Finite Group Databasebeta

← C23×#9613 #19157239

Subgroups — C23×#9613 #19157239

Order 5888.

366 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
1C23×#1948C211yesmaximal, characteristic
2C23×#1947C211yesmaximal, characteristic
3C23×#1834C211yesmaximal, characteristic
4C8×C4⋊C4×C23C211yesmaximal, characteristic
5C2×C23×#334C211yesmaximal, characteristic
6C23×#1296C211yesmaximal, characteristic
7C23×#1296C211yesmaximal, characteristic
8C2×C23×#107—21no—
9C2×C23×#106—21no—
10C2×C23×C4⋊C8C2211yescharacteristic
11C2×C23×C4⋊C8C2211yescharacteristic
12C2×C23×C4⋊C8C2211yescharacteristic
13C2×C4×C8×C23—21no—
14C2×C4×C8×C23C2211yescharacteristic
15C23×#384C2211yescharacteristic
16C23×#384C2211yescharacteristic
17C4×C4⋊C4×C23C2211yescharacteristic
18C23×#340—21no—
19C23×#340—21no—
20C23×#340—21no—
21C23×#340—21no—
22C23×#336—21no—
23C23×#334C414yescomplemented
24C22×C8×C23—21no—
25C22×C8×C23—21no—
26C22×C8×C23—21no—
27C22×C8×C23D411yescharacteristic
28C22×C8×C23D411yescharacteristic
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23D411yescharacteristic
36C2×C4⋊C4×C23D411yescharacteristic
37C2×C42×C23—21no—
38C2×C42×C23C2311yescharacteristic
39C23×#107—42no—
40C23×#106—42no—
41C23×C4⋊C8—21no—
42C23×C4⋊C8—21no—
43C23×C4⋊C8—21no—
44C23×C4⋊C8C2×C412yes—
45C23×C4⋊C8C2×C412yes—
46C23×C4⋊C8Q812yes—
47C4×C8×C23—42no—
48C4×C8×C23C2×C412yes—
49C4×C8×C23Q812yes—
50C23×#95—21no—
51C22×C4×C23—21no—
52C22×C4×C23—21no—
53C22×C4×C23—41no—
54C22×C4×C23—41no—
55C22×C4×C23C2×D411yescharacteristic
56C22×C4×C23C2×D411yescharacteristic
57C22×C4×C23#4111yescharacteristic
58C2×C8×C23—21no—
59C2×C8×C23—21no—
60C2×C8×C23—21no—
61C2×C8×C23—21no—
62C2×C8×C23—22no—
63C2×C8×C23—22no—
64C2×C8×C23—22no—
65C2×C8×C23—41no—
66C2×C8×C23—41no—
67C2×C8×C23—41no—
68C2×C8×C23—41no—
69C2×C8×C23—41no—
70C2×C8×C23—41no—
71C2×C8×C23C4⋊C412yes—
72C2×C8×C23C4⋊C412yes—
73C4⋊C4×C23—21no—
74C4⋊C4×C23—21no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—41no—
79C4⋊C4×C23—41no—
80C4⋊C4×C23—41no—
81C4⋊C4×C23—41no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23—81no—
84C4⋊C4×C23—81no—
85C4⋊C4×C23SD1612yes—
86C4⋊C4×C23SD1612yes—
87C42×C23—41no—
88C42×C23—41no—
89C42×C23C22×C411yescharacteristic
90C42×C23C2×Q811yescharacteristic
91C42×C23C2×Q811yescharacteristic
92C42×C23#4111yescharacteristic
93#9613C2311yesmaximal, characteristic, direct product factor, Sylow
94C23×C23#12111yescharacteristic
95C2×C4×C23—21no—
96C2×C4×C23—21no—
97C2×C4×C23—22no—
98C2×C4×C23—22no—
99C2×C4×C23—41no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—81no—
104C2×C4×C23—81no—
105C2×C4×C23—81no—
106C2×C4×C23—81no—
107C2×C4×C23—81no—
108C2×C4×C23—81no—
109C2×C4×C23C2×C4⋊C411yescharacteristic
110C2×C4×C23C4×D411yescharacteristic
111C2×C4×C23C4×Q811yescharacteristic
112C2×C4×C23C22⋊Q811yescharacteristic
113C2×C4×C23#12511yescharacteristic
114C2×C4×C23C4⋊Q811yescharacteristic
115C2×C4×C23C2×SD1611yescharacteristic
116C2×C4×C23C2×SD1611yescharacteristic
117C2×C4×C23#13511yescharacteristic
118C2×C4×C23#13511yescharacteristic
119C2×C4×C23#13511yescharacteristic
120C2×C4×C23#13511yescharacteristic
121C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C23#13711yescharacteristic
123C8×C23—22no—
124C8×C23—22no—
125C8×C23—42no—
126C8×C23—42no—
127C8×C23—42no—
128#1948C2×C2311yescharacteristic
129#1947C2×C2311yescharacteristic
130#1834C2×C2311yescharacteristic
131C8×C4⋊C4C2×C2311yescharacteristic
132C2×#334C2×C2311yescharacteristic
133#1296C2×C2311yescharacteristic
134#1296C2×C2311yescharacteristic
135C22×C23#38911yescharacteristic
136C22×C23#46311yescharacteristic
137C22×C23#46411yescharacteristic
138C22×C23#46511yescharacteristic
139C22×C23#46511yescharacteristic
140C22×C23#47211yescharacteristic
141C22×C23#47311yescharacteristic
142C4×C23—21no—
143C4×C23—21no—
144C4×C23—42no—
