Finite Group Databasebeta

← C23×#9982 #19157608

Subgroups — C23×#9982 #19157608

Order 5888.

722 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C23×#3007C212yesmaximal
2C23×#2817C211yesmaximal, characteristic, complemented
3C23×#1782C211yesmaximal, characteristic
4C23×#1511C212yesmaximal, complemented
5C23×#1306C211yesmaximal, characteristic, complemented
6C22×#41×C23C2211yescharacteristic
7C2×C23×#137—22no—
8C2×C23×C22⋊Q8C2212yes—
9C23×#452—24no—
10C23×#451—22no—
11C23×#450—22no—
12C23×#417—22no—
13C2×C23×#99—22no—
14C23×#407C2212yes—
15C23×#400—22no—
16C23×#397—22no—
17C23×#395—22no—
18C23×#381C2211yescharacteristic
19C23×#381C2211yescharacteristic
20C23×#328—24no—
21C2×#41×C23—22no—
22C2×#41×C23—22no—
23C2×#41×C23—44no—
24C22×Q8×C23D412yes—
25C22×Q8×C23D411yescharacteristic
26C22×D4×C23—21no—
27C22×D4×C23D411yescharacteristic
28C23×C4×C23—21no—
29C23×C4×C23C2311yescharacteristic
30C23×#137—48no—
31C2×Q16×C23—24no—
32C2×SD16×C23—24no—
33C2×M16×C23—22no—
34C23×C22⋊Q8—22no—
35C23×C22⋊Q8—22no—
36C23×C22⋊Q8—22no—
37C23×C22⋊Q8—22no—
38C2×C4⋊C4×C23—22no—
39C2×C4⋊C4×C23—22no—
40C2×C4⋊C4×C23D412yes—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23D412yes—
44C23×#103—24no—
45C23×#102—24no—
46C23×#99—44no—
47C23×C22⋊C8—22no—
48C23×C22⋊C8D414yes—
49C23×#95—24no—
50C23×#95—22no—
51C23×#95—22no—
52C23×#95—22no—
53C24×C23—21no—
54C24×C23C2×D411yescharacteristic
55#41×C23—28no—
56#41×C23—44no—
57#41×C23—44no—
58#41×C23—44no—
59C2×Q8×C23—22no—
60C2×Q8×C23—22no—
61C2×Q8×C23—22no—
62C2×Q8×C23—42no—
63C2×Q8×C23—44no—
64C2×D4×C23—24no—
65C2×D4×C23—22no—
66C2×D4×C23—42no—
67C2×D4×C23—42no—
68C2×D4×C23—41no—
69C2×D4×C23—41no—
70C22×C4×C23—21no—
71C22×C4×C23—22no—
72C22×C4×C23—22no—
73C22×C4×C23—22no—
74C22×C4×C23—42no—
75C22×C4×C23—42no—
76C22×C4×C23—42no—
77C22×C4×C23—42no—
78C22×C4×C23—41no—
79C22×C4×C23C2×D412yes—
80C22×C4×C23C2×D411yescharacteristic
81C22×C4×C23C2×D412yes—
82C22×C4×C23#4111yescharacteristic
83Q16×C23—48no—
84SD16×C23—48no—
85M16×C23—44no—
86C2×C8×C23—24no—
87C4⋊C4×C23—22no—
88C4⋊C4×C23—22no—
89C4⋊C4×C23—42no—
90C4⋊C4×C23—42no—
91C4⋊C4×C23—82no—
92C22⋊C4×C23—42no—
93C22⋊C4×C23—42no—
94C22⋊C4×C23—42no—
95C22⋊C4×C23—42no—
96#9982C2311yesmaximal, characteristic, direct product factor, Sylow
97C23×C23—22no—
98C23×C23—22no—
99C23×C23—42no—
100C23×C23—41no—
101C23×C23—41no—
102C23×C23—82no—
103C23×C23#12011yescharacteristic
104C23×C23#12111yescharacteristic
105C23×C23#12711yescharacteristic
106Q8×C23—24no—
107Q8×C23—42no—
108Q8×C23—44no—
109Q8×C23—82no—
110D4×C23—24no—
111D4×C23—42no—
112D4×C23—44no—
113D4×C23—44no—
114C2×C4×C23—24no—
115C2×C4×C23—22no—
116C2×C4×C23—41no—
117C2×C4×C23—41no—
118C2×C4×C23—42no—
119C2×C4×C23—48no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23—44no—
123C2×C4×C23—44no—
124C2×C4×C23—42no—
125C2×C4×C23—42no—
126C2×C4×C23—42no—
127C2×C4×C23—44no—
128C2×C4×C23—81no—
129C2×C4×C23—82no—
130C2×C4×C23#12012yes—
131C2×C4×C23#12112yes—
132C8×C23—44no—
133#3007C2×C2312yes—
134#2817C2×C2311yescharacteristic, complemented
135#1782C2×C2311yescharacteristic
136#1511C2×C2312yescomplemented
