Finite Group Databasebeta

← C67×#34530 #30713781

Subgroups — C67×#34530 #30713781

Order 17152.

1800 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#564×C67C211yesmaximal, characteristic
2C67×#2382C213yesmaximal
3C67×#2371C216yesmaximal
4C67×#2370C216yesmaximal
5C67×#2336C216yesmaximal
6C67×#2330C216yesmaximal
7C4×#125×C67C213yesmaximal
8#564×C67C4116yescomplemented
9C2×#125×C67C2213yes—
10C2×#125×C67C2216yes—
11C2×#125×C67C2216yes—
12#389×C67C2213yes—
13#389×C67C2213yes—
14#389×C67C2216yes—
15#389×C67C2216yes—
16#389×C67C2216yes—
17#389×C67C2216yes—
18#387×C67C2216yes—
19#387×C67C2216yes—
20#387×C67C2216yes—
21#387×C67C2216yes—
22#387×C67C2216yes—
23#387×C67C2216yes—
24#387×C67C2216yes—
25#387×C67C2216yes—
26#387×C67C2216yes—
27#387×C67C2216yes—
28#384×C67C2212yes—
29#383×C67C2212yes—
30#382×C67C2213yes—
31C4×C4⋊C4×C67C2216yes—
32C4×C4⋊C4×C67C2213yes—
33C4×C4⋊C4×C67C2216yes—
34C4×C4⋊C4×C67C2213yes—
35C4×C4⋊C4×C67C2216yes—
36C4×C4⋊C4×C67C2216yes—
37C4×C4⋊C4×C67C2216yes—
38#376×C67C2216yes—
39C43×C67C2211yescharacteristic
40#125×C67C2×C4124yes—
41#125×C67C2×C4148yes—
42#125×C67C2×C4148yes—
43C2×C4⋊C4×C67C2316yes—
44C2×C4⋊C4×C67C2313yes—
45C2×C4⋊C4×C67C2313yes—
46C2×C4⋊C4×C67C2316yes—
47C2×C4⋊C4×C67C2316yes—
48C2×C4⋊C4×C67C2316yes—
49C2×C4⋊C4×C67C2313yes—
50C2×C4⋊C4×C67C2316yes—
51C2×C4⋊C4×C67C2316yes—
52C2×C4⋊C4×C67C2313yes—
53C2×C4⋊C4×C67C2316yes—
54C2×C4⋊C4×C67C2316yes—
55C2×C42×C67C2313yes—
56C2×C42×C67C2316yes—
57C2×C42×C67C2313yes—
58C2×C42×C67C2316yes—
59C2×C42×C67C2316yes—
60C2×C42×C67C2316yes—
61C2×C42×C67C2311yescharacteristic
62C2×C42×C67C2312yes—
63C2×C42×C67C2312yes—
64#95×C67C2316yes—
65#95×C67C2316yes—
66#95×C67C2316yes—
67#95×C67C2316yes—
68#95×C67C2316yes—
69#95×C67C2316yes—
70#95×C67C2316yes—
71#95×C67C2316yes—
72#95×C67C2316yes—
73#95×C67C2316yes—
74C22×C4×C67C2416yes—
75C22×C4×C67C2411yescharacteristic
76C22×C4×C67C2416yes—
77C22×C4×C67C2413yes—
78C22×C4×C67C2416yes—
79C22×C4×C67C2416yes—
80C22×C4×C67C2413yes—
81C4⋊C4×C67—212no—
82C4⋊C4×C67—46no—
83C4⋊C4×C67—43no—
84C4⋊C4×C67—46no—
85C4⋊C4×C67—46no—
86C4⋊C4×C67—43no—
87C4⋊C4×C67C22×C4112yes—
88C4⋊C4×C67C22×C4124yes—
89C4⋊C4×C67C22×C4124yes—
90C4⋊C4×C67C22×C4112yes—
91C4⋊C4×C67C22×C4124yes—
92C4⋊C4×C67C22×C4124yes—
93C42×C67—26no—
94C42×C67—212no—
95C42×C67—46no—
96C42×C67—43no—
97C42×C67—46no—
98C42×C67—46no—
99C42×C67C22×C414yes—
100C42×C67C22×C418yes—
101C42×C67C22×C418yes—
102C23×C67C2511yescharacteristic
103C2×C4×C67—23no—
104C2×C4×C67—26no—
105C2×C4×C67—23no—
106C2×C4×C67—26no—
107C2×C4×C67—23no—
108C2×C4×C67—26no—
109C2×C4×C67—23no—
110C2×C4×C67—26no—
111C2×C4×C67—26no—
112C2×C4×C67—26no—
113C2×C4×C67—26no—
114C2×C4×C67—26no—
115C2×C4×C67—26no—
116C2×C4×C67—23no—
117C2×C4×C67—26no—
118C2×C4×C67C23×C416yes—
119C2×C4×C67C23×C4112yes—
120C2×C4×C67C23×C4112yes—
121C2×C4×C672−1+416yes—
122C22×C67C24×C411yescharacteristic
123C22×C67C2×2−1+413yes—
124C22×C67#58513yes—
125C4×C67—26no—
126C4×C67—212no—
127C4×C67—212no—
128C4×C67—26no—
129C4×C67—46no—
130C4×C67—46no—
131C4×C67—41no—
132#34530C6711yesmaximal, characteristic, direct product factor, Sylow
133C2×C67C4×2−1+413yes—
134C2×C67#356313yes—
135C2×C67#356711yescharacteristic
136C2×#564C2×C6711yescharacteristic
137#2382C2×C6713yes—
