Finite Group Databasebeta

← C23×#11532 #19159158

Subgroups — C23×#11532 #19159158

Order 5888.

412 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C4×C23×C4⋊Q8C211yesmaximal, characteristic
2C23×#1467C212yesmaximal
3C23×#1325C212yesmaximal
4C23×#1319C212yesmaximal
5C2×C23×C4⋊Q8C411yescharacteristic
6C2×C4×Q8×C23C411yescharacteristic
7C2×C4×Q8×C23C2211yescharacteristic
8C23×#432—22no—
9C23×#391C411yescharacteristic
10C23×#391C2211yescharacteristic
11C23×#389—22no—
12C23×#384C2211yescharacteristic
13C4×C4⋊C4×C23—22no—
14C43×C23C411yescharacteristic
15C23×#324—22no—
16C23×#324C2214yes—
17C22×Q8×C23C2×C411yescharacteristic
18C22×Q8×C23C2×C411yescharacteristic
19C23×C4⋊Q8—22no—
20C23×C4⋊Q8—22no—
21C4×Q8×C23—41no—
22C4×Q8×C23—41no—
23C4×Q8×C23—41no—
24C4×Q8×C23—41no—
25C2×C4⋊C4×C23—22no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23C2×C411yescharacteristic
29C2×C4⋊C4×C23D412yes—
30C2×C4⋊C4×C23C2311yescharacteristic
31C2×C42×C23—21no—
32C2×C42×C23—21no—
33C2×C42×C23—21no—
34C2×C42×C23C2×C411yescharacteristic
35C2×C42×C23C2×C411yescharacteristic
36C2×C42×C23C2×C411yescharacteristic
37C2×C42×C23D412yes—
38C23×C4⋊C8—42no—
39C23×C22⋊C8—24no—
40C23×#97—42no—
41C23×#95—22no—
42C23×#95—22no—
43C2×Q8×C23—21no—
44C2×Q8×C23—21no—
45C2×Q8×C23—21no—
46C2×Q8×C23—41no—
47C2×Q8×C23—41no—
48C2×Q8×C23—41no—
49C2×Q8×C23—41no—
50C2×Q8×C23M1612yes—
51C22×C4×C23—21no—
52C22×C4×C23—21no—
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23C22⋊C412yes—
56C22×C4×C23C22⋊C411yescharacteristic
57C22×C4×C23C22⋊C411yescharacteristic
58C22×C4×C23C22×C411yescharacteristic
59C22×C4×C23C2×D412yes—
60C2×C8×C23—44no—
61C4⋊C4×C23—28no—
62C4⋊C4×C23—42no—
63C4⋊C4×C23—41no—
64C4⋊C4×C23—41no—
65C4⋊C4×C23—44no—
66C4⋊C4×C23—42no—
67C42×C23—24no—
68C42×C23—22no—
69C42×C23—21no—
70C42×C23—21no—
71C42×C23—22no—
72C42×C23—21no—
73C42×C23—41no—
74C42×C23—42no—
75C42×C23—41no—
76C42×C23—41no—
77C42×C23M1612yes—
78#11532C2311yesmaximal, characteristic, direct product factor, Sylow
79C23×C23C2×C22⋊C411yescharacteristic
80Q8×C23—42no—
81Q8×C23—42no—
82Q8×C23—42no—
83Q8×C23—42no—
84C2×C4×C23—22no—
85C2×C4×C23—22no—
86C2×C4×C23—22no—
87C2×C4×C23—21no—
88C2×C4×C23—21no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—22no—
93C2×C4×C23—21no—
94C2×C4×C23—22no—
95C2×C4×C23—22no—
96C2×C4×C23—22no—
97C2×C4×C23—41no—
98C2×C4×C23—41no—
99C2×C4×C23—41no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—41no—
105C2×C4×C23#9911yescharacteristic
106C2×C4×C23#9911yescharacteristic
107C2×C4×C23C2×M1611yescharacteristic
108C2×C4×C23C2×M1611yescharacteristic
109C8×C23—84no—
110C4×C4⋊Q8C2×C2311yescharacteristic
111#1467C2×C2312yes—
112#1325C2×C2312yes—
113#1319C2×C2312yes—
114C22×C23—21no—
115C22×C23—21no—
116C22×C23#40711yescharacteristic
117C22×C23C2×#9911yescharacteristic
118C22×C23#41311yescharacteristic
119C4×C23—24no—
120C4×C23—21no—
121C4×C23—21no—
122C4×C23—21no—
123C4×C23—42no—
124C4×C23—42no—
125C4×C23—41no—
126C4×C23—42no—
127C4×C23—42no—
128C4×C23—81no—
129C4×C23#35612yes—
130C2×C4⋊Q8C4×C2311yescharacteristic
131C2×C4×Q8C4×C2311yescharacteristic
132C2×C4×Q8C22×C2311yescharacteristic
133#432—22no—
134#391C4×C2311yescharacteristic
135#391C22×C2311yescharacteristic
136#389—22no—
137#384C22×C2311yescharacteristic
138C4×C4⋊C4—22no—
139C43C4×C2311yescharacteristic
140#324—22no—
141#324C22×C2314yes—
142C2×C23—21no—
143C2×C23—21no—
144C2×C23#146511yescharacteristic
145C2×C23#212611yescharacteristic
146C2×C23C2×#35611yescharacteristic
147C22×Q8C2×C4×C2311yescharacteristic
148C22×Q8C2×C4×C2311yescharacteristic
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×C4×C2311yescharacteristic
159C2×C4⋊C4D4×C2312yes—
160C2×C4⋊C4C23×C2311yescharacteristic
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42C2×C4×C2311yescharacteristic
165C2×C42C2×C4×C2311yescharacteristic
166C2×C42C2×C4×C2311yescharacteristic
167C2×C42D4×C2312yes—
168C4⋊C8—42no—
169C22⋊C8—24no—
170#97—42no—
171#95—22no—
172#95—22no—
173C23#1153211yescharacteristic, direct product factor, Sylow
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8M16×C2312yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4C22⋊C4×C2312yes—
187C22×C4C22⋊C4×C2311yescharacteristic
188C22×C4C22⋊C4×C2311yescharacteristic
189C22×C4C22×C4×C2311yescharacteristic
190C22×C4C2×D4×C2312yes—
191C2×C8—44no—
192C4⋊C4—28no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C42—24no—
199C42—22no—
200C42—22no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—42no—
206C42—41no—
207C42—41no—
208C42M16×C2312yes—
209C23C2×C22⋊C4×C2311yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4C23×#9911yescharacteristic
236C2×C4C23×#9911yescharacteristic
237C2×C4C2×M16×C2311yescharacteristic
238C2×C4C2×M16×C2311yescharacteristic
239C8—84no—
240C22—21no—
241C22—21no—
242C22C23×#40711yescharacteristic
243C22C2×C23×#9911yescharacteristic
244C22C23×#41311yescharacteristic
245C4—24no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—42no—
252C4—42no—
253C4—41no—
254C4—81no—
255C4C23×#35612yes—
256C2—21no—
257C2—21no—
258C2C23×#146511yescharacteristic
259C2C23×#212611yescharacteristic
260C2C2×C23×#35611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×C23C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
78. #11532C23direct (N × H)
173. C23#11532direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C23×C4⋊Q82
C23×#14672
C23×#13252
C23×#13192
#1153223