Finite Group Databasebeta

← C23×#12971 #19160597

Subgroups — C23×#12971 #19160597

Order 5888.

424 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2381C211yesmaximal, characteristic
2C2×C23×#356C211yesmaximal, characteristic
3C23×#1886C211yesmaximal, characteristic
4C23×#1857C212yesmaximal
5C23×#1394C212yesmaximal
6C2×C23×C4⋊Q8C2211yescharacteristic
7C23×#519C2211yescharacteristic
8C23×#428—22no—
9C23×#419—21no—
10C2×C23×#101C2211yescharacteristic
11C2×C23×#101C2212yes—
12C23×#391—21no—
13C23×#391C2212yes—
14C23×#387—22no—
15C23×#387—22no—
16C23×#383—21no—
17C23×#356C414yescomplemented
18C23×#339—21no—
19C23×#339—22no—
20C22×Q8×C23D411yescharacteristic
21C22×Q8×C23C2311yescharacteristic
22C2×M16×C23—22no—
23C2×M16×C23—21no—
24C23×C4⋊Q8—22no—
25C23×C4⋊Q8C2×C412yes—
26C23×C4⋊Q8Q812yes—
27C4×Q8×C23—42no—
28C23×#117—21no—
29C23×#117—22no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23D412yes—
33C2×C42×C23—21no—
34C2×C42×C23D411yescharacteristic
35C2×C42×C23D412yes—
36C23×#107—42no—
37C23×#106—42no—
38C23×#103—42no—
39C23×#101—22no—
40C23×#101—22no—
41C23×#101C2×C414yes—
42C23×#95—21no—
43C23×#95—22no—
44C23×#95—21no—
45C23×#95—22no—
46C23×#95—22no—
47C23×#95—22no—
48C2×Q8×C23—21no—
49C2×Q8×C23—41no—
50C2×Q8×C23—41no—
51C2×Q8×C23—41no—
52C2×Q8×C23—41no—
53C2×Q8×C23C4⋊C412yes—
54C2×Q8×C23C22×C411yescharacteristic
55C2×Q8×C23C2×Q811yescharacteristic
56C2×Q8×C23#4112yes—
57C22×C4×C23—21no—
58C22×C4×C23—21no—
59C22×C4×C23—22no—
60C22×C4×C23—22no—
61C22×C4×C23C2×D411yescharacteristic
62C22×C4×C23C2×D412yes—
63M16×C23—24no—
64M16×C23—42no—
65M16×C23—41no—
66M16×C23—41no—
67C2×C8×C23—42no—
68C2×C8×C23—41no—
69C4⋊C4×C23—24no—
70C4⋊C4×C23—24no—
71C4⋊C4×C23—22no—
72C4⋊C4×C23—22no—
73C4⋊C4×C23—22no—
74C4⋊C4×C23—42no—
75C4⋊C4×C23—81no—
76C22⋊C4×C23—41no—
77C42×C23—21no—
78C42×C23—42no—
79C42×C23—42no—
80C42×C23—41no—
81C42×C23—81no—
82C42×C23C4⋊C412yes—
83#12971C2311yesmaximal, characteristic, direct product factor, Sylow
84C23×C23#12011yescharacteristic
85Q8×C23—41no—
86Q8×C23—41no—
87Q8×C23—42no—
88Q8×C23—42no—
89Q8×C23—81no—
90C2×C4×C23—22no—
91C2×C4×C23—21no—
92C2×C4×C23—22no—
93C2×C4×C23—21no—
94C2×C4×C23—22no—
95C2×C4×C23—22no—
96C2×C4×C23—42no—
97C2×C4×C23—41no—
98C2×C4×C23—41no—
99C2×C4×C23—44no—
100C2×C4×C23—44no—
101C2×C4×C23—42no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23C2×C4⋊C411yescharacteristic
105C2×C4×C23C4×D412yes—
106C2×C4×C23C22⋊Q812yes—
107C2×C4×C23#12311yescharacteristic
108C8×C23—44no—
109C8×C23—81no—
110#2381C2×C2311yescharacteristic
111C2×#356C2×C2311yescharacteristic
112#1886C2×C2311yescharacteristic
113#1857C2×C2312yes—
114#1394C2×C2312yes—
115C22×C23—22no—
116C22×C23#38511yescharacteristic
