Finite Group Databasebeta

← C23×#12251 #19159877

Subgroups — C23×#12251 #19159877

Order 5888.

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

normal subgroups only

This group has 342 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×#2822C211yesmaximal, characteristic
2C23×#2061C211yesmaximal, characteristic
3C23×#1611C212yesmaximal
4C23×#1346C212yesmaximal
5C23×#1324C211yesmaximal, characteristic
6C23×#544C2211yescharacteristic
7C2×C23×#125—21no—
8C23×#483—22no—
9C23×#428—21no—
10C23×#400—22no—
11C23×#398—22no—
12C23×#398—21no—
13C23×#398C2212yes—
14C23×#382C2211yescharacteristic
15C23×#343C2211yescharacteristic
16C23×#339—21no—
17C23×#328—22no—
18C23×#324C2212yes—
19C2×M16×C23—21no—
20C23×C4⋊Q8—41no—
21C23×#125—42no—
22C23×#125—41no—
23C23×#125—42no—
24C23×C22⋊Q8—22no—
25C23×#117—21no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23D411yescharacteristic
32C2×C4⋊C4×C23D412yes—
33C2×C4⋊C4×C23C2311yescharacteristic
34C2×C42×C23D411yescharacteristic
35C23×#107—41no—
36C23×#106—41no—
37C23×#106—42no—
38C23×#103—42no—
39C23×C22⋊C8—22no—
40C23×#95—22no—
41C23×#95—21no—
42C23×#95—22no—
43C23×#95—22no—
44C23×#95D412yes—
45C2×Q8×C23—22no—
46C22×C4×C23—21no—
47C22×C4×C23—22no—
48C22×C4×C23—22no—
49C22×C4×C23—21no—
50C22×C4×C23C2×D411yescharacteristic
51C22×C4×C23C2×D412yes—
52M16×C23—22no—
53M16×C23—41no—
54C2×C8×C23—41no—
55C2×C8×C23—42no—
56C4⋊C4×C23—21no—
57C4⋊C4×C23—22no—
58C4⋊C4×C23—21no—
59C4⋊C4×C23—41no—
60C4⋊C4×C23—42no—
61C4⋊C4×C23—41no—
62C4⋊C4×C23—42no—
63C4⋊C4×C23—44no—
64C4⋊C4×C23—42no—
65C4⋊C4×C23—42no—
66C4⋊C4×C23—42no—
67C4⋊C4×C23—42no—
68C4⋊C4×C23—81no—
69C22⋊C4×C23—41no—
70C42×C23—21no—
71C42×C23—41no—
72C42×C23SD1612yes—
73#12251C2311yesmaximal, characteristic, direct product factor, Sylow
74C23×C23#12011yescharacteristic
75Q8×C23—82no—
76C2×C4×C23—21no—
77C2×C4×C23—21no—
78C2×C4×C23—22no—
79C2×C4×C23—22no—
80C2×C4×C23—41no—
81C2×C4×C23—41no—
82C2×C4×C23—42no—
83C2×C4×C23—42no—
84C2×C4×C23—42no—
85C2×C4×C23—42no—
86C2×C4×C23—42no—
87C2×C4×C23—42no—
88C2×C4×C23—42no—
89C2×C4×C23—41no—
90C2×C4×C23—41no—
91C2×C4×C23—42no—
92C2×C4×C23—41no—
93C2×C4×C23C2×SD1611yescharacteristic
94C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
95C8×C23—42no—
96C8×C23—82no—
97#2822C2×C2311yescharacteristic
98#2061C2×C2311yescharacteristic
99#1611C2×C2312yes—
100#1346C2×C2312yes—
101#1324C2×C2311yescharacteristic
102C22×C23—21no—
103C22×C23—21no—
104C22×C23#45011yescharacteristic
105C22×C23#45611yescharacteristic
106C22×C23B(4,𝔽2)11yescharacteristic
107C4×C23—21no—
108C4×C23—21no—
109C4×C23—42no—
110C4×C23—42no—
111C4×C23—82no—
112C4×C23—82no—
113C4×C23—81no—
114C4×C23—81no—
115C4×C23—81no—
116#544C22×C2311yescharacteristic
117C2×#125—21no—
118#483—22no—
119#428—21no—
120#400—22no—
121#398—21no—
122#398—22no—
123#398C22×C2312yes—
124#382C22×C2311yescharacteristic
125#343C22×C2311yescharacteristic
126#339—21no—
127#328—22no—
128#324C22×C2312yes—
129C2×C23—21no—
130C2×C23—21no—
131C2×C23#160411yescharacteristic
132C2×C23#220111yescharacteristic
133C2×C23#220311yescharacteristic
134C2×M16—21no—
135C4⋊Q8—41no—
136#125—42no—
137#125—41no—
138#125—42no—
139C22⋊Q8—22no—
140#117—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4D4×C2311yescharacteristic
147C2×C4⋊C4D4×C2312yes—
148C2×C4⋊C4C23×C2311yescharacteristic
149C2×C42D4×C2311yescharacteristic
150#107—41no—
151#106—41no—
152#106—42no—
153#103—42no—
154C22⋊C8—22no—
155#95—22no—
156#95—21no—
157#95—22no—
158#95—22no—
159#95D4×C2312yes—
160C23#1225111yescharacteristic, direct product factor, Sylow
161C2×Q8—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C2×D4×C2311yescharacteristic
167C22×C4C2×D4×C2312yes—
168M16—22no—
169M16—41no—
170C2×C8—41no—
171C2×C8—42no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C22⋊C4—41no—
186C42—21no—
187C42—41no—
188C42SD16×C2312yes—
189C23C23×#12011yescharacteristic
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4C2×SD16×C2311yescharacteristic
209C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
210C8—42no—
211C8—82no—
212C22—21no—
213C22—21no—
214C22C23×#45011yescharacteristic
215C22C23×#45611yescharacteristic
216C22C23×B(4,𝔽2)11yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—81no—
223C4—81no—
224C4—81no—
225C4—82no—
226C2—21no—
227C2—21no—
228C2C23×#160411yescharacteristic
229C2C23×#220111yescharacteristic
230C2C23×#220311yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×C23#120
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
73. #12251C23direct (N × H)
160. C23#12251direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28222
C23×#20612
C23×#16112
C23×#13462
C23×#13242
#1225123