Finite Group Databasebeta

← C23×#13673 #19161299

Subgroups — C23×#13673 #19161299

Order 5888.

468 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3292C211yesmaximal, characteristic, complemented
2C23×#2196C212yesmaximal, complemented
3C23×#1619C211yesmaximal, characteristic, complemented
4C23×#1408C211yesmaximal, characteristic, complemented
5C23×#1402C212yesmaximal, complemented
6C23×#576—22no—
7D42×C23C2211yescharacteristic, complemented
8C23×#493C2211yescharacteristic
9C23×#472—22no—
10C23×#459—21no—
11C23×B(4,𝔽2)—22no—
12C23×#453—21no—
13C23×#453C2212yes—
14C23×#443—21no—
15C23×#353C2212yescomplemented
16C23×AΣL(1,𝔽16)—22no—
17C23×#331C2211yescharacteristic, complemented
18C23×D4⋊C8—21no—
19C23×H(5,𝔽2)—21no—
20C23×H(5,𝔽2)D412yescomplemented
21C22×D4×C23—41no—
22C23×AGL(1,ℤ/8)—22no—
23C23×AGL(1,ℤ/8)—22no—
24C23×AGL(1,ℤ/8)—42no—
25C23×AGL(1,ℤ/8)—42no—
26C23×#135—21no—
27C23×#135—42no—
28C2×D8×C23—22no—
29C2×D8×C23—42no—
30C2×D8×C23—41no—
31C23×#131—22no—
32C23×#127D411yescharacteristic, complemented
33C23×#127C2311yescharacteristic
34C23×#121—41no—
35C23×#120—22no—
36C23×#120—42no—
37C4×D4×C23—21no—
38C23×#108—21no—
39C23×C4⋊C8—21no—
40C23×#104—22no—
41C23×#102—41no—
42C23×#100—21no—
43C23×#100D412yescomplemented
44C23×#99—22no—
45C23×#99—42no—
46C4×C8×C23D411yescharacteristic, complemented
47C24×C23—22no—
48#41×C23—22no—
49#41×C23—42no—
50#41×C23—42no—
51C2×D4×C23—21no—
52C2×D4×C23—21no—
53C2×D4×C23—41no—
54C2×D4×C23—41no—
55C2×D4×C23—42no—
56C2×D4×C23—42no—
57C2×D4×C23—42no—
58C2×D4×C23—42no—
59C2×D4×C23—82no—
60C2×D4×C23—81no—
61C2×D4×C23—82no—
62C2×D4×C23—82no—
63C2×D4×C23C2×D412yes—
64C22×C4×C23—41no—
65Q16×C23—21no—
66SD16×C23—22no—
67SD16×C23—42no—
68SD16×C23—42no—
69D8×C23—21no—
70D8×C23—42no—
71D8×C23—42no—
72D8×C23—42no—
73D8×C23—42no—
74D8×C23—82no—
75M16×C23—22no—
76M16×C23—42no—
77C2×C8×C23—21no—
78C2×C8×C23—42no—
79C2×C8×C23—41no—
80C4⋊C4×C23—21no—
81C22⋊C4×C23—22no—
82C22⋊C4×C23—41no—
83C22⋊C4×C23—42no—
84C22⋊C4×C23—82no—
85C42×C23C2×D411yescharacteristic
86#13673C2311yesmaximal, characteristic, direct product factor, Sylow
87C23×C23—22no—
88C23×C23—41no—
89C23×C23—42no—
90C23×C23—41no—
91C23×C23—81no—
92C23×C23—82no—
93C23×C23—84no—
94Q8×C23—22no—
95D4×C23—22no—
96D4×C23—22no—
97D4×C23—42no—
98D4×C23—41no—
99D4×C23—42no—
100D4×C23—42no—
101D4×C23—81no—
102D4×C23—81no—
103D4×C23—81no—
104D4×C23—81no—
105D4×C23—82no—
106D4×C23—82no—
107D4×C23—84no—
108D4×C23—164no—
109C2×C4×C23—21no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—41no—
113C2×C4×C23—81no—
114C2×C4×C23—81no—
115C2×C4×C23—81no—
116C2×C4×C23—82no—
117C2×C4×C23#12011yescharacteristic
118C8×C23—21no—
119C8×C23—21no—
120C8×C23—42no—
121C8×C23—81no—
122#3292C2×C2311yescharacteristic, complemented
123#2196C2×C2312yescomplemented
124#1619C2×C2311yescharacteristic, complemented
125#1408C2×C2311yescharacteristic, complemented
126#1402C2×C2312yescomplemented
127C22×C23—42no—
128C22×C23—42no—
129C22×C23—42no—
130C22×C23—82no—
131C22×C23—81no—
132C22×C23—84no—
133C22×C23—81no—
134C22×C23—81no—
135C22×C23—161no—
136C22×C23—162no—
137C22×C23—162no—
138C22×C23—321no—
139C22×C23B(4,𝔽2)11yescharacteristic
140C4×C23—21no—
141C4×C23—21no—
142C4×C23—21no—
143C4×C23—42no—
144C4×C23—81no—
145C4×C23—162no—
146#576—22no—
147D42C22×C2311yescharacteristic, complemented
