Finite Group Databasebeta

← C2×C67×#2984 #30739495

Subgroups — C2×C67×#2984 #30739495

Order 17152.

2022 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
1C22×C4×Q8×C67C211yesmaximal, characteristic
2C67×#2984C2116yesmaximal, direct product factor
3C2×#446×C67C216yesmaximal, complemented
4C2×C8×Q8×C67C212yesmaximal, complemented
5C2×#432×C67C213yesmaximal
6C2×#431×C67C213yesmaximal
7C23×Q8×C67C411yescharacteristic
8C2×C4×Q8×C67C418yes—
9C2×C4×Q8×C67C413yes—
10C2×C4×Q8×C67C411yescharacteristic
11C2×C4×Q8×C67C2211yescharacteristic
12C2×C4×Q8×C67C2213yes—
13C2×C4×Q8×C67C2218yes—
14C22×C4⋊C4×C67C413yes—
15C22×C42×C67C2213yes—
16#446×C67C22148yescomplemented
17C8×Q8×C67C22116yescomplemented
18#432×C67C22124yes—
19#431×C67C22124yes—
20C2×C4⋊C8×C67C2216yes—
21C2×C4⋊C8×C67C2216yes—
22C2×C22⋊C8×C67C2211yescharacteristic
23C2×C22⋊C8×C67C2213yes—
24C2×#97×C67C2216yes—
25C2×C4×C8×C67C2216yes—
26C22×Q8×C67—28no—
27C22×Q8×C67C2×C418yes—
28C22×Q8×C67C2×C413yes—
29C22×Q8×C67C2×C411yescharacteristic
30C23×C4×C67C2×C413yes—
31C23×C4×C67C2×C413yes—
32C23×C4×C67C2311yescharacteristic
33C22×C8×C67C2312yes—
34C22×C8×C67C2316yes—
35C4×Q8×C67C2×C4112yes—
36C4×Q8×C67C2×C4124yes—
37C4×Q8×C67C2×C414yes—
38C4×Q8×C67C2×C418yes—
39C4×Q8×C67C2314yes—
40C4×Q8×C67C23112yes—
41C2×C4⋊C4×C67C2×C416yes—
42C2×C4⋊C4×C67C2×C4124yes—
43C2×C4⋊C4×C67C2×C413yes—
44C2×C4⋊C4×C67C2×C413yes—
45C2×C42×C67C2×C413yes—
46C2×C42×C67C2×C4112yes—
47C2×C42×C67C2×C413yes—
48C2×C42×C67C23112yes—
49C2×C42×C67C2313yes—
50C2×C42×C67C2313yes—
51C4⋊C8×C67C23124yes—
52C4⋊C8×C67C23124yes—
53C22⋊C8×C67C2314yes—
54C22⋊C8×C67C23112yes—
55#97×C67C23124yes—
56C4×C8×C67C23124yes—
57C24×C67C22×C411yescharacteristic
58C2×Q8×C67—232no—
59C2×Q8×C67—212no—
60C2×Q8×C67M1618yes—
61C2×Q8×C67C22×C4112yes—
62C2×Q8×C67C22×C414yes—
63C22×C4×C67—212no—
64C22×C4×C67—212no—
65C22×C4×C67—24no—
66C22×C4×C67C22×C411yescharacteristic
67C22×C4×C67C22×C412yes—
68C22×C4×C67C22×C4112yes—
69C22×C4×C67C22×C413yes—
70C22×C4×C67C22×C413yes—
71C22×C4×C67C22×C413yes—
72C22×C4×C67C22×C4112yes—
73C22×C4×C67C22×C413yes—
74C22×C4×C67C2412yes—
75C22×C4×C67C2411yescharacteristic
76C2×C8×C67—24no—
77C2×C8×C67—212no—
78C2×C8×C67C2414yes—
79C2×C8×C67C24112yes—
80C4⋊C4×C67C22×C4124yes—
81C4⋊C4×C67C22×C4112yes—
82C4⋊C4×C67C22×C4112yes—
83C42×C67C22×C4112yes—
84C42×C67C22×C4112yes—
85C42×C67C22×C416yes—
86C42×C67C22×C416yes—
87C42×C67C2416yes—
88C42×C67C2416yes—
89C23×C67—22no—
90C23×C67—22no—
91C23×C67#13114yes—
92C23×C67C23×C412yes—
93C23×C67C23×C411yescharacteristic
