Finite Group Databasebeta

← C23×#9590 #19157216

Subgroups — C23×#9590 #19157216

Order 5888.

364 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1980C211yesmaximal, characteristic
2C23×#1950C211yesmaximal, characteristic, complemented
3C23×#1540C211yesmaximal, characteristic
4C23×#1492C211yesmaximal, characteristic
5C8×M16×C23C211yesmaximal, characteristic, complemented
6C23×#1303C211yesmaximal, characteristic, complemented
7C23×#1279C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C23×#431C2211yescharacteristic
10C2×C23×#108—21no—
11C23×#423C2211yescharacteristic
12C2×C23×C4⋊C8C2211yescharacteristic
13C2×C23×#104—21no—
14C2×C23×#104—21no—
15C4×M16×C23C2211yescharacteristic
16C2×C4×C8×C23C2211yescharacteristic
17C4×C4⋊C4×C23C2211yescharacteristic
18C23×#341—21no—
19C23×#332C411yescharacteristic, complemented
20C23×#332C411yescharacteristic, complemented
21C23×#331C411yescharacteristic, complemented
22C23×#331C411yescharacteristic, complemented
23C23×D4⋊C8—21no—
24C23×D4⋊C8—21no—
25C23×#322—21no—
26C82×C23—21no—
27C2×#41×C23—21no—
28C2×M16×C23—21no—
29C2×M16×C23D411yescharacteristic
30C2×M16×C23D411yescharacteristic
31C22×C8×C23—21no—
32C22×C8×C23—21no—
33C23×C4⋊Q8C2×C411yescharacteristic
34C23×#127C2×C411yescharacteristic
35C23×#124—21no—
36C23×#121—21no—
37C23×#121—21no—
38C4×D4×C23—21no—
39C4×D4×C23—21no—
40C2×C4⋊C4×C23—21no—
41C2×C42×C23—21no—
42C2×C42×C23C2311yescharacteristic
43C23×#108—42no—
44C23×C4⋊C8—21no—
45C23×C4⋊C8—21no—
46C23×C4⋊C8—21no—
47C23×C4⋊C8C2×C411yescharacteristic
48C23×C4⋊C8C2×C411yescharacteristic
49C23×C4⋊C8C2×C411yescharacteristic
50C23×C4⋊C8C2×C411yescharacteristic
51C23×#104—42no—
52C23×#104—42no—
53C23×C22⋊C8—21no—
54C23×#97—21no—
55C4×C8×C23—21no—
56C4×C8×C23—21no—
57C4×C8×C23—21no—
58C4×C8×C23D411yescharacteristic
59C4×C8×C23D411yescharacteristic
60C23×#95—21no—
61#41×C23—41no—
62#41×C23—41no—
63#41×C23—41no—
64#41×C23—41no—
65C2×Q8×C23—21no—
66C2×D4×C23—21no—
67C2×D4×C23—21no—
68C2×D4×C23—21no—
69C2×D4×C23—41no—
70C22×C4×C23—41no—
71C22×C4×C23—41no—
72C22×C4×C23C2×D411yescharacteristic
73C22×C4×C23#4111yescharacteristic
74C22×C4×C23#4111yescharacteristic
75M16×C23—21no—
76M16×C23—21no—
77M16×C23—41no—
78M16×C23—41no—
79M16×C23D812yes—
80M16×C23SD1612yes—
81C2×C8×C23—21no—
82C2×C8×C23—21no—
83C2×C8×C23—21no—
84C2×C8×C23—21no—
85C2×C8×C23—21no—
86C2×C8×C23—21no—
87C2×C8×C23—21no—
88C2×C8×C23—21no—
89C2×C8×C23—42no—
90C2×C8×C23—41no—
91C2×C8×C23—41no—
92C2×C8×C23—41no—
93C4⋊C4×C23—22no—
94C4⋊C4×C23—21no—
95C4⋊C4×C23—21no—
96C4⋊C4×C23—41no—
97C22⋊C4×C23—41no—
98C22⋊C4×C23—41no—
99C42×C23—41no—
100C42×C23—41no—
101C42×C23C22×C411yescharacteristic
102C42×C23C2×D411yescharacteristic
103C42×C23#4111yescharacteristic
104C42×C23#4111yescharacteristic
105#9590C2311yesmaximal, characteristic, direct product factor, Sylow
106C23×C23—41no—
107C23×C23#12411yescharacteristic
108Q8×C23—41no—
109Q8×C23—41no—
110D4×C23—41no—
111D4×C23—41no—
112D4×C23—41no—
113D4×C23—41no—
114D4×C23—81no—
115D4×C23—81no—
116D4×C23—81no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—41no—
120C2×C4×C23—42no—
121C2×C4×C23—41no—
122C2×C4×C23—81no—
123C2×C4×C23—81no—
124C2×C4×C23—81no—
125C2×C4×C23—81no—
126C2×C4×C23#11711yescharacteristic
127C2×C4×C23C4×D411yescharacteristic
128C2×C4×C23C4×D411yescharacteristic
129C2×C4×C23#12111yescharacteristic
130C2×C4×C23#12311yescharacteristic
131C2×C4×C23#12611yescharacteristic
132C2×C4×C23C2×D811yescharacteristic
133C2×C4×C23C2×SD1611yescharacteristic
134C2×C4×C23#13511yescharacteristic
135C2×C4×C23#13511yescharacteristic
136C8×C23—22no—
137C8×C23—22no—
138C8×C23—22no—
139C8×C23—41no—
140C8×C23—41no—
141C8×C23—41no—
142C8×C23—41no—
143C8×C23—41no—
144C8×C23—81no—
145#1980C2×C2311yescharacteristic
