Finite Group Databasebeta

← C23×#12181 #19159807

Subgroups — C23×#12181 #19159807

Order 5888.

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

normal subgroups only

This group has 372 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×#2571C211yesmaximal, characteristic, complemented
2C23×#2039C211yesmaximal, characteristic
3C23×#1616C212yesmaximal
4C23×#1353C212yesmaximal, complemented
5C23×#1325C211yesmaximal, characteristic, complemented
6C23×#537C2211yescharacteristic
7C23×#481—22no—
8C2×C23×#104—21no—
9C23×#417—21no—
10C23×#400—21no—
11C23×#400C2212yes—
12C23×#389—22no—
13C23×#384C2211yescharacteristic
14C23×#382—22no—
15C4×C4⋊C4×C23—21no—
16C23×#343C2211yescharacteristic
17C23×#327—22no—
18C23×#324C2212yes—
19C2×#41×C23—21no—
20C2×M16×C23—21no—
21C23×C22⋊Q8—41no—
22C23×#121—22no—
23C23×#121—41no—
24C2×C4⋊C4×C23—21no—
25C2×C4⋊C4×C23—21no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23D411yescharacteristic
29C2×C4⋊C4×C23D412yes—
30C2×C4⋊C4×C23C2311yescharacteristic
31C2×C42×C23—22no—
32C2×C42×C23D411yescharacteristic
33C23×#106—42no—
34C23×#104—42no—
35C23×#103—41no—
36C23×#102—41no—
37C23×#102—42no—
38C23×C22⋊C8—22no—
39C23×#95—21no—
40C23×#95—22no—
41C23×#95—22no—
42C23×#95—22no—
43C23×#95D412yes—
44#41×C23—41no—
45#41×C23—41no—
46#41×C23—41no—
47#41×C23—41no—
48C2×Q8×C23—21no—
49C2×D4×C23—21no—
50C2×D4×C23—22no—
51C2×D4×C23—41no—
52C22×C4×C23—21no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—21no—
56C22×C4×C23—41no—
57C22×C4×C23C2×D411yescharacteristic
58C22×C4×C23C2×D412yes—
59M16×C23—41no—
60M16×C23—41no—
61C2×C8×C23—41no—
62C2×C8×C23—42no—
63C4⋊C4×C23—22no—
64C4⋊C4×C23—41no—
65C4⋊C4×C23—41no—
66C4⋊C4×C23—42no—
67C4⋊C4×C23—42no—
68C4⋊C4×C23—44no—
69C4⋊C4×C23—42no—
70C4⋊C4×C23—81no—
71C22⋊C4×C23—42no—
72C42×C23—21no—
73C42×C23—44no—
74C42×C23SD1612yes—
75#12181C2311yesmaximal, characteristic, direct product factor, Sylow
76C23×C23—41no—
77C23×C23#12011yescharacteristic
78Q8×C23—41no—
79Q8×C23—41no—
80D4×C23—41no—
81D4×C23—41no—
82D4×C23—82no—
83D4×C23—81no—
84D4×C23—81no—
85C2×C4×C23—22no—
86C2×C4×C23—22no—
87C2×C4×C23—22no—
88C2×C4×C23—21no—
89C2×C4×C23—21no—
90C2×C4×C23—22no—
91C2×C4×C23—42no—
92C2×C4×C23—42no—
93C2×C4×C23—42no—
94C2×C4×C23—42no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—41no—
99C2×C4×C23—41no—
100C2×C4×C23—42no—
101C2×C4×C23—81no—
102C2×C4×C23—81no—
103C2×C4×C23C2×SD1611yescharacteristic
104C2×C4×C23#13711yescharacteristic
105C8×C23—81no—
106C8×C23—82no—
107#2571C2×C2311yescharacteristic, complemented
108#2039C2×C2311yescharacteristic
109#1616C2×C2312yes—
110#1353C2×C2312yescomplemented
111#1325C2×C2311yescharacteristic, complemented
112C22×C23—21no—
113C22×C23—21no—
114C22×C23—81no—
115C22×C23—81no—
116C22×C23—81no—
117C22×C23#44811yescharacteristic
118C22×C23#45411yescharacteristic
119C22×C23B(4,𝔽2)11yescharacteristic
120C4×C23—24no—
121C4×C23—21no—
122C4×C23—21no—
123C4×C23—42no—
124C4×C23—82no—
125C4×C23—81no—
126C4×C23—81no—
127C4×C23—82no—
128#537C22×C2311yescharacteristic
129#481—22no—
130C2×#104—21no—
131#417—21no—
132#400—21no—
133#400C22×C2312yes—
134#389—22no—
135#384C22×C2311yescharacteristic
136#382—22no—
137C4×C4⋊C4—21no—
138#343C22×C2311yescharacteristic
139#327—22no—
140#324C22×C2312yes—
141C2×C23—21no—
142C2×C23—21no—
143C2×C23—161no—
144C2×C23#159911yescharacteristic
145C2×C23#219811yescharacteristic
146C2×C23#220411yescharacteristic
147C2×#41—21no—
148C2×M16—21no—
149C22⋊Q8—41no—
150#121—22no—
151#121—41no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4D4×C2311yescharacteristic
157C2×C4⋊C4D4×C2312yes—
158C2×C4⋊C4C23×C2311yescharacteristic
159C2×C42—22no—
160C2×C42D4×C2311yescharacteristic
161#106—42no—
162#104—42no—
163#103—41no—
164#102—41no—
165#102—42no—
166C22⋊C8—22no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95D4×C2312yes—
172C23#1218111yescharacteristic, direct product factor, Sylow
173#41—41no—
174#41—41no—
175#41—41no—
176#41—41no—
177C2×Q8—21no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—41no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4C2×D4×C2311yescharacteristic
187C22×C4C2×D4×C2312yes—
188M16—41no—
189M16—41no—
190C2×C8—42no—
191C2×C8—41no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C22⋊C4—42no—
201C42—21no—
202C42—44no—
203C42SD16×C2312yes—
204C23—41no—
205C23C23×#12011yescharacteristic
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—82no—
211D4—81no—
212D4—81no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4C2×SD16×C2311yescharacteristic
232C2×C4C23×#13711yescharacteristic
233C8—81no—
234C8—82no—
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22C23×#44811yescharacteristic
241C22C23×#45411yescharacteristic
242C22C23×B(4,𝔽2)11yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4—82no—
250C4—81no—
251C2—21no—
252C2—21no—
253C2—161no—
254C2C23×#159911yescharacteristic
255C2C23×#219811yescharacteristic
256C2C23×#220411yescharacteristic
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×#2571C2semidirect
4. C23×#1353C2semidirect
5. C23×#1325C2semidirect
75. #12181C23direct (N × H)
107. #2571C2×C23semidirect
110. #1353C2×C23semidirect
111. #1325C2×C23semidirect
172. C23#12181direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25712
C23×#20392
C23×#16162
C23×#13532
C23×#13252
#1218123