Finite Group Databasebeta

← C23×#13037 #19160663

Subgroups — C23×#13037 #19160663

Order 5888.

538 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#509C211yesmaximal, characteristic
2C2×C23×#442C211yesmaximal, characteristic, complemented
3C2×C23×#361C211yesmaximal, characteristic, complemented
4C2×C23×#360C211yesmaximal, characteristic, complemented
5C23×#1975C211yesmaximal, characteristic, complemented
6C23×#1383C211yesmaximal, characteristic, complemented
7C23×#1375C211yesmaximal, characteristic, complemented
8C2×C23×#135C2211yescharacteristic
9C22×D8×C23C2211yescharacteristic
10C2×C4×D4×C23—21no—
11C23×#509—22no—
12C23×#509—22no—
13C2×C23×SD32C412yescomplemented
14C2×C23×D16C412yescomplemented
15C2×C23×M32C2211yescharacteristic
16C2×C23×M32C2211yescharacteristic
17C23×#442—22no—
18C23×#442—22no—
19C2×C23×#108C2211yescharacteristic
20C2×C23×#106C2211yescharacteristic
21C2×C23×#104—21no—
22C2×C23×#102—21no—
23C23×#408—21no—
24C2×C23×#97C2211yescharacteristic
25C23×#361—22no—
26C23×#360—22no—
27C2×#41×C23—21no—
28C22×D4×C23—21no—
29C23×C4×C23—41no—
30C23×#135—21no—
31C23×#135—21no—
32C23×#135—21no—
33C23×#135—21no—
34C2×Q16×C23C2×C411yescharacteristic, complemented
35C2×SD16×C23—21no—
36C2×D8×C23—21no—
37C2×D8×C23—21no—
38C2×D8×C23—21no—
39C2×D8×C23C2×C412yes—
40C2×D8×C23C2×C411yescharacteristic, complemented
41C2×D8×C23D412yes—
42C2×D8×C23D412yes—
43C2×M16×C23—21no—
44C22×C8×C23D411yescharacteristic
45C22×C8×C23C2311yescharacteristic
46C4×D4×C23—24no—
47C4×D4×C23—42no—
48C2×C4⋊C4×C23—21no—
49C2×C22⋊C4×C23—41no—
50C2×C42×C23D411yescharacteristic
51C23×SD32—24no—
52C23×D16—24no—
53C23×M32—21no—
54C23×M32—21no—
55C23×M32—21no—
56C23×M32—21no—
57C2×C16×C23—21no—
58C2×C16×C23C2×C412yescomplemented
59C23×#108—21no—
60C23×#108—21no—
61C23×#106—21no—
62C23×#106D412yes—
63C23×#104—42no—
64C23×#102—24no—
65C23×C22⋊C8—42no—
66C23×#97—21no—
67C23×#97—21no—
68C24×C23—41no—
69#41×C23—41no—
70#41×C23—41no—
71#41×C23—41no—
72#41×C23—41no—
73C2×Q8×C23—21no—
74C2×D4×C23—22no—
75C2×D4×C23—21no—
76C2×D4×C23—22no—
77C2×D4×C23—22no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C22×C4×C23—21no—
83C22×C4×C23—44no—
84C22×C4×C23—41no—
85C22×C4×C23—41no—
86C22×C4×C23—84no—
87C22×C4×C23C2×D411yescharacteristic
88Q16×C23—21no—
89Q16×C23—21no—
90SD16×C23—41no—
91SD16×C23—41no—
92D8×C23—21no—
93D8×C23—22no—
94D8×C23—21no—
95D8×C23—22no—
96D8×C23—22no—
97D8×C23C22⋊C414yes—
98M16×C23—41no—
99M16×C23—41no—
100C2×C8×C23—21no—
101C2×C8×C23—21no—
102C2×C8×C23—21no—
103C2×C8×C23—41no—
104C2×C8×C23C22×C411yescharacteristic
105C2×C8×C23C2×D411yescharacteristic
106C2×C8×C23C2×D411yescharacteristic
107C2×C8×C23C2×D411yescharacteristic
108C2×C8×C23#4111yescharacteristic
109C2×C8×C23#4111yescharacteristic
110C4⋊C4×C23—22no—
111C4⋊C4×C23—41no—
112C22⋊C4×C23—44no—
113C42×C23—21no—
114C42×C23D812yes—
115C16×C23—22no—
116C16×C23—41no—
117#13037C2311yesmaximal, characteristic, direct product factor, Sylow
118C23×C23—42no—
119C23×C23—42no—
120C23×C23—42no—
121C23×C23—41no—
