Finite Group Databasebeta

← C2×C23×#2198 #19181171

Subgroups — C2×C23×#2198 #19181171

Order 5888.

1244 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 222 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×#559C211yesmaximal, characteristic, complemented
2C2×C23×#458C212yesmaximal, complemented
3C2×C23×#453C211yesmaximal, characteristic
4C23×#2198C218yesmaximal, direct product factor
5C2×C23×#353C211yesmaximal, characteristic, complemented
6C2×C23×#352C212yesmaximal, complemented
7C2×C23×H(5,𝔽2)C2211yescharacteristic
8C2×C23×AGL(1,ℤ/8)—21no—
9C23×#559—22no—
10C23×#559—24no—
11C23×#559C2214yescomplemented
12C2×C23×#127C2211yescharacteristic
13C2×C23×#124—22no—
14C2×C23×#123—21no—
15C2×C23×#123—22no—
16C2×C23×#123C2212yes—
17C2×C4×Q8×C23—21no—
18C2×C4×D4×C23—22no—
19C23×#458—24no—
20C23×#458C2218yescomplemented
21C23×#453—22no—
22C23×#453C2214yes—
23C2×C23×#104—21no—
24C2×C23×#100C2211yescharacteristic
25C2×C23×#99—22no—
26C2×C23×#99C2212yes—
27C23×#353C2214yescomplemented
28C23×#352C2218yescomplemented
29C23×H(5,𝔽2)—22no—
30C23×H(5,𝔽2)—21no—
31C23×H(5,𝔽2)—42no—
32C23×H(5,𝔽2)C2312yes—
33C2×#41×C23—21no—
34C2×#41×C23—41no—
35C22×Q8×C23—21no—
36C22×D4×C23—22no—
37C22×D4×C23—22no—
38C22×D4×C23—41no—
39C22×D4×C23D411yescharacteristic
40C22×D4×C23C2311yescharacteristic
41C23×C4×C23—21no—
42C23×C4×C23D412yes—
43C23×AGL(1,ℤ/8)—24no—
44C23×AGL(1,ℤ/8)—42no—
45C2×SD16×C23—41no—
46C2×D8×C23—41no—
47C2×M16×C23—21no—
48C23×#127—22no—
49C23×#127D412yes—
50C23×#127C2312yes—
51C23×#124—28no—
52C23×#124—28no—
53C23×#123—24no—
54C23×#123—22no—
55C23×#123—22no—
56C23×#123—24no—
57C23×#123—28no—
58C23×#123—44no—
59C23×#123D414yes—
60C23×#123C2314yes—
61C4×Q8×C23—24no—
62C4×Q8×C23—24no—
63C4×D4×C23—28no—
64C4×D4×C23—28no—
65C2×C4⋊C4×C23—21no—
66C2×C4⋊C4×C23—22no—
67C2×C4⋊C4×C23—42no—
68C2×C22⋊C4×C23—22no—
69C2×C22⋊C4×C23—21no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—22no—
72C2×C22⋊C4×C23—42no—
73C2×C22⋊C4×C23D412yes—
74C2×C42×C23—22no—
75C2×C42×C23D411yescharacteristic
76C23×#104—24no—
77C23×#100—21no—
78C23×#100C2312yes—
79C23×#99—22no—
80C23×#99—28no—
81C23×#99C2314yes—
82C24×C23—22no—
83C24×C23—21no—
84C24×C23C2×D412yes—
85#41×C23—22no—
86#41×C23—22no—
87#41×C23—42no—
88#41×C23—41no—
89#41×C23—44no—
90#41×C23—42no—
91#41×C23—41no—
92C2×Q8×C23—24no—
93C2×Q8×C23—21no—
94C2×Q8×C23—44no—
95C2×D4×C23—21no—
96C2×D4×C23—24no—
97C2×D4×C23—24no—
98C2×D4×C23—24no—
99C2×D4×C23—24no—
100C2×D4×C23—21no—
101C2×D4×C23—41no—
102C2×D4×C23—41no—
103C2×D4×C23—48no—
104C2×D4×C23—41no—
105C2×D4×C23—41no—
106C2×D4×C23—42no—
107C2×D4×C23—42no—
108C2×D4×C23—42no—
109C2×D4×C23—42no—
110C2×D4×C23—44no—
111C2×D4×C23—42no—
112C2×D4×C23—41no—
113C2×D4×C23C2×D412yes—
114C2×D4×C23C2×D412yes—
115C2×D4×C23C2×D411yescharacteristic
116C2×D4×C23C2411yescharacteristic
117C22×C4×C23—21no—
118C22×C4×C23—22no—
119C22×C4×C23—22no—
120C22×C4×C23—22no—
121C22×C4×C23—24no—
122C22×C4×C23—22no—
123C22×C4×C23—41no—
124C22×C4×C23—42no—
125C22×C4×C23—42no—
126C22×C4×C23—42no—
127C22×C4×C23—81no—
128C22×C4×C23—81no—
129C22×C4×C23C2×D411yescharacteristic
130C22×C4×C23C2×D414yes—
131SD16×C23—44no—
132D8×C23—44no—
133M16×C23—22no—
134M16×C23—41no—
135C2×C8×C23—41no—
136C4⋊C4×C23—24no—
137C4⋊C4×C23—24no—
138C4⋊C4×C23—24no—
139C4⋊C4×C23—48no—
140C22⋊C4×C23—28no—
