Finite Group Databasebeta

← C23×#12097 #19159723

Subgroups — C23×#12097 #19159723

Order 5888.

466 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 74 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, complemented
2C23×#2006C211yesmaximal, characteristic
3C23×#1610C212yesmaximal, complemented
4C23×#1346C212yesmaximal, complemented
5C23×#1318C211yesmaximal, characteristic, complemented
6C2×C23×AGL(1,ℤ/8)—21no—
7C23×#537C2211yescharacteristic
8C2×C23×#123—21no—
9C23×#480—22no—
10C2×C23×#104—21no—
11C23×#398C2212yes—
12C23×#397—22no—
13C23×#390C2211yescharacteristic
14C23×#389—21no—
15C23×#385—22no—
16C23×#343C2211yescharacteristic
17C23×#327—22no—
18C23×#324C2212yes—
19C2×#41×C23—21no—
20C22×D4×C23D411yescharacteristic
21C23×C4×C23—21no—
22C23×AGL(1,ℤ/8)—42no—
23C23×AGL(1,ℤ/8)—42no—
24C2×SD16×C23—41no—
25C2×D8×C23—41no—
26C2×M16×C23—21no—
27C23×#123—42no—
28C23×#123—42no—
29C23×C22⋊Q8—41no—
30C23×#121—22no—
31C23×#121—41no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23D412yes—
36C2×C4⋊C4×C23C2311yescharacteristic
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—22no—
40C2×C42×C23D411yescharacteristic
41C23×#107—42no—
42C23×#104—42no—
43C23×#102—42no—
44C23×C22⋊C8—22no—
45C23×#95—22no—
46C23×#95—22no—
47C23×#95D412yes—
48C24×C23—21no—
49#41×C23—41no—
50#41×C23—41no—
51#41×C23—41no—
52#41×C23—41no—
53C2×Q8×C23—21no—
54C2×D4×C23—21no—
55C2×D4×C23—22no—
56C2×D4×C23—21no—
57C2×D4×C23—41no—
58C2×D4×C23—41no—
59C2×D4×C23—41no—
60C2×D4×C23D812yes—
61C22×C4×C23—21no—
62C22×C4×C23—22no—
63C22×C4×C23—22no—
64C22×C4×C23—41no—
65C22×C4×C23—42no—
66C22×C4×C23—81no—
67C22×C4×C23—81no—
68C22×C4×C23C2×D411yescharacteristic
69C22×C4×C23C2×D412yes—
70SD16×C23—44no—
71D8×C23—44no—
72M16×C23—22no—
73M16×C23—41no—
74C2×C8×C23—41no—
75C2×C8×C23—42no—
76C4⋊C4×C23—22no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—42no—
79C4⋊C4×C23—42no—
80C4⋊C4×C23—42no—
81C4⋊C4×C23—42no—
82C4⋊C4×C23—81no—
83C22⋊C4×C23—44no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—42no—
86C22⋊C4×C23—44no—
87C42×C23—41no—
88#12097C2311yesmaximal, characteristic, direct product factor, Sylow
89C23×C23—22no—
90C23×C23—42no—
91C23×C23—42no—
92C23×C23—41no—
93C23×C23—41no—
94C23×C23—41no—
95C23×C23#12011yescharacteristic
96Q8×C23—41no—
97Q8×C23—41no—
98D4×C23—42no—
99D4×C23—41no—
100D4×C23—41no—
101D4×C23—81no—
102D4×C23—82no—
103D4×C23—81no—
104D4×C23—81no—
105C2×C4×C23—21no—
106C2×C4×C23—21no—
107C2×C4×C23—22no—
108C2×C4×C23—22no—
109C2×C4×C23—41no—
110C2×C4×C23—41no—
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—41no—
118C2×C4×C23—42no—
119C2×C4×C23—81no—
120C2×C4×C23—81no—
121C2×C4×C23—84no—
122C2×C4×C23C2×D811yescharacteristic
123C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
124C8×C23—42no—
125C8×C23—82no—
126#2724C2×C2311yescharacteristic, complemented
127#2006C2×C2311yescharacteristic
128#1610C2×C2312yescomplemented
129#1346C2×C2312yescomplemented
130#1318C2×C2311yescharacteristic, complemented
131C22×C23—21no—
132C22×C23—21no—
133C22×C23—44no—
134C22×C23—42no—
135C22×C23—82no—
136C22×C23—81no—
137C22×C23—81no—
