Finite Group Databasebeta

← C23×#12126 #19159752

Subgroups — C23×#12126 #19159752

Order 5888.

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

normal subgroups only

This group has 522 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×#2683C211yesmaximal, characteristic, complemented
2C23×#2006C211yesmaximal, characteristic
3C23×#1604C212yesmaximal, complemented
4C23×#1354C212yesmaximal, complemented
5C23×#1318C211yesmaximal, characteristic, complemented
6C2×C23×AGL(1,ℤ/8)—21no—
7C23×#537C2211yescharacteristic
8C2×C23×C22⋊Q8—22no—
9C23×#448—22no—
10C2×C23×#104—21no—
11C23×#396—22no—
12C23×#395C2212yes—
13C23×#390C2211yescharacteristic
14C23×#389—21no—
15C23×#386—21no—
16C23×#343C2211yescharacteristic
17C23×#327—22no—
18C23×#324C2212yes—
19C2×#41×C23—21no—
20C22×Q8×C23D412yes—
21C22×D4×C23D411yescharacteristic
22C23×C4×C23—21no—
23C23×AGL(1,ℤ/8)—42no—
24C23×AGL(1,ℤ/8)—42no—
25C2×SD16×C23—41no—
26C2×SD16×C23—42no—
27C2×D8×C23—41no—
28C2×M16×C23—21no—
29C23×C22⋊Q8—41no—
30C23×C22⋊Q8—48no—
31C23×#121—22no—
32C23×#121—41no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23—22no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23C2311yescharacteristic
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23—21no—
41C2×C42×C23D411yescharacteristic
42C23×#104—42no—
43C23×#103—42no—
44C23×C22⋊C8—22no—
45C23×#95—22no—
46C23×#95D412yes—
47C24×C23—21no—
48#41×C23—41no—
49#41×C23—41no—
50#41×C23—41no—
51#41×C23—41no—
52C2×Q8×C23—21no—
53C2×Q8×C23—44no—
54C2×Q8×C23—42no—
55C2×D4×C23—21no—
56C2×D4×C23—22no—
57C2×D4×C23—21no—
58C2×D4×C23—41no—
59C2×D4×C23—41no—
60C2×D4×C23—41no—
61C2×D4×C23SD1612yescomplemented
62C22×C4×C23—21no—
63C22×C4×C23—22no—
64C22×C4×C23—22no—
65C22×C4×C23—41no—
66C22×C4×C23—44no—
67C22×C4×C23—81no—
68C22×C4×C23C2×D411yescharacteristic
69C22×C4×C23C2×D412yes—
70SD16×C23—44no—
71SD16×C23—84no—
72D8×C23—44no—
73M16×C23—22no—
74M16×C23—41no—
75C2×C8×C23—41no—
76C2×C8×C23—42no—
77C4⋊C4×C23—24no—
78C4⋊C4×C23—22no—
79C4⋊C4×C23—41no—
80C4⋊C4×C23—44no—
81C4⋊C4×C23—44no—
82C4⋊C4×C23—81no—
83C22⋊C4×C23—44no—
84C22⋊C4×C23—44no—
85C22⋊C4×C23—42no—
86C22⋊C4×C23—81no—
87C42×C23—41no—
88#12126C2311yesmaximal, characteristic, direct product factor, Sylow
89C23×C23—22no—
90C23×C23—42no—
91C23×C23—42no—
92C23×C23—41no—
93C23×C23—41no—
94C23×C23—41no—
95C23×C23#12011yescharacteristic
96Q8×C23—41no—
97Q8×C23—41no—
98Q8×C23—44no—
99Q8×C23—82no—
100D4×C23—42no—
101D4×C23—41no—
102D4×C23—41no—
103D4×C23—81no—
104D4×C23—82no—
105D4×C23—81no—
106D4×C23—81no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—22no—
110C2×C4×C23—24no—
111C2×C4×C23—22no—
112C2×C4×C23—42no—
113C2×C4×C23—41no—
114C2×C4×C23—41no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—41no—
121C2×C4×C23—81no—
122C2×C4×C23—81no—
123C2×C4×C23—84no—
124C2×C4×C23C2×SD1611yescharacteristic
125C2×C4×C23#13711yescharacteristic
126C8×C23—42no—
127C8×C23—82no—
128#2683C2×C2311yescharacteristic, complemented
129#2006C2×C2311yescharacteristic
130#1604C2×C2312yescomplemented
131#1354C2×C2312yescomplemented
132#1318C2×C2311yescharacteristic, complemented
133C22×C23—21no—
134C22×C23—21no—
135C22×C23—44no—
136C22×C23—42no—
137C22×C23—82no—
138C22×C23—81no—
