Finite Group Databasebeta

← C23×#13072 #19160698

Subgroups — C23×#13072 #19160698

Order 5888.

458 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#504C211yesmaximal, characteristic
2C2×C23×#473C211yesmaximal, characteristic
3C2×C23×#362C211yesmaximal, characteristic, complemented
4C2×C23×#358C211yesmaximal, characteristic, complemented
5C23×#1816C211yesmaximal, characteristic
6C23×#1387C211yesmaximal, characteristic, complemented
7C23×#1381C211yesmaximal, characteristic, complemented
8C2×C23×#137—21no—
9C22×Q16×C23C2211yescharacteristic
10C2×C23×#131C2211yescharacteristic
11C23×#504—22no—
12C23×#504—22no—
13C2×C23×M32—21no—
14C2×C23×M32C2211yescharacteristic
15C22×C16×C23—21no—
16C22×C16×C23C2211yescharacteristic
17C23×#473—22no—
18C23×#473C414yescomplemented
19C2×C23×#108C2211yescharacteristic
20C2×C23×#108C2211yescharacteristic
21C2×C23×#107C2211yescharacteristic
22C23×#417—21no—
23C2×C23×#101—21no—
24C23×#362—22no—
25C23×#358—21no—
26C23×#358—21no—
27C23×#343—21no—
28C2×#41×C23D411yescharacteristic
29C22×Q8×C23—21no—
30C23×#137—24no—
31C23×#137—42no—
32C2×Q16×C23—21no—
33C2×Q16×C23—21no—
34C2×Q16×C23—21no—
35C2×Q16×C23—21no—
36C2×Q16×C23—21no—
37C2×Q16×C23—41no—
38C2×Q16×C23C2×C412yes—
39C2×SD16×C23—41no—
40C23×#131—21no—
41C23×#131—21no—
42C23×#131C2×C412yes—
43C23×#131D412yes—
44C2×M16×C23—21no—
45C2×M16×C23—21no—
46C2×M16×C23—21no—
47C2×M16×C23D411yescharacteristic
48C22×C8×C23—21no—
49C22×C8×C23C2311yescharacteristic
50C2×C4⋊C4×C23—21no—
51C23×M32—24no—
52C23×M32—21no—
53C23×M32D412yes—
54C2×C16×C23—24no—
55C2×C16×C23—21no—
56C2×C16×C23—21no—
57C2×C16×C23—21no—
58C2×C16×C23D411yescharacteristic
59C2×C16×C23D411yescharacteristic
60C23×#108—21no—
61C23×#108—21no—
62C23×#108—21no—
63C23×#108C2×C412yes—
64C23×#107—21no—
65C23×#107—21no—
66C23×#103—41no—
67C23×#102—41no—
68C23×#101—24no—
69#41×C23—21no—
70#41×C23—21no—
71#41×C23C22⋊C412yes—
72#41×C23SD1612yes—
73C2×Q8×C23—22no—
74C2×Q8×C23—41no—
75C2×Q8×C23—41no—
76C2×Q8×C23—41no—
77C2×Q8×C23—41no—
78C2×Q8×C23—41no—
79C2×Q8×C23D811yescharacteristic
80C2×D4×C23—21no—
81C2×D4×C23D811yescharacteristic
82C22×C4×C23—21no—
83C22×C4×C23—41no—
84C22×C4×C23—41no—
85C22×C4×C23C2×D411yescharacteristic
86Q16×C23—22no—
87Q16×C23—22no—
88Q16×C23—44no—
89Q16×C23—41no—
90Q16×C23—41no—
91SD16×C23—44no—
92M16×C23—21no—
93M16×C23—22no—
94M16×C23—22no—
95M16×C23—22no—
96M16×C23—41no—
97M16×C23—41no—
98M16×C23—41no—
99M16×C23C22⋊C412yes—
100C2×C8×C23—21no—
101C2×C8×C23—22no—
102C2×C8×C23—22no—
103C2×C8×C23—41no—
104C2×C8×C23—41no—
105C2×C8×C23C22×C411yescharacteristic
106C2×C8×C23C2×D411yescharacteristic
107C2×C8×C23C2×D411yescharacteristic
108C2×C8×C23C2×D411yescharacteristic
109C2×C8×C23#4111yescharacteristic
110C2×C8×C23#4111yescharacteristic
111C4⋊C4×C23—41no—
112C4⋊C4×C23—41no—
113C16×C23—21no—
