Finite Group Databasebeta

← C2×C23×#2201 #19181175

Subgroups — C2×C23×#2201 #19181175

Order 5888.

1136 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1136 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×#563C211yesmaximal, characteristic, complemented
2C2×C23×#458C212yesmaximal
3C2×C23×#455C211yesmaximal, characteristic
4C23×#2201C218yesmaximal, direct product factor
5C2×C23×#354C211yesmaximal, characteristic, complemented
6C2×C23×#352C212yesmaximal, complemented
7C2×C23×H(5,𝔽2)C2211yescharacteristic
8C2×C23×#137—21no—
9C23×#563—22no—
10C23×#563—24no—
11C23×#563C2214yescomplemented
12C2×C23×C4⋊Q8—21no—
13C2×C23×#126—22no—
14C2×C23×#125—21no—
15C2×C23×#124C2211yescharacteristic
16C2×C23×#123—22no—
17C2×C23×#123C2212yes—
18C2×C23×C22⋊Q8—22no—
19C23×#458—24no—
20C23×#458C2218yes—
21C23×#455—22no—
22C23×#455C2214yes—
23C2×C23×#104—21no—
24C2×C23×#100C2211yescharacteristic
25C2×C23×#99—22no—
26C2×C23×#99C2212yes—
27C23×#354C2214yescomplemented
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×Q8×C23D411yescharacteristic
37C22×D4×C23—22no—
38C22×D4×C23—41no—
39C22×D4×C23C2311yescharacteristic
40C23×C4×C23D412yes—
41C23×#137—24no—
42C23×#137—42no—
43C2×Q16×C23—41no—
44C2×SD16×C23—41no—
45C2×M16×C23—21no—
46C23×C4⋊Q8—24no—
47C23×C4⋊Q8—24no—
48C23×#126—28no—
49C23×#126—28no—
50C23×#125—24no—
51C23×#125—24no—
52C23×#124—22no—
53C23×#124D412yes—
54C23×#124C2312yes—
55C23×#123—22no—
56C23×#123—22no—
57C23×#123—28no—
58C23×#123—44no—
59C23×#123D414yes—
60C23×#123C2314yes—
61C23×C22⋊Q8—28no—
62C23×C22⋊Q8—28no—
63C2×C4⋊C4×C23—21no—
64C2×C4⋊C4×C23—22no—
65C2×C4⋊C4×C23—22no—
66C2×C4⋊C4×C23—21no—
67C2×C4⋊C4×C23—22no—
68C2×C4⋊C4×C23—42no—
69C2×C22⋊C4×C23—22no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—22no—
72C2×C22⋊C4×C23—42no—
73C2×C22⋊C4×C23D412yes—
74C2×C42×C23—21no—
75C2×C42×C23D411yescharacteristic
76C23×#104—24no—
77C23×#100—21no—
78C23×#100C2312yes—
79C23×#99—22no—
80C23×#99—28no—
81C23×#99C2314yes—
82C24×C23—22no—
83C24×C23C2×D412yes—
84#41×C23—22no—
85#41×C23—22no—
86#41×C23—42no—
87#41×C23—41no—
88#41×C23—44no—
89#41×C23—42no—
90#41×C23—41no—
91C2×Q8×C23—24no—
92C2×Q8×C23—21no—
93C2×Q8×C23—21no—
94C2×Q8×C23—41no—
95C2×Q8×C23—41no—
96C2×Q8×C23—81no—
97C2×Q8×C23—81no—
98C2×Q8×C23C2×D412yes—
99C2×D4×C23—21no—
100C2×D4×C23—24no—
101C2×D4×C23—24no—
102C2×D4×C23—41no—
103C2×D4×C23—41no—
104C2×D4×C23—42no—
105C2×D4×C23—42no—
106C2×D4×C23—42no—
107C2×D4×C23—42no—
108C2×D4×C23—44no—
109C2×D4×C23—42no—
110C2×D4×C23—41no—
111C2×D4×C23C2×D412yes—
112C2×D4×C23C2×D411yescharacteristic
113C2×D4×C23C2411yescharacteristic
114C22×C4×C23—22no—
115C22×C4×C23—21no—
116C22×C4×C23—22no—
117C22×C4×C23—21no—
118C22×C4×C23—22no—
119C22×C4×C23—22no—
120C22×C4×C23—41no—
121C22×C4×C23—42no—
122C22×C4×C23—42no—
123C22×C4×C23—42no—
124C22×C4×C23C2×D411yescharacteristic
125C22×C4×C23C2×D414yes—
126Q16×C23—44no—
127SD16×C23—44no—
128M16×C23—22no—
129M16×C23—41no—
130C2×C8×C23—41no—
131C4⋊C4×C23—24no—
132C4⋊C4×C23—28no—
133C4⋊C4×C23—28no—
134C4⋊C4×C23—24no—
135C4⋊C4×C23—24no—
