Finite Group Databasebeta

← C23×#13047 #19160673

Subgroups — C23×#13047 #19160673

Order 5888.

888 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 118 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×C23×#453C211yesmaximal, characteristic
2C23×#2366C211yesmaximal, characteristic, complemented
3C2×C23×#353C211yesmaximal, characteristic, complemented
4C23×#1790C212yesmaximal, complemented
5C23×#1390C212yesmaximal, complemented
6C2×C23×H(5,𝔽2)C2211yescharacteristic
7C2×C23×AGL(1,ℤ/8)—21no—
8C2×C23×#127C2211yescharacteristic
9C2×C4×D4×C23—22no—
10C23×#453—22no—
11C23×#453C414yescomplemented
12C2×C23×#104—21no—
13C2×C23×#100C2211yescharacteristic
14C2×C23×#99—22no—
15C2×C23×#99C2212yes—
16C23×#390—22no—
17C23×#386—21no—
18C23×#386—22no—
19C23×#386C2212yes—
20C4×C4⋊C4×C23—21no—
21C23×#353—22no—
22C23×H(5,𝔽2)—22no—
23C23×H(5,𝔽2)—21no—
24C23×H(5,𝔽2)—42no—
25C23×H(5,𝔽2)C2×C412yescomplemented
26C2×#41×C23—21no—
27C2×#41×C23—41no—
28C22×D4×C23—22no—
29C22×D4×C23—22no—
30C22×D4×C23—41no—
31C22×D4×C23D411yescharacteristic
32C22×D4×C23C2311yescharacteristic
33C23×C4×C23—21no—
34C23×C4×C23D412yes—
35C23×AGL(1,ℤ/8)—24no—
36C23×AGL(1,ℤ/8)—42no—
37C2×SD16×C23—41no—
38C2×D8×C23—41no—
39C2×M16×C23—21no—
40C23×#127—22no—
41C23×#127C2×C412yes—
42C23×#127D412yes—
43C4×D4×C23—48no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C22⋊C4×C23—22no—
47C2×C22⋊C4×C23—22no—
48C2×C22⋊C4×C23—22no—
49C2×C22⋊C4×C23—21no—
50C2×C22⋊C4×C23—42no—
51C2×C22⋊C4×C23D412yes—
52C2×C42×C23—22no—
53C2×C42×C23D411yescharacteristic
54C23×#104—24no—
55C23×#100—21no—
56C23×#100C2×C412yescomplemented
57C23×#99—22no—
58C23×#99—22no—
59C23×#99—44no—
60C23×#95—22no—
61C23×#95—42no—
62C24×C23—22no—
63C24×C23—21no—
64C24×C23C2×D412yes—
65#41×C23—22no—
66#41×C23—22no—
67#41×C23—42no—
68#41×C23—41no—
69#41×C23—44no—
70#41×C23—42no—
71#41×C23—41no—
72C2×Q8×C23—21no—
73C2×D4×C23—21no—
74C2×D4×C23—24no—
75C2×D4×C23—24no—
76C2×D4×C23—24no—
77C2×D4×C23—24no—
78C2×D4×C23—21no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—44no—
82C2×D4×C23—44no—
83C2×D4×C23—41no—
84C2×D4×C23—41no—
85C2×D4×C23—42no—
86C2×D4×C23—42no—
87C2×D4×C23—42no—
88C2×D4×C23—42no—
89C2×D4×C23—44no—
90C2×D4×C23—41no—
91C2×D4×C23—42no—
92C2×D4×C23C22⋊C412yescomplemented
93C2×D4×C23C22×C411yescharacteristic
94C2×D4×C23C2×D411yescharacteristic
95C2×D4×C23#4112yes—
96C22×C4×C23—21no—
97C22×C4×C23—22no—
98C22×C4×C23—22no—
99C22×C4×C23—22no—
100C22×C4×C23—41no—
101C22×C4×C23—44no—
102C22×C4×C23—44no—
103C22×C4×C23—42no—
104C22×C4×C23—42no—
105C22×C4×C23—81no—
106C22×C4×C23C2×D411yescharacteristic
107SD16×C23—44no—
108D8×C23—44no—
109M16×C23—22no—
110M16×C23—41no—
111C2×C8×C23—41no—
112C4⋊C4×C23—24no—
113C4⋊C4×C23—42no—
114C22⋊C4×C23—44no—
115C22⋊C4×C23—44no—
116C22⋊C4×C23—42no—
117C22⋊C4×C23—42no—
118C22⋊C4×C23—42no—
119C22⋊C4×C23—81no—
120C22⋊C4×C23—84no—
121C42×C23—21no—
122C42×C23—44no—
123C42×C23C22⋊C412yescomplemented