114C16×C23—22no—
115C16×C23—21no—
116#13072C2311yesmaximal, characteristic, direct product factor, Sylow
117C23×C23—21no—
118C23×C23#13511yescharacteristic
119Q8×C23—21no—
120Q8×C23—42no—
121Q8×C23—42no—
122Q8×C23—81no—
123Q8×C23—81no—
124Q8×C23#10212yes—
125D4×C23—22no—
126D4×C23—22no—
127D4×C23—21no—
128D4×C23#10212yes—
129C2×C4×C23—21no—
130C2×C4×C23—22no—
131C2×C4×C23—22no—
132C2×C4×C23—42no—
133C2×C4×C23—81no—
134C2×C4×C23—81no—
135C2×C4×C23—81no—
136C2×C4×C23—161no—
137C2×C4×C23C2×C22⋊C411yescharacteristic
138C2×C4×C23#12111yescharacteristic
139C2×C4×C23#12111yescharacteristic
140C2×C4×C23C2×D811yescharacteristic
141C2×C4×C23C2×SD1611yescharacteristic
142C2×C4×C23#13511yescharacteristic
143C8×C23—22no—
144C8×C23—42no—
145C8×C23—81no—
146C8×C23C4×D411yescharacteristic
147C8×C23C4×D411yescharacteristic
148C8×C23#12011yescharacteristic
149C8×C23#12311yescharacteristic
150C2×#504C2×C2311yescharacteristic
151C2×#473C2×C2311yescharacteristic
152C2×#362C2×C2311yescharacteristic, complemented
153C2×#358C2×C2311yescharacteristic, complemented
154#1816C2×C2311yescharacteristic
155#1387C2×C2311yescharacteristic, complemented
156#1381C2×C2311yescharacteristic, complemented
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—22no—
160C22×C23—41no—
161C22×C23—41no—
162C22×C23#41611yescharacteristic
163C22×C23#46511yescharacteristic
164C22×C23#46611yescharacteristic
165C4×C23—22no—
166C4×C23—82no—
167C4×C23—161no—
168C4×C23#38611yescharacteristic
169C4×C23C2×#10211yescharacteristic
170C4×C23#46511yescharacteristic
171C4×C23#46611yescharacteristic
172C2×#137—21no—
173C22×Q16C22×C2311yescharacteristic
174C2×#131C22×C2311yescharacteristic
175#504—22no—
176#504—22no—
177C2×M32—21no—
178C2×M32C22×C2311yescharacteristic
179C22×C16—21no—
180C22×C16C22×C2311yescharacteristic
181#473—22no—
182#473C4×C2314yescomplemented
183C2×#108C22×C2311yescharacteristic
184C2×#108C22×C2311yescharacteristic
185C2×#107C22×C2311yescharacteristic
186#417—21no—
187C2×#101—21no—
188#362—22no—
189#358—21no—
190#358—21no—
191#343—21no—
192C2×C23—21no—
193C2×C23—21no—
194C2×C23—42no—
195C2×C23#189311yescharacteristic
196C2×C23#222211yescharacteristic
197C2×C23#222311yescharacteristic
198C2×#41D4×C2311yescharacteristic
199C22×Q8—21no—
200#137—24no—
201#137—42no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×Q16—21no—
207C2×Q16—41no—
208C2×Q16C2×C4×C2312yes—
209C2×SD16—41no—
210#131—21no—
211#131—21no—
212#131C2×C4×C2312yes—
213#131D4×C2312yes—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16D4×C2311yescharacteristic
218C22×C8—21no—
219C22×C8C23×C2311yescharacteristic
220C2×C4⋊C4—21no—
221M32—24no—
222M32—21no—
223M32D4×C2312yes—
224C2×C16—24no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16D4×C2311yescharacteristic
229C2×C16D4×C2311yescharacteristic
230#108—21no—