141C22⋊C4×C23—24no—
142C22⋊C4×C23—24no—
143C22⋊C4×C23—22no—
144C22⋊C4×C23—24no—
145C22⋊C4×C23—24no—
146C22⋊C4×C23—42no—
147C22⋊C4×C23—48no—
148C22⋊C4×C23C2×D414yes—
149C42×C23—28no—
150C42×C23—21no—
151C42×C23C2×D412yes—
152C2×#2198C2311yesmaximal, characteristic, direct product factor, Sylow
153C23×C23—22no—
154C23×C23—22no—
155C23×C23—24no—
156C23×C23—22no—
157C23×C23—44no—
158C23×C23—44no—
159C23×C23—42no—
160C23×C23—41no—
161C23×C23—41no—
162C23×C23—41no—
163C23×C23—84no—
164C23×C23#12012yes—
165C23×C23#12011yescharacteristic
166C23×C23C22×D412yes—
167Q8×C23—22no—
168Q8×C23—48no—
169Q8×C23—41no—
170D4×C23—22no—
171D4×C23—41no—
172D4×C23—42no—
173D4×C23—41no—
174D4×C23—42no—
175D4×C23—48no—
176D4×C23—48no—
177D4×C23—41no—
178D4×C23—44no—
179D4×C23—44no—
180D4×C23—42no—
181D4×C23—42no—
182D4×C23—81no—
183C2×C4×C23—24no—
184C2×C4×C23—22no—
185C2×C4×C23—21no—
186C2×C4×C23—21no—
187C2×C4×C23—24no—
188C2×C4×C23—24no—
189C2×C4×C23—24no—
190C2×C4×C23—42no—
191C2×C4×C23—42no—
192C2×C4×C23—42no—
193C2×C4×C23—44no—
194C2×C4×C23—42no—
195C2×C4×C23—41no—
196C2×C4×C23—42no—
197C2×C4×C23—44no—
198C2×C4×C23—44no—
199C2×C4×C23—82no—
200C2×C4×C23—84no—
201C2×C4×C23—84no—
202C2×C4×C23#12011yescharacteristic
203C2×C4×C23C22×D411yescharacteristic
204C8×C23—42no—
205C2×#559C2×C2311yescharacteristic, complemented
206C2×#458C2×C2312yescomplemented
207C2×#453C2×C2311yescharacteristic
208#2198C2×C2318yesdirect product factor
209C2×#353C2×C2311yescharacteristic, complemented
210C2×#352C2×C2312yescomplemented
211C22×C23—22no—
212C22×C23—22no—
213C22×C23—22no—
214C22×C23—42no—
215C22×C23—42no—
216C22×C23—44no—
217C22×C23—44no—
218C22×C23—42no—
219C22×C23—42no—
220C22×C23—82no—
221C22×C23—82no—
222C22×C23—84no—
223C22×C23—82no—
224C22×C23—82no—
225C22×C23—162no—
226C22×C23B(4,𝔽2)11yescharacteristic
227C22×C23B(4,𝔽2)11yescharacteristic
228C22×C23C2×#12011yescharacteristic
229C4×C23—24no—
230C4×C23—21no—
231C4×C23—21no—
232C4×C23—44no—
233C4×C23—42no—
234C4×C23—44no—
235C4×C23—84no—
236C2×H(5,𝔽2)C22×C2311yescharacteristic
237C2×AGL(1,ℤ/8)—21no—
238#559—22no—
239#559—24no—
240#559C22×C2314yescomplemented
241C2×#127C22×C2311yescharacteristic
242C2×#124—22no—
243C2×#123—21no—
244C2×#123—22no—
245C2×#123C22×C2312yes—
246C2×C4×Q8—21no—
247C2×C4×D4—22no—
248#458—24no—
249#458C22×C2318yescomplemented
250#453—22no—
251#453C22×C2314yes—
252C2×#104—21no—
253C2×#100C22×C2311yescharacteristic
254C2×#99—22no—
255C2×#99C22×C2312yes—
256#353C22×C2314yescomplemented
257#352C22×C2318yescomplemented
258C2×C23—21no—
259C2×C23—21no—
260C2×C23—42no—
261C2×C23—42no—
262C2×C23—82no—
263C2×C23—82no—
264C2×C23#219812yesdirect product factor
265C2×C23C2×B(4,𝔽2)11yescharacteristic
266H(5,𝔽2)—22no—
267H(5,𝔽2)—21no—
268H(5,𝔽2)—42no—
269H(5,𝔽2)C23×C2312yes—
270C2×#41—21no—
271C2×#41—41no—
272C22×Q8—21no—
273C22×D4—22no—
274C22×D4—22no—
275C22×D4—41no—
276C22×D4D4×C2311yescharacteristic
277C22×D4C23×C2311yescharacteristic
278C23×C4—21no—
279C23×C4D4×C2312yes—
280AGL(1,ℤ/8)—24no—
281AGL(1,ℤ/8)—42no—
282C2×SD16—41no—
283C2×D8—41no—
284C2×M16—21no—
285#127—22no—
286#127D4×C2312yes—
287#127C23×C2312yes—
288#124—28no—
289#124—28no—
290#123—24no—
291#123—24no—
292#123—22no—
293#123—22no—
294#123—28no—
295#123—44no—
296#123D4×C2314yes—
