Finite Group Databasebeta

← C2×C61×#2979 #29563660

Subgroups — C2×C61×#2979 #29563660

Order 15616.

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×C61×#532C211yesmaximal, characteristic
2C61×#2979C2116yesmaximal, direct product factor
3C2×C61×#436C214yesmaximal, complemented
4C2×C8×D4×C61C214yesmaximal, complemented
5C2×C61×#431C212yesmaximal, complemented
6C2×C61×#408C214yesmaximal, complemented
7C22×#41×C61C2212yes—
8C22×M16×C61C2214yes—
9C23×C8×C61C2214yes—
10C61×#532C418yes—
11C61×#532C2218yes—
12C2×C4⋊Q8×C61C411yescharacteristic
13C2×#127×C61C411yescharacteristic
14C2×#124×C61C412yes—
15C2×#121×C61C414yes—
16C2×C4×D4×C61C2214yes—
17C22×C42×C61C2211yescharacteristic
18C61×#436C22132yescomplemented
19C8×D4×C61C22132yescomplemented
20C61×#431C22116yescomplemented
21C2×C4⋊C8×C61C2212yes—
22C2×C4⋊C8×C61C2212yes—
23C61×#408C22132yescomplemented
24C2×C22⋊C8×C61C2214yes—
25C2×C22⋊C8×C61C2218yes—
26C2×C4×C8×C61C2214yes—
27C2×#41×C61—44no—
28C2×#41×C61—44no—
29C2×#41×C61C2×C418yes—
30C2×#41×C61C2318yes—
31C22×Q8×C61C2×C412yes—
32C22×D4×C61C2×C414yes—
33C22×D4×C61C2×C412yes—
34C22×D4×C61C2×C414yes—
35C23×C4×C61C2×C412yes—
36C23×C4×C61C2314yes—
37C23×C4×C61C2311yescharacteristic
38C2×M16×C61—48no—
39C2×M16×C61C23116yes—
40C22×C8×C61—48no—
41C22×C8×C61C2314yes—
42C22×C8×C61C2314yes—
43C22×C8×C61C23116yes—
44C4⋊Q8×C61C2×C418yes—
45#127×C61C2×C418yes—
46#124×C61C2×C4116yes—
47#121×C61C2×C4132yes—
48C4×D4×C61C2×C4116yes—
49C4×D4×C61C23116yes—
50C2×C4⋊C4×C61C2×C414yes—
51C2×C22⋊C4×C61C2×C418yes—
52C2×C42×C61C2×C411yescharacteristic
53C2×C42×C61C2×C414yes—
54C2×C42×C61C2×C411yescharacteristic
55C2×C42×C61C2314yes—
56C2×C42×C61C2312yes—
57C4⋊C8×C61C2318yes—
58C4⋊C8×C61C2318yes—
59C22⋊C8×C61C23116yes—
60C22⋊C8×C61C23132yes—
61C4×C8×C61C23116yes—
62C24×C61C22×C414yes—
63C24×C61C22×C411yescharacteristic
64#41×C61—416no—
65#41×C61—48no—
66#41×C61—48no—
67C2×Q8×C61—44no—
68C2×Q8×C61C22×C418yes—
69C2×D4×C61—216no—
70C2×D4×C61—44no—
71C2×D4×C61—48no—
72C2×D4×C61C22×C4116yes—
73C2×D4×C61C22×C4116yes—
74C2×D4×C61C22×C418yes—
75C22×C4×C61—28no—
76C22×C4×C61—28no—
77C22×C4×C61—28no—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C22×C412yes—
82C22×C4×C61C22×C412yes—
83C22×C4×C61C22×C418yes—
84C22×C4×C61C22×C414yes—
85C22×C4×C61C22×C418yes—
86C22×C4×C61C22×C412yes—
87C22×C4×C61C2412yes—
88C22×C4×C61C2418yes—
89C22×C4×C61C2411yescharacteristic
90M16×C61—416no—
91C2×C8×C61—28no—
92C2×C8×C61—28no—
93C2×C8×C61—416no—
94C2×C8×C61C2418yes—
95C2×C8×C61C2418yes—
96C4⋊C4×C61C22×C4116yes—
97C22⋊C4×C61C22×C4132yes—
98C42×C61C22×C414yes—
99C42×C61C22×C414yes—
100C42×C61C22×C414yes—
101C42×C61C2414yes—
102C23×C61—22no—
103C23×C61—28no—
104C23×C61—24no—
105C23×C61—216no—
106C23×C61C23×C418yes—
107C23×C61C23×C412yes—
108C23×C61C23×C411yescharacteristic
109Q8×C61—48no—
