Finite Group Databasebeta

← C23×#18551 #19166177

Subgroups — C23×#18551 #19166177

Order 5888.

918 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 246 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
1C2×Q82×C23C211yesmaximal, characteristic
2C23×#3139C211yesmaximal, characteristic
3C23×#3113C212yesmaximal
4C23×#1645C218yesmaximal, complemented
5C23×#1558C211yesmaximal, characteristic
6C23×#1491C212yesmaximal
7Q82×C23—22no—
8Q82×C23—24no—
9Q82×C23C2214yes—
10C23×#533C2211yescharacteristic
11C2×C23×C4⋊Q8—22no—
12C2×C23×C4⋊Q8—22no—
13C2×C23×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8×C23—22no—
15C2×C4×Q8×C23C2212yes—
16C23×#491—22no—
17C23×#487C2214yes—
18C23×#483—22no—
19C23×#475C2214yes—
20C23×#462C2214yes—
21C23×#451—22no—
22C23×#441—24no—
23C23×#432C2212yes—
24C23×#419—22no—
25C4×M16×C23C2211yescharacteristic
26C23×#333C2214yes—
27C23×Q8⋊C8C2218yes—
28C22×Q8×C23—22no—
29C22×Q8×C23D412yes—
30C22×Q8×C23D412yes—
31C2×Q16×C23—24no—
32C2×M16×C23—21no—
33C23×C4⋊Q8—28no—
34C23×C4⋊Q8—28no—
35C23×C4⋊Q8—28no—
36C23×C4⋊Q8—28no—
37C23×C4⋊Q8—22no—
38C23×C4⋊Q8D412yes—
39C23×C4⋊Q8C2312yes—
40C23×C4⋊Q8C2312yes—
41C23×#125—21no—
42C23×C22⋊Q8—22no—
43C4×Q8×C23—28no—
44C4×Q8×C23—28no—
45C4×Q8×C23—22no—
46C4×Q8×C23—22no—
47C4×Q8×C23—22no—
48C4×Q8×C23D414yes—
49C4×Q8×C23C2314yes—
50C23×#117—21no—
51C2×C4⋊C4×C23—24no—
52C2×C4⋊C4×C23—22no—
53C2×C4⋊C4×C23—22no—
54C2×C4⋊C4×C23D412yes—
55C2×C42×C23—22no—
56C2×C42×C23—22no—
57C2×C42×C23C2311yescharacteristic
58C23×#106—24no—
59C23×C4⋊C8C2314yes—
60C23×#103—28no—
61C23×#103—24no—
62C23×#103—24no—
63C23×C22⋊C8—22no—
64C23×#97—21no—
65C23×#97—22no—
66C4×C8×C23C2312yes—
67C2×Q8×C23—28no—
68C2×Q8×C23—28no—
69C2×Q8×C23—22no—
70C2×Q8×C23—22no—
71C2×Q8×C23—22no—
72C2×Q8×C23—44no—
73C2×Q8×C23—84no—
74C2×Q8×C23C2×D414yes—
75C2×Q8×C23C2×D414yes—
76C22×C4×C23—22no—
77C22×C4×C23—22no—
78C22×C4×C23—22no—
79C22×C4×C23C2×D412yes—
80C22×C4×C23C2×D411yescharacteristic
81Q16×C23—48no—
82M16×C23—42no—
83C2×C8×C23—22no—
84C2×C8×C23—24no—
85C4⋊C4×C23—216no—
86C4⋊C4×C23—24no—
87C4⋊C4×C23—24no—
88C4⋊C4×C23—22no—
89C4⋊C4×C23—24no—
90C4⋊C4×C23—24no—
91C4⋊C4×C23—22no—
92C4⋊C4×C23—22no—
93C4⋊C4×C23—42no—
94C4⋊C4×C23C2×D414yes—
95C22⋊C4×C23—41no—
96C42×C23—28no—
97C42×C23—24no—
98C42×C23—24no—
99C42×C23—41no—
100C42×C23C2×D412yes—
101C42×C23C2×D411yescharacteristic
102C42×C23C2411yescharacteristic
103#18551C2311yesmaximal, characteristic, direct product factor, Sylow
104C23×C23#12011yescharacteristic
105Q8×C23—28no—
106Q8×C23—24no—
107Q8×C23—44no—
108Q8×C23—44no—
109Q8×C23—82no—
110Q8×C23—88no—
111Q8×C23#13718yes—
112C2×C4×C23—28no—
113C2×C4×C23—24no—
114C2×C4×C23—22no—
115C2×C4×C23—24no—
116C2×C4×C23—24no—
117C2×C4×C23—24no—
118C2×C4×C23—44no—
119C2×C4×C23—44no—
120C2×C4×C23—42no—
121C2×C4×C23#12012yes—
122C2×C4×C23#12011yescharacteristic
123C2×C4×C23C22×D411yescharacteristic
124C2×C4×C23C22×D412yes—
125C8×C23—42no—
126C8×C23—44no—
127C2×Q82C2×C2311yescharacteristic
128#3139C2×C2311yescharacteristic