297#123C23×C2314yes—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×D4—28no—
301C4×D4—28no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—42no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4D4×C2312yes—
311C2×C42—22no—
312C2×C42D4×C2311yescharacteristic
313#104—24no—
314#100—21no—
315#100C23×C2312yes—
316#99—22no—
317#99—28no—
318#99C23×C2314yes—
319C23C2×#219811yescharacteristic, direct product factor, Sylow
320C24—22no—
321C24—21no—
322C24C2×D4×C2312yes—
323#41—22no—
324#41—22no—
325#41—42no—
326#41—41no—
327#41—44no—
328#41—42no—
329#41—41no—
330C2×Q8—24no—
331C2×Q8—21no—
332C2×Q8—44no—
333C2×D4—21no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—21no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—48no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—44no—
349C2×D4—41no—
350C2×D4—42no—
351C2×D4C2×D4×C2312yes—
352C2×D4C2×D4×C2312yes—
353C2×D4C2×D4×C2311yescharacteristic
354C2×D4C24×C2311yescharacteristic
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4C2×D4×C2311yescharacteristic
368C22×C4C2×D4×C2314yes—
369SD16—44no—
370D8—44no—
371M16—22no—
372M16—41no—
373C2×C8—41no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—48no—
378C22⋊C4—28no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—48no—
386C22⋊C4C2×D4×C2314yes—
387C42—28no—
388C42—21no—
389C42C2×D4×C2312yes—
390C23—22no—
391C23—22no—
392C23—24no—
393C23—22no—
394C23—44no—
395C23—44no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—84no—
401C23C23×#12011yescharacteristic
402C23C23×#12012yes—
403C23C22×D4×C2312yes—
404Q8—22no—
405Q8—48no—
406Q8—41no—
407D4—22no—
408D4—41no—
409D4—42no—
410D4—41no—
411D4—42no—
412D4—41no—
413D4—48no—
414D4—48no—
415D4—44no—
416D4—44no—
417D4—42no—
418D4—42no—
419D4—81no—
420C2×C4—24no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—82no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4C23×#12011yescharacteristic
440C2×C4C22×D4×C2311yescharacteristic
441C8—42no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—42no—
446C22—42no—
447C22—44no—
448C22—44no—
449C22—42no—
450C22—42no—
451C22—82no—
452C22—82no—
453C22—84no—
454C22—82no—
455C22—82no—
456C22—162no—
457C22C23×B(4,𝔽2)11yescharacteristic
458C22C23×B(4,𝔽2)11yescharacteristic
459C22C2×C23×#12011yescharacteristic
460C4—24no—
461C4—21no—
462C4—21no—
463C4—44no—
464C4—42no—
465C4—44no—
466C4—84no—
467C2—21no—
468C2—21no—
469C2—42no—
470C2—42no—
471C2—82no—
472C2—82no—
473C2C23×#219812yesdirect product factor
474C2C2×C23×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×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
1. C2×C23×#559C2semidirect
2. C2×C23×#458C2semidirect
4. C23×#2198C2direct (N × H)
5. C2×C23×#353C2semidirect
6. C2×C23×#352C2semidirect
11. C23×#559C22semidirect
20. C23×#458C22semidirect
27. C23×#353C22semidirect
28. C23×#352C22semidirect
152. C2×#2198C23direct (N × H)
205. C2×#559C2×C23semidirect
206. C2×#458C2×C23semidirect
208. #2198C2×C23direct (N × H)
209. C2×#353C2×C23semidirect
210. C2×#352C2×C23semidirect
240. #559C22×C23semidirect
249. #458C22×C23semidirect
256. #353C22×C23semidirect
257. #352C22×C23semidirect
264. C2×C23#2198direct (N × H)
319. C23C2×#2198direct (N × H)
473. C2C23×#2198direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5592
C2×C23×#4582
C2×C23×#4532
C23×#21982
C2×C23×#3532
C2×C23×#3522
C2×#219823