110D4×C61—232no—
111D4×C61—48no—
112D4×C61—416no—
113C2×C4×C61—22no—
114C2×C4×C61—28no—
115C2×C4×C61—24no—
116C2×C4×C61—216no—
117C2×C4×C61—24no—
118C2×C4×C61—216no—
119C2×C4×C61—28no—
120C2×C4×C61—28no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61#13118yes—
124C2×C4×C61C23×C412yes—
125C2×C4×C61C23×C414yes—
126C2×C4×C61C23×C418yes—
127C2×C4×C61C23×C411yescharacteristic
128C2×C4×C61C23×C414yes—
129C2×C4×C61H(5,𝔽2)12yes—
130C2×C4×C61H(5,𝔽2)12yes—
131C2×C4×C61C2511yescharacteristic
132C8×C61—28no—
133C8×C61—28no—
134C2×#2979C6111yesmaximal, characteristic, direct product factor, Sylow
135C22×C61—22no—
136C22×C61—28no—
137C22×C61—24no—
138C22×C61—216no—
139C22×C61—44no—
140C22×C61—416no—
141C22×C61#51812yes—
142C22×C61C2×#13114yes—
143C22×C61C24×C411yescharacteristic
144C4×C61—22no—
145C4×C61—28no—
146C4×C61—24no—
147C4×C61#51812yes—
148C4×C61C2×#13118yes—
149C4×C61C2×H(5,𝔽2)11yescharacteristic
150C4×C61C2×H(5,𝔽2)11yescharacteristic
151C2×#532C2×C6111yescharacteristic
152#2979C2×C61116yesdirect product factor
153C2×#436C2×C6114yescomplemented
154C2×C8×D4C2×C6114yescomplemented
155C2×#431C2×C6112yescomplemented
156C2×#408C2×C6114yescomplemented
157C2×C61—42no—
158C2×C61—48no—
159C2×C61#297914yesdirect product factor
160C2×C61C2×#51811yescharacteristic
161C2×C61C22×#13112yes—
162C22×#41C22×C6112yes—
163C22×M16C22×C6114yes—
164C23×C8C22×C6114yes—
165#532C4×C6118yes—
166#532C22×C6118yes—
167C2×C4⋊Q8C4×C6111yescharacteristic
168C2×#127C4×C6111yescharacteristic
169C2×#124C4×C6112yes—
170C2×#121C4×C6114yes—
171C2×C4×D4C22×C6114yes—
172C22×C42C22×C6111yescharacteristic
173#436C22×C61132yescomplemented
174C8×D4C22×C61132yescomplemented
175#431C22×C61116yescomplemented
176C2×C4⋊C8C22×C6112yes—
177C2×C4⋊C8C22×C6112yes—
178#408C22×C61132yescomplemented
179C2×C22⋊C8C22×C6114yes—
180C2×C22⋊C8C22×C6118yes—
181C2×C4×C8C22×C6114yes—
182C61C2×#297911yescharacteristic, direct product factor, Sylow
183C2×#41—44no—
184C2×#41—44no—
185C2×#41C2×C4×C6118yes—
186C2×#41C23×C6118yes—
187C22×Q8C2×C4×C6112yes—
188C22×D4C2×C4×C6114yes—
189C22×D4C2×C4×C6114yes—
190C22×D4C2×C4×C6112yes—
191C23×C4C2×C4×C6112yes—
192C23×C4C23×C6114yes—
193C23×C4C23×C6111yescharacteristic
194C2×M16—48no—
195C2×M16C23×C61116yes—
196C22×C8—48no—
197C22×C8C23×C6114yes—
198C22×C8C23×C6114yes—
199C22×C8C23×C61116yes—
200C4⋊Q8C2×C4×C6118yes—
201#127C2×C4×C6118yes—
202#124C2×C4×C61116yes—
203#121C2×C4×C61132yes—
204C4×D4C2×C4×C61116yes—
205C4×D4C23×C61116yes—
206C2×C4⋊C4C2×C4×C6114yes—
207C2×C22⋊C4C2×C4×C6118yes—
208C2×C42C2×C4×C6111yescharacteristic
209C2×C42C2×C4×C6111yescharacteristic
210C2×C42C2×C4×C6114yes—
211C2×C42C23×C6112yes—
212C2×C42C23×C6114yes—
213C4⋊C8C23×C6118yes—
214C4⋊C8C23×C6118yes—
215C22⋊C8C23×C61116yes—
216C22⋊C8C23×C61132yes—
217C4×C8C23×C61116yes—
218C24C22×C4×C6114yes—
