Finite Group Databasebeta

← C2×C67×#2979 #30739478

Subgroups — C2×C67×#2979 #30739478

Order 17152.

2274 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#532×C67C211yesmaximal, characteristic
2C67×#2979C2116yesmaximal, direct product factor
3C2×#436×C67C214yesmaximal, complemented
4C2×C8×D4×C67C214yesmaximal, complemented
5C2×#431×C67C212yesmaximal, complemented
6C2×#408×C67C214yesmaximal, complemented
7C22×#41×C67C2212yes—
8C22×M16×C67C2214yes—
9C23×C8×C67C2214yes—
10#532×C67C418yes—
11#532×C67C2218yes—
12C2×C4⋊Q8×C67C411yescharacteristic
13C2×#127×C67C411yescharacteristic
14C2×#124×C67C412yes—
15C2×#121×C67C414yes—
16C2×C4×D4×C67C2214yes—
17C22×C42×C67C2211yescharacteristic
18#436×C67C22132yescomplemented
19C8×D4×C67C22132yescomplemented
20#431×C67C22116yescomplemented
21C2×C4⋊C8×C67C2212yes—
22C2×C4⋊C8×C67C2212yes—
23#408×C67C22132yescomplemented
24C2×C22⋊C8×C67C2214yes—
25C2×C22⋊C8×C67C2218yes—
26C2×C4×C8×C67C2214yes—
27C2×#41×C67—44no—
28C2×#41×C67—44no—
29C2×#41×C67C2×C418yes—
30C2×#41×C67C2318yes—
31C22×Q8×C67C2×C412yes—
32C22×D4×C67C2×C414yes—
33C22×D4×C67C2×C414yes—
34C22×D4×C67C2×C412yes—
35C23×C4×C67C2×C412yes—
36C23×C4×C67C2311yescharacteristic
37C23×C4×C67C2314yes—
38C2×M16×C67—48no—
39C2×M16×C67C23116yes—
40C22×C8×C67—48no—
41C22×C8×C67C2314yes—
42C22×C8×C67C2314yes—
43C22×C8×C67C23116yes—
44C4⋊Q8×C67C2×C418yes—
45#127×C67C2×C418yes—
46#124×C67C2×C4116yes—
47#121×C67C2×C4132yes—
48C4×D4×C67C2×C4116yes—
49C4×D4×C67C23116yes—
50C2×C4⋊C4×C67C2×C414yes—
51C2×C22⋊C4×C67C2×C418yes—
52C2×C42×C67C2×C411yescharacteristic
53C2×C42×C67C2×C411yescharacteristic
54C2×C42×C67C2×C414yes—
55C2×C42×C67C2314yes—
56C2×C42×C67C2312yes—
57C4⋊C8×C67C2318yes—
58C4⋊C8×C67C2318yes—
59C22⋊C8×C67C23116yes—
60C22⋊C8×C67C23132yes—
61C4×C8×C67C23116yes—
62C24×C67C22×C414yes—
63C24×C67C22×C411yescharacteristic
64#41×C67—416no—
65#41×C67—48no—
66#41×C67—48no—
67C2×Q8×C67—44no—
68C2×Q8×C67C22×C418yes—
69C2×D4×C67—216no—
70C2×D4×C67—44no—
71C2×D4×C67—48no—
72C2×D4×C67C22×C4116yes—
73C2×D4×C67C22×C4116yes—
74C2×D4×C67C22×C418yes—
75C22×C4×C67—28no—
76C22×C4×C67—28no—
77C22×C4×C67—28no—
78C22×C4×C67—22no—
79C22×C4×C67—22no—
80C22×C4×C67C22×C411yescharacteristic
81C22×C4×C67C22×C412yes—
82C22×C4×C67C22×C412yes—
83C22×C4×C67C22×C418yes—
84C22×C4×C67C22×C414yes—
85C22×C4×C67C22×C418yes—
86C22×C4×C67C22×C412yes—
87C22×C4×C67C2412yes—
88C22×C4×C67C2418yes—
89C22×C4×C67C2411yescharacteristic
90M16×C67—416no—
91C2×C8×C67—28no—
92C2×C8×C67—28no—
93C2×C8×C67—416no—
94C2×C8×C67C2418yes—
95C2×C8×C67C2418yes—
96C4⋊C4×C67C22×C4116yes—
97C22⋊C4×C67C22×C4132yes—
98C42×C67C22×C414yes—
99C42×C67C22×C414yes—
100C42×C67C22×C414yes—
101C42×C67C2414yes—
102C23×C67—22no—
103C23×C67—28no—
104C23×C67—24no—
105C23×C67—216no—
106C23×C67C23×C418yes—
107C23×C67C23×C412yes—