139C22×C23—81no—
140C22×C23—81no—
141C22×C23—81no—
142C22×C23—81no—
143C22×C23#44811yescharacteristic
144C22×C23#45311yescharacteristic
145C22×C23B(4,𝔽2)11yescharacteristic
146C4×C23—21no—
147C4×C23—21no—
148C4×C23—42no—
149C4×C23—42no—
150C4×C23—44no—
151C4×C23—82no—
152C4×C23—81no—
153C2×AGL(1,ℤ/8)—21no—
154#537C22×C2311yescharacteristic
155C2×C22⋊Q8—22no—
156#448—22no—
157C2×#104—21no—
158#396—22no—
159#395C22×C2312yes—
160#390C22×C2311yescharacteristic
161#389—21no—
162#386—21no—
163#343C22×C2311yescharacteristic
164#327—22no—
165#324C22×C2312yes—
166C2×C23—21no—
167C2×C23—21no—
168C2×C23—82no—
169C2×C23—161no—
170C2×C23#159611yescharacteristic
171C2×C23#219611yescharacteristic
172C2×C23#220311yescharacteristic
173C2×#41—21no—
174C22×Q8D4×C2312yes—
175C22×D4D4×C2311yescharacteristic
176C23×C4—21no—
177AGL(1,ℤ/8)—42no—
178AGL(1,ℤ/8)—42no—
179C2×SD16—41no—
180C2×SD16—42no—
181C2×D8—41no—
182C2×M16—21no—
183C22⋊Q8—41no—
184C22⋊Q8—48no—
185#121—22no—
186#121—41no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C23×C2311yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C42D4×C2311yescharacteristic
196#104—42no—
197#103—42no—
198C22⋊C8—22no—
199#95—22no—
200#95D4×C2312yes—
201C23#1212611yescharacteristic, direct product factor, Sylow
202C24—21no—
203#41—41no—
204#41—41no—
205#41—41no—
206#41—41no—
207C2×Q8—21no—
208C2×Q8—44no—
209C2×Q8—42no—
210C2×D4—21no—
211C2×D4—22no—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4SD16×C2312yescomplemented
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4—44no—
222C22×C4—81no—
223C22×C4C2×D4×C2311yescharacteristic
224C22×C4C2×D4×C2312yes—
225SD16—44no—
226SD16—84no—
227D8—44no—
228M16—22no—
229M16—41no—
230C2×C8—41no—
231C2×C8—42no—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—81no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—42no—
241C22⋊C4—81no—
242C42—41no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23C23×#12011yescharacteristic
250Q8—41no—
251Q8—41no—
252Q8—44no—
253Q8—82no—
254D4—42no—
255D4—41no—
256D4—41no—
257D4—81no—
258D4—82no—
259D4—81no—
260D4—81no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—84no—
278C2×C4C2×SD16×C2311yescharacteristic
279C2×C4C23×#13711yescharacteristic
280C8—42no—
281C8—82no—
282C22—21no—
283C22—21no—
284C22—44no—
285C22—42no—
286C22—82no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22C23×#44811yescharacteristic
293C22C23×#45311yescharacteristic
294C22C23×B(4,𝔽2)11yescharacteristic
295C4—21no—
296C4—21no—
297C4—42no—
298C4—42no—
299C4—44no—
300C4—82no—
301C4—81no—
302C2—21no—
303C2—21no—
304C2—82no—
305C2—161no—
306C2C23×#159611yescharacteristic
307C2C23×#219611yescharacteristic
308C2C23×#220311yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×#2006C2
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×#2683C2semidirect
3. C23×#1604C2semidirect
4. C23×#1354C2semidirect
5. C23×#1318C2semidirect
61. C2×D4×C23SD16semidirect
88. #12126C23direct (N × H)
128. #2683C2×C23semidirect
130. #1604C2×C23semidirect
131. #1354C2×C23semidirect
132. #1318C2×C23semidirect
201. C23#12126direct (N × H)
216. C2×D4SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#26832
C23×#20062
C23×#16042
C23×#13542
C23×#13182
#1212623