94Q8×C67—224no—
95Q8×C67C2×M16116yes—
96C2×C4×C67—22no—
97C2×C4×C67—26no—
98C2×C4×C67—224no—
99C2×C4×C67—26no—
100C2×C4×C67—26no—
101C2×C4×C67—24no—
102C2×C4×C67—224no—
103C2×C4×C67—24no—
104C2×C4×C67C2×M16112yes—
105C2×C4×C67C23×C412yes—
106C2×C4×C67C23×C416yes—
107C2×C4×C67C23×C416yes—
108C2×C4×C67C23×C416yes—
109C2×C4×C67C23×C411yescharacteristic
110C2×C4×C67C23×C416yes—
111C2×C4×C672−1+414yes—
112C2×C4×C67C2511yescharacteristic
113C8×C67—24no—
114C8×C67—212no—
115C22×C67—22no—
116C22×C67—28no—
117C22×C67—22no—
118C22×C67#51912yes—
119C22×C67C22×M1612yes—
120C22×C67C2×#13114yes—
121C22×C67C2×#13112yes—
122C22×C67C24×C411yescharacteristic
123C4×C67—22no—
124C4×C67—26no—
125C4×C67—26no—
126C4×C67—26no—
127C4×C67#51912yes—
128C4×C67C22×M16112yes—
129C4×C67C2×2−1+412yes—
130C2×#2984C6711yesmaximal, characteristic, direct product factor, Sylow
131C2×C67—24no—
132C2×C67#298414yesdirect product factor
133C2×C67C2×#51911yescharacteristic
134C2×C67C23×M1611yescharacteristic
135C2×C67C22×#13111yescharacteristic
136C22×C4×Q8C2×C6711yescharacteristic
137#2984C2×C67116yesdirect product factor
138C2×#446C2×C6716yescomplemented
139C2×C8×Q8C2×C6712yescomplemented
140C2×#432C2×C6713yes—
141C2×#431C2×C6713yes—
142C67C2×#298411yescharacteristic, direct product factor, Sylow
143C23×Q8C4×C6711yescharacteristic
144C2×C4×Q8C4×C6718yes—
145C2×C4×Q8C4×C6713yes—
146C2×C4×Q8C4×C6711yescharacteristic
147C2×C4×Q8C22×C6711yescharacteristic
148C2×C4×Q8C22×C6713yes—
149C2×C4×Q8C22×C6718yes—
150C22×C4⋊C4C4×C6713yes—
151C22×C42C22×C6713yes—
152#446C22×C67148yescomplemented
153C8×Q8C22×C67116yescomplemented
154#432C22×C67124yes—
155#431C22×C67124yes—
156C2×C4⋊C8C22×C6716yes—
157C2×C4⋊C8C22×C6716yes—
158C2×C22⋊C8C22×C6711yescharacteristic
159C2×C22⋊C8C22×C6713yes—
160C2×#97C22×C6716yes—
161C2×C4×C8C22×C6716yes—
162C22×Q8—28no—
163C22×Q8C2×C4×C6718yes—
164C22×Q8C2×C4×C6713yes—
165C22×Q8C2×C4×C6711yescharacteristic
166C23×C4C2×C4×C6713yes—
167C23×C4C2×C4×C6713yes—
168C23×C4C23×C6711yescharacteristic
169C22×C8C23×C6712yes—
170C22×C8C23×C6716yes—
171C4×Q8C2×C4×C67112yes—
172C4×Q8C2×C4×C67124yes—
173C4×Q8C2×C4×C6714yes—
174C4×Q8C2×C4×C6718yes—
175C4×Q8C23×C6714yes—
176C4×Q8C23×C67112yes—
177C2×C4⋊C4C2×C4×C6716yes—
178C2×C4⋊C4C2×C4×C67124yes—
179C2×C4⋊C4C2×C4×C6713yes—
180C2×C4⋊C4C2×C4×C6713yes—
181C2×C42C2×C4×C6713yes—
182C2×C42C2×C4×C67112yes—
183C2×C42C2×C4×C6713yes—
184C2×C42C23×C6713yes—
185C2×C42C23×C6713yes—
186C2×C42C23×C67112yes—
187C4⋊C8C23×C67124yes—
188C4⋊C8C23×C67124yes—
189C22⋊C8C23×C6714yes—