138C22×C23—81no—
139C22×C23—81no—
140C22×C23—81no—
141C22×C23#44711yescharacteristic
142C22×C23#45311yescharacteristic
143C22×C23B(4,𝔽2)11yescharacteristic
144C4×C23—21no—
145C4×C23—21no—
146C4×C23—42no—
147C4×C23—42no—
148C4×C23—82no—
149C4×C23—82no—
150C4×C23—81no—
151C2×AGL(1,ℤ/8)—21no—
152#537C22×C2311yescharacteristic
153C2×#123—21no—
154#480—22no—
155C2×#104—21no—
156#398C22×C2312yes—
157#397—22no—
158#390C22×C2311yescharacteristic
159#389—21no—
160#385—22no—
161#343C22×C2311yescharacteristic
162#327—22no—
163#324C22×C2312yes—
164C2×C23—21no—
165C2×C23—21no—
166C2×C23—82no—
167C2×C23—161no—
168C2×C23#159511yescharacteristic
169C2×C23#219811yescharacteristic
170C2×C23#220311yescharacteristic
171C2×#41—21no—
172C22×D4D4×C2311yescharacteristic
173C23×C4—21no—
174AGL(1,ℤ/8)—42no—
175AGL(1,ℤ/8)—42no—
176C2×SD16—41no—
177C2×D8—41no—
178C2×M16—21no—
179#123—42no—
180#123—42no—
181C22⋊Q8—41no—
182#121—22no—
183#121—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4D4×C2312yes—
188C2×C4⋊C4C23×C2311yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C42D4×C2311yescharacteristic
193#107—42no—
194#104—42no—
195#102—42no—
196C22⋊C8—22no—
197#95—22no—
198#95—22no—
199#95D4×C2312yes—
200C23#1209711yescharacteristic, direct product factor, Sylow
201C24—21no—
202#41—41no—
203#41—41no—
204#41—41no—
205#41—41no—
206C2×Q8—21no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—21no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4D8×C2312yes—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4C2×D4×C2311yescharacteristic
222C22×C4C2×D4×C2312yes—
223SD16—44no—
224D8—44no—
225M16—22no—
226M16—41no—
227C2×C8—41no—
228C2×C8—42no—
229C4⋊C4—22no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—81no—
236C22⋊C4—44no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C42—41no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C23×#12011yescharacteristic
248Q8—41no—
249Q8—41no—
250D4—42no—
251D4—41no—
252D4—41no—
253D4—81no—
254D4—82no—
255D4—81no—
256D4—81no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4C2×D8×C2311yescharacteristic
275C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
276C8—42no—
277C8—82no—
278C22—21no—
279C22—21no—
280C22—44no—
281C22—42no—
282C22—82no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22C23×#44711yescharacteristic
289C22C23×#45311yescharacteristic
290C22C23×B(4,𝔽2)11yescharacteristic
291C4—21no—
292C4—21no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—82no—
297C4—81no—
298C2—21no—
299C2—21no—
300C2—82no—
301C2—161no—
302C2C23×#159511yescharacteristic
303C2C23×#219811yescharacteristic
304C2C23×#220311yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×#2006C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2724C2semidirect
3. C23×#1610C2semidirect
4. C23×#1346C2semidirect
5. C23×#1318C2semidirect
88. #12097C23direct (N × H)
126. #2724C2×C23semidirect
128. #1610C2×C23semidirect
129. #1346C2×C23semidirect
130. #1318C2×C23semidirect
200. C23#12097direct (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×#20062
C23×#16102
C23×#13462
C23×#13182
#1209723