Finite Group Databasebeta

← C23×#12130 #19159756

Subgroups — C23×#12130 #19159756

Order 5888.

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

normal subgroups only

This group has 442 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×#2811C211yesmaximal, characteristic, complemented
2C23×#2007C211yesmaximal, characteristic
3C23×#1604C212yesmaximal
4C23×#1354C212yesmaximal, complemented
5C23×#1319C211yesmaximal, characteristic, complemented
6C2×C23×#137—21no—
7C23×#537C2211yescharacteristic
8C23×#448—22no—
9C2×C23×#104—21no—
10C23×#395—22no—
11C23×#395C2212yes—
12C23×#391—22no—
13C23×#391—21no—
14C23×#391—21no—
15C23×#391C2211yescharacteristic
16C23×#343C2211yescharacteristic
17C23×#327—22no—
18C23×#324C2212yes—
19C2×#41×C23—21no—
20C22×Q8×C23—21no—
21C22×Q8×C23D411yescharacteristic
22C22×Q8×C23D412yes—
23C23×#137—42no—
24C23×#137—42no—
25C2×Q16×C23—41no—
26C2×SD16×C23—41no—
27C2×SD16×C23—42no—
28C2×M16×C23—21no—
29C23×C22⋊Q8—41no—
30C23×#121—22no—
31C23×#121—41no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23C2311yescharacteristic
36C2×C42×C23—21no—
37C2×C42×C23D411yescharacteristic
38C23×#104—42no—
39C23×#103—42no—
40C23×C22⋊C8—22no—
41C23×#95—22no—
42C23×#95—22no—
43C23×#95—22no—
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—21no—
53C2×Q8×C23—42no—
54C2×Q8×C23—41no—
55C2×Q8×C23—41no—
56C2×Q8×C23—82no—
57C2×Q8×C23—81no—
58C2×Q8×C23—81no—
59C2×Q8×C23SD1612yescomplemented
60C2×D4×C23—21no—
61C2×D4×C23—22no—
62C2×D4×C23—41no—
63C22×C4×C23—22no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—22no—
67C22×C4×C23—41no—
68C22×C4×C23C2×D411yescharacteristic
69C22×C4×C23C2×D412yes—
70Q16×C23—44no—
71SD16×C23—44no—
72SD16×C23—84no—
73M16×C23—22no—
74M16×C23—41no—
75C2×C8×C23—41no—
76C2×C8×C23—42no—
77C4⋊C4×C23—22no—
78C4⋊C4×C23—41no—
79C4⋊C4×C23—82no—
80C4⋊C4×C23—81no—
81C4⋊C4×C23—81no—
82C22⋊C4×C23—42no—
83C42×C23—41no—
84C42×C23—81no—
85#12130C2311yesmaximal, characteristic, direct product factor, Sylow
86C23×C23—41no—
87C23×C23#12011yescharacteristic
88Q8×C23—42no—
89Q8×C23—41no—
90Q8×C23—41no—
91Q8×C23—44no—
92Q8×C23—81no—
93Q8×C23—84no—
94Q8×C23—82no—
95D4×C23—41no—
96D4×C23—41no—
97D4×C23—82no—
98D4×C23—81no—
99D4×C23—81no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—24no—
103C2×C4×C23—22no—
104C2×C4×C23—22no—
105C2×C4×C23—22no—
106C2×C4×C23—42no—
107C2×C4×C23—41no—
108C2×C4×C23—41no—
109C2×C4×C23—41no—
110C2×C4×C23—42no—
111C2×C4×C23—41no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—41no—
116C2×C4×C23—42no—
117C2×C4×C23—81no—
118C2×C4×C23—81no—
119C2×C4×C23C2×SD1611yescharacteristic
120C2×C4×C23#13711yescharacteristic
121C8×C23—42no—
122C8×C23—82no—
123#2811C2×C2311yescharacteristic, complemented
124#2007C2×C2311yescharacteristic
125#1604C2×C2312yes—
126#1354C2×C2312yescomplemented
127#1319C2×C2311yescharacteristic, complemented
128C22×C23—21no—
129C22×C23—21no—
130C22×C23—81no—
131C22×C23—81no—
132C22×C23—81no—
133C22×C23#44811yescharacteristic
134C22×C23#45311yescharacteristic
135C22×C23B(4,𝔽2)11yescharacteristic
136C4×C23—21no—
137C4×C23—21no—
138C4×C23—42no—
139C4×C23—44no—
140C4×C23—42no—
141C4×C23—42no—
142C4×C23—81no—
143C4×C23—82no—
144C2×#137—21no—
145#537C22×C2311yescharacteristic
146#448—22no—
147C2×#104—21no—
148#395—22no—
149#395C22×C2312yes—
150#391—21no—
151#391—21no—
152#391—22no—
153#391C22×C2311yescharacteristic
154#343C22×C2311yescharacteristic
155#327—22no—
156#324C22×C2312yes—
157C2×C23—21no—
158C2×C23—21no—
159C2×C23—161no—
160C2×C23#159611yescharacteristic
161C2×C23#219911yescharacteristic
162C2×C23#220511yescharacteristic
163C2×#41—21no—
164C22×Q8—21no—
165C22×Q8D4×C2311yescharacteristic
166C22×Q8D4×C2312yes—
167#137—42no—
168#137—42no—
169C2×Q16—41no—
170C2×SD16—41no—
171C2×SD16—42no—
172C2×M16—21no—
173C22⋊Q8—41no—
174#121—22no—
175#121—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C23×C2311yescharacteristic
180C2×C42—21no—
181C2×C42D4×C2311yescharacteristic
182#104—42no—
183#103—42no—
184C22⋊C8—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95D4×C2312yes—
190C23#1213011yescharacteristic, direct product factor, Sylow
191#41—41no—
192#41—41no—
193#41—41no—
194#41—41no—
195C2×Q8—24no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—82no—
202C2×Q8—81no—
203C2×Q8—81no—
204C2×Q8SD16×C2312yescomplemented
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4C2×D4×C2311yescharacteristic
214C22×C4C2×D4×C2312yes—
215Q16—44no—
216SD16—44no—
217SD16—84no—
218M16—22no—
219M16—41no—
220C2×C8—41no—
221C2×C8—42no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—82no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C22⋊C4—42no—
228C42—41no—
229C42—81no—
230C23—41no—
231C23C23×#12011yescharacteristic
232Q8—42no—
233Q8—41no—
234Q8—41no—
235Q8—44no—
236Q8—81no—
237Q8—84no—
238Q8—82no—
239D4—41no—
240D4—41no—
241D4—82no—
242D4—81no—
243D4—81no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4C2×SD16×C2311yescharacteristic
264C2×C4C23×#13711yescharacteristic
265C8—42no—
266C8—82no—
267C22—21no—
268C22—21no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22C23×#44811yescharacteristic
273C22C23×#45311yescharacteristic
274C22C23×B(4,𝔽2)11yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—44no—
279C4—42no—
280C4—42no—
281C4—82no—
282C4—81no—
283C2—21no—
284C2—21no—
285C2—161no—
286C2C23×#159611yescharacteristic
287C2C23×#219911yescharacteristic
288C2C23×#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×#2811C2semidirect
4. C23×#1354C2semidirect
5. C23×#1319C2semidirect
59. C2×Q8×C23SD16semidirect
85. #12130C23direct (N × H)
123. #2811C2×C23semidirect
126. #1354C2×C23semidirect
127. #1319C2×C23semidirect
190. C23#12130direct (N × H)
204. C2×Q8SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#28112
C23×#20072
C23×#16042
C23×#13542
C23×#13192
#1213023