Finite Group Databasebeta

← C23×#12206 #19159832

Subgroups — C23×#12206 #19159832

Order 5888.

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

normal subgroups only

This group has 500 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×#2815C211yesmaximal, characteristic
2C23×#2030C211yesmaximal, characteristic, complemented
3C23×#1596C212yesmaximal, complemented
4C23×#1343C212yesmaximal, complemented
5C23×#1317C211yesmaximal, characteristic, complemented
6C23×#543C2211yescharacteristic
7C2×C23×#121—21no—
8C23×#450—22no—
9C23×#397—22no—
10C23×#392—22no—
11C23×#392—21no—
12C23×#392C2212yes—
13C23×#381—21no—
14C23×#381C2211yescharacteristic
15C23×#342—21no—
16C23×#342C2211yescharacteristic
17C23×#328—22no—
18C23×#323C2212yes—
19C22×D4×C23—21no—
20C22×D4×C23—22no—
21C22×D4×C23—21no—
22C22×D4×C23D412yes—
23C23×C4×C23D411yescharacteristic
24C2×SD16×C23—42no—
25C23×C22⋊Q8—22no—
26C23×C22⋊Q8—42no—
27C23×#121—42no—
28C23×#121—42no—
29C2×C4⋊C4×C23—21no—
30C2×C22⋊C4×C23—22no—
31C2×C22⋊C4×C23—21no—
32C2×C22⋊C4×C23—21no—
33C2×C22⋊C4×C23—22no—
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23D411yescharacteristic
36C2×C22⋊C4×C23C2311yescharacteristic
37C23×#102—42no—
38C23×C22⋊C8—22no—
39C23×#95—21no—
40C23×#95—22no—
41C23×#95—42no—
42C23×#95D412yes—
43C24×C23—22no—
44C24×C23C2×D411yescharacteristic
45C2×Q8×C23—22no—
46C2×D4×C23—42no—
47C2×D4×C23—42no—
48C2×D4×C23—42no—
49C2×D4×C23—42no—
50C2×D4×C23—82no—
51C2×D4×C23—81no—
52C2×D4×C23—81no—
53C2×D4×C23—82no—
54C2×D4×C23—82no—
55C2×D4×C23—82no—
56C2×D4×C23—81no—
57C22×C4×C23—21no—
58C22×C4×C23—21no—
59C22×C4×C23—22no—
60C22×C4×C23—21no—
61C22×C4×C23—22no—
62C22×C4×C23—41no—
63C22×C4×C23—41no—
64C22×C4×C23—41no—
65C22×C4×C23SD1612yescomplemented
66C22×C4×C23C2×D412yes—
67SD16×C23—84no—
68C2×C8×C23—42no—
69C4⋊C4×C23—22no—
70C4⋊C4×C23—42no—
71C4⋊C4×C23—42no—
72C22⋊C4×C23—41no—
73C22⋊C4×C23—44no—
74C22⋊C4×C23—41no—
75C22⋊C4×C23—41no—
76C22⋊C4×C23—41no—
77C22⋊C4×C23—41no—
78C22⋊C4×C23—41no—
79C22⋊C4×C23—82no—
80C22⋊C4×C23—82no—
81#12206C2311yesmaximal, characteristic, direct product factor, Sylow
82C23×C23—21no—
83C23×C23—21no—
84C23×C23—42no—
85C23×C23—42no—
86C23×C23—42no—
87C23×C23—42no—
88C23×C23—42no—
89C23×C23—42no—
90C23×C23—42no—
91C23×C23—42no—
92C23×C23#12011yescharacteristic
93C23×C23C2×SD1611yescharacteristic
94C23×C23AGL(1,ℤ/8)11yescharacteristic
95Q8×C23—82no—
96D4×C23—44no—
97D4×C23—82no—
98D4×C23—84no—
99D4×C23—84no—
100D4×C23—84no—
101D4×C23—84no—
102C2×C4×C23—22no—
103C2×C4×C23—22no—
104C2×C4×C23—41no—
105C2×C4×C23—41no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—41no—
113C2×C4×C23—42no—
114C2×C4×C23—81no—
115C2×C4×C23—82no—
116C2×C4×C23—81no—
117C2×C4×C23—82no—
118C8×C23—82no—
119#2815C2×C2311yescharacteristic
120#2030C2×C2311yescharacteristic, complemented
121#1596C2×C2312yescomplemented
122#1343C2×C2312yescomplemented
123#1317C2×C2311yescharacteristic, complemented
