Finite Group Databasebeta

← C23×#13679 #19161305

Subgroups — C23×#13679 #19161305

Order 5888.

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

normal subgroups only

This group has 386 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
2C23×#2203C211yesmaximal, characteristic, complemented
3C23×#2201C211yesmaximal, characteristic, complemented
4C23×#1657C211yesmaximal, characteristic, complemented
5C23×#1406C211yesmaximal, characteristic, complemented
6C23×#1404C211yesmaximal, characteristic, complemented
7C23×#1403C211yesmaximal, characteristic, complemented
8C23×#577—21no—
9C23×#576—21no—
10C23×#563C2211yescharacteristic, complemented
11C23×#496C2211yescharacteristic
12C23×#490—21no—
13C23×#472—21no—
14C23×#471—21no—
15C23×#458—21no—
16C23×#456—21no—
17C23×#455C2211yescharacteristic
18C23×#454C2211yescharacteristic
19C23×#453—21no—
20C23×#444—21no—
21C23×#356—21no—
22C23×#355C2211yescharacteristic, complemented
23C23×#354C2211yescharacteristic, complemented
24C23×#352—21no—
25C23×#330—21no—
26C23×#329C2211yescharacteristic, complemented
27C23×2−1+4D411yescharacteristic
28C23×H(5,𝔽2)—21no—
29C23×H(5,𝔽2)D411yescharacteristic
30C23×#137—21no—
31C23×#137—41no—
32C23×#137—41no—
33C23×AGL(1,ℤ/8)—21no—
34C23×AGL(1,ℤ/8)—21no—
35C23×AGL(1,ℤ/8)—21no—
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×D8×C23—21no—
45C2×D8×C23—41no—
46C23×#131—21no—
47C23×#131—21no—
48C23×C4⋊Q8—21no—
49C23×#127D411yescharacteristic
50C23×#126—41no—
51C23×#125—21no—
52C23×#124C2311yescharacteristic
53C23×#123—21no—
54C23×#123—41no—
55C23×C22⋊Q8—41no—
56C23×#108—21no—
57C23×#104—21no—
58C23×#104—21no—
59C23×#104—41no—
60C23×#102—41no—
61C23×#102—41no—
62C23×#101—21no—
63C23×#101D411yescharacteristic
64C23×#100—21no—
65C23×#100D411yescharacteristic
66C23×#99—21no—
67C23×#99—41no—
68C23×#97—21no—
69C23×#97D411yescharacteristic
70#41×C23—21no—
71#41×C23—21no—
72#41×C23—41no—
73#41×C23—41no—
74#41×C23—41no—
75#41×C23—41no—
76#41×C23—41no—
77C2×Q8×C23—21no—
78C2×Q8×C23—41no—
79C2×Q8×C23C2×D411yescharacteristic
80C2×D4×C23—21no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—41no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C2×D4×C23—81no—
87C2×D4×C23C2×D411yescharacteristic
88C22×C4×C23—21no—
89Q16×C23—21no—
90Q16×C23—41no—
91Q16×C23—81no—
92SD16×C23—21no—
93SD16×C23—21no—
94SD16×C23—41no—
95SD16×C23—41no—
96SD16×C23—41no—
97SD16×C23—41no—
98SD16×C23—41no—
99SD16×C23—41no—
100SD16×C23—81no—
101D8×C23—21no—
102D8×C23—41no—
103D8×C23—41no—
104D8×C23—41no—
105D8×C23—41no—
106D8×C23—41no—
107M16×C23—21no—
108M16×C23—21no—
109M16×C23—41no—
110M16×C23—41no—
111M16×C23—41no—
112C2×C8×C23—21no—
113C2×C8×C23—41no—
114C2×C8×C23—41no—
115C2×C8×C23—41no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23—21no—
118C4⋊C4×C23—41no—
119C4⋊C4×C23—41no—
120C4⋊C4×C23—41no—
121C4⋊C4×C23—81no—
122C22⋊C4×C23—21no—
123C22⋊C4×C23—41no—
124C22⋊C4×C23—41no—
125C22⋊C4×C23—41no—
126C22⋊C4×C23—81no—
127C42×C23—21no—
128C42×C23C2×D411yescharacteristic
129#13679C2311yesmaximal, characteristic, direct product factor, Sylow
130C23×C23—21no—
131C23×C23—41no—
132C23×C23—41no—
133C23×C23—81no—
134Q8×C23—21no—
135Q8×C23—21no—
136Q8×C23—41no—
137Q8×C23—41no—
138Q8×C23—81no—
139D4×C23—21no—
140D4×C23—21no—
141D4×C23—41no—
142D4×C23—41no—
143D4×C23—41no—
144D4×C23—41no—
145D4×C23—41no—
146D4×C23—41no—
147D4×C23—81no—
148D4×C23—81no—
149D4×C23—81no—
150D4×C23—82no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—81no—
159C2×C4×C23—81no—
160C2×C4×C23—81no—
161C2×C4×C23—81no—
162C2×C4×C23#12011yescharacteristic
163C8×C23—21no—
164C8×C23—21no—
165C8×C23—41no—
166C8×C23—41no—
167C8×C23—81no—
168C8×C23—81no—
169#3288C2×C2311yescharacteristic
170#2203C2×C2311yescharacteristic, complemented
171#2201C2×C2311yescharacteristic, complemented
172#1657C2×C2311yescharacteristic, complemented
173#1406C2×C2311yescharacteristic, complemented
174#1404C2×C2311yescharacteristic, complemented
175#1403C2×C2311yescharacteristic, complemented
