Finite Group Databasebeta

← C23×#12184 #19159810

Subgroups — C23×#12184 #19159810

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×#2844C211yesmaximal, characteristic, complemented
2C23×#2040C211yesmaximal, characteristic
3C23×#1604C212yesmaximal
4C23×#1354C212yesmaximal, complemented
5C23×#1324C211yesmaximal, characteristic, complemented
6C23×#537C2211yescharacteristic
7C23×#448—22no—
8C2×C23×#104—21no—
9C23×#417—21no—
10C23×#400—22no—
11C23×#400—21no—
12C23×#395—22no—
13C23×#395C2212yes—
14C23×#382—21no—
15C23×#382C2211yescharacteristic
16C23×#343C2211yescharacteristic
17C23×#327—22no—
18C23×#324C2212yes—
19C2×#41×C23—21no—
20C22×Q8×C23D412yes—
21C2×SD16×C23—42no—
22C2×M16×C23—21no—
23C23×C22⋊Q8—41no—
24C23×#121—22no—
25C23×#121—41no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23D411yescharacteristic
31C2×C4⋊C4×C23C2311yescharacteristic
32C2×C42×C23D411yescharacteristic
33C23×#104—42no—
34C23×#103—42no—
35C23×#103—41no—
36C23×#102—41no—
37C23×C22⋊C8—22no—
38C23×#95—22no—
39C23×#95—22no—
40C23×#95—21no—
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×Q8×C23—42no—
50C2×Q8×C23—82no—
51C2×D4×C23—21no—
52C2×D4×C23—22no—
53C2×D4×C23—41no—
54C22×C4×C23—21no—
55C22×C4×C23—22no—
56C22×C4×C23—22no—
57C22×C4×C23—21no—
58C22×C4×C23—41no—
59C22×C4×C23C2×D411yescharacteristic
60C22×C4×C23C2×D412yes—
61SD16×C23—84no—
62M16×C23—41no—
63M16×C23—41no—
64C2×C8×C23—41no—
65C2×C8×C23—42no—
66C4⋊C4×C23—22no—
67C4⋊C4×C23—41no—
68C4⋊C4×C23—41no—
69C4⋊C4×C23—81no—
70C4⋊C4×C23—81no—
71C4⋊C4×C23—82no—
72C4⋊C4×C23—81no—
73C22⋊C4×C23—42no—
74C42×C23—21no—
75C42×C23SD1612yescomplemented
76#12184C2311yesmaximal, characteristic, direct product factor, Sylow
77C23×C23—41no—
78C23×C23#12011yescharacteristic
79Q8×C23—41no—
80Q8×C23—41no—
81Q8×C23—44no—
82Q8×C23—82no—
83D4×C23—41no—
84D4×C23—41no—
85D4×C23—82no—
86D4×C23—81no—
87D4×C23—81no—
88C2×C4×C23—21no—
89C2×C4×C23—21no—
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—41no—
93C2×C4×C23—41no—
94C2×C4×C23—42no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—42no—
104C2×C4×C23—81no—
105C2×C4×C23—81no—
106C2×C4×C23C2×SD1611yescharacteristic
107C2×C4×C23#13711yescharacteristic
108C8×C23—81no—
109C8×C23—82no—
110#2844C2×C2311yescharacteristic, complemented
111#2040C2×C2311yescharacteristic
112#1604C2×C2312yes—
113#1354C2×C2312yescomplemented
114#1324C2×C2311yescharacteristic, complemented
115C22×C23—21no—
116C22×C23—21no—
117C22×C23—81no—
118C22×C23—81no—
119C22×C23—81no—
120C22×C23#44811yescharacteristic
121C22×C23#45411yescharacteristic
122C22×C23B(4,𝔽2)11yescharacteristic
123C4×C23—21no—
124C4×C23—21no—
125C4×C23—42no—
126C4×C23—42no—
127C4×C23—82no—
128C4×C23—81no—
129C4×C23—82no—
130C4×C23—81no—
131#537C22×C2311yescharacteristic
132#448—22no—
133C2×#104—21no—
134#417—21no—
135#400—21no—
136#400—22no—
137#395—22no—
138#395C22×C2312yes—
139#382—21no—
140#382C22×C2311yescharacteristic
141#343C22×C2311yescharacteristic
142#327—22no—
143#324C22×C2312yes—
144C2×C23—21no—
145C2×C23—21no—
146C2×C23—161no—
147C2×C23#159911yescharacteristic
148C2×C23#219911yescharacteristic
149C2×C23#220311yescharacteristic
150C2×#41—21no—
151C22×Q8D4×C2312yes—
152C2×SD16—42no—
153C2×M16—21no—
154C22⋊Q8—41no—
155#121—22no—
156#121—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4D4×C2311yescharacteristic
162C2×C4⋊C4C23×C2311yescharacteristic
163C2×C42D4×C2311yescharacteristic
164#104—42no—
165#103—42no—
166#103—41no—
167#102—41no—
168C22⋊C8—22no—
169#95—22no—
170#95—22no—
171#95—21no—
172#95—22no—
173#95—22no—
174#95D4×C2312yes—
175C23#1218411yescharacteristic, direct product factor, Sylow
176#41—41no—
177#41—41no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—42no—
182C2×Q8—82no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—41no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4C2×D4×C2311yescharacteristic
192C22×C4C2×D4×C2312yes—
193SD16—84no—
194M16—41no—
195M16—41no—
196C2×C8—42no—
197C2×C8—41no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—82no—
204C4⋊C4—81no—
205C22⋊C4—42no—
206C42—21no—
207C42SD16×C2312yescomplemented
208C23—41no—
209C23C23×#12011yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—44no—
213Q8—82no—
214D4—41no—
215D4—41no—
216D4—82no—
217D4—81no—
218D4—81no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4C2×SD16×C2311yescharacteristic
238C2×C4C23×#13711yescharacteristic
239C8—81no—
240C8—82no—
241C22—21no—
242C22—21no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22C23×#44811yescharacteristic
247C22C23×#45411yescharacteristic
248C22C23×B(4,𝔽2)11yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C4—82no—
255C4—81no—
256C4—81no—
257C2—21no—
258C2—21no—
259C2—161no—
260C2C23×#159911yescharacteristic
261C2C23×#219911yescharacteristic
262C2C23×#220311yescharacteristic
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×#2844C2semidirect
4. C23×#1354C2semidirect
5. C23×#1324C2semidirect
75. C42×C23SD16semidirect
76. #12184C23direct (N × H)
110. #2844C2×C23semidirect
113. #1354C2×C23semidirect
114. #1324C2×C23semidirect
175. C23#12184direct (N × H)
207. C42SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#28442
C23×#20402
C23×#16042
C23×#13542
C23×#13242
#1218423