124C22×C23—22no—
125C22×C23—21no—
126C22×C23—21no—
127C22×C23—41no—
128C22×C23—41no—
129C22×C23—42no—
130C22×C23—81no—
131C22×C23—82no—
132C22×C23—82no—
133C22×C23—82no—
134C22×C23—82no—
135C22×C23—82no—
136C22×C23—82no—
137C22×C23—82no—
138C22×C23#45011yescharacteristic
139C22×C23#45511yescharacteristic
140C22×C23B(4,𝔽2)11yescharacteristic
141C4×C23—42no—
142C4×C23—42no—
143C4×C23—82no—
144C4×C23—81no—
145C4×C23—82no—
146C4×C23—161no—
147#543C22×C2311yescharacteristic
148C2×#121—21no—
149#450—22no—
150#397—22no—
151#392—22no—
152#392—21no—
153#392C22×C2312yes—
154#381—21no—
155#381C22×C2311yescharacteristic
156#342—21no—
157#342C22×C2311yescharacteristic
158#328—22no—
159#323C22×C2312yes—
160C2×C23—21no—
161C2×C23—21no—
162C2×C23—41no—
163C2×C23—41no—
164C2×C23—162no—
165C2×C23#160111yescharacteristic
166C2×C23#220012yes—
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4D4×C2312yes—
171C23×C4D4×C2311yescharacteristic
172C2×SD16—42no—
173C22⋊Q8—22no—
174C22⋊Q8—42no—
175#121—42no—
176#121—42no—
177C2×C4⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4D4×C2311yescharacteristic
184C2×C22⋊C4C23×C2311yescharacteristic
185#102—42no—
186C22⋊C8—22no—
187#95—21no—
188#95—22no—
189#95—42no—
190#95D4×C2312yes—
191C23#1220611yescharacteristic, direct product factor, Sylow
192C24—22no—
193C24C2×D4×C2311yescharacteristic
194C2×Q8—22no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—82no—
205C2×D4—81no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4SD16×C2312yescomplemented
215C22×C4C2×D4×C2312yes—
216SD16—84no—
217C2×C8—42no—
218C4⋊C4—22no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—44no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23C23×#12011yescharacteristic
241C23C2×SD16×C2311yescharacteristic
242C23C23×AGL(1,ℤ/8)11yescharacteristic
243Q8—82no—
244D4—44no—
245D4—82no—
246D4—84no—
247D4—84no—
248D4—84no—
249D4—84no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—82no—
266C8—82no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22C23×#45011yescharacteristic
282C22C23×#45511yescharacteristic
283C22C23×B(4,𝔽2)11yescharacteristic
284C4—42no—
285C4—42no—
286C4—82no—
287C4—81no—
288C4—82no—
289C4—161no—
290C2—21no—
291C2—21no—
292C2—41no—
293C2—41no—
294C2—162no—
295C2C23×#160111yescharacteristic
296C2C23×#220012yes—
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#2815C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C22⋊C4×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C23×#2030C2semidirect
3. C23×#1596C2semidirect
4. C23×#1343C2semidirect
5. C23×#1317C2semidirect
65. C22×C4×C23SD16semidirect
81. #12206C23direct (N × H)
120. #2030C2×C23semidirect
121. #1596C2×C23semidirect
122. #1343C2×C23semidirect
123. #1317C2×C23semidirect
191. C23#12206direct (N × H)
214. C22×C4SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#28152
C23×#20302
C23×#15962
C23×#13432
C23×#13172
#1220623