Finite Group Databasebeta

← C23×#12727 #19160353

Subgroups — C23×#12727 #19160353

Order 5888.

810 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3508C211yesmaximal, characteristic
2C23×#2127C212yesmaximal, complemented
3C2×C23×#353C212yesmaximal, complemented
4C23×#1828C212yesmaximal, complemented
5C23×#559—22no—
6C23×#550—22no—
7C23×#546—24no—
8C23×#540—22no—
9C23×#535C411yescharacteristic, complemented
10C23×#535C2211yescharacteristic
11C23×#533C411yescharacteristic, complemented
12C23×#532—22no—
13C2×C23×#127C2211yescharacteristic
14C23×#518C411yescharacteristic, complemented
15C23×#518C2211yescharacteristic
16C4×#41×C23C411yescharacteristic, complemented
17C2×C23×#99C2214yes—
18C23×#354—22no—
19C23×#353—22no—
20C23×#353—22no—
21C23×#353—22no—
22C23×#342—24no—
23C2×#41×C23—24no—
24C2×#41×C23C2×C411yescharacteristic
25C22×D4×C23—22no—
26C22×D4×C23C2×C411yescharacteristic
27C22×D4×C23C2311yescharacteristic
28C23×C4⋊Q8—21no—
29C23×#127—21no—
30C23×#127—22no—
31C23×#127D412yes—
32C23×#127D412yes—
33C23×#127D412yes—
34C23×#125—21no—
35C23×#124—24no—
36C23×#124—21no—
37C23×#124D412yes—
38C23×#123—24no—
39C23×#123—24no—
40C23×C22⋊Q8—22no—
41C23×#121—24no—
42C23×#121—24no—
43C23×#121—24no—
44C23×#121—22no—
45C23×#121—22no—
46C23×#120—22no—
47C23×#120—22no—
48C4×Q8×C23—21no—
49C4×Q8×C23—22no—
50C4×D4×C23—24no—
51C4×D4×C23—21no—
52C4×D4×C23—22no—
53C4×D4×C23—22no—
54C4×D4×C23—22no—
55C4×D4×C23—22no—
56C4×D4×C23—22no—
57C23×#117—21no—
58C23×#117C2×C411yescharacteristic
59C23×#117C2×C411yescharacteristic
60C23×#117C2×C411yescharacteristic
61C2×C22⋊C4×C23—22no—
62C2×C22⋊C4×C23—24no—
63C2×C22⋊C4×C23D414yes—
64C2×C42×C23—21no—
65C2×C42×C23C2×C411yescharacteristic
66C23×#99—24no—
67C23×#99—24no—
68C24×C23—21no—
69C24×C23C2×D412yes—
70#41×C23—42no—
71#41×C23—88no—
72C2×Q8×C23—22no—
73C2×Q8×C23C22⋊C411yescharacteristic
74C2×D4×C23—21no—
75C2×D4×C23—24no—
76C2×D4×C23—24no—
77C2×D4×C23—24no—
78C2×D4×C23—24no—
79C2×D4×C23—21no—
80C2×D4×C23—42no—
81C2×D4×C23—42no—
82C2×D4×C23—84no—
83C2×D4×C23C22⋊C412yes—
84C2×D4×C23C22⋊C411yescharacteristic
85C2×D4×C23C2×D412yes—
86C2×D4×C23C2×D412yes—
87C22×C4×C23—22no—
88C22×C4×C23—21no—
89C22×C4×C23—24no—
90C22×C4×C23—24no—
91C22×C4×C23—21no—
92C22×C4×C23—44no—
93C22×C4×C23C22×C411yescharacteristic
94C4⋊C4×C23—22no—
95C4⋊C4×C23—22no—
96C4⋊C4×C23—22no—
97C4⋊C4×C23—22no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C22⋊C4×C23—24no—
102C22⋊C4×C23—22no—
103C22⋊C4×C23—24no—
104C22⋊C4×C23—21no—
105C22⋊C4×C23—24no—
106C22⋊C4×C23—24no—
107C22⋊C4×C23—44no—
108C22⋊C4×C23—44no—
109C22⋊C4×C23C22⋊C412yes—
110C22⋊C4×C23C22⋊C412yes—
111C42×C23—22no—
112C42×C23—21no—
113C42×C23—21no—
114C42×C23—21no—
115C42×C23C22⋊C411yescharacteristic
116C42×C23C22⋊C411yescharacteristic
117C42×C23C22⋊C412yes—
118#12727C2311yesmaximal, characteristic, direct product factor, Sylow
119C23×C23—24no—
120C23×C23—22no—
121C23×C23—21no—
122C23×C23—42no—
123C23×C23—82no—
124C23×C23—82no—
125C23×C23C2×C22⋊C411yescharacteristic
126C23×C23#12014yes—
127Q8×C23—41no—
128Q8×C23—44no—
129D4×C23—42no—
130D4×C23—42no—
131D4×C23—42no—
132D4×C23—41no—
133D4×C23—42no—
134D4×C23—48no—
135D4×C23—42no—
136D4×C23—84no—
137D4×C23—84no—
138D4×C23—84no—
139C2×C4×C23—21no—
140C2×C4×C23—22no—
141C2×C4×C23—21no—
142C2×C4×C23—22no—
143C2×C4×C23—21no—
144C2×C4×C23—22no—
145C2×C4×C23—22no—
146C2×C4×C23—42no—
147C2×C4×C23—42no—
148C2×C4×C23—44no—
149C2×C4×C23—84no—
150C2×C4×C23—84no—
151C2×C4×C23—84no—
152C2×C4×C23—164no—
153C2×C4×C23C2×C22⋊C411yescharacteristic
154C2×C4×C23C2×C22⋊C411yescharacteristic
155#3508C2×C2311yescharacteristic
156#2127C2×C2312yescomplemented
157C2×#353C2×C2312yescomplemented
158#1828C2×C2312yescomplemented
159C22×C23—24no—
160C22×C23—41no—
161C22×C23—44no—
162C22×C23—42no—
163C22×C23—44no—
164C22×C23—41no—
165C22×C23—42no—
166C22×C23—84no—
167C22×C23—162no—