219C24C22×C4×C6111yescharacteristic
220#41—416no—
221#41—48no—
222#41—48no—
223C2×Q8—44no—
224C2×Q8C22×C4×C6118yes—
225C2×D4—216no—
226C2×D4—44no—
227C2×D4—48no—
228C2×D4C22×C4×C61116yes—
229C2×D4C22×C4×C61116yes—
230C2×D4C22×C4×C6118yes—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4C22×C4×C6111yescharacteristic
237C22×C4C22×C4×C6112yes—
238C22×C4C22×C4×C6112yes—
239C22×C4C22×C4×C6118yes—
240C22×C4C22×C4×C6114yes—
241C22×C4C22×C4×C6118yes—
242C22×C4C22×C4×C6112yes—
243C22×C4C24×C6112yes—
244C22×C4C24×C6118yes—
245C22×C4C24×C6111yescharacteristic
246M16—416no—
247C2×C8—28no—
248C2×C8—28no—
249C2×C8—416no—
250C2×C8C24×C6118yes—
251C2×C8C24×C6118yes—
252C4⋊C4C22×C4×C61116yes—
253C22⋊C4C22×C4×C61132yes—
254C42C22×C4×C6114yes—
255C42C22×C4×C6114yes—
256C42C22×C4×C6114yes—
257C42C24×C6114yes—
258C23—24no—
259C23—216no—
260C23—22no—
261C23—28no—
262C23C23×C4×C6118yes—
263C23C23×C4×C6112yes—
264C23C23×C4×C6111yescharacteristic
265Q8—48no—
266D4—232no—
267D4—48no—
268D4—416no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—28no—
272C2×C4—216no—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—216no—
278C2×C4—24no—
279C2×C4#131×C6118yes—
280C2×C4C23×C4×C6112yes—
281C2×C4C23×C4×C6114yes—
282C2×C4C23×C4×C6118yes—
283C2×C4C23×C4×C6114yes—
284C2×C4C23×C4×C6111yescharacteristic
285C2×C4H(5,𝔽2)×C6112yes—
286C2×C4H(5,𝔽2)×C6112yes—
287C2×C4C25×C6111yescharacteristic
288C8—28no—
289C8—28no—
290C22—24no—
291C22—216no—
292C22—22no—
293C22—28no—
294C22—44no—
295C22—416no—
296C22C61×#51812yes—
297C22C2×#131×C6114yes—
298C22C24×C4×C6111yescharacteristic
299C4—24no—
300C4—22no—
301C4—28no—
302C4C61×#51812yes—
303C4C2×#131×C6118yes—
304C4C2×H(5,𝔽2)×C6111yescharacteristic
305C4C2×H(5,𝔽2)×C6111yescharacteristic
306C2—42no—
307C2—48no—
308C2C61×#297914yesdirect product factor
309C2C2×C61×#51811yescharacteristic
310C2C22×#131×C6112yes—
subgroupG/H
centerC22×C4×C61C24
commutatorC22C24×C4×C61
Frattini, cosocleC2×C4C25×C61
socleC23×C61C23×C4
prime radicalC2×C61×#532C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2979C2direct (N × H)
3. C2×C61×#436C2semidirect
4. C2×C8×D4×C61C2semidirect
5. C2×C61×#431C2semidirect
6. C2×C61×#408C2semidirect
18. C61×#436C22semidirect
19. C8×D4×C61C22semidirect
20. C61×#431C22semidirect
23. C61×#408C22semidirect
134. C2×#2979C61direct (N × H)
152. #2979C2×C61direct (N × H)
153. C2×#436C2×C61semidirect
154. C2×C8×D4C2×C61semidirect
155. C2×#431C2×C61semidirect
156. C2×#408C2×C61semidirect
159. C2×C61#2979direct (N × H)
173. #436C22×C61semidirect
174. C8×D4C22×C61semidirect
175. #431C22×C61semidirect
178. #408C22×C61semidirect
182. C61C2×#2979direct (N × H)
308. C2C61×#2979direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#5322
C61×#29792
C2×C61×#4362
C2×C8×D4×C612
C2×C61×#4312
C2×C61×#4082
C2×#297961