231#108—21no—
232#108—21no—
233#108C2×C4×C2312yes—
234#107—21no—
235#107—21no—
236#103—41no—
237#102—41no—
238#101—24no—
239C23#1307211yescharacteristic, direct product factor, Sylow
240#41—21no—
241#41—21no—
242#41C22⋊C4×C2312yes—
243#41SD16×C2312yes—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8D8×C2311yescharacteristic
251C2×D4—21no—
252C2×D4D8×C2311yescharacteristic
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×D4×C2311yescharacteristic
257Q16—22no—
258Q16—22no—
259Q16—44no—
260Q16—41no—
261Q16—41no—
262SD16—44no—
263M16—21no—
264M16—22no—
265M16—22no—
266M16—22no—
267M16—41no—
268M16—41no—
269M16—41no—
270M16C22⋊C4×C2312yes—
271C2×C8—21no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8C22×C4×C2311yescharacteristic
277C2×C8C2×D4×C2311yescharacteristic
278C2×C8C2×D4×C2311yescharacteristic
279C2×C8C2×D4×C2311yescharacteristic
280C2×C8#41×C2311yescharacteristic
281C2×C8#41×C2311yescharacteristic
282C4⋊C4—41no—
283C4⋊C4—41no—
284C16—21no—
285C16—22no—
286C16—21no—
287C23—21no—
288C23C23×#13511yescharacteristic
289Q8—21no—
290Q8—42no—
291Q8—42no—
292Q8—81no—
293Q8—81no—
294Q8C23×#10212yes—
295D4—22no—
296D4—22no—
297D4—21no—
298D4C23×#10212yes—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—161no—
307C2×C4C2×C22⋊C4×C2311yescharacteristic
308C2×C4C23×#12111yescharacteristic
309C2×C4C23×#12111yescharacteristic
310C2×C4C2×D8×C2311yescharacteristic
311C2×C4C2×SD16×C2311yescharacteristic
312C2×C4C23×#13511yescharacteristic
313C8—22no—
314C8—42no—
315C8—81no—
316C8C4×D4×C2311yescharacteristic
317C8C4×D4×C2311yescharacteristic
318C8C23×#12011yescharacteristic
319C8C23×#12311yescharacteristic
320C22—21no—
321C22—21no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22C23×#41611yescharacteristic
326C22C23×#46511yescharacteristic
327C22C23×#46611yescharacteristic
328C4—22no—
329C4—82no—
330C4—161no—
331C4C23×#38611yescharacteristic
332C4C2×C23×#10211yescharacteristic
333C4C23×#46511yescharacteristic
334C4C23×#46611yescharacteristic
335C2—21no—
336C2—21no—
337C2—42no—
338C2C23×#189311yescharacteristic
339C2C23×#222211yescharacteristic
340C2C23×#222311yescharacteristic
subgroupG/H
center, socleC22×C23#466
commutatorC2×C8C22×C4×C23
Frattini, cosocleC22×C8C23×C23
prime radicalC2×#41×C23D4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×#131 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#362C2semidirect
4. C2×C23×#358C2semidirect
6. C23×#1387C2semidirect
7. C23×#1381C2semidirect
18. C23×#473C4semidirect
116. #13072C23direct (N × H)
152. C2×#362C2×C23semidirect
153. C2×#358C2×C23semidirect
155. #1387C2×C23semidirect
156. #1381C2×C23semidirect
182. #473C4×C23semidirect
239. C23#13072direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5042
C2×C23×#4732
C2×C23×#3622
C2×C23×#3582
C23×#18162
C23×#13872
C23×#13812
#1307223