Finite Group Databasebeta

← C23×#11803 #19159429

Subgroups — C23×#11803 #19159429

Order 5888.

478 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#398C211yesmaximal, characteristic
2C23×#1821C211yesmaximal, characteristic
3C23×#1463C211yesmaximal, characteristic
4C23×#1346C214yesmaximal, complemented
5C22×C4⋊C4×C23—21no—
6C22×C4⋊C4×C23C411yescharacteristic
7C22×C4⋊C4×C23C2211yescharacteristic
8C23×#407—21no—
9C23×#407—21no—
10C23×#398—21no—
11C23×#398—21no—
12C23×#398C412yes—
13C23×#398C2212yes—
14C2×C23×#95—21no—
15C2×C23×#95C411yescharacteristic
16C23×#343—21no—
17C23×#343C2212yes—
18C23×#324—21no—
19C23×#324C2212yes—
20C23×C4×C23—21no—
21C23×C4×C23—21no—
22C23×C4×C23C2×C411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23D411yescharacteristic
25C2×M16×C23—22no—
26C2×M16×C23—41no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—41no—
31C2×C4⋊C4×C23—41no—
32C2×C4⋊C4×C23—41no—
33C2×C4⋊C4×C23—41no—
34C2×C4⋊C4×C23—44no—
35C2×C4⋊C4×C23C2×C412yes—
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C4⋊C4×C23D411yescharacteristic
38C2×C4⋊C4×C23D411yescharacteristic
39C2×C4⋊C4×C23C2311yescharacteristic
40C23×C22⋊C8—22no—
41C23×C22⋊C8—41no—
42C23×C22⋊C8—41no—
43C23×C22⋊C8—41no—
44C23×#95—22no—
45C23×#95—21no—
46C23×#95—22no—
47C23×#95C2×C412yes—
48C24×C23C22⋊C411yescharacteristic
49C22×C4×C23—24no—
50C22×C4×C23—21no—
51C22×C4×C23—21no—
52C22×C4×C23—21no—
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23—22no—
56C22×C4×C23—21no—
57C22×C4×C23—22no—
58C22×C4×C23—22no—
59C22×C4×C23—41no—
60C22×C4×C23—42no—
61C22×C4×C23—42no—
62C22×C4×C23—42no—
63C22×C4×C23—42no—
64C22×C4×C23—42no—
65C22×C4×C23—42no—
66C22×C4×C23—41no—
67C22×C4×C23—41no—
68C22×C4×C23C22⋊C411yescharacteristic
69C22×C4×C23C22⋊C411yescharacteristic
70C22×C4×C23C22⋊C411yescharacteristic
71C22×C4×C23C22×C411yescharacteristic
72C22×C4×C23C2×D411yescharacteristic
73C22×C4×C23C2×D411yescharacteristic
74M16×C23—42no—
75M16×C23—42no—
76M16×C23—82no—
77C2×C8×C23—42no—
78C2×C8×C23—42no—
79C4⋊C4×C23—41no—
80C4⋊C4×C23—41no—
81C4⋊C4×C23—41no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23—42no—
84C4⋊C4×C23—42no—
85C4⋊C4×C23—44no—
86C4⋊C4×C23—44no—
87#11803C2311yesmaximal, characteristic, direct product factor, Sylow
88C23×C23—22no—
89C23×C23—22no—
90C23×C23—21no—
91C23×C23—21no—
92C23×C23#9911yescharacteristic
93C23×C23#9911yescharacteristic
94C23×C23C2×C22⋊C411yescharacteristic
95C2×C4×C23—21no—
96C2×C4×C23—21no—
97C2×C4×C23—24no—
98C2×C4×C23—21no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—48no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—42no—
113C2×C4×C23—41no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—41no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—81no—
121C2×C4×C23—82no—
122C2×C4×C23—82no—
123C2×C4×C23AGL(1,ℤ/8)12yes—
124C8×C23—82no—
125C8×C23—82no—
126C2×#398C2×C2311yescharacteristic
127#1821C2×C2311yescharacteristic
128#1463C2×C2311yescharacteristic
129#1346C2×C2314yescomplemented
130C22×C23—24no—
131C22×C23—24no—
132C22×C23—21no—
133C22×C23—21no—
134C22×C23—21no—
135C22×C23—21no—
136C22×C23—21no—
137C22×C23—21no—
138C22×C23#35212yes—
139C22×C23#35612yes—
140C22×C23C2×#9911yescharacteristic
141C22×C23#41811yescharacteristic
142C22×C23#42111yescharacteristic
143C4×C23—22no—
144C4×C23—44no—
145C4×C23—41no—
146C4×C23—41no—
147C4×C23—41no—
148C4×C23—41no—
149C4×C23—41no—
150C4×C23—82no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4C4×C2311yescharacteristic
153C22×C4⋊C4C22×C2311yescharacteristic
154#407—21no—
155#407—21no—
156#398—21no—
157#398—21no—
158#398C4×C2312yes—
159#398C22×C2312yes—
160C2×#95—21no—
161C2×#95C4×C2311yescharacteristic
162#343—21no—
163#343C22×C2312yes—
164#324—21no—
165#324C22×C2312yes—
166C2×C23—21no—
167C2×C23—22no—
168C2×C23—22no—
169C2×C23—21no—
170C2×C23#151011yescharacteristic
171C2×C23C2×#35211yescharacteristic
172C2×C23C2×#35611yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4C2×C4×C2311yescharacteristic
176C23×C4D4×C2311yescharacteristic
177C23×C4D4×C2311yescharacteristic
178C2×M16—22no—
179C2×M16—41no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4C2×C4×C2312yes—
189C2×C4⋊C4C2×C4×C2311yescharacteristic
190C2×C4⋊C4D4×C2311yescharacteristic
191C2×C4⋊C4D4×C2311yescharacteristic
192C2×C4⋊C4C23×C2311yescharacteristic
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#95—22no—
198#95—21no—
199#95—22no—
200#95C2×C4×C2312yes—
201C23#1180311yescharacteristic, direct product factor, Sylow
202C24C22⋊C4×C2311yescharacteristic
203C22×C4—24no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4C22⋊C4×C2311yescharacteristic
223C22×C4C22⋊C4×C2311yescharacteristic
224C22×C4C22⋊C4×C2311yescharacteristic
225C22×C4C22×C4×C2311yescharacteristic
226C22×C4C2×D4×C2311yescharacteristic
227C22×C4C2×D4×C2311yescharacteristic
228M16—42no—
229M16—42no—
230M16—82no—
231C2×C8—42no—
232C2×C8—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23C23×#9911yescharacteristic
246C23C23×#9911yescharacteristic
247C23C2×C22⋊C4×C2311yescharacteristic
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4C23×AGL(1,ℤ/8)12yes—
277C8—82no—
278C8—82no—
279C22—24no—
280C22—24no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22C23×#35212yes—
288C22C23×#35612yes—
289C22C2×C23×#9911yescharacteristic
290C22C23×#41811yescharacteristic
291C22C23×#42111yescharacteristic
292C4—22no—
293C4—44no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C2—21no—
301C2—22no—
302C2—22no—
303C2—21no—
304C2C23×#151011yescharacteristic
305C2C2×C23×#35211yescharacteristic
306C2C2×C23×#35611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C22×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1346C2semidirect
87. #11803C23direct (N × H)
129. #1346C2×C23semidirect
201. C23#11803direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3982
C23×#18212
C23×#14632
C23×#13462
#1180323