Finite Group Databasebeta

← C23×#13677 #19161303

Subgroups — C23×#13677 #19161303

Order 5888.

420 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3288C211yesmaximal, characteristic, complemented
2C23×#2202C211yesmaximal, characteristic, complemented
3C23×#2199C211yesmaximal, characteristic, complemented
4C23×#1655C211yesmaximal, characteristic, complemented
5C23×#1406C211yesmaximal, characteristic, complemented
6C23×#1404C211yesmaximal, characteristic, complemented
7C23×#1403C211yesmaximal, characteristic, complemented
8C23×#577—21no—
9C23×#576—21no—
10C23×H(3,𝔽4)C2211yescharacteristic, complemented
11C23×#496—21no—
12C23×#496C2211yescharacteristic
13C23×#472—21no—
14C23×#471—21no—
15C23×B(4,𝔽2)—21no—
16C23×#455C2211yescharacteristic
17C23×#454—21no—
18C23×#454C2211yescharacteristic
19C23×#453—21no—
20C23×#444—21no—
21C23×#355—21no—
22C23×#355C2211yescharacteristic, complemented
23C23×#354C2211yescharacteristic, complemented
24C23×AΣL(1,𝔽16)—21no—
25C23×#329—21no—
26C23×#329C2211yescharacteristic, complemented
27C23×2−1+4D411yescharacteristic, complemented
28C23×H(5,𝔽2)—21no—
29C23×H(5,𝔽2)D411yescharacteristic
30C23×#137—21no—
31C23×#137—41no—
32C23×AGL(1,ℤ/8)—21no—
33C23×AGL(1,ℤ/8)—21no—
34C23×AGL(1,ℤ/8)—21no—
35C23×AGL(1,ℤ/8)—41no—
36C23×AGL(1,ℤ/8)—41no—
37C23×AGL(1,ℤ/8)—41no—
38C23×AGL(1,ℤ/8)—41no—
39C23×#135—21no—
40C23×#135—41no—
41C23×#135—41no—
42C2×SD16×C23—21no—
43C2×SD16×C23—41no—
44C2×SD16×C23—41no—
45C2×D8×C23—21no—
46C2×D8×C23—41no—
47C2×D8×C23—41no—
48C23×#131—21no—
49C23×#131—21no—
50C23×#127D411yescharacteristic, complemented
51C23×#124—21no—
52C23×#124—21no—
53C23×#124—41no—
54C23×#124C2311yescharacteristic
55C23×#120—21no—
56C23×#120—41no—
57C23×#120—41no—
58C23×#108—21no—
59C23×#104—21no—
60C23×#104—21no—
61C23×#104—41no—
62C23×#101—21no—
63C23×#101D411yescharacteristic, complemented
64C23×#100—21no—
65C23×#100D411yescharacteristic
66C23×#99—21no—
67C23×#99—41no—
68C23×#97—21no—
69C23×#97D411yescharacteristic, complemented
70C24×C23—21no—
71#41×C23—21no—
72#41×C23—21no—
73#41×C23—41no—
74#41×C23—41no—
75#41×C23—41no—
76#41×C23—41no—
77#41×C23—41no—
78C2×Q8×C23—21no—
79C2×Q8×C23—41no—
80C2×Q8×C23C2×D411yescharacteristic
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—41no—
88C2×D4×C23—41no—
89C2×D4×C23—41no—
90C2×D4×C23—81no—
91C2×D4×C23—81no—
92C2×D4×C23C2×D411yescharacteristic
93Q16×C23—21no—
94Q16×C23—41no—
95SD16×C23—21no—
96SD16×C23—21no—
97SD16×C23—41no—
98SD16×C23—41no—
99SD16×C23—41no—
100SD16×C23—41no—
101SD16×C23—41no—
102SD16×C23—41no—
103SD16×C23—82no—
104SD16×C23—81no—
105D8×C23—21no—
106D8×C23—41no—
107D8×C23—41no—
108D8×C23—41no—
109D8×C23—41no—
110D8×C23—41no—
111D8×C23—82no—
112D8×C23—81no—
113M16×C23—21no—
114M16×C23—21no—
115M16×C23—41no—
116M16×C23—41no—
117M16×C23—41no—
118C2×C8×C23—21no—
119C2×C8×C23—41no—
120C2×C8×C23—41no—
121C2×C8×C23—41no—
122C22⋊C4×C23—21no—
123C22⋊C4×C23—41no—
124C22⋊C4×C23—41no—
125C22⋊C4×C23—41no—
126C22⋊C4×C23—41no—