108C23×C67C23×C411yescharacteristic
109Q8×C67—48no—
110D4×C67—232no—
111D4×C67—48no—
112D4×C67—416no—
113C2×C4×C67—22no—
114C2×C4×C67—28no—
115C2×C4×C67—24no—
116C2×C4×C67—216no—
117C2×C4×C67—24no—
118C2×C4×C67—216no—
119C2×C4×C67—28no—
120C2×C4×C67—28no—
121C2×C4×C67—22no—
122C2×C4×C67—22no—
123C2×C4×C67#13118yes—
124C2×C4×C67C23×C412yes—
125C2×C4×C67C23×C414yes—
126C2×C4×C67C23×C418yes—
127C2×C4×C67C23×C411yescharacteristic
128C2×C4×C67C23×C414yes—
129C2×C4×C67H(5,𝔽2)12yes—
130C2×C4×C67H(5,𝔽2)12yes—
131C2×C4×C67C2511yescharacteristic
132C8×C67—28no—
133C8×C67—28no—
134C22×C67—22no—
135C22×C67—28no—
136C22×C67—24no—
137C22×C67—216no—
138C22×C67—44no—
139C22×C67—416no—
140C22×C67#51812yes—
141C22×C67C2×#13114yes—
142C22×C67C24×C411yescharacteristic
143C4×C67—22no—
144C4×C67—28no—
145C4×C67—24no—
146C4×C67#51812yes—
147C4×C67C2×#13118yes—
148C4×C67C2×H(5,𝔽2)11yescharacteristic
149C4×C67C2×H(5,𝔽2)11yescharacteristic
150C2×#2979C6711yesmaximal, characteristic, direct product factor, Sylow
151C2×C67—42no—
152C2×C67—48no—
153C2×C67#297914yesdirect product factor
154C2×C67C2×#51811yescharacteristic
155C2×C67C22×#13112yes—
156C2×#532C2×C6711yescharacteristic
157#2979C2×C67116yesdirect product factor
158C2×#436C2×C6714yescomplemented
159C2×C8×D4C2×C6714yescomplemented
160C2×#431C2×C6712yescomplemented
161C2×#408C2×C6714yescomplemented
162C67C2×#297911yescharacteristic, direct product factor, Sylow
163C22×#41C22×C6712yes—
164C22×M16C22×C6714yes—
165C23×C8C22×C6714yes—
166#532C4×C6718yes—
167#532C22×C6718yes—
168C2×C4⋊Q8C4×C6711yescharacteristic
169C2×#127C4×C6711yescharacteristic
170C2×#124C4×C6712yes—
171C2×#121C4×C6714yes—
172C2×C4×D4C22×C6714yes—
173C22×C42C22×C6711yescharacteristic
174#436C22×C67132yescomplemented
175C8×D4C22×C67132yescomplemented
176#431C22×C67116yescomplemented
177C2×C4⋊C8C22×C6712yes—
178C2×C4⋊C8C22×C6712yes—
179#408C22×C67132yescomplemented
180C2×C22⋊C8C22×C6714yes—
181C2×C22⋊C8C22×C6718yes—
182C2×C4×C8C22×C6714yes—
183C2×#41—44no—
184C2×#41—44no—
185C2×#41C2×C4×C6718yes—
186C2×#41C23×C6718yes—
187C22×Q8C2×C4×C6712yes—
188C22×D4C2×C4×C6714yes—
189C22×D4C2×C4×C6714yes—
190C22×D4C2×C4×C6712yes—
191C23×C4C2×C4×C6712yes—
192C23×C4C23×C6714yes—
193C23×C4C23×C6711yescharacteristic
194C2×M16—48no—
195C2×M16C23×C67116yes—
196C22×C8—48no—
197C22×C8C23×C6714yes—
198C22×C8C23×C6714yes—
199C22×C8C23×C67116yes—
200C4⋊Q8C2×C4×C6718yes—
201#127C2×C4×C6718yes—
202#124C2×C4×C67116yes—
203#121C2×C4×C67132yes—
204C4×D4C2×C4×C67116yes—
205C4×D4C23×C67116yes—
206C2×C4⋊C4C2×C4×C6714yes—
207C2×C22⋊C4C2×C4×C6718yes—
208C2×C42C2×C4×C6711yescharacteristic
209C2×C42C2×C4×C6711yescharacteristic
210C2×C42C2×C4×C6714yes—
211C2×C42C23×C6712yes—
212C2×C42C23×C6714yes—
213C4⋊C8C23×C6718yes—
214C4⋊C8C23×C6718yes—
215C22⋊C8C23×C67116yes—
216C22⋊C8C23×C67132yes—
