Finite Group Databasebeta

← C23×#12224 #19159850

Subgroups — C23×#12224 #19159850

Order 5888.

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

normal subgroups only

This group has 380 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
2C23×#2028C211yesmaximal, characteristic
3C23×#1605C212yesmaximal
4C23×#1354C212yesmaximal
5C23×#1319C211yesmaximal, characteristic
6C23×#544C2211yescharacteristic
7C23×#451—22no—
8C23×#419—21no—
9C23×#395—22no—
10C23×#395C2212yes—
11C23×#391—22no—
12C23×#391—21no—
13C23×#391—21no—
14C23×#391C2211yescharacteristic
15C23×#343C2211yescharacteristic
16C23×#339—21no—
17C23×#328—22no—
18C23×#324C2212yes—
19C22×Q8×C23—21no—
20C22×Q8×C23D411yescharacteristic
21C22×Q8×C23D412yes—
22C2×Q16×C23—42no—
23C2×M16×C23—21no—
24C23×C4⋊Q8—41no—
25C23×#125—41no—
26C23×C22⋊Q8—22no—
27C23×#117—21no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23C2311yescharacteristic
32C2×C42×C23—21no—
33C2×C42×C23D411yescharacteristic
34C23×#103—41no—
35C23×#103—41no—
36C23×#103—42no—
37C23×C22⋊C8—22no—
38C23×#95—22no—
39C23×#95—22no—
40C23×#95—22no—
41C23×#95—22no—
42C23×#95D412yes—
43C2×Q8×C23—24no—
44C2×Q8×C23—21no—
45C2×Q8×C23—22no—
46C2×Q8×C23—41no—
47C2×Q8×C23—42no—
48C2×Q8×C23—41no—
49C2×Q8×C23—82no—
50C2×Q8×C23—81no—
51C2×Q8×C23—81no—
52C2×Q8×C23Q1612yescomplemented
53C22×C4×C23—22no—
54C22×C4×C23—21no—
55C22×C4×C23—21no—
56C22×C4×C23—22no—
57C22×C4×C23C2×D411yescharacteristic
58C22×C4×C23C2×D412yes—
59Q16×C23—84no—
60M16×C23—41no—
61M16×C23—41no—
62C2×C8×C23—41no—
63C2×C8×C23—42no—
64C4⋊C4×C23—21no—
65C4⋊C4×C23—21no—
66C4⋊C4×C23—22no—
67C4⋊C4×C23—41no—
68C4⋊C4×C23—42no—
69C4⋊C4×C23—82no—
70C4⋊C4×C23—81no—
71C4⋊C4×C23—81no—
72C22⋊C4×C23—41no—
73C42×C23—41no—
74C42×C23—41no—
75C42×C23—81no—
76#12224C2311yesmaximal, characteristic, direct product factor, Sylow
77C23×C23#12011yescharacteristic
78Q8×C23—42no—
79Q8×C23—44no—
80Q8×C23—81no—
81Q8×C23—84no—
82Q8×C23—82no—
83Q8×C23—82no—
84C2×C4×C23—21no—
85C2×C4×C23—21no—
86C2×C4×C23—24no—
87C2×C4×C23—22no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—42no—
91C2×C4×C23—41no—
92C2×C4×C23—41no—
93C2×C4×C23—41no—
94C2×C4×C23—42no—
95C2×C4×C23—41no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—41no—
100C2×C4×C23—41no—
101C2×C4×C23—42no—
102C2×C4×C23C2×Q1611yescharacteristic
103C2×C4×C23#13711yescharacteristic
104C8×C23—81no—
105C8×C23—82no—
106#2811C2×C2311yescharacteristic
107#2028C2×C2311yescharacteristic
108#1605C2×C2312yes—
109#1354C2×C2312yes—
110#1319C2×C2311yescharacteristic
111C22×C23—21no—
112C22×C23—21no—
113C22×C23#45111yescharacteristic
114C22×C23#45511yescharacteristic
115C22×C23B(4,𝔽2)11yescharacteristic
116C4×C23—21no—
117C4×C23—21no—
118C4×C23—42no—
119C4×C23—44no—
120C4×C23—42no—
121C4×C23—42no—
122C4×C23—82no—
123C4×C23—81no—
124C4×C23—81no—
125#544C22×C2311yescharacteristic
126#451—22no—
127#419—21no—
128#395—22no—
129#395C22×C2312yes—
130#391—21no—
131#391—21no—
132#391—22no—
133#391C22×C2311yescharacteristic
134#343C22×C2311yescharacteristic
135#339—21no—
136#328—22no—
137#324C22×C2312yes—
138C2×C23—21no—
139C2×C23—21no—
140C2×C23#160211yescharacteristic
141C2×C23#219911yescharacteristic
142C2×C23#220511yescharacteristic
143C22×Q8—21no—
144C22×Q8D4×C2311yescharacteristic
145C22×Q8D4×C2312yes—
146C2×Q16—42no—
147C2×M16—21no—
148C4⋊Q8—41no—
149#125—41no—
150C22⋊Q8—22no—
151#117—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C23×C2311yescharacteristic
156C2×C42—21no—
157C2×C42D4×C2311yescharacteristic
158#103—41no—
159#103—41no—
160#103—42no—
161C22⋊C8—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95D4×C2312yes—
167C23#1222411yescharacteristic, direct product factor, Sylow
168C2×Q8—24no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—41no—
172C2×Q8—42no—
173C2×Q8—41no—
174C2×Q8—82no—
175C2×Q8—81no—
176C2×Q8—81no—
177C2×Q8Q16×C2312yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×D4×C2311yescharacteristic
183C22×C4C2×D4×C2312yes—
184Q16—84no—
185M16—41no—
186M16—41no—
187C2×C8—42no—
188C2×C8—41no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—82no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C22⋊C4—41no—
198C42—41no—
199C42—41no—
200C42—81no—
201C23C23×#12011yescharacteristic
202Q8—42no—
203Q8—44no—
204Q8—81no—
205Q8—84no—
206Q8—82no—
207Q8—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4C2×Q16×C2311yescharacteristic
227C2×C4C23×#13711yescharacteristic
228C8—81no—
229C8—82no—
230C22—21no—
231C22—21no—
232C22C23×#45111yescharacteristic
233C22C23×#45511yescharacteristic
234C22C23×B(4,𝔽2)11yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—81no—
243C4—81no—
244C2—21no—
245C2—21no—
246C2C23×#160211yescharacteristic
247C2C23×#219911yescharacteristic
248C2C23×#220511yescharacteristic
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
52. C2×Q8×C23Q16semidirect
76. #12224C23direct (N × H)
167. C23#12224direct (N × H)
177. C2×Q8Q16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#28112
C23×#20282
C23×#16052
C23×#13542
C23×#13192
#1222423