127C22⋊C4×C23—41no—
128C22⋊C4×C23—81no—
129C42×C23—21no—
130C42×C23C2×D411yescharacteristic
131#13677C2311yesmaximal, characteristic, direct product factor, Sylow
132C23×C23—21no—
133C23×C23—41no—
134C23×C23—41no—
135C23×C23—41no—
136C23×C23—81no—
137C23×C23—81no—
138C23×C23—161no—
139Q8×C23—21no—
140Q8×C23—21no—
141Q8×C23—41no—
142Q8×C23—41no—
143Q8×C23—81no—
144D4×C23—21no—
145D4×C23—21no—
146D4×C23—41no—
147D4×C23—41no—
148D4×C23—41no—
149D4×C23—41no—
150D4×C23—41no—
151D4×C23—41no—
152D4×C23—81no—
153D4×C23—81no—
154D4×C23—81no—
155D4×C23—81no—
156D4×C23—81no—
157D4×C23—81no—
158D4×C23—81no—
159D4×C23—161no—
160D4×C23—162no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—41no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23—41no—
167C2×C4×C23—81no—
168C2×C4×C23—81no—
169C2×C4×C23—81no—
170C2×C4×C23#12011yescharacteristic
171C8×C23—21no—
172C8×C23—21no—
173C8×C23—41no—
174C8×C23—41no—
175C8×C23—42no—
176C8×C23—81no—
177#3288C2×C2311yescharacteristic, complemented
178#2202C2×C2311yescharacteristic, complemented
179#2199C2×C2311yescharacteristic, complemented
180#1655C2×C2311yescharacteristic, complemented
181#1406C2×C2311yescharacteristic, complemented
182#1404C2×C2311yescharacteristic, complemented
183#1403C2×C2311yescharacteristic, complemented
184C22×C23—41no—
185C22×C23—41no—
186C22×C23—41no—
187C22×C23—81no—
188C22×C23—81no—
189C22×C23—81no—
190C22×C23—82no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23—161no—
194C22×C23—161no—
195C22×C23—161no—
196C22×C23—321no—
197C22×C23B(4,𝔽2)11yescharacteristic
198C4×C23—21no—
199C4×C23—41no—
200C4×C23—41no—
201C4×C23—41no—
202C4×C23—41no—
203C4×C23—81no—
204C4×C23—161no—
205#577—21no—
206#576—21no—
207H(3,𝔽4)C22×C2311yescharacteristic, complemented
208#496—21no—
209#496C22×C2311yescharacteristic
210#472—21no—
211#471—21no—
212B(4,𝔽2)—21no—
213#455C22×C2311yescharacteristic
214#454—21no—
215#454C22×C2311yescharacteristic
216#453—21no—
217#444—21no—
218#355—21no—
219#355C22×C2311yescharacteristic, complemented
220#354C22×C2311yescharacteristic, complemented
221AΣL(1,𝔽16)—21no—
222#329—21no—
223#329C22×C2311yescharacteristic, complemented
224C2×C23—21no—
225C2×C23—81no—
226C2×C23—81no—
227C2×C23—81no—
228C2×C23—161no—
229C2×C23—161no—
230C2×C23#219611yescharacteristic
2312−1+4D4×C2311yescharacteristic, complemented
232H(5,𝔽2)—21no—
233H(5,𝔽2)D4×C2311yescharacteristic
234#137—21no—
235#137—41no—
236AGL(1,ℤ/8)—21no—
237AGL(1,ℤ/8)—21no—
238AGL(1,ℤ/8)—21no—
239AGL(1,ℤ/8)—41no—
240AGL(1,ℤ/8)—41no—
241AGL(1,ℤ/8)—41no—
242AGL(1,ℤ/8)—41no—
243#135—21no—
244#135—41no—
245#135—41no—
246C2×SD16—21no—
247C2×SD16—41no—
248C2×SD16—41no—
249C2×D8—21no—
250C2×D8—41no—
251C2×D8—41no—
252#131—21no—
253#131—21no—
254#127D4×C2311yescharacteristic, complemented
255#124—21no—
256#124—21no—
257#124—41no—
258#124C23×C2311yescharacteristic
259#120—21no—
260#120—41no—
261#120—41no—
262#108—21no—
263#104—21no—
264#104—21no—
265#104—41no—
266#101—21no—
