Finite Group Databasebeta

← C2×C23×#2979 #19207853

Subgroups — C2×C23×#2979 #19207853

Order 5888.

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×C23×#532C211yesmaximal, characteristic
2C23×#2979C2116yesmaximal, direct product factor
3C2×C23×#436C214yesmaximal, complemented
4C2×C8×D4×C23C214yesmaximal, complemented
5C2×C23×#431C212yesmaximal, complemented
6C2×C23×#408C214yesmaximal, complemented
7C22×#41×C23C2212yes—
8C22×M16×C23C2214yes—
9C23×C8×C23C2214yes—
10C23×#532C418yes—
11C23×#532C2218yes—
12C2×C23×C4⋊Q8C411yescharacteristic
13C2×C23×#127C411yescharacteristic
14C2×C23×#124C412yes—
15C2×C23×#121C414yes—
16C2×C4×D4×C23C2214yes—
17C22×C42×C23C2211yescharacteristic
18C23×#436C22132yescomplemented
19C8×D4×C23C22132yescomplemented
20C23×#431C22116yescomplemented
21C2×C23×C4⋊C8C2212yes—
22C2×C23×C4⋊C8C2212yes—
23C23×#408C22132yescomplemented
24C2×C23×C22⋊C8C2214yes—
25C2×C23×C22⋊C8C2218yes—
26C2×C4×C8×C23C2214yes—
27C2×#41×C23—44no—
28C2×#41×C23—44no—
29C2×#41×C23C2×C418yes—
30C2×#41×C23C2318yes—
31C22×Q8×C23C2×C412yes—
32C22×D4×C23C2×C414yes—
33C22×D4×C23C2×C414yes—
34C22×D4×C23C2×C412yes—
35C23×C4×C23C2×C412yes—
36C23×C4×C23C2314yes—
37C23×C4×C23C2311yescharacteristic
38C2×M16×C23—48no—
39C2×M16×C23C23116yes—
40C22×C8×C23—48no—
41C22×C8×C23C2314yes—
42C22×C8×C23C2314yes—
43C22×C8×C23C23116yes—
44C23×C4⋊Q8C2×C418yes—
45C23×#127C2×C418yes—
46C23×#124C2×C4116yes—
47C23×#121C2×C4132yes—
48C4×D4×C23C2×C4116yes—
49C4×D4×C23C23116yes—
50C2×C4⋊C4×C23C2×C414yes—
51C2×C22⋊C4×C23C2×C418yes—
52C2×C42×C23C2×C411yescharacteristic
53C2×C42×C23C2×C411yescharacteristic
54C2×C42×C23C2×C414yes—
55C2×C42×C23C2314yes—
56C2×C42×C23C2312yes—
57C23×C4⋊C8C2318yes—
58C23×C4⋊C8C2318yes—
59C23×C22⋊C8C23116yes—
60C23×C22⋊C8C23132yes—
61C4×C8×C23C23116yes—
62C24×C23C22×C414yes—
63C24×C23C22×C411yescharacteristic
64#41×C23—416no—
65#41×C23—48no—
66#41×C23—48no—
67C2×Q8×C23—44no—
68C2×Q8×C23C22×C418yes—
69C2×D4×C23—216no—
70C2×D4×C23—44no—
71C2×D4×C23—48no—
72C2×D4×C23C22×C4116yes—
73C2×D4×C23C22×C4116yes—
74C2×D4×C23C22×C418yes—
75C22×C4×C23—28no—
76C22×C4×C23—28no—
77C22×C4×C23—28no—
78C22×C4×C23—22no—
79C22×C4×C23—22no—
80C22×C4×C23C22×C411yescharacteristic
81C22×C4×C23C22×C412yes—
82C22×C4×C23C22×C412yes—
83C22×C4×C23C22×C418yes—
84C22×C4×C23C22×C414yes—
85C22×C4×C23C22×C418yes—
86C22×C4×C23C22×C412yes—
87C22×C4×C23C2412yes—
88C22×C4×C23C2418yes—
89C22×C4×C23C2411yescharacteristic
90M16×C23—416no—
91C2×C8×C23—28no—
92C2×C8×C23—28no—
93C2×C8×C23—416no—
94C2×C8×C23C2418yes—
95C2×C8×C23C2418yes—
96C4⋊C4×C23C22×C4116yes—
97C22⋊C4×C23C22×C4132yes—
98C42×C23C22×C414yes—
99C42×C23C22×C414yes—
100C42×C23C22×C414yes—
101C42×C23C2414yes—
102C2×#2979C2311yesmaximal, characteristic, direct product factor, Sylow
103C23×C23—22no—