122C23×C23—82no—
123C23×C23—82no—
124C23×C23AGL(1,ℤ/8)11yescharacteristic
125Q8×C23—41no—
126Q8×C23—41no—
127D4×C23—24no—
128D4×C23—41no—
129D4×C23—42no—
130D4×C23—41no—
131D4×C23—42no—
132D4×C23—42no—
133D4×C23—81no—
134D4×C23—81no—
135C2×C4×C23—22no—
136C2×C4×C23—42no—
137C2×C4×C23—41no—
138C2×C4×C23—42no—
139C2×C4×C23—88no—
140C2×C4×C23—82no—
141C2×C4×C23—81no—
142C2×C4×C23—81no—
143C2×C4×C23C4×D411yescharacteristic
144C2×C4×C23#12011yescharacteristic
145C2×C4×C23#12111yescharacteristic
146C2×C4×C23C2×D811yescharacteristic
147C2×C4×C23#13511yescharacteristic
148C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
149C8×C23—41no—
150C8×C23—81no—
151C8×C23C2×C22⋊C411yescharacteristic
152C8×C23C4×D411yescharacteristic
153C8×C23#12111yescharacteristic
154C8×C23#12311yescharacteristic
155C2×#509C2×C2311yescharacteristic
156C2×#442C2×C2311yescharacteristic, complemented
157C2×#361C2×C2311yescharacteristic, complemented
158C2×#360C2×C2311yescharacteristic, complemented
159#1975C2×C2311yescharacteristic, complemented
160#1383C2×C2311yescharacteristic, complemented
161#1375C2×C2311yescharacteristic, complemented
162C22×C23—22no—
163C22×C23—44no—
164C22×C23—82no—
165C22×C23—82no—
166C22×C23—84no—
167C22×C23—84no—
168C22×C23—81no—
169C22×C23—82no—
170C22×C23#44211yescharacteristic
171C22×C23#44711yescharacteristic
172C22×C23#46311yescharacteristic
173C4×C23—42no—
174C4×C23—82no—
175C4×C23—81no—
176C4×C23#38611yescharacteristic
177C4×C23C4×D811yescharacteristic
178C4×C23#44911yescharacteristic
179C4×C23#45911yescharacteristic
180C2×#135C22×C2311yescharacteristic
181C22×D8C22×C2311yescharacteristic
182C2×C4×D4—21no—
183#509—22no—
184#509—22no—
185C2×SD32C4×C2312yescomplemented
186C2×D16C4×C2312yescomplemented
187C2×M32C22×C2311yescharacteristic
188C2×M32C22×C2311yescharacteristic
189#442—22no—
190#442—22no—
191C2×#108C22×C2311yescharacteristic
192C2×#106C22×C2311yescharacteristic
193C2×#104—21no—
194C2×#102—21no—
195#408—21no—
196C2×#97C22×C2311yescharacteristic
197#361—22no—
198#360—22no—
199C2×C23—21no—
200C2×C23—21no—
201C2×C23—84no—
202C2×C23—161no—
203C2×C23#187911yescharacteristic
204C2×C23AGL(1,ℤ/16)12yes—
205C2×#41—21no—
206C22×D4—21no—
207C23×C4—41no—
208#135—21no—
209#135—21no—
210#135—21no—
211#135—21no—
212C2×Q16C2×C4×C2311yescharacteristic, complemented
213C2×SD16—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8C2×C4×C2312yes—
218C2×D8C2×C4×C2311yescharacteristic, complemented
219C2×D8D4×C2312yes—
220C2×D8D4×C2312yes—
221C2×M16—21no—
222C22×C8D4×C2311yescharacteristic
223C22×C8C23×C2311yescharacteristic
224C4×D4—24no—
225C4×D4—42no—
226C2×C4⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C42D4×C2311yescharacteristic
229SD32—24no—
230D16—24no—
231M32—21no—
232M32—21no—
233M32—21no—
234M32—21no—
235C2×C16—21no—
236C2×C16C2×C4×C2312yescomplemented
237#108—21no—
238#108—21no—
239#106—21no—
240#106D4×C2312yes—
241#104—42no—
242#102—24no—
243C22⋊C8—42no—
244#97—21no—
245#97—21no—
246C23#1303711yescharacteristic, direct product factor, Sylow
