Finite Group Databasebeta

← C2×C67×#2906 #30739349

Subgroups — C2×C67×#2906 #30739349

Order 17152.

2810 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#131×C67C212yesmaximal
2C2×C4×#41×C67C211yesmaximal, characteristic
3C67×#2906C2116yesmaximal, direct product factor
4C2×#424×C67C216yesmaximal, complemented
5C2×#423×C67C213yesmaximal, complemented
6C22×C4⋊C8×C67C213yesmaximal, complemented
7C22×#41×C67C2211yescharacteristic
8C2×#131×C67—216no—
9C2×#131×C67C22116yes—
10C22×M16×C67C2216yes—
11C23×C8×C67C2216yes—
12C4×#41×C67C418yes—
13C4×#41×C67C2218yes—
14C2×C4×Q8×C67C411yescharacteristic
15C2×C4×Q8×C67C411yescharacteristic
16C2×C4×D4×C67C413yes—
17C2×C4×D4×C67C413yes—
18C2×#117×C67C2213yes—
19C22×C42×C67C2213yes—
20#424×C67C22148yescomplemented
21#423×C67C22124yescomplemented
22C2×C4⋊C8×C67C2211yescharacteristic
23C2×C4⋊C8×C67C2213yes—
24C2×C4⋊C8×C67C2216yes—
25C2×C4⋊C8×C67C2216yes—
26C2×C4⋊C8×C67C22124yescomplemented
27C2×#41×C67C2×C414yes—
28C2×#41×C67D418yes—
29C2×#41×C67Q818yes—
30C2×#41×C67C2314yes—
31C22×Q8×C67C2×C411yescharacteristic
32C22×D4×C67C2×C413yes—
33C23×C4×C67C2×C413yes—
34C23×C4×C67C2×C413yes—
35C23×C4×C67C2313yes—
36#131×C67—264no—
37C2×M16×C67—224no—
38C2×M16×C67C23124yes—
39C22×C8×C67—224no—
40C22×C8×C67C23124yes—
41C22×C8×C67C2312yes—
42C22×C8×C67C2316yes—
43C4×Q8×C67C2×C418yes—
44C4×Q8×C67C2×C418yes—
45C4×D4×C67C2×C4124yes—
46C4×D4×C67C2×C4124yes—
47#117×C67C2×C4112yes—
48#117×C67C23112yes—
49C2×C4⋊C4×C67C2×C413yes—
50C2×C4⋊C4×C67C2×C413yes—
51C2×C22⋊C4×C67C2×C416yes—
52C2×C42×C67C2×C413yes—
53C2×C42×C67C2×C413yes—
54C2×C42×C67C2×C4112yes—
55C2×C42×C67C23112yes—
56C2×C42×C67C2311yescharacteristic
57C2×C42×C67C2313yes—
58C4⋊C8×C67C2314yes—
59C4⋊C8×C67C23112yes—
60C4⋊C8×C67C23124yes—
61C4⋊C8×C67C23124yes—
62C24×C67C22×C413yes—
63#41×C67C4⋊C4132yes—
64#41×C67C2×D4116yes—
65#41×C67C2×Q8116yes—
66C2×Q8×C67C4⋊C418yes—
67C2×Q8×C67C22×C414yes—
68C2×D4×C67C4⋊C4124yes—
69C2×D4×C67C22×C4112yes—
70C22×C4×C67—212no—
71C22×C4×C67—46no—
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C22×C413yes—
74C22×C4×C67C22×C413yes—
75C22×C4×C67C22×C4112yes—
76C22×C4×C67C22×C4112yes—
77C22×C4×C67C22×C413yes—
78C22×C4×C67C22×C416yes—
79C22×C4×C67C22×C411yescharacteristic
80C22×C4×C67C2×D4112yes—
81C22×C4×C67C2×Q8112yes—
82C22×C4×C67C2416yes—
83C22×C4×C67C2411yescharacteristic
84M16×C67—248no—
85C2×C8×C67—24no—
86C2×C8×C67—212no—
87C2×C8×C67—248no—
88C2×C8×C67C2414yes—
89C2×C8×C67C24112yes—
90C4⋊C4×C67C22×C4112yes—
91C4⋊C4×C67C22×C4112yes—
92C22⋊C4×C67C22×C4124yes—
93C42×C67C22×C412yes—
94C42×C67C22×C416yes—
95C42×C67C22×C4112yes—
96C42×C67C22×C4112yes—
97C42×C67C2412yes—
98C42×C67C2416yes—
99C23×C67—26no—
100C23×C67—26no—
101C23×C67C2×C4⋊C4112yes—
102C23×C67C23×C416yes—
103C23×C67C23×C411yescharacteristic
104Q8×C67C2×C4⋊C4116yes—
105D4×C67C2×C4⋊C4148yes—
106C2×C4×C67—26no—
107C2×C4×C67—22no—
108C2×C4×C67—26no—
109C2×C4×C67—224no—
