Finite Group Databasebeta

← C23×#8554 #19156180

Subgroups — C23×#8554 #19156180

Order 5888.

778 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×C23×#125C211yesmaximal, characteristic
2C23×#1821C212yesmaximal
3C23×#1463C212yesmaximal
4C23×#1463C212yesmaximal
5C2×C23×#125—24no—
6C2×C23×#125—24no—
7C2×C23×#125—24no—
8C22×C4⋊C4×C23C411yescharacteristic
9C22×C4⋊C4×C23C411yescharacteristic
10C22×C4⋊C4×C23C411yescharacteristic
11C22×C4⋊C4×C23C2211yescharacteristic
12C22×C4⋊C4×C23C2211yescharacteristic
13C22×C4⋊C4×C23C2211yescharacteristic
14C22×C42×C23C411yescharacteristic
15C23×#407C2214yes—
16C23×#343—24no—
17C23×#324—24no—
18C23×#324—24no—
19C23×C4×C23C2×C411yescharacteristic
20C23×C4×C23C2×C411yescharacteristic
21C23×C4×C23C2×C411yescharacteristic
22C23×C4×C23C2×C411yescharacteristic
23C23×C4×C23C2×C411yescharacteristic
24C23×C4×C23C2×C411yescharacteristic
25C23×C4×C23C2311yescharacteristic
26C2×M16×C23—24no—
27C23×#125—232no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23—22no—
36C2×C4⋊C4×C23—22no—
37C2×C4⋊C4×C23—22no—
38C2×C4⋊C4×C23—22no—
39C2×C4⋊C4×C23—22no—
40C2×C4⋊C4×C23—22no—
41C2×C4⋊C4×C23—22no—
42C2×C4⋊C4×C23—22no—
43C2×C4⋊C4×C23D414yes—
44C2×C4⋊C4×C23D414yes—
45C2×C4⋊C4×C23D414yes—
46C2×C42×C23—22no—
47C2×C42×C23—22no—
48C2×C42×C23—22no—
49C23×C22⋊C8—24no—
50C23×C22⋊C8—24no—
51C24×C23C22×C411yescharacteristic
52C22×C4×C23—21no—
53C22×C4×C23—21no—
54C22×C4×C23—24no—
55C22×C4×C23—21no—
56C22×C4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—42no—
66C22×C4×C23—42no—
67C22×C4×C23—42no—
68C22×C4×C23—42no—
69C22×C4×C23—42no—
70C22×C4×C23—42no—
71C22×C4×C23C22⋊C412yes—
72C22×C4×C23C22⋊C412yes—
73C22×C4×C23C22⋊C412yes—
74C22×C4×C23C22⋊C412yes—
75C22×C4×C23C22⋊C412yes—
76C22×C4×C23C22⋊C412yes—
77C22×C4×C23C2×D412yes—
78C22×C4×C23C2×D412yes—
79C22×C4×C23C2×D412yes—
80M16×C23—48no—
81C2×C8×C23—44no—
82C4⋊C4×C23—28no—
83C4⋊C4×C23—28no—
84C4⋊C4×C23—28no—
85C4⋊C4×C23—28no—
86C4⋊C4×C23—28no—
87C4⋊C4×C23—28no—
88C42×C23—28no—
89#8554C2311yesmaximal, characteristic, direct product factor, Sylow
90C23×C23—22no—
91C23×C23—22no—
92C23×C23—22no—
93C23×C23C2×C22⋊C411yescharacteristic
94C23×C23C2×C22⋊C411yescharacteristic
95C23×C23C2×C22⋊C411yescharacteristic
96C2×C4×C23—24no—
97C2×C4×C23—24no—
98C2×C4×C23—22no—
99C2×C4×C23—24no—
100C2×C4×C23—22no—
101C2×C4×C23—22no—
102C2×C4×C23—22no—
103C2×C4×C23—22no—
104C2×C4×C23—22no—
105C2×C4×C23—22no—
106C2×C4×C23—24no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—42no—
123C2×C4×C23#10114yes—
124C2×C4×C23#12014yes—
125C8×C23—84no—
126C22×#125C2×C2311yescharacteristic
127#1821C2×C2312yes—
128#1463C2×C2312yes—
129#1463C2×C2312yes—
130C22×C23—28no—
131C22×C23—21no—
132C22×C23—21no—
133C22×C23—21no—
134C22×C23—21no—
135C22×C23—21no—
136C22×C23—21no—
137C22×C23#37911yescharacteristic
138C22×C23#41012yes—
139C22×C23C2×#10112yes—
140C22×C23#41312yes—
141C4×C23—24no—
142C4×C23—24no—
143C4×C23—42no—
144C4×C23—42no—
145C4×C23—42no—
146C4×C23—42no—
147C4×C23—42no—
148C2×#125—24no—
149C2×#125—24no—
150C2×#125—24no—
151C22×C4⋊C4C4×C2311yescharacteristic
152C22×C4⋊C4C4×C2311yescharacteristic
153C22×C4⋊C4C4×C2311yescharacteristic
154C22×C4⋊C4C22×C2311yescharacteristic
155C22×C4⋊C4C22×C2311yescharacteristic
156C22×C4⋊C4C22×C2311yescharacteristic
157C22×C42C4×C2311yescharacteristic
158#407C22×C2314yes—
159#343—24no—
160#324—24no—
161#324—24no—
162C2×C23—22no—
163C2×C23—22no—
164C2×C23—22no—
165C2×C23#178211yescharacteristic
166C2×C23#178411yescharacteristic
167C2×C23#178511yescharacteristic
168C23×C4C2×C4×C2311yescharacteristic
169C23×C4C2×C4×C2311yescharacteristic
170C23×C4C2×C4×C2311yescharacteristic
171C23×C4C2×C4×C2311yescharacteristic
172C23×C4C2×C4×C2311yescharacteristic
173C23×C4C2×C4×C2311yescharacteristic
174C23×C4C23×C2311yescharacteristic
175C2×M16—24no—
176#125—232no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4D4×C2314yes—
193C2×C4⋊C4D4×C2314yes—
194C2×C4⋊C4D4×C2314yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C22⋊C8—24no—
199C22⋊C8—24no—
200C23#855411yescharacteristic, direct product factor, Sylow
201C24C22×C4×C2311yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4C22⋊C4×C2312yes—
222C22×C4C22⋊C4×C2312yes—
223C22×C4C22⋊C4×C2312yes—
224C22×C4C22⋊C4×C2312yes—
225C22×C4C22⋊C4×C2312yes—
226C22×C4C22⋊C4×C2312yes—
227C22×C4C2×D4×C2312yes—
228C22×C4C2×D4×C2312yes—
229C22×C4C2×D4×C2312yes—
230M16—48no—
231C2×C8—44no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C42—28no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23C2×C22⋊C4×C2311yescharacteristic
243C23C2×C22⋊C4×C2311yescharacteristic
244C23C2×C22⋊C4×C2311yescharacteristic
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4C23×#10114yes—
273C2×C4C23×#12014yes—
274C8—84no—
275C22—28no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22C23×#37911yescharacteristic
283C22C23×#41012yes—
284C22C2×C23×#10112yes—
285C22C23×#41312yes—
286C4—24no—
287C4—24no—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—42no—
292C4—42no—
293C2—22no—
294C2—22no—
295C2—22no—
296C2C23×#178211yescharacteristic
297C2C23×#178411yescharacteristic
298C2C23×#178511yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
89. #8554C23direct (N × H)
200. C23#8554direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×#1252
C23×#18212
C23×#14632
#855423