Finite Group Databasebeta

← C23×#8462 #19156088

Subgroups — C23×#8462 #19156088

Order 5888.

1014 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 150 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×#3485C211yesmaximal, characteristic
2C23×#1782C211yesmaximal, characteristic, complemented
3C23×#1780C211yesmaximal, characteristic, complemented
4C23×#1516C214yesmaximal, complemented
5C2×C23×H(5,𝔽2)C411yescharacteristic, complemented
6C22×#41×C23C2211yescharacteristic
7C23×#547—24no—
8C23×#546—24no—
9C23×#536C412yescomplemented
10C23×#534—22no—
11C23×#525C2212yes—
12C2×C23×#123—22no—
13C23×#518C411yescharacteristic, complemented
14C2×C23×#99—22no—
15C2×C23×C22⋊C8—22no—
16C2×C23×C22⋊C8C2212yescomplemented
17C23×#381C2212yescomplemented
18C23×#328—24no—
19C23×#327—24no—
20C23×H(5,𝔽2)—44no—
21C2×#41×C23—22no—
22C2×#41×C23—24no—
23C2×#41×C23—44no—
24C2×#41×C23C2×C412yes—
25C22×Q8×C23C2×C411yescharacteristic
26C22×D4×C23—22no—
27C22×D4×C23—22no—
28C22×D4×C23C2×C411yescharacteristic
29C22×D4×C23D412yes—
30C23×C4×C23D412yes—
31C23×C4×C23C2311yescharacteristic
32C22×C8×C23—22no—
33C23×C4⋊Q8—22no—
34C23×#124—22no—
35C23×#123—24no—
36C23×#123—28no—
37C23×#123—24no—
38C23×#123—22no—
39C23×C22⋊Q8—24no—
40C23×C22⋊Q8D414yes—
41C23×#121—28no—
42C23×#121D414yes—
43C23×#120—24no—
44C23×#120—22no—
45C4×D4×C23—24no—
46C4×D4×C23—24no—
47C23×#117C2×C412yes—
48C2×C4⋊C4×C23—24no—
49C2×C22⋊C4×C23—24no—
50C2×C22⋊C4×C23—22no—
51C2×C22⋊C4×C23—22no—
52C23×#99—44no—
53C23×C22⋊C8—24no—
54C23×C22⋊C8—44no—
55C23×#95—24no—
56C24×C23—22no—
57C24×C23C22⋊C412yes—
58C24×C23C2×D411yescharacteristic
59#41×C23—44no—
60#41×C23—416no—
61#41×C23—48no—
62C2×Q8×C23—22no—
63C2×Q8×C23—42no—
64C2×Q8×C23C22⋊C412yes—
65C2×D4×C23—24no—
66C2×D4×C23—24no—
67C2×D4×C23—24no—
68C2×D4×C23—22no—
69C2×D4×C23—24no—
70C2×D4×C23—24no—
71C2×D4×C23—22no—
72C2×D4×C23—42no—
73C2×D4×C23—48no—
74C2×D4×C23—44no—
75C2×D4×C23—44no—
76C2×D4×C23C22⋊C412yes—
77C22×C4×C23—22no—
78C22×C4×C23—24no—
79C22×C4×C23—24no—
80C22×C4×C23—22no—
81C22×C4×C23—24no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23—42no—
85C22×C4×C23C22⋊C412yes—
86C22×C4×C23C22×C411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23C2×D414yes—
89C2×C8×C23—44no—
90C2×C8×C23—42no—
91C4⋊C4×C23—28no—
92C4⋊C4×C23—24no—
93C4⋊C4×C23C22⋊C414yes—
94C22⋊C4×C23—24no—
95C22⋊C4×C23—28no—
96C22⋊C4×C23—22no—
97C22⋊C4×C23—24no—
98C22⋊C4×C23—24no—
99C22⋊C4×C23—44no—
100C42×C23—22no—
101#8462C2311yesmaximal, characteristic, direct product factor, Sylow
102C23×C23—22no—
103C23×C23—22no—
104C23×C23—22no—
105C23×C23—22no—
106C23×C23—42no—
107C23×C23—44no—
108C23×C23—44no—
109C23×C23—84no—
110C23×C23#9914yes—
111C23×C23C2×C22⋊C411yescharacteristic
112C23×C23#12012yes—
113Q8×C23—42no—
114Q8×C23—44no—
115D4×C23—28no—
116D4×C23—42no—
117D4×C23—48no—
118D4×C23—48no—
119D4×C23—44no—
120D4×C23—84no—
121C2×C4×C23—22no—
122C2×C4×C23—22no—
123C2×C4×C23—24no—
124C2×C4×C23—22no—
125C2×C4×C23—22no—
126C2×C4×C23—22no—
127C2×C4×C23—22no—
128C2×C4×C23—44no—
129C2×C4×C23—48no—
130C2×C4×C23—44no—
131C2×C4×C23—44no—
132C2×C4×C23—84no—
133C2×C4×C23—82no—
134C2×C4×C23—84no—
135C2×C4×C23C2×C22⋊C412yes—
136C2×C4×C23#12012yes—
137C8×C23—82no—
138#3485C2×C2311yescharacteristic
139#1782C2×C2311yescharacteristic, complemented
140#1780C2×C2311yescharacteristic, complemented
141#1516C2×C2314yescomplemented
142C22×C23—22no—
143C22×C23—24no—
144C22×C23—24no—
145C22×C23—22no—
146C22×C23—44no—
147C22×C23—44no—
148C22×C23—44no—
149C22×C23—42no—
150C22×C23—82no—
151C22×C23—82no—
