416no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—28no—
218C2×C4—28no—
219C2×C4—24no—
220C2×C4—28no—
221C2×C4—216no—
222C2×C4—216no—
223C2×C4—216no—
224C2×C4—48no—
225C2×C4C23×C4×C6712yes—
226C2×C4C23×C4×C6714yes—
227C2×C4C23×C4×C6714yes—
228C2×C4C23×C4×C67116yes—
229C2×C4C23×C4×C6712yes—
230C2×C42−1+4×C6714yes—
231C22—22no—
232C22—24no—
233C22—44no—
234C22C24×C4×C6711yescharacteristic
235C22C2×2−1+4×C6712yes—
236C22#585×C6714yes—
237C4—22no—
238C4—24no—
239C4—24no—
240C4—216no—
241C4—416no—
242C4#519×C6714yes—
243C2—42no—
244C2C2×#519×C6711yescharacteristic
245C2C67×#342812yes—
246C2C67×#355714yes—
subgroupG/H
center, socleC23×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC23C25×C67
prime radicalC24×C67C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2381C2semidirect
3. C67×#2371C2semidirect
4. C67×#2355C2semidirect
8. #533×C67C4semidirect
119. #34375C67direct (N × H)
125. #2381C2×C67semidirect
126. #2371C2×C67semidirect
127. #2355C2×C67semidirect
131. C67#34375direct (N × H)
132. #533C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#533×C672
C67×#23812
C67×#23712
C67×#23552
C67×#23322
C67×#23142
C67×#22832
#3437567