267#101D4×C2311yescharacteristic, complemented
268#100—21no—
269#100D4×C2311yescharacteristic
270#99—21no—
271#99—41no—
272#97—21no—
273#97D4×C2311yescharacteristic, complemented
274C23#1367711yescharacteristic, direct product factor, Sylow
275C24—21no—
276#41—21no—
277#41—21no—
278#41—41no—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
283C2×Q8—21no—
284C2×Q8—41no—
285C2×Q8C2×D4×C2311yescharacteristic
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—81no—
296C2×D4—81no—
297C2×D4C2×D4×C2311yescharacteristic
298Q16—21no—
299Q16—41no—
300SD16—21no—
301SD16—21no—
302SD16—41no—
303SD16—41no—
304SD16—41no—
305SD16—41no—
306SD16—41no—
307SD16—41no—
308SD16—82no—
309SD16—81no—
310D8—21no—
311D8—41no—
312D8—41no—
313D8—41no—
314D8—41no—
315D8—41no—
316D8—82no—
317D8—81no—
318M16—21no—
319M16—21no—
320M16—41no—
321M16—41no—
322M16—41no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C22⋊C4—21no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—81no—
334C42—21no—
335C42C2×D4×C2311yescharacteristic
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—81no—
341C23—81no—
342C23—161no—
343Q8—21no—
344Q8—21no—
345Q8—41no—
346Q8—41no—
347Q8—81no—
348D4—21no—
349D4—21no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—161no—
364D4—162no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4C23×#12011yescharacteristic
375C8—21no—
376C8—21no—
377C8—41no—
378C8—41no—
379C8—42no—
380C8—81no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—82no—
388C22—81no—
389C22—81no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22—321no—
394C22C23×B(4,𝔽2)11yescharacteristic
395C4—21no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—81no—
401C4—161no—
402C2—21no—
403C2—81no—
404C2—81no—
405C2—81no—
406C2—161no—
407C2—161no—
408C2C23×#219611yescharacteristic
subgroupG/H
center, socleC2×C23#2196
commutator, Frattini, cosocle#124C23×C23
LCSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C23 ◁ C2×C4×C23 ◁ C23×#124 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3288C2semidirect
2. C23×#2202C2semidirect
3. C23×#2199C2semidirect
4. C23×#1655C2semidirect
5. C23×#1406C2semidirect
6. C23×#1404C2semidirect
7. C23×#1403C2semidirect
10. C23×H(3,𝔽4)C22semidirect
22. C23×#355C22semidirect
23. C23×#354C22semidirect
26. C23×#329C22semidirect
27. C23×2−1+4D4semidirect
50. C23×#127D4semidirect
63. C23×#101D4semidirect
69. C23×#97D4semidirect
131. #13677C23direct (N × H)
177. #3288C2×C23semidirect
178. #2202C2×C23semidirect
179. #2199C2×C23semidirect
180. #1655C2×C23semidirect
181. #1406C2×C23semidirect
182. #1404C2×C23semidirect
183. #1403C2×C23semidirect
207. H(3,𝔽4)C22×C23semidirect
219. #355C22×C23semidirect
220. #354C22×C23semidirect
223. #329C22×C23semidirect
231. 2−1+4D4×C23semidirect
254. #127D4×C23semidirect
267. #101D4×C23semidirect
273. #97D4×C23semidirect
274. C23#13677direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32882
C23×#22022
C23×#21992
C23×#16552
C23×#14062
C23×#14042
C23×#14032
#1367723