117C22×C23B(4,𝔽2)11yescharacteristic
118C22×C23#45811yescharacteristic
119C4×C23—22no—
120C4×C23—21no—
121C4×C23—21no—
122C4×C23—22no—
123C4×C23—42no—
124C4×C23—42no—
125C4×C23—82no—
126C4×C23—82no—
127C4×C23—82no—
128C2×C4⋊Q8C22×C2311yescharacteristic
129#519C22×C2311yescharacteristic
130#428—22no—
131#419—21no—
132C2×#101C22×C2311yescharacteristic
133C2×#101C22×C2312yes—
134#391—21no—
135#391C22×C2312yes—
136#387—22no—
137#387—22no—
138#383—21no—
139#356C4×C2314yescomplemented
140#339—22no—
141#339—21no—
142C2×C23—21no—
143C2×C23—21no—
144C2×C23#186711yescharacteristic
145C2×C23#220312yes—
146C22×Q8D4×C2311yescharacteristic
147C22×Q8C23×C2311yescharacteristic
148C2×M16—22no—
149C2×M16—21no—
150C4⋊Q8—22no—
151C4⋊Q8C2×C4×C2312yes—
152C4⋊Q8Q8×C2312yes—
153C4×Q8—42no—
154#117—21no—
155#117—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4D4×C2312yes—
159C2×C42—21no—
160C2×C42D4×C2311yescharacteristic
161C2×C42D4×C2312yes—
162#107—42no—
163#106—42no—
164#103—42no—
165#101—22no—
166#101—22no—
167#101C2×C4×C2314yes—
168#95—21no—
169#95—22no—
170#95—21no—
171#95—22no—
172#95—22no—
173#95—22no—
174C23#1297111yescharacteristic, direct product factor, Sylow
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C4⋊C4×C2312yes—
181C2×Q8C22×C4×C2311yescharacteristic
182C2×Q8C2×Q8×C2311yescharacteristic
183C2×Q8#41×C2312yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4C2×D4×C2311yescharacteristic
189C22×C4C2×D4×C2312yes—
190M16—24no—
191M16—42no—
192M16—41no—
193M16—41no—
194C2×C8—42no—
195C2×C8—41no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—81no—
203C22⋊C4—41no—
204C42—21no—
205C42—42no—
206C42—42no—
207C42—41no—
208C42—81no—
209C42C4⋊C4×C2312yes—
210C23C23×#12011yescharacteristic
211Q8—41no—
212Q8—41no—
213Q8—42no—
214Q8—42no—
215Q8—81no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4C2×C4⋊C4×C2311yescharacteristic
231C2×C4C4×D4×C2312yes—
232C2×C4C23×C22⋊Q812yes—
233C2×C4C23×#12311yescharacteristic
234C8—44no—
235C8—81no—
236C22—22no—
237C22C23×#38511yescharacteristic
238C22C23×B(4,𝔽2)11yescharacteristic
239C22C23×#45811yescharacteristic
240C4—22no—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—82no—
248C4—82no—
249C2—21no—
250C2—21no—
251C2C23×#186711yescharacteristic
252C2C23×#220312yes—
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutatorC2×Q8C22×C4×C23
Frattini, cosocleC22×Q8C23×C23
prime radicalC23×C23#120
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
17. C23×#356C4semidirect
83. #12971C23direct (N × H)
139. #356C4×C23semidirect
174. C23#12971direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23812
C2×C23×#3562
C23×#18862
C23×#18572
C23×#13942
#1297123