190C22⋊C8C23×C67112yes—
191#97C23×C67124yes—
192C4×C8C23×C67124yes—
193C24C22×C4×C6711yescharacteristic
194C2×Q8—232no—
195C2×Q8—212no—
196C2×Q8M16×C6718yes—
197C2×Q8C22×C4×C67112yes—
198C2×Q8C22×C4×C6714yes—
199C22×C4—212no—
200C22×C4—212no—
201C22×C4—24no—
202C22×C4C22×C4×C6711yescharacteristic
203C22×C4C22×C4×C6712yes—
204C22×C4C22×C4×C67112yes—
205C22×C4C22×C4×C6713yes—
206C22×C4C22×C4×C6713yes—
207C22×C4C22×C4×C6713yes—
208C22×C4C22×C4×C67112yes—
209C22×C4C22×C4×C6713yes—
210C22×C4C24×C6712yes—
211C22×C4C24×C6711yescharacteristic
212C2×C8—24no—
213C2×C8—212no—
214C2×C8C24×C6714yes—
215C2×C8C24×C67112yes—
216C4⋊C4C22×C4×C67124yes—
217C4⋊C4C22×C4×C67112yes—
218C4⋊C4C22×C4×C67112yes—
219C42C22×C4×C67112yes—
220C42C22×C4×C67112yes—
221C42C22×C4×C6716yes—
222C42C22×C4×C6716yes—
223C42C24×C6716yes—
224C42C24×C6716yes—
225C23—22no—
226C23—22no—
227C23#131×C6714yes—
228C23C23×C4×C6712yes—
229C23C23×C4×C6711yescharacteristic
230Q8—224no—
231Q8C2×M16×C67116yes—
232C2×C4—22no—
233C2×C4—26no—
234C2×C4—224no—
235C2×C4—26no—
236C2×C4—26no—
237C2×C4—24no—
238C2×C4—224no—
239C2×C4—24no—
240C2×C4C2×M16×C67112yes—
241C2×C4C23×C4×C6712yes—
242C2×C4C23×C4×C6716yes—
243C2×C4C23×C4×C6716yes—
244C2×C4C23×C4×C6716yes—
245C2×C4C23×C4×C6711yescharacteristic
246C2×C4C23×C4×C6716yes—
247C2×C42−1+4×C6714yes—
248C2×C4C25×C6711yescharacteristic
249C8—24no—
250C8—212no—
251C22—22no—
252C22—22no—
253C22—28no—
254C22#519×C6712yes—
255C22C22×M16×C6712yes—
256C22C2×#131×C6714yes—
257C22C2×#131×C6712yes—
258C22C24×C4×C6711yescharacteristic
259C4—26no—
260C4—26no—
261C4—26no—
262C4—22no—
263C4#519×C6712yes—
264C4C22×M16×C67112yes—
265C4C2×2−1+4×C6712yes—
266C2—24no—
267C2C67×#298414yesdirect product factor
268C2C2×#519×C6711yescharacteristic
269C2C23×M16×C6711yescharacteristic
270C2C22×#131×C6711yescharacteristic
subgroupG/H
centerC22×C4×C67C24
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC23×C67C23×C4
prime radicalC24×C67C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2984C2direct (N × H)
3. C2×#446×C67C2semidirect
4. C2×C8×Q8×C67C2semidirect
16. #446×C67C22semidirect
17. C8×Q8×C67C22semidirect
130. C2×#2984C67direct (N × H)
132. C2×C67#2984direct (N × H)
137. #2984C2×C67direct (N × H)
138. C2×#446C2×C67semidirect
139. C2×C8×Q8C2×C67semidirect
142. C67C2×#2984direct (N × H)
152. #446C22×C67semidirect
153. C8×Q8C22×C67semidirect
267. C2C67×#2984direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Q8×C672
C67×#29842
C2×#446×C672
C2×C8×Q8×C672
C2×#432×C672
C2×#431×C672
C2×#298467