217C4×C8C23×C67116yes—
218C24C22×C4×C6714yes—
219C24C22×C4×C6711yescharacteristic
220#41—416no—
221#41—48no—
222#41—48no—
223C2×Q8—44no—
224C2×Q8C22×C4×C6718yes—
225C2×D4—216no—
226C2×D4—44no—
227C2×D4—48no—
228C2×D4C22×C4×C67116yes—
229C2×D4C22×C4×C67116yes—
230C2×D4C22×C4×C6718yes—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4C22×C4×C6711yescharacteristic
237C22×C4C22×C4×C6712yes—
238C22×C4C22×C4×C6712yes—
239C22×C4C22×C4×C6718yes—
240C22×C4C22×C4×C6714yes—
241C22×C4C22×C4×C6718yes—
242C22×C4C22×C4×C6712yes—
243C22×C4C24×C6712yes—
244C22×C4C24×C6718yes—
245C22×C4C24×C6711yescharacteristic
246M16—416no—
247C2×C8—28no—
248C2×C8—28no—
249C2×C8—416no—
250C2×C8C24×C6718yes—
251C2×C8C24×C6718yes—
252C4⋊C4C22×C4×C67116yes—
253C22⋊C4C22×C4×C67132yes—
254C42C22×C4×C6714yes—
255C42C22×C4×C6714yes—
256C42C22×C4×C6714yes—
257C42C24×C6714yes—
258C23—22no—
259C23—28no—
260C23—24no—
261C23—216no—
262C23C23×C4×C6718yes—
263C23C23×C4×C6712yes—
264C23C23×C4×C6711yescharacteristic
265Q8—48no—
266D4—232no—
267D4—48no—
268D4—416no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—216no—
274C2×C4—216no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4#131×C6718yes—
280C2×C4C23×C4×C6712yes—
281C2×C4C23×C4×C6714yes—
282C2×C4C23×C4×C6718yes—
283C2×C4C23×C4×C6711yescharacteristic
284C2×C4C23×C4×C6714yes—
285C2×C4H(5,𝔽2)×C6712yes—
286C2×C4H(5,𝔽2)×C6712yes—
287C2×C4C25×C6711yescharacteristic
288C8—28no—
289C8—28no—
290C22—24no—
291C22—216no—
292C22—22no—
293C22—28no—
294C22—44no—
295C22—416no—
296C22#518×C6712yes—
297C22C2×#131×C6714yes—
298C22C24×C4×C6711yescharacteristic
299C4—24no—
300C4—22no—
301C4—28no—
302C4#518×C6712yes—
303C4C2×#131×C6718yes—
304C4C2×H(5,𝔽2)×C6711yescharacteristic
305C4C2×H(5,𝔽2)×C6711yescharacteristic
306C2—42no—
307C2—48no—
308C2C67×#297914yesdirect product factor
309C2C2×#518×C6711yescharacteristic
310C2C22×#131×C6712yes—
subgroupG/H
centerC22×C4×C67C24
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC23×C67C23×C4
prime radicalC2×#532×C67C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2979C2direct (N × H)
3. C2×#436×C67C2semidirect
4. C2×C8×D4×C67C2semidirect
5. C2×#431×C67C2semidirect
6. C2×#408×C67C2semidirect
18. #436×C67C22semidirect
19. C8×D4×C67C22semidirect
20. #431×C67C22semidirect
23. #408×C67C22semidirect
150. C2×#2979C67direct (N × H)
153. C2×C67#2979direct (N × H)
157. #2979C2×C67direct (N × H)
158. C2×#436C2×C67semidirect
159. C2×C8×D4C2×C67semidirect
160. C2×#431C2×C67semidirect
161. C2×#408C2×C67semidirect
162. C67C2×#2979direct (N × H)
174. #436C22×C67semidirect
175. C8×D4C22×C67semidirect
176. #431C22×C67semidirect
179. #408C22×C67semidirect
308. C2C67×#2979direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#532×C672
C67×#29792
C2×#436×C672
C2×C8×D4×C672
C2×#431×C672
C2×#408×C672
C2×#297967