Finite Group Databasebeta

← C23×#7751 #19155377

Subgroups — C23×#7751 #19155377

Order 5888.

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

normal subgroups only

This group has 492 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×#2724C211yesmaximal, characteristic
2C23×#2057C212yesmaximal, complemented
3C23×#2013C211yesmaximal, characteristic
4C23×#1271C212yesmaximal, complemented
5C23×#1270C211yesmaximal, characteristic
6C22×SD16×C23—21no—
7C2×C23×C4⋊Q8C2211yescharacteristic
8C2×C23×#123—21no—
9C2×C23×#107—22no—
10C2×C23×#103—21no—
11C23×#398—21no—
12C23×#397—22no—
13C23×#390C2211yescharacteristic
14C23×#389C2212yes—
15C23×#385—22no—
16C23×#340—22no—
17C23×#336C2211yescharacteristic
18C23×#336C2212yes—
19C22×Q8×C23—21no—
20C22×D4×C23D411yescharacteristic
21C23×C4×C23—21no—
22C2×SD16×C23—42no—
23C2×SD16×C23—42no—
24C2×SD16×C23—42no—
25C22×C8×C23—21no—
26C22×C8×C23—22no—
27C23×C4⋊Q8—42no—
28C23×#123—42no—
29C23×#123—42no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23D412yes—
34C2×C4⋊C4×C23D412yes—
35C2×C4⋊C4×C23D411yescharacteristic
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—22no—
39C2×C42×C23C2311yescharacteristic
40C23×#107—44no—
41C23×#103—42no—
42C23×#95—22no—
43C23×#95—21no—
44C23×#95—22no—
45C24×C23—21no—
46C2×Q8×C23—22no—
47C2×Q8×C23—41no—
48C2×Q8×C23—41no—
49C2×Q8×C23—41no—
50C2×Q8×C23—41no—
51C2×Q8×C23—41no—
52C2×D4×C23—21no—
53C2×D4×C23—41no—
54C2×D4×C23—41no—
55C2×D4×C23D812yes—
56C22×C4×C23—22no—
57C22×C4×C23—21no—
58C22×C4×C23—22no—
59C22×C4×C23—42no—
60C22×C4×C23—41no—
61C22×C4×C23—81no—
62C22×C4×C23—81no—
63C22×C4×C23C2×D411yescharacteristic
64C22×C4×C23C2×D412yes—
65SD16×C23—48no—
66C2×C8×C23—22no—
67C2×C8×C23—24no—
68C2×C8×C23—42no—
69C2×C8×C23—41no—
70C2×C8×C23—41no—
71C2×C8×C23—42no—
72C4⋊C4×C23—24no—
73C4⋊C4×C23—22no—
74C4⋊C4×C23—22no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—42no—
77C4⋊C4×C23—42no—
78C4⋊C4×C23—42no—
79C4⋊C4×C23—42no—
80C4⋊C4×C23—81no—
81C22⋊C4×C23—42no—
82C22⋊C4×C23—44no—
83C22⋊C4×C23—42no—
84C22⋊C4×C23—42no—
85C42×C23—41no—
86#7751C2311yesmaximal, characteristic, direct product factor, Sylow
87C23×C23—22no—
88C23×C23—42no—
89C23×C23—41no—
90C23×C23—42no—
91C23×C23—41no—
92C23×C23#12011yescharacteristic
93Q8×C23—42no—
94Q8×C23—42no—
95Q8×C23—81no—
96Q8×C23—81no—
97D4×C23—42no—
98D4×C23—81no—
99C2×C4×C23—21no—
100C2×C4×C23—22no—
101C2×C4×C23—22no—
102C2×C4×C23—21no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—41no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—84no—
113C2×C4×C23—81no—
114C2×C4×C23—81no—
115C2×C4×C23C2×D811yescharacteristic
116C2×C4×C23#13512yes—
117C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C23#13712yes—
119C8×C23—44no—
120C8×C23—42no—
121#2724C2×C2311yescharacteristic
122#2057C2×C2312yescomplemented
123#2013C2×C2311yescharacteristic
124#1271C2×C2312yescomplemented
125#1270C2×C2311yescharacteristic
126C22×C23—42no—
127C22×C23—44no—
128C22×C23—82no—