129#3113C2×C2312yes—
130#1645C2×C2318yescomplemented
131#1558C2×C2311yescharacteristic
132#1491C2×C2312yes—
133C22×C23—22no—
134C22×C23#45512yes—
135C22×C23C2×#12011yescharacteristic
136C4×C23—28no—
137C4×C23—22no—
138C4×C23—21no—
139C4×C23—21no—
140C4×C23—44no—
141C4×C23—44no—
142C4×C23—82no—
143C4×C23C2×#13714yes—
144Q82—22no—
145Q82—24no—
146Q82C22×C2314yes—
147#533C22×C2311yescharacteristic
148C2×C4⋊Q8—22no—
149C2×C4⋊Q8—22no—
150C2×C4⋊Q8C22×C2311yescharacteristic
151C2×C4×Q8—22no—
152C2×C4×Q8C22×C2312yes—
153#491—22no—
154#487C22×C2314yes—
155#483—22no—
156#475C22×C2314yes—
157#462C22×C2314yes—
158#451—22no—
159#441—24no—
160#432C22×C2312yes—
161#419—22no—
162C4×M16C22×C2311yescharacteristic
163#333C22×C2314yes—
164Q8⋊C8C22×C2318yes—
165C2×C23—22no—
166C2×C23#300412yes—
167C2×C23C2×#45511yescharacteristic
168C22×Q8—22no—
169C22×Q8D4×C2312yes—
170C22×Q8D4×C2312yes—
171C2×Q16—24no—
172C2×M16—21no—
173C4⋊Q8—28no—
174C4⋊Q8—28no—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—22no—
178C4⋊Q8D4×C2312yes—
179C4⋊Q8C23×C2312yes—
180C4⋊Q8C23×C2312yes—
181#125—21no—
182C22⋊Q8—22no—
183C4×Q8—28no—
184C4×Q8—28no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8D4×C2314yes—
189C4×Q8C23×C2314yes—
190#117—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4D4×C2312yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42C23×C2311yescharacteristic
198#106—24no—
199C4⋊C8C23×C2314yes—
200#103—28no—
201#103—24no—
202#103—24no—
203C22⋊C8—22no—
204#97—21no—
205#97—22no—
206C4×C8C23×C2312yes—
207C23#1855111yescharacteristic, direct product factor, Sylow
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—44no—
214C2×Q8—84no—
215C2×Q8C2×D4×C2314yes—
216C2×Q8C2×D4×C2314yes—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4C2×D4×C2312yes—
221C22×C4C2×D4×C2311yescharacteristic
222Q16—48no—
223M16—42no—
224C2×C8—22no—
225C2×C8—24no—
226C4⋊C4—216no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—42no—
235C4⋊C4C2×D4×C2314yes—
236C22⋊C4—41no—
237C42—28no—
238C42—24no—
239C42—24no—
240C42—41no—
241C42C2×D4×C2312yes—
242C42C2×D4×C2311yescharacteristic
243C42C24×C2311yescharacteristic
244C23C23×#12011yescharacteristic
245Q8—28no—
246Q8—24no—
247Q8—44no—
248Q8—44no—
249Q8—82no—
250Q8—88no—
251Q8C23×#13718yes—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4C23×#12012yes—
262C2×C4C23×#12011yescharacteristic
263C2×C4C22×D4×C2311yescharacteristic
264C2×C4C22×D4×C2312yes—
265C8—42no—
266C8—44no—
267C22—22no—
268C22C23×#45512yes—
269C22C2×C23×#12011yescharacteristic
270C4—28no—
271C4—22no—
272C4—21no—
273C4—21no—
274C4—44no—
275C4—44no—
276C4—82no—
277C4C2×C23×#13714yes—
278C2—22no—
279C2C23×#300412yes—
280C2C2×C23×#45511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×C23#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1645C2semidirect
103. #18551C23direct (N × H)
130. #1645C2×C23semidirect
207. C23#18551direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q82×C232
C23×#31392
C23×#31132
C23×#16452
C23×#15582
C23×#14912
#1855123