104C23×C23—28no—
105C23×C23—24no—
106C23×C23—216no—
107C23×C23C23×C418yes—
108C23×C23C23×C412yes—
109C23×C23C23×C411yescharacteristic
110Q8×C23—48no—
111D4×C23—232no—
112D4×C23—48no—
113D4×C23—416no—
114C2×C4×C23—22no—
115C2×C4×C23—28no—
116C2×C4×C23—24no—
117C2×C4×C23—216no—
118C2×C4×C23—24no—
119C2×C4×C23—216no—
120C2×C4×C23—28no—
121C2×C4×C23—28no—
122C2×C4×C23—22no—
123C2×C4×C23—22no—
124C2×C4×C23#13118yes—
125C2×C4×C23C23×C412yes—
126C2×C4×C23C23×C414yes—
127C2×C4×C23C23×C418yes—
128C2×C4×C23C23×C411yescharacteristic
129C2×C4×C23C23×C414yes—
130C2×C4×C23H(5,𝔽2)12yes—
131C2×C4×C23H(5,𝔽2)12yes—
132C2×C4×C23C2511yescharacteristic
133C8×C23—28no—
134C8×C23—28no—
135C2×#532C2×C2311yescharacteristic
136#2979C2×C23116yesdirect product factor
137C2×#436C2×C2314yescomplemented
138C2×C8×D4C2×C2314yescomplemented
139C2×#431C2×C2312yescomplemented
140C2×#408C2×C2314yescomplemented
141C22×C23—22no—
142C22×C23—28no—
143C22×C23—24no—
144C22×C23—216no—
145C22×C23—44no—
146C22×C23—416no—
147C22×C23#51812yes—
148C22×C23C2×#13114yes—
149C22×C23C24×C411yescharacteristic
150C4×C23—22no—
151C4×C23—28no—
152C4×C23—24no—
153C4×C23#51812yes—
154C4×C23C2×#13118yes—
155C4×C23C2×H(5,𝔽2)11yescharacteristic
156C4×C23C2×H(5,𝔽2)11yescharacteristic
157C22×#41C22×C2312yes—
158C22×M16C22×C2314yes—
159C23×C8C22×C2314yes—
160#532C4×C2318yes—
161#532C22×C2318yes—
162C2×C4⋊Q8C4×C2311yescharacteristic
163C2×#127C4×C2311yescharacteristic
164C2×#124C4×C2312yes—
165C2×#121C4×C2314yes—
166C2×C4×D4C22×C2314yes—
167C22×C42C22×C2311yescharacteristic
168#436C22×C23132yescomplemented
169C8×D4C22×C23132yescomplemented
170#431C22×C23116yescomplemented
171C2×C4⋊C8C22×C2312yes—
172C2×C4⋊C8C22×C2312yes—
173#408C22×C23132yescomplemented
174C2×C22⋊C8C22×C2314yes—
175C2×C22⋊C8C22×C2318yes—
176C2×C4×C8C22×C2314yes—
177C2×C23—42no—
178C2×C23—48no—
179C2×C23#297914yesdirect product factor
180C2×C23C2×#51811yescharacteristic
181C2×C23C22×#13112yes—
182C2×#41—44no—
183C2×#41—44no—
184C2×#41C2×C4×C2318yes—
185C2×#41C23×C2318yes—
186C22×Q8C2×C4×C2312yes—
187C22×D4C2×C4×C2314yes—
188C22×D4C2×C4×C2314yes—
189C22×D4C2×C4×C2312yes—
190C23×C4C2×C4×C2312yes—
191C23×C4C23×C2314yes—
192C23×C4C23×C2311yescharacteristic
193C2×M16—48no—
194C2×M16C23×C23116yes—
195C22×C8—48no—
196C22×C8C23×C2314yes—
197C22×C8C23×C2314yes—
198C22×C8C23×C23116yes—
199C4⋊Q8C2×C4×C2318yes—
200#127C2×C4×C2318yes—
201#124C2×C4×C23116yes—
202#121C2×C4×C23132yes—
203C4×D4C2×C4×C23116yes—
204C4×D4C23×C23116yes—
205C2×C4⋊C4C2×C4×C2314yes—
206C2×C22⋊C4C2×C4×C2318yes—
207C2×C42C2×C4×C2311yescharacteristic
208C2×C42C2×C4×C2311yescharacteristic
209C2×C42C2×C4×C2314yes—
210C2×C42C23×C2312yes—
211C2×C42C23×C2314yes—
212C4⋊C8C23×C2318yes—
213C4⋊C8C23×C2318yes—
214C22⋊C8C23×C23116yes—
215C22⋊C8C23×C23132yes—