136C4⋊C4×C23—24no—
137C4⋊C4×C23—48no—
138C22⋊C4×C23—24no—
139C22⋊C4×C23—24no—
140C22⋊C4×C23—22no—
141C22⋊C4×C23—24no—
142C22⋊C4×C23—24no—
143C22⋊C4×C23—24no—
144C22⋊C4×C23—42no—
145C22⋊C4×C23—48no—
146C22⋊C4×C23C2×D414yes—
147C42×C23—24no—
148C42×C23—21no—
149C42×C23C2×D412yes—
150C2×#2201C2311yesmaximal, characteristic, direct product factor, Sylow
151C23×C23—22no—
152C23×C23—22no—
153C23×C23—24no—
154C23×C23—44no—
155C23×C23—42no—
156C23×C23—41no—
157C23×C23—84no—
158C23×C23#12012yes—
159C23×C23#12011yescharacteristic
160C23×C23C22×D412yes—
161Q8×C23—22no—
162Q8×C23—42no—
163Q8×C23—41no—
164Q8×C23—81no—
165Q8×C23—84no—
166D4×C23—22no—
167D4×C23—41no—
168D4×C23—42no—
169D4×C23—41no—
170D4×C23—41no—
171D4×C23—44no—
172D4×C23—44no—
173D4×C23—42no—
174D4×C23—42no—
175C2×C4×C23—21no—
176C2×C4×C23—21no—
177C2×C4×C23—24no—
178C2×C4×C23—22no—
179C2×C4×C23—24no—
180C2×C4×C23—22no—
181C2×C4×C23—24no—
182C2×C4×C23—42no—
183C2×C4×C23—42no—
184C2×C4×C23—44no—
185C2×C4×C23—42no—
186C2×C4×C23—42no—
187C2×C4×C23—42no—
188C2×C4×C23—41no—
189C2×C4×C23—41no—
190C2×C4×C23—42no—
191C2×C4×C23—41no—
192C2×C4×C23—44no—
193C2×C4×C23—44no—
194C2×C4×C23—82no—
195C2×C4×C23—84no—
196C2×C4×C23#12011yescharacteristic
197C2×C4×C23C22×D411yescharacteristic
198C8×C23—42no—
199C2×#563C2×C2311yescharacteristic, complemented
200C2×#458C2×C2312yes—
201C2×#455C2×C2311yescharacteristic
202#2201C2×C2318yesdirect product factor
203C2×#354C2×C2311yescharacteristic, complemented
204C2×#352C2×C2312yescomplemented
205C22×C23—22no—
206C22×C23—22no—
207C22×C23—22no—
208C22×C23—42no—
209C22×C23—42no—
210C22×C23—44no—
211C22×C23—42no—
212C22×C23—82no—
213C22×C23—84no—
214C22×C23—82no—
215C22×C23—162no—
216C22×C23B(4,𝔽2)11yescharacteristic
217C22×C23B(4,𝔽2)11yescharacteristic
218C22×C23C2×#12011yescharacteristic
219C4×C23—21no—
220C4×C23—21no—
221C4×C23—44no—
222C4×C23—42no—
223C4×C23—42no—
224C4×C23—44no—
225C4×C23—42no—
226C4×C23—84no—
227C2×H(5,𝔽2)C22×C2311yescharacteristic
228C2×#137—21no—
229#563—22no—
230#563—24no—
231#563C22×C2314yescomplemented
232C2×C4⋊Q8—21no—
233C2×#126—22no—
234C2×#125—21no—
235C2×#124C22×C2311yescharacteristic
236C2×#123—22no—
237C2×#123C22×C2312yes—
238C2×C22⋊Q8—22no—
239#458—24no—
240#458C22×C2318yes—
241#455—22no—
242#455C22×C2314yes—
243C2×#104—21no—
244C2×#100C22×C2311yescharacteristic
245C2×#99—22no—
246C2×#99C22×C2312yes—
247#354C22×C2314yescomplemented
248#352C22×C2318yescomplemented
249C2×C23—21no—
250C2×C23—21no—
251C2×C23—42no—
252C2×C23—42no—
253C2×C23—82no—
254C2×C23#220112yesdirect product factor
255C2×C23C2×B(4,𝔽2)11yescharacteristic
256H(5,𝔽2)—22no—
257H(5,𝔽2)—21no—
258H(5,𝔽2)—42no—
259H(5,𝔽2)C23×C2312yes—
260C2×#41—21no—
261C2×#41—41no—
262C22×Q8—21no—
263C22×Q8D4×C2311yescharacteristic
264C22×D4—22no—
265C22×D4—41no—
266C22×D4C23×C2311yescharacteristic
267C23×C4D4×C2312yes—
268#137—24no—
269#137—42no—
270C2×Q16—41no—
271C2×SD16—41no—
272C2×M16—21no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275#126—28no—
276#126—28no—
277#125—24no—
278#125—24no—