124#13047C2311yesmaximal, characteristic, direct product factor, Sylow
125C23×C23—22no—
126C23×C23—22no—
127C23×C23—24no—
128C23×C23—22no—
129C23×C23—42no—
130C23×C23—42no—
131C23×C23—42no—
132C23×C23—42no—
133C23×C23—42no—
134C23×C23—41no—
135C23×C23—41no—
136C23×C23—41no—
137C23×C23—82no—
138C23×C23—82no—
139C23×C23C4×D412yes—
140C23×C23#12011yescharacteristic
141C23×C23#12112yes—
142Q8×C23—22no—
143Q8×C23—41no—
144D4×C23—22no—
145D4×C23—41no—
146D4×C23—41no—
147D4×C23—42no—
148D4×C23—42no—
149D4×C23—48no—
150D4×C23—48no—
151D4×C23—41no—
152D4×C23—44no—
153D4×C23—44no—
154D4×C23—42no—
155D4×C23—42no—
156D4×C23—81no—
157C2×C4×C23—22no—
158C2×C4×C23—22no—
159C2×C4×C23—22no—
160C2×C4×C23—21no—
161C2×C4×C23—21no—
162C2×C4×C23—24no—
163C2×C4×C23—24no—
164C2×C4×C23—42no—
165C2×C4×C23—42no—
166C2×C4×C23—42no—
167C2×C4×C23—42no—
168C2×C4×C23—41no—
169C2×C4×C23—42no—
170C2×C4×C23—44no—
171C2×C4×C23—42no—
172C2×C4×C23—42no—
173C2×C4×C23—42no—
174C2×C4×C23—82no—
175C2×C4×C23—84no—
176C2×C4×C23—82no—
177C2×C4×C23—82no—
178C2×C4×C23—82no—
179C2×C4×C23C2×C22⋊C411yescharacteristic
180C2×C4×C23#12311yescharacteristic
181C8×C23—42no—
182C2×#453C2×C2311yescharacteristic
183#2366C2×C2311yescharacteristic, complemented
184C2×#353C2×C2311yescharacteristic, complemented
185#1790C2×C2312yescomplemented
186#1390C2×C2312yescomplemented
187C22×C23—21no—
188C22×C23—21no—
189C22×C23—22no—
190C22×C23—22no—
191C22×C23—42no—
192C22×C23—42no—
193C22×C23—44no—
194C22×C23—42no—
195C22×C23—42no—
196C22×C23—42no—
197C22×C23—42no—
198C22×C23—82no—
199C22×C23—82no—
200C22×C23—84no—
201C22×C23—81no—
202C22×C23—81no—
203C22×C23—81no—
204C22×C23—81no—
205C22×C23—162no—
206C22×C23#38611yescharacteristic
207C22×C23B(4,𝔽2)11yescharacteristic
208C22×C23B(4,𝔽2)11yescharacteristic
209C4×C23—24no—
210C4×C23—21no—
211C4×C23—21no—
212C4×C23—42no—
213C4×C23—44no—
214C4×C23—82no—
215C4×C23—162no—
216C2×H(5,𝔽2)C22×C2311yescharacteristic
217C2×AGL(1,ℤ/8)—21no—
218C2×#127C22×C2311yescharacteristic
219C2×C4×D4—22no—
220#453—22no—
221#453C4×C2314yescomplemented
222C2×#104—21no—
223C2×#100C22×C2311yescharacteristic
224C2×#99—22no—
225C2×#99C22×C2312yes—
226#390—22no—
227#386—21no—
228#386—22no—
229#386C22×C2312yes—
230C4×C4⋊C4—21no—
231#353—22no—
232C2×C23—21no—
233C2×C23—21no—
234C2×C23—42no—
235C2×C23—42no—
236C2×C23—82no—
237C2×C23—82no—
238C2×C23#188911yescharacteristic
239C2×C23#219611yescharacteristic
240C2×C23#219811yescharacteristic
241H(5,𝔽2)—22no—
242H(5,𝔽2)—21no—
243H(5,𝔽2)—42no—
244H(5,𝔽2)C2×C4×C2312yescomplemented
245C2×#41—21no—
246C2×#41—41no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—41no—
250C22×D4D4×C2311yescharacteristic
251C22×D4C23×C2311yescharacteristic
252C23×C4—21no—
253C23×C4D4×C2312yes—
254AGL(1,ℤ/8)—24no—
255AGL(1,ℤ/8)—42no—
256C2×SD16—41no—
257C2×D8—41no—
258C2×M16—21no—
259#127—22no—
260#127C2×C4×C2312yes—
261#127D4×C2312yes—