148#493C22×C2311yescharacteristic
149#472—22no—
150#459—21no—
151B(4,𝔽2)—22no—
152#453—21no—
153#453C22×C2312yes—
154#443—21no—
155#353C22×C2312yescomplemented
156AΣL(1,𝔽16)—22no—
157#331C22×C2311yescharacteristic, complemented
158D4⋊C8—21no—
159C2×C23—21no—
160C2×C23—82no—
161C2×C23—82no—
162C2×C23—82no—
163C2×C23—161no—
164C2×C23#219611yescharacteristic
165H(5,𝔽2)—21no—
166H(5,𝔽2)D4×C2312yescomplemented
167C22×D4—41no—
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—22no—
170AGL(1,ℤ/8)—42no—
171AGL(1,ℤ/8)—42no—
172#135—21no—
173#135—42no—
174C2×D8—22no—
175C2×D8—42no—
176C2×D8—41no—
177#131—22no—
178#127D4×C2311yescharacteristic, complemented
179#127C23×C2311yescharacteristic
180#121—41no—
181#120—22no—
182#120—42no—
183C4×D4—21no—
184#108—21no—
185C4⋊C8—21no—
186#104—22no—
187#102—41no—
188#100—21no—
189#100D4×C2312yescomplemented
190#99—22no—
191#99—42no—
192C4×C8D4×C2311yescharacteristic, complemented
193C23#1367311yescharacteristic, direct product factor, Sylow
194C24—22no—
195#41—22no—
196#41—42no—
197#41—42no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—82no—
207C2×D4—81no—
208C2×D4—82no—
209C2×D4—82no—
210C2×D4C2×D4×C2312yes—
211C22×C4—41no—
212Q16—21no—
213SD16—22no—
214SD16—42no—
215SD16—42no—
216D8—21no—
217D8—42no—
218D8—42no—
219D8—42no—
220D8—42no—
221D8—82no—
222M16—22no—
223M16—42no—
224C2×C8—21no—
225C2×C8—42no—
226C2×C8—41no—
227C4⋊C4—21no—
228C22⋊C4—22no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—82no—
232C42C2×D4×C2311yescharacteristic
233C23—22no—
234C23—41no—
235C23—42no—
236C23—41no—
237C23—81no—
238C23—82no—
239C23—84no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—41no—
245D4—42no—
246D4—42no—
247D4—81no—
248D4—81no—
249D4—81no—
250D4—81no—
251D4—82no—
252D4—82no—
253D4—84no—
254D4—164no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—82no—
263C2×C4C23×#12011yescharacteristic
264C8—21no—
265C8—21no—
266C8—42no—
267C8—81no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—82no—
272C22—81no—
273C22—84no—
274C22—81no—
275C22—81no—
276C22—161no—
277C22—162no—
278C22—162no—
279C22—321no—
280C22C23×B(4,𝔽2)11yescharacteristic
281C4—21no—
282C4—21no—
283C4—21no—
284C4—42no—
285C4—81no—
286C4—162no—
287C2—21no—
288C2—82no—
289C2—82no—
290C2—82no—
291C2—161no—
292C2C23×#219611yescharacteristic
subgroupG/H
center, socleC2×C23#2196
commutator, Frattini, cosocle#127C23×C23
LCSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C23 ◁ C2×C4×C23 ◁ C23×#127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3292C2semidirect
2. C23×#2196C2semidirect
3. C23×#1619C2semidirect
4. C23×#1408C2semidirect
5. C23×#1402C2semidirect
7. D42×C23C22semidirect
15. C23×#353C22semidirect
17. C23×#331C22semidirect
20. C23×H(5,𝔽2)D4semidirect
32. C23×#127D4semidirect
43. C23×#100D4semidirect
46. C4×C8×C23D4semidirect
86. #13673C23direct (N × H)
122. #3292C2×C23semidirect
123. #2196C2×C23semidirect
124. #1619C2×C23semidirect
125. #1408C2×C23semidirect
126. #1402C2×C23semidirect
147. D42C22×C23semidirect
155. #353C22×C23semidirect
157. #331C22×C23semidirect
166. H(5,𝔽2)D4×C23semidirect
178. #127D4×C23semidirect
189. #100D4×C23semidirect
192. C4×C8D4×C23semidirect
193. C23#13673direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32922
C23×#21962
C23×#16192
C23×#14082
C23×#14022
#1367323