279#124—22no—
280#124D4×C2312yes—
281#124C23×C2312yes—
282#123—22no—
283#123—22no—
284#123—28no—
285#123—44no—
286#123D4×C2314yes—
287#123C23×C2314yes—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—42no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4D4×C2312yes—
301C2×C42—21no—
302C2×C42D4×C2311yescharacteristic
303#104—24no—
304#100—21no—
305#100C23×C2312yes—
306#99—22no—
307#99—28no—
308#99C23×C2314yes—
309C23C2×#220111yescharacteristic, direct product factor, Sylow
310C24—22no—
311C24C2×D4×C2312yes—
312#41—22no—
313#41—22no—
314#41—42no—
315#41—41no—
316#41—44no—
317#41—42no—
318#41—41no—
319C2×Q8—24no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—81no—
325C2×Q8—81no—
326C2×Q8C2×D4×C2312yes—
327C2×D4—21no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4C2×D4×C2312yes—
340C2×D4C2×D4×C2311yescharacteristic
341C2×D4C24×C2311yescharacteristic
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4C2×D4×C2311yescharacteristic
353C22×C4C2×D4×C2314yes—
354Q16—44no—
355SD16—44no—
356M16—22no—
357M16—41no—
358C2×C8—41no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—48no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—22no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—48no—
374C22⋊C4C2×D4×C2314yes—
375C42—24no—
376C42—21no—
377C42C2×D4×C2312yes—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—44no—
382C23—42no—
383C23—41no—
384C23—84no—
385C23C23×#12011yescharacteristic
386C23C23×#12012yes—
387C23C22×D4×C2312yes—
388Q8—22no—
389Q8—42no—
390Q8—41no—
391Q8—81no—
392Q8—84no—
393D4—22no—
394D4—41no—
395D4—42no—
396D4—41no—
397D4—41no—
398D4—44no—
399D4—44no—
400D4—42no—
401D4—42no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4C23×#12011yescharacteristic
424C2×C4C22×D4×C2311yescharacteristic
425C8—42no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—42no—
430C22—42no—
431C22—44no—
432C22—42no—
433C22—82no—
434C22—84no—
435C22—82no—
436C22—162no—
437C22C23×B(4,𝔽2)11yescharacteristic
438C22C23×B(4,𝔽2)11yescharacteristic
439C22C2×C23×#12011yescharacteristic
440C4—21no—
441C4—21no—
442C4—44no—
443C4—42no—
444C4—42no—
445C4—44no—
446C4—42no—
447C4—84no—
448C2—21no—
449C2—21no—
450C2—42no—
451C2—42no—
452C2—82no—
453C2C23×#220112yesdirect product factor
454C2C2×C23×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C23
prime radicalC2×C23×H(5,𝔽2)C22
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×#563C2semidirect
4. C23×#2201C2direct (N × H)
5. C2×C23×#354C2semidirect
6. C2×C23×#352C2semidirect
11. C23×#563C22semidirect
27. C23×#354C22semidirect
28. C23×#352C22semidirect
150. C2×#2201C23direct (N × H)
199. C2×#563C2×C23semidirect
202. #2201C2×C23direct (N × H)
203. C2×#354C2×C23semidirect
204. C2×#352C2×C23semidirect
231. #563C22×C23semidirect
247. #354C22×C23semidirect
248. #352C22×C23semidirect
254. C2×C23#2201direct (N × H)
309. C23C2×#2201direct (N × H)
453. C2C23×#2201direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5632
C2×C23×#4582
C2×C23×#4552
C23×#22012
C2×C23×#3542
C2×C23×#3522
C2×#220123