152C22×C23—164no—
153C22×C23#37911yescharacteristic
154C22×C23C2×#9912yes—
155C4×C23—22no—
156C4×C23—22no—
157C4×C23—44no—
158C4×C23—42no—
159C4×C23—42no—
160C4×C23—162no—
161C2×H(5,𝔽2)C4×C2311yescharacteristic, complemented
162C22×#41C22×C2311yescharacteristic
163#547—24no—
164#546—24no—
165#536C4×C2312yescomplemented
166#534—22no—
167#525C22×C2312yes—
168C2×#123—22no—
169#518C4×C2311yescharacteristic, complemented
170C2×#99—22no—
171C2×C22⋊C8—22no—
172C2×C22⋊C8C22×C2312yescomplemented
173#381C22×C2312yescomplemented
174#328—24no—
175#327—24no—
176C2×C23—22no—
177C2×C23—42no—
178C2×C23—44no—
179C2×C23—82no—
180C2×C23#178111yescharacteristic
181C2×C23#179412yes—
182H(5,𝔽2)—44no—
183C2×#41—22no—
184C2×#41—24no—
185C2×#41—44no—
186C2×#41C2×C4×C2312yes—
187C22×Q8C2×C4×C2311yescharacteristic
188C22×D4—22no—
189C22×D4—22no—
190C22×D4C2×C4×C2311yescharacteristic
191C22×D4D4×C2312yes—
192C23×C4D4×C2312yes—
193C23×C4C23×C2311yescharacteristic
194C22×C8—22no—
195C4⋊Q8—22no—
196#124—22no—
197#123—28no—
198#123—24no—
199#123—24no—
200#123—22no—
201C22⋊Q8—24no—
202C22⋊Q8D4×C2314yes—
203#121—28no—
204#121D4×C2314yes—
205#120—24no—
206#120—22no—
207C4×D4—24no—
208C4×D4—24no—
209#117C2×C4×C2312yes—
210C2×C4⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214#99—44no—
215C22⋊C8—24no—
216C22⋊C8—44no—
217#95—24no—
218C23#846211yescharacteristic, direct product factor, Sylow
219C24—22no—
220C24C22⋊C4×C2312yes—
221C24C2×D4×C2311yescharacteristic
222#41—44no—
223#41—416no—
224#41—48no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×Q8C22⋊C4×C2312yes—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—48no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4C22⋊C4×C2312yes—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C22⋊C4×C2312yes—
249C22×C4C22×C4×C2311yescharacteristic
250C22×C4C2×D4×C2311yescharacteristic
251C22×C4C2×D4×C2314yes—
252C2×C8—44no—
253C2×C8—42no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4C22⋊C4×C2314yes—
257C22⋊C4—24no—
258C22⋊C4—28no—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—44no—
263C42—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—42no—
269C23—44no—
270C23—44no—
271C23—84no—
272C23C23×#9914yes—
273C23C2×C22⋊C4×C2311yescharacteristic
274C23C23×#12012yes—
275Q8—42no—
276Q8—44no—
277D4—28no—
278D4—42no—
279D4—48no—
280D4—48no—
281D4—44no—
282D4—84no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—84no—
297C2×C4C2×C22⋊C4×C2312yes—
298C2×C4C23×#12012yes—
299C8—82no—
300C22—22no—
301C22—24no—
302C22—22no—
303C22—24no—
304C22—44no—
305C22—44no—
306C22—42no—
307C22—44no—
308C22—82no—
309C22—82no—
310C22—164no—
311C22C23×#37911yescharacteristic
312C22C2×C23×#9912yes—
313C4—22no—
314C4—22no—
315C4—42no—
316C4—42no—
317C4—44no—
318C4—162no—
319C2—22no—
320C2—42no—
321C2—44no—
322C2—82no—
323C2C23×#178111yescharacteristic
324C2C23×#179412yes—
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#3485C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1782C2semidirect
3. C23×#1780C2semidirect
4. C23×#1516C2semidirect
5. C2×C23×H(5,𝔽2)C4semidirect
9. C23×#536C4semidirect
13. C23×#518C4semidirect
16. C2×C23×C22⋊C8C22semidirect
17. C23×#381C22semidirect
101. #8462C23direct (N × H)
139. #1782C2×C23semidirect
140. #1780C2×C23semidirect
141. #1516C2×C23semidirect
161. C2×H(5,𝔽2)C4×C23semidirect
165. #536C4×C23semidirect
169. #518C4×C23semidirect
172. C2×C22⋊C8C22×C23semidirect
173. #381C22×C23semidirect
218. C23#8462direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34852
C23×#17822
C23×#17802
C23×#15162
#846223