138#2371C2×C6716yes—
139#2370C2×C6716yes—
140#2336C2×C6716yes—
141#2330C2×C6716yes—
142C4×#125C2×C6713yes—
143C67#3453011yescharacteristic, direct product factor, Sylow
144#564C4×C67116yescomplemented
145C2×#125C22×C6713yes—
146C2×#125C22×C6716yes—
147C2×#125C22×C6716yes—
148#389C22×C6713yes—
149#389C22×C6713yes—
150#389C22×C6716yes—
151#389C22×C6716yes—
152#389C22×C6716yes—
153#389C22×C6716yes—
154#387C22×C6716yes—
155#387C22×C6716yes—
156#387C22×C6716yes—
157#387C22×C6716yes—
158#387C22×C6716yes—
159#387C22×C6716yes—
160#387C22×C6716yes—
161#387C22×C6716yes—
162#387C22×C6716yes—
163#387C22×C6716yes—
164#384C22×C6712yes—
165#383C22×C6712yes—
166#382C22×C6713yes—
167C4×C4⋊C4C22×C6716yes—
168C4×C4⋊C4C22×C6713yes—
169C4×C4⋊C4C22×C6716yes—
170C4×C4⋊C4C22×C6713yes—
171C4×C4⋊C4C22×C6716yes—
172C4×C4⋊C4C22×C6716yes—
173C4×C4⋊C4C22×C6716yes—
174#376C22×C6716yes—
175C43C22×C6711yescharacteristic
176#125C2×C4×C67124yes—
177#125C2×C4×C67148yes—
178#125C2×C4×C67148yes—
179C2×C4⋊C4C23×C6716yes—
180C2×C4⋊C4C23×C6713yes—
181C2×C4⋊C4C23×C6713yes—
182C2×C4⋊C4C23×C6716yes—
183C2×C4⋊C4C23×C6716yes—
184C2×C4⋊C4C23×C6716yes—
185C2×C4⋊C4C23×C6713yes—
186C2×C4⋊C4C23×C6716yes—
187C2×C4⋊C4C23×C6716yes—
188C2×C4⋊C4C23×C6713yes—
189C2×C4⋊C4C23×C6716yes—
190C2×C4⋊C4C23×C6716yes—
191C2×C42C23×C6713yes—
192C2×C42C23×C6716yes—
193C2×C42C23×C6713yes—
194C2×C42C23×C6716yes—
195C2×C42C23×C6716yes—
196C2×C42C23×C6716yes—
197C2×C42C23×C6711yescharacteristic
198C2×C42C23×C6712yes—
199C2×C42C23×C6712yes—
200#95C23×C6716yes—
201#95C23×C6716yes—
202#95C23×C6716yes—
203#95C23×C6716yes—
204#95C23×C6716yes—
205#95C23×C6716yes—
206#95C23×C6716yes—
207#95C23×C6716yes—
208#95C23×C6716yes—
209#95C23×C6716yes—
210C22×C4C24×C6716yes—
211C22×C4C24×C6716yes—
212C22×C4C24×C6711yescharacteristic
213C22×C4C24×C6716yes—
214C22×C4C24×C6713yes—
215C22×C4C24×C6716yes—
216C22×C4C24×C6713yes—
217C4⋊C4—212no—
218C4⋊C4—46no—
219C4⋊C4—43no—
220C4⋊C4—46no—
221C4⋊C4—46no—
222C4⋊C4—43no—
223C4⋊C4C22×C4×C67112yes—
224C4⋊C4C22×C4×C67124yes—
225C4⋊C4C22×C4×C67124yes—
226C4⋊C4C22×C4×C67112yes—
227C4⋊C4C22×C4×C67124yes—
228C4⋊C4C22×C4×C67124yes—
229C42—26no—
230C42—212no—
231C42—46no—
232C42—43no—
233C42—46no—
234C42—46no—
235C42C22×C4×C6714yes—
236C42C22×C4×C6718yes—
237C42C22×C4×C6718yes—
238C23C25×C6711yescharacteristic
239C2×C4—23no—
240C2×C4—23no—
241C2×C4—26no—
242C2×C4—26no—
243C2×C4—23no—
244C2×C4—26no—
245C2×C4—23no—
246C2×C4—26no—
247C2×C4—26no—
248C2×C4—26no—
249C2×C4—26no—
250C2×C4—26no—
251C2×C4—26no—
252C2×C4—23no—
253C2×C4—26no—
254C2×C4C23×C4×C6716yes—
255C2×C4C23×C4×C67112yes—
256C2×C4C23×C4×C67112yes—
257C2×C42−1+4×C6716yes—
258C22C24×C4×C6711yescharacteristic
259C22C2×2−1+4×C6713yes—
260C22#585×C6713yes—
261C4—26no—
262C4—26no—
263C4—212no—
264C4—212no—
265C4—46no—
266C4—46no—
267C4—41no—
268C2C4×2−1+4×C6713yes—
269C2C67×#356313yes—
270C2C67×#356711yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC23C25×C67
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
8. #564×C67C4semidirect
132. #34530C67direct (N × H)
143. C67#34530direct (N × H)
144. #564C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#564×C672
C67×#23822
C67×#23712
C67×#23702
C67×#23362
C67×#23302
C4×#125×C672
#3453067