262C4×D4—48no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4D4×C2312yes—
271C2×C42—22no—
272C2×C42D4×C2311yescharacteristic
273#104—24no—
274#100—21no—
275#100C2×C4×C2312yescomplemented
276#99—22no—
277#99—22no—
278#99—44no—
279#95—22no—
280#95—42no—
281C23#1304711yescharacteristic, direct product factor, Sylow
282C24—22no—
283C24—21no—
284C24C2×D4×C2312yes—
285#41—22no—
286#41—22no—
287#41—42no—
288#41—41no—
289#41—44no—
290#41—42no—
291#41—41no—
292C2×Q8—21no—
293C2×D4—21no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—21no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4C22⋊C4×C2312yescomplemented
313C2×D4C22×C4×C2311yescharacteristic
314C2×D4C2×D4×C2311yescharacteristic
315C2×D4#41×C2312yes—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—81no—
326C22×C4C2×D4×C2311yescharacteristic
327SD16—44no—
328D8—44no—
329M16—22no—
330M16—41no—
331C2×C8—41no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—81no—
340C22⋊C4—84no—
341C42—21no—
342C42—44no—
343C42C22⋊C4×C2312yescomplemented
344C23—22no—
345C23—22no—
346C23—24no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—82no—
357C23—82no—
358C23C4×D4×C2312yes—
359C23C23×#12011yescharacteristic
360C23C23×#12112yes—
361Q8—22no—
362Q8—41no—
363D4—22no—
364D4—41no—
365D4—41no—
366D4—42no—
367D4—42no—
368D4—41no—
369D4—48no—
370D4—48no—
371D4—44no—
372D4—44no—
373D4—42no—
374D4—42no—
375D4—81no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—82no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4C2×C22⋊C4×C2311yescharacteristic
399C2×C4C23×#12311yescharacteristic
400C8—42no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—42no—
406C22—42no—
407C22—44no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—82no—
413C22—82no—
414C22—84no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—162no—
420C22C23×#38611yescharacteristic
421C22C23×B(4,𝔽2)11yescharacteristic
422C22C23×B(4,𝔽2)11yescharacteristic
423C4—24no—
424C4—21no—
425C4—21no—
426C4—42no—
427C4—44no—
428C4—82no—
429C4—162no—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—42no—
434C2—82no—
435C2—82no—
436C2C23×#188911yescharacteristic
437C2C23×#219611yescharacteristic
438C2C23×#219811yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×#453C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2366C2semidirect
3. C2×C23×#353C2semidirect
4. C23×#1790C2semidirect
5. C23×#1390C2semidirect
11. C23×#453C4semidirect
25. C23×H(5,𝔽2)C2×C4semidirect
56. C23×#100C2×C4semidirect
92. C2×D4×C23C22⋊C4semidirect
123. C42×C23C22⋊C4semidirect
124. #13047C23direct (N × H)
183. #2366C2×C23semidirect
184. C2×#353C2×C23semidirect
185. #1790C2×C23semidirect
186. #1390C2×C23semidirect
221. #453C4×C23semidirect
244. H(5,𝔽2)C2×C4×C23semidirect
275. #100C2×C4×C23semidirect
281. C23#13047direct (N × H)
312. C2×D4C22⋊C4×C23semidirect
343. C42C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#4532
C23×#23662
C2×C23×#3532
C23×#17902
C23×#13902
#1304723