110C2×C4×C67—412no—
111C2×C4×C67C2×C4⋊C4112yes—
112C2×C4×C67C2×C4⋊C4124yes—
113C2×C4×C67#13114yescomplemented
114C2×C4×C67C23×C412yes—
115C2×C4×C67C23×C416yes—
116C2×C4×C67C23×C416yes—
117C2×C4×C67C23×C411yescharacteristic
118C2×C4×C67C23×C416yes—
119C2×C4×C67C23×C412yes—
120C2×C4×C67C22×D412yes—
121C2×C4×C67C22×D4112yes—
122C2×C4×C67C22×Q812yes—
123C2×C4×C67C22×Q8112yes—
124C2×C4×C67C2511yescharacteristic
125C8×C67—24no—
126C8×C67—212no—
127C22×C67—224no—
128C22×C67—26no—
129C22×C67—26no—
130C22×C67C22×C4⋊C4112yes—
131C22×C67C22×C4⋊C412yes—
132C22×C67C2×#13112yes—
133C22×C67#56812yes—
134C22×C67C24×C411yescharacteristic
135C4×C67—26no—
136C4×C67—22no—
137C4×C67—26no—
138C4×C67C22×C4⋊C412yes—
139C4×C67C22×C4⋊C4112yes—
140C4×C67C2×#13114yescomplemented
141C4×C67C23×D411yescharacteristic
142C4×C67C23×Q811yescharacteristic
143C2×#2906C6711yesmaximal, characteristic, direct product factor, Sylow
144C2×C67—212no—
145C2×C67#290614yesdirect product factor
146C2×C67C23×C4⋊C411yescharacteristic
147C2×C67C22×#13111yescharacteristic
148C2×C67C2×#56811yescharacteristic
149C22×#131C2×C6712yes—
150C2×C4×#41C2×C6711yescharacteristic
151#2906C2×C67116yesdirect product factor
152C2×#424C2×C6716yescomplemented
153C2×#423C2×C6713yescomplemented
154C22×C4⋊C8C2×C6713yescomplemented
155C67C2×#290611yescharacteristic, direct product factor, Sylow
156C22×#41C22×C6711yescharacteristic
157C2×#131—216no—
158C2×#131C22×C67116yes—
159C22×M16C22×C6716yes—
160C23×C8C22×C6716yes—
161C4×#41C4×C6718yes—
162C4×#41C22×C6718yes—
163C2×C4×Q8C4×C6711yescharacteristic
164C2×C4×Q8C4×C6711yescharacteristic
165C2×C4×D4C4×C6713yes—
166C2×C4×D4C4×C6713yes—
167C2×#117C22×C6713yes—
168C22×C42C22×C6713yes—
169#424C22×C67148yescomplemented
170#423C22×C67124yescomplemented
171C2×C4⋊C8C22×C6711yescharacteristic
172C2×C4⋊C8C22×C6713yes—
173C2×C4⋊C8C22×C6716yes—
174C2×C4⋊C8C22×C6716yes—
175C2×C4⋊C8C22×C67124yescomplemented
176C2×#41C2×C4×C6714yes—
177C2×#41D4×C6718yes—
178C2×#41Q8×C6718yes—
179C2×#41C23×C6714yes—
180C22×Q8C2×C4×C6711yescharacteristic
181C22×D4C2×C4×C6713yes—
182C23×C4C2×C4×C6713yes—
183C23×C4C2×C4×C6713yes—
184C23×C4C23×C6713yes—
185#131—264no—
186C2×M16—224no—
187C2×M16C23×C67124yes—
188C22×C8—224no—
189C22×C8C23×C6712yes—
190C22×C8C23×C6716yes—
191C22×C8C23×C67124yes—
192C4×Q8C2×C4×C6718yes—
193C4×Q8C2×C4×C6718yes—
194C4×D4C2×C4×C67124yes—
195C4×D4C2×C4×C67124yes—
196#117C2×C4×C67112yes—
197#117C23×C67112yes—
198C2×C4⋊C4C2×C4×C6713yes—
199C2×C4⋊C4C2×C4×C6713yes—
200C2×C22⋊C4C2×C4×C6716yes—
201C2×C42C2×C4×C6713yes—
202C2×C42C2×C4×C6713yes—
203C2×C42C2×C4×C67112yes—
204C2×C42C23×C6713yes—
205C2×C42C23×C67112yes—
206C2×C42C23×C6711yescharacteristic
207C4⋊C8C23×C6714yes—
208C4⋊C8C23×C67112yes—
209C4⋊C8C23×C67124yes—
210C4⋊C8C23×C67124yes—
211C24C22×C4×C6713yes—
212#41C4⋊C4×C67132yes—
213#41C2×D4×C67116yes—
214#41C2×Q8×C67116yes—
215C2×Q8C4⋊C4×C6718yes—
216C2×Q8C22×C4×C6714yes—
