Finite Group Databasebeta

← C23×#12721 #19160347

Subgroups — C23×#12721 #19160347

Order 5888.

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

normal subgroups only

This group has 694 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×#3510C211yesmaximal, characteristic
2C23×#2135C211yesmaximal, characteristic, complemented
3C23×#2129C211yesmaximal, characteristic, complemented
4C23×#2122C212yesmaximal
5C23×#1827C212yesmaximal, complemented
6C23×#559—22no—
7C23×#556—22no—
8C23×#551—24no—
9C23×#542—22no—
10C23×#536C2211yescharacteristic
11C23×#535C2211yescharacteristic
12C23×#533C411yescharacteristic, complemented
13C23×#529—22no—
14C2×C23×#124C411yescharacteristic, complemented
15C23×#519C411yescharacteristic, complemented
16C23×#518C411yescharacteristic, complemented
17C4×#41×C23C2211yescharacteristic
18C23×#413C2212yes—
19C23×#410C2212yes—
20C23×#356—22no—
21C23×#354—22no—
22C23×AΣL(1,𝔽16)—24no—
23C23×#339—22no—
24C23×H(3,ℤ/4)—22no—
25C2×#41×C23C2311yescharacteristic
26C22×Q8×C23C2×C411yescharacteristic
27C22×D4×C23C2×C411yescharacteristic
28C2×M16×C23—22no—
29C23×C4⋊Q8—21no—
30C23×C4⋊Q8D412yes—
31C23×#127—21no—
32C23×#126—24no—
33C23×#126—24no—
34C23×#125—21no—
35C23×#124—24no—
36C23×#124—22no—
37C23×#124—21no—
38C23×#124—21no—
39C23×#124—21no—
40C23×#124D412yes—
41C23×#123—24no—
42C23×#123—24no—
43C23×#123—22no—
44C23×C22⋊Q8—24no—
45C23×C22⋊Q8—22no—
46C23×C22⋊Q8—22no—
47C23×#121—22no—
48C23×#120D414yes—
49C4×Q8×C23—22no—
50C4×Q8×C23—21no—
51C4×Q8×C23—22no—
52C4×Q8×C23—22no—
53C4×D4×C23—24no—
54C4×D4×C23—22no—
55C4×D4×C23—21no—
56C4×D4×C23—22no—
57C4×D4×C23—22no—
58C23×#117—24no—
59C23×#117—22no—
60C23×#117—22no—
61C23×#117C2×C411yescharacteristic
62C23×#117C2×C411yescharacteristic
63C23×#117C2×C411yescharacteristic
64C23×#117D412yes—
65C2×C22⋊C4×C23—22no—
66C2×C22⋊C4×C23—22no—
67C2×C42×C23C2×C411yescharacteristic
68C2×C42×C23D412yes—
69C23×#101—22no—
70C23×#100—22no—
71C23×#99—22no—
72C23×#99—22no—
73C24×C23C22⋊C412yes—
74#41×C23—42no—
75C2×Q8×C23—22no—
76C2×Q8×C23—21no—
77C2×Q8×C23—42no—
78C2×Q8×C23C22⋊C412yes—
79C2×Q8×C23C2×D411yescharacteristic
80C2×D4×C23—24no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—42no—
84C2×D4×C23C2×D411yescharacteristic
85C2×D4×C23C2×D412yes—
86C22×C4×C23—21no—
87C22×C4×C23—24no—
88C22×C4×C23—21no—
89C22×C4×C23—22no—
90C22×C4×C23C22×C411yescharacteristic
91C22×C4×C23C2×D412yes—
92M16×C23—42no—
93M16×C23—42no—
94C2×C8×C23—42no—
95C4⋊C4×C23—22no—
96C4⋊C4×C23—24no—
97C4⋊C4×C23—22no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—24no—
103C4⋊C4×C23—22no—
104C4⋊C4×C23—42no—
105C22⋊C4×C23—24no—
106C22⋊C4×C23—24no—
107C22⋊C4×C23—24no—
108C22⋊C4×C23—21no—
109C22⋊C4×C23—21no—
110C22⋊C4×C23—22no—
111C22⋊C4×C23—24no—
112C22⋊C4×C23—24no—
113C22⋊C4×C23—42no—
114C22⋊C4×C23C22⋊C412yes—
115C42×C23—24no—
116C42×C23—21no—
117C42×C23—21no—
118C42×C23—22no—
119C42×C23—42no—
120C42×C23C22⋊C412yes—
121C42×C23C22⋊C411yescharacteristic
122C42×C23C22⋊C411yescharacteristic
123C42×C23C22⋊C412yes—
124#12721C2311yesmaximal, characteristic, direct product factor, Sylow
125C23×C23—22no—
126C23×C23—42no—
127C23×C23—82no—
128C23×C23C2×C22⋊C411yescharacteristic
129C23×C23#12012yes—
130Q8×C23—41no—
131Q8×C23—42no—
132Q8×C23—42no—
133Q8×C23—82no—
134D4×C23—42no—
135D4×C23—41no—
136D4×C23—42no—
137D4×C23—42no—
138D4×C23—84no—
139C2×C4×C23—22no—
140C2×C4×C23—22no—
141C2×C4×C23—22no—
142C2×C4×C23—22no—
143C2×C4×C23—21no—
144C2×C4×C23—22no—
145C2×C4×C23—22no—
146C2×C4×C23—21no—
147C2×C4×C23—22no—
148C2×C4×C23—42no—
149C2×C4×C23—82no—
150C2×C4×C23—84no—
151C2×C4×C23—81no—
152C2×C4×C23—82no—
153C2×C4×C23C2×C22⋊C411yescharacteristic
154C2×C4×C23C2×C22⋊C411yescharacteristic
155C2×C4×C23#12012yes—
156C8×C23—82no—
157#3510C2×C2311yescharacteristic
158#2135C2×C2311yescharacteristic, complemented
159#2129C2×C2311yescharacteristic, complemented
160#2122C2×C2312yes—
161#1827C2×C2312yescomplemented
162C22×C23—21no—
163C22×C23—22no—