137#1306C2×C2311yescharacteristic, complemented
138C22×C23—22no—
139C22×C23—22no—
140C22×C23—22no—
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—42no—
144C22×C23—82no—
145C22×C23—82no—
146C22×C23—82no—
147C22×C23—82no—
148C22×C23#39211yescharacteristic
149C22×C23B(4,𝔽2)12yes—
150C4×C23—22no—
151C4×C23—22no—
152C4×C23—42no—
153C4×C23—82no—
154C4×C23—82no—
155C4×C23—82no—
156C4×C23—161no—
157C22×#41C22×C2311yescharacteristic
158C2×#137—22no—
159C2×C22⋊Q8C22×C2312yes—
160#452—24no—
161#451—22no—
162#450—22no—
163#417—22no—
164C2×#99—22no—
165#407C22×C2312yes—
166#400—22no—
167#397—22no—
168#395—22no—
169#381C22×C2311yescharacteristic
170#381C22×C2311yescharacteristic
171#328—24no—
172C2×C23—22no—
173C2×C23—41no—
174C2×C23—41no—
175C2×C23—82no—
176C2×C23#200912yes—
177C2×C23#202111yescharacteristic
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—44no—
181C22×Q8D4×C2312yes—
182C22×Q8D4×C2311yescharacteristic
183C22×D4—21no—
184C22×D4D4×C2311yescharacteristic
185C23×C4—21no—
186C23×C4C23×C2311yescharacteristic
187#137—48no—
188C2×Q16—24no—
189C2×SD16—24no—
190C2×M16—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4D4×C2312yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4D4×C2312yes—
201#103—24no—
202#102—24no—
203#99—44no—
204C22⋊C8—22no—
205C22⋊C8D4×C2314yes—
206#95—24no—
207#95—22no—
208#95—22no—
209#95—22no—
210C23#998211yescharacteristic, direct product factor, Sylow
211C24—21no—
212C24C2×D4×C2311yescharacteristic
213#41—28no—
214#41—44no—
215#41—44no—
216#41—44no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—41no—
227C2×D4—41no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4C2×D4×C2312yes—
238C22×C4C2×D4×C2311yescharacteristic
239C22×C4C2×D4×C2312yes—
240C22×C4#41×C2311yescharacteristic
241Q16—48no—
242SD16—48no—
243M16—44no—
244C2×C8—24no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—82no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23—82no—
260C23C23×#12011yescharacteristic
261C23C23×#12111yescharacteristic
262C23C23×#12711yescharacteristic
263Q8—24no—
264Q8—42no—
265Q8—44no—
266Q8—82no—
267D4—24no—
268D4—42no—
269D4—44no—
270D4—44no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—48no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4C23×#12012yes—
288C2×C4C23×#12112yes—
289C8—44no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—21no—
294C22—21no—
295C22—42no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22C23×#39211yescharacteristic
301C22C23×B(4,𝔽2)12yes—
302C4—22no—
303C4—22no—
304C4—42no—
305C4—82no—
306C4—82no—
307C4—82no—
308C4—161no—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—82no—
313C2C23×#200912yes—
314C2C23×#202111yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC22×#41×C23C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2817C2semidirect
4. C23×#1511C2semidirect
5. C23×#1306C2semidirect
96. #9982C23direct (N × H)
134. #2817C2×C23semidirect
136. #1511C2×C23semidirect
137. #1306C2×C23semidirect
210. C23#9982direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30072
C23×#28172
C23×#17822
C23×#15112
C23×#13062
#998223