176C22×C23—41no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23—81no—
180C22×C23—81no—
181C22×C23—161no—
182C22×C23—161no—
183C22×C23—321no—
184C22×C23B(4,𝔽2)11yescharacteristic
185C4×C23—21no—
186C4×C23—41no—
187C4×C23—41no—
188C4×C23—41no—
189C4×C23—41no—
190C4×C23—81no—
191C4×C23—81no—
192C4×C23—161no—
193#577—21no—
194#576—21no—
195#563C22×C2311yescharacteristic, complemented
196#496C22×C2311yescharacteristic
197#490—21no—
198#472—21no—
199#471—21no—
200#458—21no—
201#456—21no—
202#455C22×C2311yescharacteristic
203#454C22×C2311yescharacteristic
204#453—21no—
205#444—21no—
206#356—21no—
207#355C22×C2311yescharacteristic, complemented
208#354C22×C2311yescharacteristic, complemented
209#352—21no—
210#330—21no—
211#329C22×C2311yescharacteristic, complemented
212C2×C23—21no—
213C2×C23—81no—
214C2×C23—81no—
215C2×C23—81no—
216C2×C23—161no—
217C2×C23#219611yescharacteristic
2182−1+4D4×C2311yescharacteristic
219H(5,𝔽2)—21no—
220H(5,𝔽2)D4×C2311yescharacteristic
221#137—21no—
222#137—41no—
223#137—41no—
224AGL(1,ℤ/8)—21no—
225AGL(1,ℤ/8)—21no—
226AGL(1,ℤ/8)—21no—
227AGL(1,ℤ/8)—41no—
228AGL(1,ℤ/8)—41no—
229AGL(1,ℤ/8)—41no—
230#135—21no—
231#135—41no—
232#135—41no—
233C2×SD16—21no—
234C2×SD16—41no—
235C2×D8—21no—
236C2×D8—41no—
237#131—21no—
238#131—21no—
239C4⋊Q8—21no—
240#127D4×C2311yescharacteristic
241#126—41no—
242#125—21no—
243#124C23×C2311yescharacteristic
244#123—21no—
245#123—41no—
246C22⋊Q8—41no—
247#108—21no—
248#104—21no—
249#104—21no—
250#104—41no—
251#102—41no—
252#102—41no—
253#101—21no—
254#101D4×C2311yescharacteristic
255#100—21no—
256#100D4×C2311yescharacteristic
257#99—21no—
258#99—41no—
259#97—21no—
260#97D4×C2311yescharacteristic
261C23#1367911yescharacteristic, direct product factor, Sylow
262#41—21no—
263#41—21no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—41no—
271C2×Q8C2×D4×C2311yescharacteristic
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—81no—
279C2×D4C2×D4×C2311yescharacteristic
280C22×C4—21no—
281Q16—21no—
282Q16—41no—
283Q16—81no—
284SD16—21no—
285SD16—21no—
286SD16—41no—
287SD16—41no—
288SD16—41no—
289SD16—41no—
290SD16—41no—
291SD16—41no—
292SD16—81no—
293D8—21no—
294D8—41no—
295D8—41no—
296D8—41no—
297D8—41no—
298D8—41no—
299M16—21no—
300M16—21no—
301M16—41no—
302M16—41no—
303M16—41no—
304C2×C8—21no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C22⋊C4—21no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—81no—
319C42—21no—
320C42C2×D4×C2311yescharacteristic
321C23—21no—
322C23—41no—
323C23—41no—
324C23—81no—
325Q8—21no—
326Q8—21no—
327Q8—41no—
328Q8—41no—
329Q8—81no—
330D4—21no—
331D4—21no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—82no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4C23×#12011yescharacteristic
354C8—21no—
355C8—21no—
356C8—41no—
357C8—41no—
358C8—81no—
359C8—81no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
364C22—81no—
365C22—161no—
366C22—161no—
367C22—321no—
368C22C23×B(4,𝔽2)11yescharacteristic
369C4—21no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—81no—
375C4—81no—
376C4—161no—
377C2—21no—
378C2—81no—
379C2—81no—
380C2—81no—
381C2—161no—
382C2C23×#219611yescharacteristic
subgroupG/H
center, socleC2×C23#2196
commutator, Frattini, cosocle#124C23×C23
prime radicalC23×#3288C2
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
2. C23×#2203C2semidirect
3. C23×#2201C2semidirect
4. C23×#1657C2semidirect
5. C23×#1406C2semidirect
6. C23×#1404C2semidirect
7. C23×#1403C2semidirect
10. C23×#563C22semidirect
22. C23×#355C22semidirect
23. C23×#354C22semidirect
26. C23×#329C22semidirect
129. #13679C23direct (N × H)
170. #2203C2×C23semidirect
171. #2201C2×C23semidirect
172. #1657C2×C23semidirect
173. #1406C2×C23semidirect
174. #1404C2×C23semidirect
175. #1403C2×C23semidirect
195. #563C22×C23semidirect
207. #355C22×C23semidirect
208. #354C22×C23semidirect
211. #329C22×C23semidirect
261. C23#13679direct (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×#22032
C23×#22012
C23×#16572
C23×#14062
C23×#14042
C23×#14032
#1367923