164C22×C23—42no—
165C22×C23—41no—
166C22×C23—44no—
167C22×C23—42no—
168C22×C23—162no—
169C22×C23#37911yescharacteristic
170C4×C23—22no—
171C4×C23—21no—
172C4×C23—42no—
173C4×C23—41no—
174C4×C23—42no—
175C4×C23—41no—
176C4×C23—42no—
177C4×C23—42no—
178C4×C23—162no—
179C4×C23#41012yes—
180#559—22no—
181#556—22no—
182#551—24no—
183#542—22no—
184#536C22×C2311yescharacteristic
185#535C22×C2311yescharacteristic
186#533C4×C2311yescharacteristic, complemented
187#529—22no—
188C2×#124C4×C2311yescharacteristic, complemented
189#519C4×C2311yescharacteristic, complemented
190#518C4×C2311yescharacteristic, complemented
191C4×#41C22×C2311yescharacteristic
192#413C22×C2312yes—
193#410C22×C2312yes—
194#356—22no—
195#354—22no—
196AΣL(1,𝔽16)—24no—
197#339—22no—
198H(3,ℤ/4)—22no—
199C2×C23—21no—
200C2×C23—42no—
201C2×C23—41no—
202C2×C23—82no—
203C2×C23#178211yescharacteristic
204C2×#41C23×C2311yescharacteristic
205C22×Q8C2×C4×C2311yescharacteristic
206C22×D4C2×C4×C2311yescharacteristic
207C2×M16—22no—
208C4⋊Q8—21no—
209C4⋊Q8D4×C2312yes—
210#127—21no—
211#126—24no—
212#126—24no—
213#125—21no—
214#124—24no—
215#124—22no—
216#124—21no—
217#124—21no—
218#124—21no—
219#124D4×C2312yes—
220#123—24no—
221#123—24no—
222#123—22no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226#121—22no—
227#120D4×C2314yes—
228C4×Q8—22no—
229C4×Q8—21no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×D4—24no—
233C4×D4—22no—
234C4×D4—21no—
235C4×D4—22no—
236C4×D4—22no—
237#117—24no—
238#117—22no—
239#117—22no—
240#117C2×C4×C2311yescharacteristic
241#117C2×C4×C2311yescharacteristic
242#117C2×C4×C2311yescharacteristic
243#117D4×C2312yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C42C2×C4×C2311yescharacteristic
247C2×C42D4×C2312yes—
248#101—22no—
249#100—22no—
250#99—22no—
251#99—22no—
252C23#1272111yescharacteristic, direct product factor, Sylow
253C24C22⋊C4×C2312yes—
254#41—42no—
255C2×Q8—22no—
256C2×Q8—21no—
257C2×Q8—42no—
258C2×Q8C22⋊C4×C2312yes—
259C2×Q8C2×D4×C2311yescharacteristic
260C2×D4—24no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4C2×D4×C2311yescharacteristic
265C2×D4C2×D4×C2312yes—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4C22×C4×C2311yescharacteristic
271C22×C4C2×D4×C2312yes—
272M16—42no—
273M16—42no—
274C2×C8—42no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4C22⋊C4×C2312yes—
295C42—24no—
296C42—21no—
297C42—21no—
298C42—22no—
299C42—42no—
300C42C22⋊C4×C2311yescharacteristic
301C42C22⋊C4×C2311yescharacteristic
302C42C22⋊C4×C2312yes—
303C42C22⋊C4×C2312yes—
304C23—22no—
305C23—42no—
306C23—82no—
307C23C2×C22⋊C4×C2311yescharacteristic
308C23C23×#12012yes—
309Q8—41no—
310Q8—42no—
311Q8—42no—
312Q8—82no—
313D4—42no—
314D4—41no—
315D4—42no—
316D4—42no—
317D4—84no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4C2×C22⋊C4×C2311yescharacteristic
333C2×C4C2×C22⋊C4×C2311yescharacteristic
334C2×C4C23×#12012yes—
335C8—82no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—41no—
340C22—44no—
341C22—42no—
342C22—162no—
343C22C23×#37911yescharacteristic
344C4—22no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—42no—
352C4—162no—
353C4C23×#41012yes—
354C2—21no—
355C2—42no—
356C2—41no—
357C2—82no—
358C2C23×#178211yescharacteristic
subgroupG/H
center, socleC2×C23#1782
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC23×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C4×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2135C2semidirect
3. C23×#2129C2semidirect
5. C23×#1827C2semidirect
12. C23×#533C4semidirect
14. C2×C23×#124C4semidirect
15. C23×#519C4semidirect
16. C23×#518C4semidirect
124. #12721C23direct (N × H)
158. #2135C2×C23semidirect
159. #2129C2×C23semidirect
161. #1827C2×C23semidirect
186. #533C4×C23semidirect
188. C2×#124C4×C23semidirect
189. #519C4×C23semidirect
190. #518C4×C23semidirect
252. C23#12721direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35102
C23×#21352
C23×#21292
C23×#21222
C23×#18272
#1272123