Finite Group Databasebeta

← C23×#12153 #19159779

Subgroups — C23×#12153 #19159779

Order 5888.

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

normal subgroups only

This group has 424 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×#2582C211yesmaximal, characteristic, complemented
2C23×#2039C211yesmaximal, characteristic
3C23×#1610C212yesmaximal
4C23×#1346C212yesmaximal, complemented
5C23×#1325C211yesmaximal, characteristic, complemented
6C23×#537C2211yescharacteristic
7C2×C4×Q8×C23—21no—
8C23×#480—22no—
9C2×C23×#104—21no—
10C23×#417—21no—
11C23×#398—21no—
12C23×#398C2212yes—
13C23×#391—22no—
14C23×#389—22no—
15C23×#384C2211yescharacteristic
16C23×#343C2211yescharacteristic
17C23×#327—22no—
18C23×#324C2212yes—
19C2×#41×C23—21no—
20C22×Q8×C23—21no—
21C2×M16×C23—21no—
22C23×C22⋊Q8—41no—
23C23×#121—22no—
24C23×#121—41no—
25C4×Q8×C23—42no—
26C4×Q8×C23—42no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23D411yescharacteristic
32C2×C4⋊C4×C23D412yes—
33C2×C4⋊C4×C23C2311yescharacteristic
34C2×C42×C23—22no—
35C2×C42×C23D411yescharacteristic
36C23×#107—42no—
37C23×#104—42no—
38C23×#103—41no—
39C23×#102—41no—
40C23×#102—42no—
41C23×C22⋊C8—22no—
42C23×#95—22no—
43C23×#95—21no—
44C23×#95—22no—
45C23×#95D412yes—
46#41×C23—41no—
47#41×C23—41no—
48#41×C23—41no—
49#41×C23—41no—
50C2×Q8×C23—24no—
51C2×Q8×C23—21no—
52C2×Q8×C23—42no—
53C2×Q8×C23—42no—
54C2×D4×C23—21no—
55C2×D4×C23—22no—
56C2×D4×C23—41no—
57C22×C4×C23—21no—
58C22×C4×C23—22no—
59C22×C4×C23—22no—
60C22×C4×C23—21no—
61C22×C4×C23—41no—
62C22×C4×C23C2×D411yescharacteristic
63C22×C4×C23C2×D412yes—
64M16×C23—41no—
65M16×C23—41no—
66C2×C8×C23—41no—
67C2×C8×C23—42no—
68C4⋊C4×C23—22no—
69C4⋊C4×C23—41no—
70C4⋊C4×C23—41no—
71C4⋊C4×C23—42no—
72C4⋊C4×C23—44no—
73C4⋊C4×C23—42no—
74C4⋊C4×C23—42no—
75C4⋊C4×C23—42no—
76C4⋊C4×C23—81no—
77C22⋊C4×C23—42no—
78C42×C23—21no—
79C42×C23—44no—
80C42×C23D812yes—
81#12153C2311yesmaximal, characteristic, direct product factor, Sylow
82C23×C23—41no—
83C23×C23#12011yescharacteristic
84Q8×C23—48no—
85Q8×C23—41no—
86Q8×C23—41no—
87D4×C23—41no—
88D4×C23—41no—
89D4×C23—82no—
90D4×C23—81no—
91D4×C23—81no—
92C2×C4×C23—22no—
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—21no—
96C2×C4×C23—21no—
97C2×C4×C23—24no—
98C2×C4×C23—22no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—42no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—41no—
106C2×C4×C23—42no—
107C2×C4×C23—41no—
108C2×C4×C23—81no—
109C2×C4×C23—81no—
110C2×C4×C23C2×D811yescharacteristic
111C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
112C8×C23—81no—
113C8×C23—82no—
114#2582C2×C2311yescharacteristic, complemented
115#2039C2×C2311yescharacteristic
116#1610C2×C2312yes—
117#1346C2×C2312yescomplemented
118#1325C2×C2311yescharacteristic, complemented
119C22×C23—21no—
120C22×C23—21no—
121C22×C23—81no—
122C22×C23—81no—
123C22×C23—81no—
124C22×C23#44711yescharacteristic
125C22×C23#45411yescharacteristic
126C22×C23B(4,𝔽2)11yescharacteristic
127C4×C23—24no—
128C4×C23—21no—
129C4×C23—21no—
130C4×C23—44no—
131C4×C23—42no—
132C4×C23—82no—
133C4×C23—81no—
134C4×C23—81no—
135#537C22×C2311yescharacteristic
136C2×C4×Q8—21no—
137#480—22no—
138C2×#104—21no—
139#417—21no—
140#398—21no—
141#398C22×C2312yes—
142#391—22no—
143#389—22no—
144#384C22×C2311yescharacteristic
145#343C22×C2311yescharacteristic
146#327—22no—
147#324C22×C2312yes—
148C2×C23—21no—
149C2×C23—21no—
150C2×C23—161no—
151C2×C23#159811yescharacteristic
152C2×C23#219811yescharacteristic
153C2×C23#220511yescharacteristic
154C2×#41—21no—
155C22×Q8—21no—
156C2×M16—21no—
157C22⋊Q8—41no—
158#121—22no—
159#121—41no—
160C4×Q8—42no—
161C4×Q8—42no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4D4×C2311yescharacteristic
167C2×C4⋊C4D4×C2312yes—
168C2×C4⋊C4C23×C2311yescharacteristic
169C2×C42—22no—
170C2×C42D4×C2311yescharacteristic
171#107—42no—
172#104—42no—
173#103—41no—
174#102—41no—
175#102—42no—
176C22⋊C8—22no—
177#95—22no—
178#95—21no—
179#95—22no—
180#95D4×C2312yes—
181C23#1215311yescharacteristic, direct product factor, Sylow
182#41—41no—
183#41—41no—
184#41—41no—
185#41—41no—
186C2×Q8—24no—
187C2×Q8—21no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—41no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4C2×D4×C2311yescharacteristic
199C22×C4C2×D4×C2312yes—
200M16—41no—
201M16—41no—
202C2×C8—42no—
203C2×C8—41no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—81no—
213C22⋊C4—42no—
214C42—21no—
215C42—44no—
216C42D8×C2312yes—
217C23—41no—
218C23C23×#12011yescharacteristic
219Q8—48no—
220Q8—41no—
221Q8—41no—
222D4—41no—
223D4—41no—
224D4—82no—
225D4—81no—
226D4—81no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4C2×D8×C2311yescharacteristic
246C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
247C8—81no—
248C8—82no—
249C22—21no—
250C22—21no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22C23×#44711yescharacteristic
255C22C23×#45411yescharacteristic
256C22C23×B(4,𝔽2)11yescharacteristic
257C4—24no—
258C4—21no—
259C4—21no—
260C4—44no—
261C4—42no—
262C4—81no—
263C4—82no—
264C4—81no—
265C2—21no—
266C2—21no—
267C2—161no—
268C2C23×#159811yescharacteristic
269C2C23×#219811yescharacteristic
270C2C23×#220511yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×#537C22
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×#2582C2semidirect
4. C23×#1346C2semidirect
5. C23×#1325C2semidirect
81. #12153C23direct (N × H)
114. #2582C2×C23semidirect
117. #1346C2×C23semidirect
118. #1325C2×C23semidirect
181. C23#12153direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25822
C23×#20392
C23×#16102
C23×#13462
C23×#13252
#1215323