216C4×C8C23×C23116yes—
217C23C2×#297911yescharacteristic, direct product factor, Sylow
218C24C22×C4×C2314yes—
219C24C22×C4×C2311yescharacteristic
220#41—416no—
221#41—48no—
222#41—48no—
223C2×Q8—44no—
224C2×Q8C22×C4×C2318yes—
225C2×D4—216no—
226C2×D4—44no—
227C2×D4—48no—
228C2×D4C22×C4×C23116yes—
229C2×D4C22×C4×C23116yes—
230C2×D4C22×C4×C2318yes—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4C22×C4×C2311yescharacteristic
237C22×C4C22×C4×C2312yes—
238C22×C4C22×C4×C2312yes—
239C22×C4C22×C4×C2318yes—
240C22×C4C22×C4×C2314yes—
241C22×C4C22×C4×C2318yes—
242C22×C4C22×C4×C2312yes—
243C22×C4C24×C2312yes—
244C22×C4C24×C2318yes—
245C22×C4C24×C2311yescharacteristic
246M16—416no—
247C2×C8—28no—
248C2×C8—28no—
249C2×C8—416no—
250C2×C8C24×C2318yes—
251C2×C8C24×C2318yes—
252C4⋊C4C22×C4×C23116yes—
253C22⋊C4C22×C4×C23132yes—
254C42C22×C4×C2314yes—
255C42C22×C4×C2314yes—
256C42C22×C4×C2314yes—
257C42C24×C2314yes—
258C23—24no—
259C23—216no—
260C23—22no—
261C23—28no—
262C23C23×C4×C2318yes—
263C23C23×C4×C2312yes—
264C23C23×C4×C2311yescharacteristic
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—24no—
278C2×C4—216no—
279C2×C4C23×#13118yes—
280C2×C4C23×C4×C2312yes—
281C2×C4C23×C4×C2314yes—
282C2×C4C23×C4×C2318yes—
283C2×C4C23×C4×C2314yes—
284C2×C4C23×C4×C2311yescharacteristic
285C2×C4C23×H(5,𝔽2)12yes—
286C2×C4C23×H(5,𝔽2)12yes—
287C2×C4C25×C2311yescharacteristic
288C8—28no—
289C8—28no—
290C22—22no—
291C22—28no—
292C22—24no—
293C22—216no—
294C22—44no—
295C22—416no—
296C22C23×#51812yes—
297C22C2×C23×#13114yes—
298C22C24×C4×C2311yescharacteristic
299C4—22no—
300C4—28no—
301C4—24no—
302C4C23×#51812yes—
303C4C2×C23×#13118yes—
304C4C2×C23×H(5,𝔽2)11yescharacteristic
305C4C2×C23×H(5,𝔽2)11yescharacteristic
306C2—42no—
307C2—48no—
308C2C23×#297914yesdirect product factor
309C2C2×C23×#51811yescharacteristic
310C2C22×C23×#13112yes—
subgroupG/H
centerC22×C4×C23C24
commutatorC22C24×C4×C23
Frattini, cosocleC2×C4C25×C23
socleC23×C23C23×C4
prime radicalC2×C23×#532C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2979C2direct (N × H)
3. C2×C23×#436C2semidirect
4. C2×C8×D4×C23C2semidirect
5. C2×C23×#431C2semidirect
6. C2×C23×#408C2semidirect
18. C23×#436C22semidirect
19. C8×D4×C23C22semidirect
20. C23×#431C22semidirect
23. C23×#408C22semidirect
102. C2×#2979C23direct (N × H)
136. #2979C2×C23direct (N × H)
137. C2×#436C2×C23semidirect
138. C2×C8×D4C2×C23semidirect
139. C2×#431C2×C23semidirect
140. C2×#408C2×C23semidirect
168. #436C22×C23semidirect
169. C8×D4C22×C23semidirect
170. #431C22×C23semidirect
173. #408C22×C23semidirect
179. C2×C23#2979direct (N × H)
217. C23C2×#2979direct (N × H)
308. C2C23×#2979direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5322
C23×#29792
C2×C23×#4362
C2×C8×D4×C232
C2×C23×#4312
C2×C23×#4082
C2×#297923