Finite Group Databasebeta

← C23×#14301 #19161927

Subgroups — C23×#14301 #19161927

Order 5888.

1830 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q8×C23C211yesmaximal, characteristic, complemented
2C2×C23×#393C211yesmaximal, characteristic
3C2×C23×#388C213yesmaximal
4C23×#1446C211yesmaximal, characteristic, complemented
5C23×#1444C213yesmaximal, complemented
6C23×#1439C213yesmaximal
7C23×#1434C213yesmaximal, complemented
8C23×Q8×C23C2211yescharacteristic
9C2×C4×Q8×C23—212no—
10C22×C4⋊C4×C23C2213yes—
11C22×C4⋊C4×C23C2213yes—
12C22×C22⋊C4×C23C2211yescharacteristic
13C22×C22⋊C4×C23C2213yes—
14C22×C22⋊C4×C23C2213yes—
15C22×C42×C23C2213yes—
16C23×#393C418yescomplemented
17C23×#388—212no—
18C2×C23×#95C2213yes—
19C2×C23×#95C2213yes—
20C2×C23×#95C2213yes—
21C2×C23×#95C2216yes—
22C2×C23×#95C2213yes—
23C25×C23C2311yescharacteristic
24C22×Q8×C23—26no—
25C22×Q8×C23—26no—
26C22×Q8×C23C2×C414yes—
27C23×C4×C23C2313yes—
28C23×C4×C23C2311yescharacteristic
29C23×C4×C23C2313yes—
30C23×C4×C23C2313yes—
31C23×C4×C23C2313yes—
32C23×C4×C23C2311yescharacteristic
33C4×Q8×C23—232no—
34C2×C4⋊C4×C23—212no—
35C2×C4⋊C4×C23—26no—
36C2×C4⋊C4×C23—26no—
37C2×C4⋊C4×C23—46no—
38C2×C22⋊C4×C23—26no—
39C2×C22⋊C4×C23—26no—
40C2×C22⋊C4×C23—26no—
41C2×C22⋊C4×C23—26no—
42C2×C22⋊C4×C23—46no—
43C2×C22⋊C4×C23C2×C4112yes—
44C2×C42×C23—212no—
45C2×C42×C23D4112yes—
46C23×#95—26no—
47C23×#95—26no—
48C23×#95—26no—
49C23×#95—46no—
50C23×#95C2×C4112yes—
51C24×C23—23no—
52C24×C23—23no—
53C24×C23—23no—
54C24×C23—21no—
55C24×C23—21no—
56C24×C23—23no—
57C24×C23C22×C412yes—
58C24×C23C2411yescharacteristic
59C2×Q8×C23—28no—
60C2×Q8×C23—224no—
61C2×Q8×C23—224no—
62C22×C4×C23—26no—
63C22×C4×C23—26no—
64C22×C4×C23—21no—
65C22×C4×C23—23no—
66C22×C4×C23—26no—
67C22×C4×C23—23no—
68C22×C4×C23—23no—
69C22×C4×C23—26no—
70C22×C4×C23—26no—
71C22×C4×C23—23no—
72C22×C4×C23—26no—
73C22×C4×C23—23no—
74C22×C4×C23—23no—
75C22×C4×C23—23no—
76C22×C4×C23—26no—
77C22×C4×C23—26no—
78C22×C4×C23—23no—
79C22×C4×C23—23no—
80C22×C4×C23—23no—
81C22×C4×C23—21no—
82C22×C4×C23C22×C416yes—
83C22×C4×C23C22×C416yes—
84C22×C4×C23C2×D416yes—
85C22×C4×C23C2×D416yes—
86C22×C4×C23C2×D416yes—
87C22×C4×C23#4116yes—
88C4⋊C4×C23—224no—
89C4⋊C4×C23—412no—
90C22⋊C4×C23—224no—
91C22⋊C4×C23—412no—
92C22⋊C4×C23—44no—
93C42×C23—224no—
94#14301C2311yesmaximal, characteristic, direct product factor, Sylow
95C23×C23—26no—
96C23×C23—22no—
97C23×C23—26no—
98C23×C23—41no—
99C23×C23—43no—
100C23×C23—43no—
101C23×C23—43no—
102C23×C23—43no—
103C23×C23—43no—
104C23×C23—43no—
105C23×C23—43no—
106C23×C23—46no—
107C23×C23C23×C411yescharacteristic
108C23×C23C22×D413yes—
109C23×C23C2×#4113yes—
110C23×C23H(5,𝔽2)13yes—