168C22×C23#37911yescharacteristic
169C22×C23#41012yes—
170C4×C23—22no—
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—42no—
174C4×C23—41no—
175C4×C23—42no—
176C4×C23—42no—
177C4×C23—41no—
178C4×C23—164no—
179#559—22no—
180#550—22no—
181#546—24no—
182#540—22no—
183#535C4×C2311yescharacteristic, complemented
184#535C22×C2311yescharacteristic
185#533C4×C2311yescharacteristic, complemented
186#532—22no—
187C2×#127C22×C2311yescharacteristic
188#518C4×C2311yescharacteristic, complemented
189#518C22×C2311yescharacteristic
190C4×#41C4×C2311yescharacteristic, complemented
191C2×#99C22×C2314yes—
192#354—22no—
193#353—22no—
194#353—22no—
195#353—22no—
196#342—24no—
197C2×C23—21no—
198C2×C23—22no—
199C2×C23—44no—
200C2×C23—81no—
201C2×C23—82no—
202C2×C23#178211yescharacteristic
203C2×#41—24no—
204C2×#41C2×C4×C2311yescharacteristic
205C22×D4—22no—
206C22×D4C2×C4×C2311yescharacteristic
207C22×D4C23×C2311yescharacteristic
208C4⋊Q8—21no—
209#127—21no—
210#127—22no—
211#127D4×C2312yes—
212#127D4×C2312yes—
213#127D4×C2312yes—
214#125—21no—
215#124—24no—
216#124—21no—
217#124D4×C2312yes—
218#123—24no—
219#123—24no—
220C22⋊Q8—22no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—22no—
225#121—22no—
226#120—22no—
227#120—22no—
228C4×Q8—21no—
229C4×Q8—22no—
230C4×D4—24no—
231C4×D4—21no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237#117—21no—
238#117C2×C4×C2311yescharacteristic
239#117C2×C4×C2311yescharacteristic
240#117C2×C4×C2311yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4D4×C2314yes—
244C2×C42—21no—
245C2×C42C2×C4×C2311yescharacteristic
246#99—24no—
247#99—24no—
248C23#1272711yescharacteristic, direct product factor, Sylow
249C24—21no—
250C24C2×D4×C2312yes—
251#41—42no—
252#41—88no—
253C2×Q8—22no—
254C2×Q8C22⋊C4×C2311yescharacteristic
255C2×D4—21no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—84no—
264C2×D4C22⋊C4×C2312yes—
265C2×D4C22⋊C4×C2311yescharacteristic
266C2×D4C2×D4×C2312yes—
267C2×D4C2×D4×C2312yes—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—44no—
274C22×C4C22×C4×C2311yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—21no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4C22⋊C4×C2312yes—
291C22⋊C4C22⋊C4×C2312yes—
292C42—22no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22⋊C4×C2311yescharacteristic
297C42C22⋊C4×C2311yescharacteristic
298C42C22⋊C4×C2312yes—
299C23—24no—
300C23—22no—
301C23—21no—
302C23—42no—
303C23—82no—
304C23—82no—
305C23C2×C22⋊C4×C2311yescharacteristic
306C23C23×#12014yes—
307Q8—41no—
308Q8—44no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—41no—
313D4—42no—
314D4—48no—
315D4—42no—
316D4—84no—
317D4—84no—
318D4—84no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—164no—
333C2×C4C2×C22⋊C4×C2311yescharacteristic
334C2×C4C2×C22⋊C4×C2311yescharacteristic
335C22—24no—
336C22—41no—
337C22—44no—
338C22—42no—
339C22—44no—
340C22—41no—
341C22—42no—
342C22—84no—
343C22—162no—
344C22C23×#37911yescharacteristic
345C22C23×#41012yes—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—41no—
351C4—42no—
352C4—42no—
353C4—41no—
354C4—164no—
355C2—21no—
356C2—22no—
357C2—44no—
358C2—82no—
359C2—81no—
360C2C23×#178211yescharacteristic
subgroupG/H
center, socleC2×C23#1782
commutatorC22×C4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C23×#518 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2127C2semidirect
3. C2×C23×#353C2semidirect
4. C23×#1828C2semidirect
9. C23×#535C4semidirect
11. C23×#533C4semidirect
14. C23×#518C4semidirect
16. C4×#41×C23C4semidirect
118. #12727C23direct (N × H)
156. #2127C2×C23semidirect
157. C2×#353C2×C23semidirect
158. #1828C2×C23semidirect
183. #535C4×C23semidirect
185. #533C4×C23semidirect
188. #518C4×C23semidirect
190. C4×#41C4×C23semidirect
248. C23#12727direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35082
C23×#21272
C2×C23×#3532
C23×#18282
#1272723