145C4×C23—42no—
146C4×C23—161no—
147C4×C23—161no—
148C4×C23#42711yescharacteristic
149C4×C23#42811yescharacteristic
150C4×C23C4×SD1611yescharacteristic
151C4×C23C4×SD1611yescharacteristic
152C4×C23#47811yescharacteristic
153C4×C23#47911yescharacteristic
154C4×C23#49811yescharacteristic
155C4×C23#49911yescharacteristic
156C2×#107—21no—
157C2×#106—21no—
158C2×C4⋊C8C22×C2311yescharacteristic
159C2×C4⋊C8C22×C2311yescharacteristic
160C2×C4⋊C8C22×C2311yescharacteristic
161C2×C4×C8—21no—
162C2×C4×C8C22×C2311yescharacteristic
163#384C22×C2311yescharacteristic
164#384C22×C2311yescharacteristic
165C4×C4⋊C4C22×C2311yescharacteristic
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#336—21no—
171#334C4×C2314yescomplemented
172C2×C23#167711yescharacteristic
173C2×C23#167811yescharacteristic
174C2×C23#167911yescharacteristic
175C2×C23#168011yescharacteristic
176C2×C23#194111yescharacteristic
177C2×C23#194911yescharacteristic
178C2×C23#195211yescharacteristic
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8D4×C2311yescharacteristic
183C22×C8D4×C2311yescharacteristic
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4D4×C2311yescharacteristic
191C2×C4⋊C4D4×C2311yescharacteristic
192C2×C42—21no—
193C2×C42C23×C2311yescharacteristic
194#107—42no—
195#106—42no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8C2×C4×C2312yes—
200C4⋊C8C2×C4×C2312yes—
201C4⋊C8Q8×C2312yes—
202C4×C8—42no—
203C4×C8C2×C4×C2312yes—
204C4×C8Q8×C2312yes—
205#95—21no—
206C23#961311yescharacteristic, direct product factor, Sylow
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4C2×D4×C2311yescharacteristic
212C22×C4C2×D4×C2311yescharacteristic
213C22×C4#41×C2311yescharacteristic
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8C4⋊C4×C2312yes—
228C2×C8C4⋊C4×C2312yes—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4SD16×C2312yes—
242C4⋊C4SD16×C2312yes—
243C42—41no—
244C42—41no—
245C42C22×C4×C2311yescharacteristic
246C42C2×Q8×C2311yescharacteristic
247C42C2×Q8×C2311yescharacteristic
248C42#41×C2311yescharacteristic
249C23C23×#12111yescharacteristic
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4C2×C4⋊C4×C2311yescharacteristic
265C2×C4C4×D4×C2311yescharacteristic
266C2×C4C4×Q8×C2311yescharacteristic
267C2×C4C23×C22⋊Q811yescharacteristic
268C2×C4C23×#12511yescharacteristic
269C2×C4C23×C4⋊Q811yescharacteristic
270C2×C4C2×SD16×C2311yescharacteristic
271C2×C4C2×SD16×C2311yescharacteristic
272C2×C4C23×#13511yescharacteristic
273C2×C4C23×#13511yescharacteristic
274C2×C4C23×#13511yescharacteristic
275C2×C4C23×#13511yescharacteristic
276C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
277C2×C4C23×#13711yescharacteristic
278C8—22no—
279C8—22no—
280C8—42no—
281C8—42no—
282C8—42no—
283C22C23×#38911yescharacteristic
284C22C23×#46311yescharacteristic
285C22C23×#46411yescharacteristic
286C22C23×#46511yescharacteristic
287C22C23×#46511yescharacteristic
288C22C23×#47211yescharacteristic
289C22C23×#47311yescharacteristic
290C4—21no—
291C4—21no—
292C4—42no—
293C4—42no—
294C4—161no—
295C4—161no—
296C4C23×#42711yescharacteristic
297C4C23×#42811yescharacteristic
298C4C4×SD16×C2311yescharacteristic
299C4C4×SD16×C2311yescharacteristic
300C4C23×#47811yescharacteristic
301C4C23×#47911yescharacteristic
302C4C23×#49811yescharacteristic
303C4C23×#49911yescharacteristic
304C2C23×#167711yescharacteristic
305C2C23×#167811yescharacteristic
306C2C23×#167911yescharacteristic
307C2C23×#168011yescharacteristic
308C2C23×#194111yescharacteristic
309C2C23×#194911yescharacteristic
310C2C23×#195211yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
23. C23×#334C4semidirect
93. #9613C23direct (N × H)
171. #334C4×C23semidirect
206. C23#9613direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#19482
C23×#19472
C23×#18342
C8×C4⋊C4×C232
C2×C23×#3342
C23×#12962
#961323