111C23×C232−1+411yescharacteristic
112C23×C232−1+411yescharacteristic
113C23×C232−1+413yes—
114Q8×C23—232no—
115C2×C4×C23—212no—
116C2×C4×C23—26no—
117C2×C4×C23—26no—
118C2×C4×C23—212no—
119C2×C4×C23—26no—
120C2×C4×C23—26no—
121C2×C4×C23—26no—
122C2×C4×C23—212no—
123C2×C4×C23—26no—
124C2×C4×C23—26no—
125C2×C4×C23—26no—
126C2×C4×C23—43no—
127C2×C4×C23—46no—
128C2×C4×C23—43no—
129C2×C4×C23—46no—
130C2×C4×C23—46no—
131C2×C4×C23—41no—
132C2×C4×C23—43no—
133C2×C4×C23—43no—
134C2×C4×C23—43no—
135C2×C4×C23—43no—
136C2×C4×C23—43no—
137C2×C4×C23—46no—
138C2×C4×C23—46no—
139C2×C4×C23C4×D4112yes—
140C2×C4×C23#12014yes—
141C22×C4×Q8C2×C2311yescharacteristic, complemented
142C2×#393C2×C2311yescharacteristic
143C2×#388C2×C2313yes—
144#1446C2×C2311yescharacteristic, complemented
145#1444C2×C2313yescomplemented
146#1439C2×C2313yes—
147#1434C2×C2313yescomplemented
148C22×C23—46no—
149C22×C23—46no—
150C22×C23—42no—
151C22×C23—81no—
152C22×C23—81no—
153C22×C23—83no—
154C22×C23—83no—
155C22×C23C2×C4×D413yes—
156C22×C23#51911yescharacteristic
157C22×C23#52013yes—
158C22×C23C2×#12011yescharacteristic
159C22×C23#52513yes—
160C22×C23#53513yes—
161C22×C23#53613yes—
162C22×C23#53613yes—
163C22×C23#53713yes—
164C22×C23#54113yes—
165C22×C23#54213yes—
166C22×C23#54216yes—
167C4×C23—24no—
168C4×C23—212no—
169C4×C23—46no—
170C4×C23—46no—
171C4×C23—81no—
172C4×C23—83no—
173C23×Q8C22×C2311yescharacteristic
174C2×C4×Q8—212no—
175C22×C4⋊C4C22×C2313yes—
176C22×C4⋊C4C22×C2313yes—
177C22×C22⋊C4C22×C2311yescharacteristic
178C22×C22⋊C4C22×C2313yes—
179C22×C22⋊C4C22×C2313yes—
180C22×C42C22×C2313yes—
181#393C4×C2318yescomplemented
182#388—212no—
183C2×#95C22×C2313yes—
184C2×#95C22×C2313yes—
185C2×#95C22×C2313yes—
186C2×#95C22×C2316yes—
187C2×#95C22×C2313yes—
188C2×C23—82no—
189C2×C23C4×#12011yescharacteristic
190C2×C23#232313yes—
191C2×C23#232613yes—
192C2×C23#261411yescharacteristic
193C2×C23#261511yescharacteristic
194C2×C23#263213yes—
195C2×C23#263713yes—
196C25C23×C2311yescharacteristic
197C22×Q8—26no—
198C22×Q8—26no—
199C22×Q8C2×C4×C2314yes—
200C23×C4C23×C2311yescharacteristic
201C23×C4C23×C2313yes—
202C23×C4C23×C2313yes—
203C23×C4C23×C2313yes—
204C23×C4C23×C2313yes—
205C23×C4C23×C2311yescharacteristic
206C4×Q8—232no—
207C2×C4⋊C4—212no—
208C2×C4⋊C4—26no—
209C2×C4⋊C4—26no—
210C2×C4⋊C4—46no—
211C2×C22⋊C4—26no—
212C2×C22⋊C4—26no—
213C2×C22⋊C4—26no—
214C2×C22⋊C4—26no—
215C2×C22⋊C4—46no—
216C2×C22⋊C4C2×C4×C23112yes—
217C2×C42—212no—
218C2×C42D4×C23112yes—
219#95—26no—
220#95—26no—
221#95—26no—
222#95—46no—
223#95C2×C4×C23112yes—
224C23#1430111yescharacteristic, direct product factor, Sylow
225C24—23no—
226C24—23no—
227C24—23no—
228C24—21no—
229C24—21no—