217C2×D4C4⋊C4×C67124yes—
218C2×D4C22×C4×C67112yes—
219C22×C4—212no—
220C22×C4—46no—
221C22×C4C22×C4×C6711yescharacteristic
222C22×C4C22×C4×C6713yes—
223C22×C4C22×C4×C6713yes—
224C22×C4C22×C4×C67112yes—
225C22×C4C22×C4×C67112yes—
226C22×C4C22×C4×C6713yes—
227C22×C4C22×C4×C6716yes—
228C22×C4C22×C4×C6711yescharacteristic
229C22×C4C2×D4×C67112yes—
230C22×C4C2×Q8×C67112yes—
231C22×C4C24×C6711yescharacteristic
232C22×C4C24×C6716yes—
233M16—248no—
234C2×C8—24no—
235C2×C8—212no—
236C2×C8—248no—
237C2×C8C24×C6714yes—
238C2×C8C24×C67112yes—
239C4⋊C4C22×C4×C67112yes—
240C4⋊C4C22×C4×C67112yes—
241C22⋊C4C22×C4×C67124yes—
242C42C22×C4×C6712yes—
243C42C22×C4×C6716yes—
244C42C22×C4×C67112yes—
245C42C22×C4×C67112yes—
246C42C24×C6712yes—
247C42C24×C6716yes—
248C23—26no—
249C23—26no—
250C23C2×C4⋊C4×C67112yes—
251C23C23×C4×C6716yes—
252C23C23×C4×C6711yescharacteristic
253Q8C2×C4⋊C4×C67116yes—
254D4C2×C4⋊C4×C67148yes—
255C2×C4—26no—
256C2×C4—22no—
257C2×C4—26no—
258C2×C4—224no—
259C2×C4—412no—
260C2×C4C2×C4⋊C4×C67112yes—
261C2×C4C2×C4⋊C4×C67124yes—
262C2×C4#131×C6714yescomplemented
263C2×C4C23×C4×C6712yes—
264C2×C4C23×C4×C6716yes—
265C2×C4C23×C4×C6716yes—
266C2×C4C23×C4×C6711yescharacteristic
267C2×C4C23×C4×C6716yes—
268C2×C4C23×C4×C6712yes—
269C2×C4C22×D4×C6712yes—
270C2×C4C22×D4×C67112yes—
271C2×C4C22×Q8×C6712yes—
272C2×C4C22×Q8×C67112yes—
273C2×C4C25×C6711yescharacteristic
274C8—24no—
275C8—212no—
276C22—26no—
277C22—26no—
278C22—224no—
279C22C22×C4⋊C4×C67112yes—
280C22C22×C4⋊C4×C6712yes—
281C22C2×#131×C6712yes—
282C22#568×C6712yes—
283C22C24×C4×C6711yescharacteristic
284C4—26no—
285C4—22no—
286C4—26no—
287C4C22×C4⋊C4×C6712yes—
288C4C22×C4⋊C4×C67112yes—
289C4C2×#131×C6714yescomplemented
290C4C23×D4×C6711yescharacteristic
291C4C23×Q8×C6711yescharacteristic
292C2—212no—
293C2C67×#290614yesdirect product factor
294C2C23×C4⋊C4×C6711yescharacteristic
295C2C22×#131×C6711yescharacteristic
296C2C2×#568×C6711yescharacteristic
subgroupG/H
centerC22×C4×C67C24
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC23×C67C23×C4
prime radicalC22×#41×C67C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C67×#2906C2direct (N × H)
4. C2×#424×C67C2semidirect
5. C2×#423×C67C2semidirect
6. C22×C4⋊C8×C67C2semidirect
20. #424×C67C22semidirect
21. #423×C67C22semidirect
26. C2×C4⋊C8×C67C22semidirect
113. C2×C4×C67#131semidirect
140. C4×C67C2×#131semidirect
143. C2×#2906C67direct (N × H)
145. C2×C67#2906direct (N × H)
151. #2906C2×C67direct (N × H)
152. C2×#424C2×C67semidirect
153. C2×#423C2×C67semidirect
154. C22×C4⋊C8C2×C67semidirect
155. C67C2×#2906direct (N × H)
169. #424C22×C67semidirect
170. #423C22×C67semidirect
175. C2×C4⋊C8C22×C67semidirect
262. C2×C4#131×C67semidirect
289. C4C2×#131×C67semidirect
293. C2C67×#2906direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131×C672
C2×C4×#41×C672
C67×#29062
C2×#424×C672
C2×#423×C672
C22×C4⋊C8×C672
C2×#290667