129C22×C23—81no—
130C22×C23—81no—
131C22×C23#44711yescharacteristic
132C22×C23#45212yes—
133C22×C23#45311yescharacteristic
134C22×C23#45612yes—
135C22×C23B(4,𝔽2)11yescharacteristic
136C4×C23—21no—
137C4×C23—21no—
138C4×C23—22no—
139C4×C23—22no—
140C4×C23—82no—
141C4×C23—82no—
142C4×C23—81no—
143C4×C23—82no—
144C22×SD16—21no—
145C2×C4⋊Q8C22×C2311yescharacteristic
146C2×#123—21no—
147C2×#107—22no—
148C2×#103—21no—
149#398—21no—
150#397—22no—
151#390C22×C2311yescharacteristic
152#389C22×C2312yes—
153#385—22no—
154#340—22no—
155#336C22×C2312yes—
156#336C22×C2311yescharacteristic
157C2×C23—82no—
158C2×C23#159511yescharacteristic
159C2×C23#160312yes—
160C2×C23#163011yescharacteristic
161C2×C23#164712yes—
162C2×C23#166011yescharacteristic
163C22×Q8—21no—
164C22×D4D4×C2311yescharacteristic
165C23×C4—21no—
166C2×SD16—42no—
167C2×SD16—42no—
168C2×SD16—42no—
169C22×C8—21no—
170C22×C8—22no—
171C4⋊Q8—42no—
172#123—42no—
173#123—42no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4D4×C2312yes—
178C2×C4⋊C4D4×C2312yes—
179C2×C4⋊C4D4×C2311yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C42C23×C2311yescharacteristic
184#107—44no—
185#103—42no—
186#95—22no—
187#95—21no—
188#95—22no—
189C23#775111yescharacteristic, direct product factor, Sylow
190C24—21no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4D8×C2312yes—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4C2×D4×C2311yescharacteristic
209C22×C4C2×D4×C2312yes—
210SD16—48no—
211C2×C8—22no—
212C2×C8—24no—
213C2×C8—42no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—42no—
217C4⋊C4—24no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—81no—
226C22⋊C4—42no—
227C22⋊C4—44no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C42—41no—
231C23—22no—
232C23—42no—
233C23—41no—
234C23—42no—
235C23—41no—
236C23C23×#12011yescharacteristic
237Q8—42no—
238Q8—42no—
239Q8—81no—
240Q8—81no—
241D4—42no—
242D4—81no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—84no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4C2×D8×C2311yescharacteristic
260C2×C4C23×#13512yes—
261C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
262C2×C4C23×#13712yes—
263C8—42no—
264C8—44no—
265C22—44no—
266C22—42no—
267C22—81no—
268C22—81no—
269C22—82no—
270C22C23×#44711yescharacteristic
271C22C23×#45212yes—
272C22C23×#45311yescharacteristic
273C22C23×#45612yes—
274C22C23×B(4,𝔽2)11yescharacteristic
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—81no—
280C4—82no—
281C4—82no—
282C4—82no—
283C2—82no—
284C2C23×#159511yescharacteristic
285C2C23×#160312yes—
286C2C23×#163011yescharacteristic
287C2C23×#164712yes—
288C2C23×#166011yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2057C2semidirect
4. C23×#1271C2semidirect
86. #7751C23direct (N × H)
122. #2057C2×C23semidirect
124. #1271C2×C23semidirect
189. C23#7751direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#27242
C23×#20572
C23×#20132
C23×#12712
C23×#12702
#775123