Finite Group Databasebeta

← C23×#12180 #19159806

Subgroups — C23×#12180 #19159806

Order 5888.

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

normal subgroups only

This group has 472 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×#2551C211yesmaximal, characteristic, complemented
2C23×#2039C211yesmaximal, characteristic
3C23×#1604C212yesmaximal
4C23×#1354C212yesmaximal, complemented
5C23×#1325C211yesmaximal, characteristic, complemented
6C23×#537C2211yescharacteristic
7C2×C23×C4⋊Q8—22no—
8C23×#448—22no—
9C2×C23×#104—21no—
10C23×#417—21no—
11C23×#395—21no—
12C23×#395C2212yes—
13C23×#389—22no—
14C23×#384C2211yescharacteristic
15C4×C4⋊C4×C23—21no—
16C23×#343C2211yescharacteristic
17C23×#327—22no—
18C23×#324C2212yes—
19C2×#41×C23—21no—
20C22×Q8×C23—21no—
21C22×Q8×C23D412yes—
22C2×SD16×C23—42no—
23C2×M16×C23—21no—
24C23×C4⋊Q8—48no—
25C23×C22⋊Q8—41no—
26C23×#121—22no—
27C23×#121—41no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23D411yescharacteristic
34C2×C4⋊C4×C23C2311yescharacteristic
35C2×C42×C23—22no—
36C2×C42×C23D411yescharacteristic
37C23×#104—42no—
38C23×#103—42no—
39C23×#103—41no—
40C23×#102—41no—
41C23×C22⋊C8—22no—
42C23×#95—21no—
43C23×#95—22no—
44C23×#95D412yes—
45#41×C23—41no—
46#41×C23—41no—
47#41×C23—41no—
48#41×C23—41no—
49C2×Q8×C23—21no—
50C2×Q8×C23—44no—
51C2×Q8×C23—44no—
52C2×Q8×C23—42no—
53C2×Q8×C23—81no—
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—
64SD16×C23—84no—
65M16×C23—41no—
66M16×C23—41no—
67C2×C8×C23—41no—
68C2×C8×C23—42no—
69C4⋊C4×C23—24no—
70C4⋊C4×C23—24no—
71C4⋊C4×C23—22no—
72C4⋊C4×C23—41no—
73C4⋊C4×C23—41no—
74C4⋊C4×C23—44no—
75C4⋊C4×C23—44no—
76C4⋊C4×C23—44no—
77C22⋊C4×C23—42no—
78C42×C23—21no—
79C42×C23—44no—
80C42×C23SD1612yescomplemented
81#12180C2311yesmaximal, characteristic, direct product factor, Sylow
82C23×C23—41no—
83C23×C23#12011yescharacteristic
84Q8×C23—41no—
85Q8×C23—41no—
86Q8×C23—44no—
87Q8×C23—82no—
88Q8×C23—84no—
89D4×C23—41no—
90D4×C23—41no—
91D4×C23—82no—
92D4×C23—81no—
93D4×C23—81no—
94C2×C4×C23—22no—
95C2×C4×C23—22no—
96C2×C4×C23—22no—
97C2×C4×C23—21no—
98C2×C4×C23—21no—
99C2×C4×C23—24no—
100C2×C4×C23—24no—
101C2×C4×C23—22no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—41no—
109C2×C4×C23—41no—
110C2×C4×C23—81no—
111C2×C4×C23—81no—
112C2×C4×C23C2×SD1611yescharacteristic
113C2×C4×C23#13711yescharacteristic
114C8×C23—81no—
115C8×C23—82no—
116#2551C2×C2311yescharacteristic, complemented
117#2039C2×C2311yescharacteristic
118#1604C2×C2312yes—
119#1354C2×C2312yescomplemented
120#1325C2×C2311yescharacteristic, complemented
121C22×C23—21no—
122C22×C23—21no—
123C22×C23—81no—
124C22×C23—81no—
125C22×C23—81no—
126C22×C23#44811yescharacteristic
127C22×C23#45411yescharacteristic
128C22×C23B(4,𝔽2)11yescharacteristic
129C4×C23—24no—
130C4×C23—21no—
131C4×C23—21no—
132C4×C23—44no—
133C4×C23—42no—
134C4×C23—44no—
135C4×C23—81no—
136C4×C23—81no—
137#537C22×C2311yescharacteristic
138C2×C4⋊Q8—22no—
139#448—22no—
140C2×#104—21no—
141#417—21no—
142#395—21no—
143#395C22×C2312yes—
144#389—22no—
145#384C22×C2311yescharacteristic
146C4×C4⋊C4—21no—
147#343C22×C2311yescharacteristic
148#327—22no—
149#324C22×C2312yes—
150C2×C23—21no—
151C2×C23—21no—
152C2×C23—161no—
153C2×C23#159911yescharacteristic
154C2×C23#219611yescharacteristic
155C2×C23#220511yescharacteristic
156C2×#41—21no—
157C22×Q8—21no—
158C22×Q8D4×C2312yes—
159C2×SD16—42no—
160C2×M16—21no—
161C4⋊Q8—48no—
162C22⋊Q8—41no—
163#121—22no—
164#121—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4D4×C2311yescharacteristic
171C2×C4⋊C4C23×C2311yescharacteristic
172C2×C42—22no—
173C2×C42D4×C2311yescharacteristic
174#104—42no—
175#103—42no—
176#103—41no—
177#102—41no—
178C22⋊C8—22no—
179#95—21no—
180#95—22no—
181#95D4×C2312yes—
182C23#1218011yescharacteristic, direct product factor, Sylow
183#41—41no—
184#41—41no—
185#41—41no—
186#41—41no—
187C2×Q8—21no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—42no—
191C2×Q8—81no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—41no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—41no—
200C22×C4C2×D4×C2311yescharacteristic
201C22×C4C2×D4×C2312yes—
202SD16—84no—
203M16—41no—
204M16—41no—
205C2×C8—42no—
206C2×C8—41no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C22⋊C4—42no—
216C42—21no—
217C42—44no—
218C42SD16×C2312yescomplemented
219C23—41no—
220C23C23×#12011yescharacteristic
221Q8—41no—
222Q8—41no—
223Q8—44no—
224Q8—82no—
225Q8—84no—
226D4—41no—
227D4—41no—
228D4—82no—
229D4—81no—
230D4—81no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4C2×SD16×C2311yescharacteristic
250C2×C4C23×#13711yescharacteristic
251C8—81no—
252C8—82no—
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22C23×#44811yescharacteristic
259C22C23×#45411yescharacteristic
260C22C23×B(4,𝔽2)11yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—44no—
265C4—42no—
266C4—44no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—161no—
272C2C23×#159911yescharacteristic
273C2C23×#219611yescharacteristic
274C2C23×#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×#2551C2semidirect
4. C23×#1354C2semidirect
5. C23×#1325C2semidirect
80. C42×C23SD16semidirect
81. #12180C23direct (N × H)
116. #2551C2×C23semidirect
119. #1354C2×C23semidirect
120. #1325C2×C23semidirect
182. C23#12180direct (N × H)
218. C42SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#25512
C23×#20392
C23×#16042
C23×#13542
C23×#13252
#1218023