247C24—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251#41—41no—
252C2×Q8—21no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—84no—
266C22×C4C2×D4×C2311yescharacteristic
267Q16—21no—
268Q16—21no—
269SD16—41no—
270SD16—41no—
271D8—21no—
272D8—22no—
273D8—21no—
274D8—22no—
275D8—22no—
276D8C22⋊C4×C2314yes—
277M16—41no—
278M16—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8C22×C4×C2311yescharacteristic
284C2×C8C2×D4×C2311yescharacteristic
285C2×C8C2×D4×C2311yescharacteristic
286C2×C8C2×D4×C2311yescharacteristic
287C2×C8#41×C2311yescharacteristic
288C2×C8#41×C2311yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—41no—
291C22⋊C4—44no—
292C42—21no—
293C42D8×C2312yes—
294C16—22no—
295C16—41no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—82no—
301C23—82no—
302C23C23×AGL(1,ℤ/8)11yescharacteristic
303Q8—41no—
304Q8—41no—
305D4—24no—
306D4—41no—
307D4—42no—
308D4—41no—
309D4—42no—
310D4—42no—
311D4—81no—
312D4—81no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—88no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4C4×D4×C2311yescharacteristic
322C2×C4C23×#12011yescharacteristic
323C2×C4C23×#12111yescharacteristic
324C2×C4C2×D8×C2311yescharacteristic
325C2×C4C23×#13511yescharacteristic
326C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
327C8—41no—
328C8—81no—
329C8C2×C22⋊C4×C2311yescharacteristic
330C8C4×D4×C2311yescharacteristic
331C8C23×#12111yescharacteristic
332C8C23×#12311yescharacteristic
333C22—22no—
334C22—44no—
335C22—82no—
336C22—82no—
337C22—84no—
338C22—84no—
339C22—82no—
340C22—81no—
341C22C23×#44211yescharacteristic
342C22C23×#44711yescharacteristic
343C22C23×#46311yescharacteristic
344C4—42no—
345C4—82no—
346C4—81no—
347C4C23×#38611yescharacteristic
348C4C4×D8×C2311yescharacteristic
349C4C23×#44911yescharacteristic
350C4C23×#45911yescharacteristic
351C2—21no—
352C2—21no—
353C2—84no—
354C2—161no—
355C2C23×#187911yescharacteristic
356C2C23×AGL(1,ℤ/16)12yes—
subgroupG/H
center, socleC22×C23#442
commutatorC2×C8C22×C4×C23
Frattini, cosocleC22×C8C23×C23
prime radicalC2×C23×#509C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×#97 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#442C2semidirect
3. C2×C23×#361C2semidirect
4. C2×C23×#360C2semidirect
5. C23×#1975C2semidirect
6. C23×#1383C2semidirect
7. C23×#1375C2semidirect
13. C2×C23×SD32C4semidirect
14. C2×C23×D16C4semidirect
34. C2×Q16×C23C2×C4semidirect
40. C2×D8×C23C2×C4semidirect
58. C2×C16×C23C2×C4semidirect
117. #13037C23direct (N × H)
156. C2×#442C2×C23semidirect
157. C2×#361C2×C23semidirect
158. C2×#360C2×C23semidirect
159. #1975C2×C23semidirect
160. #1383C2×C23semidirect
161. #1375C2×C23semidirect
185. C2×SD32C4×C23semidirect
186. C2×D16C4×C23semidirect
212. C2×Q16C2×C4×C23semidirect
218. C2×D8C2×C4×C23semidirect
236. C2×C16C2×C4×C23semidirect
246. C23#13037direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5092
C2×C23×#4422
C2×C23×#3612
C2×C23×#3602
C23×#19752
C23×#13832
C23×#13752
#1303723