230C24—23no—
231C24C22×C4×C2312yes—
232C24C24×C2311yescharacteristic
233C2×Q8—28no—
234C2×Q8—224no—
235C2×Q8—224no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—21no—
239C22×C4—23no—
240C22×C4—23no—
241C22×C4—26no—
242C22×C4—23no—
243C22×C4—23no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—23no—
253C22×C4—23no—
254C22×C4—23no—
255C22×C4—21no—
256C22×C4C22×C4×C2316yes—
257C22×C4C22×C4×C2316yes—
258C22×C4C2×D4×C2316yes—
259C22×C4C2×D4×C2316yes—
260C22×C4C2×D4×C2316yes—
261C22×C4#41×C2316yes—
262C4⋊C4—224no—
263C4⋊C4—412no—
264C22⋊C4—224no—
265C22⋊C4—412no—
266C22⋊C4—44no—
267C42—224no—
268C23—26no—
269C23—22no—
270C23—26no—
271C23—41no—
272C23—43no—
273C23—43no—
274C23—43no—
275C23—43no—
276C23—43no—
277C23—43no—
278C23—43no—
279C23—46no—
280C23C23×C4×C2311yescharacteristic
281C23C22×D4×C2313yes—
282C23C2×#41×C2313yes—
283C23C23×H(5,𝔽2)13yes—
284C23C23×2−1+411yescharacteristic
285C23C23×2−1+411yescharacteristic
286C23C23×2−1+413yes—
287Q8—232no—
288C2×C4—212no—
289C2×C4—26no—
290C2×C4—26no—
291C2×C4—212no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—26no—
295C2×C4—212no—
296C2×C4—26no—
297C2×C4—26no—
298C2×C4—26no—
299C2×C4—43no—
300C2×C4—46no—
301C2×C4—43no—
302C2×C4—46no—
303C2×C4—46no—
304C2×C4—41no—
305C2×C4—43no—
306C2×C4—43no—
307C2×C4—43no—
308C2×C4—43no—
309C2×C4—43no—
310C2×C4—46no—
311C2×C4—46no—
312C2×C4C4×D4×C23112yes—
313C2×C4C23×#12014yes—
314C22—42no—
315C22—46no—
316C22—46no—
317C22—81no—
318C22—81no—
319C22—83no—
320C22—83no—
321C22C2×C4×D4×C2313yes—
322C22C23×#51911yescharacteristic
323C22C23×#52013yes—
324C22C2×C23×#12011yescharacteristic
325C22C23×#52513yes—
326C22C23×#53513yes—
327C22C23×#53613yes—
328C22C23×#53613yes—
329C22C23×#53713yes—
330C22C23×#54113yes—
331C22C23×#54213yes—
332C22C23×#54216yes—
333C4—212no—
334C4—24no—
335C4—46no—
336C4—46no—
337C4—81no—
338C4—83no—
339C2—82no—
340C2C4×C23×#12011yescharacteristic
341C2C23×#232313yes—
342C2C23×#232613yes—
343C2C23×#261411yescharacteristic
344C2C23×#261511yescharacteristic
345C2C23×#263213yes—
346C2C23×#263713yes—
subgroupG/H
center, socleC24×C23C24
commutatorC23C23×C4×C23
Frattini, cosocleC24C24×C23
prime radicalC25×C23C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C4×Q8×C23C2semidirect
4. C23×#1446C2semidirect
5. C23×#1444C2semidirect
7. C23×#1434C2semidirect
16. C23×#393C4semidirect
94. #14301C23direct (N × H)
141. C22×C4×Q8C2×C23semidirect
144. #1446C2×C23semidirect
145. #1444C2×C23semidirect
147. #1434C2×C23semidirect
181. #393C4×C23semidirect
224. C23#14301direct (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×C232
C2×C23×#3932
C2×C23×#3882
C23×#14462
C23×#14442
C23×#14392
C23×#14342
#1430123