146#1950C2×C2311yescharacteristic, complemented
147#1540C2×C2311yescharacteristic
148#1492C2×C2311yescharacteristic
149C8×M16C2×C2311yescharacteristic, complemented
150#1303C2×C2311yescharacteristic, complemented
151#1279C2×C2311yescharacteristic, complemented
152C22×C23—21no—
153C22×C23—21no—
154C22×C23—81no—
155C22×C23—81no—
156C22×C23—81no—
157C22×C23#38811yescharacteristic
158C22×C23#48611yescharacteristic
159C22×C23#48811yescharacteristic
160C4×C23—21no—
161C4×C23—21no—
162C4×C23—21no—
163C4×C23—82no—
164C4×C23—81no—
165C4×C23C4×D811yescharacteristic
166C4×C23C4×SD1611yescharacteristic
167C4×C23#44311yescharacteristic
168C4×C23#44411yescharacteristic
169C4×C23#46511yescharacteristic
170C4×C23#46611yescharacteristic
171#532C22×C2311yescharacteristic
172#431C22×C2311yescharacteristic
173C2×#108—21no—
174#423C22×C2311yescharacteristic
175C2×C4⋊C8C22×C2311yescharacteristic
176C2×#104—21no—
177C2×#104—21no—
178C4×M16C22×C2311yescharacteristic
179C2×C4×C8C22×C2311yescharacteristic
180C4×C4⋊C4C22×C2311yescharacteristic
181#341—21no—
182#332C4×C2311yescharacteristic, complemented
183#332C4×C2311yescharacteristic, complemented
184#331C4×C2311yescharacteristic, complemented
185#331C4×C2311yescharacteristic, complemented
186D4⋊C8—21no—
187D4⋊C8—21no—
188#322—21no—
189C82—21no—
190C2×C23—21no—
191C2×C23—21no—
192C2×C23—161no—
193C2×C23#192911yescharacteristic
194C2×C23#193211yescharacteristic
195C2×C23#193411yescharacteristic
196C2×#41—21no—
197C2×M16—21no—
198C2×M16D4×C2311yescharacteristic
199C2×M16D4×C2311yescharacteristic
200C22×C8—21no—
201C22×C8—21no—
202C4⋊Q8C2×C4×C2311yescharacteristic
203#127C2×C4×C2311yescharacteristic
204#124—21no—
205#121—21no—
206#121—21no—
207C4×D4—21no—
208C4×D4—21no—
209C2×C4⋊C4—21no—
210C2×C42—21no—
211C2×C42C23×C2311yescharacteristic
212#108—42no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8C2×C4×C2311yescharacteristic
217C4⋊C8C2×C4×C2311yescharacteristic
218C4⋊C8C2×C4×C2311yescharacteristic
219C4⋊C8C2×C4×C2311yescharacteristic
220#104—42no—
221#104—42no—
222C22⋊C8—21no—
223#97—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8D4×C2311yescharacteristic
228C4×C8D4×C2311yescharacteristic
229#95—21no—
230C23#959011yescharacteristic, direct product factor, Sylow
231#41—41no—
232#41—41no—
233#41—41no—
234#41—41no—
235C2×Q8—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4C2×D4×C2311yescharacteristic
243C22×C4#41×C2311yescharacteristic
244C22×C4#41×C2311yescharacteristic
245M16—21no—
246M16—21no—
247M16—41no—
248M16—41no—
249M16D8×C2312yes—
250M16SD16×C2312yes—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C4⋊C4—22no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C42—41no—
270C42—41no—
271C42C22×C4×C2311yescharacteristic
272C42C2×D4×C2311yescharacteristic
273C42#41×C2311yescharacteristic
274C42#41×C2311yescharacteristic
275C23—41no—
276C23C23×#12411yescharacteristic
277Q8—41no—
278Q8—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—81no—
284D4—81no—
285D4—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4C23×#11711yescharacteristic
296C2×C4C4×D4×C2311yescharacteristic
297C2×C4C4×D4×C2311yescharacteristic
298C2×C4C23×#12111yescharacteristic
299C2×C4C23×#12311yescharacteristic
300C2×C4C23×#12611yescharacteristic
301C2×C4C2×D8×C2311yescharacteristic
302C2×C4C2×SD16×C2311yescharacteristic
303C2×C4C23×#13511yescharacteristic
304C2×C4C23×#13511yescharacteristic
305C8—22no—
306C8—22no—
307C8—22no—
308C8—41no—
309C8—41no—
310C8—41no—
311C8—41no—
312C8—41no—
313C8—81no—
314C22—21no—
315C22—21no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22C23×#38811yescharacteristic
320C22C23×#48611yescharacteristic
321C22C23×#48811yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—82no—
326C4—81no—
327C4C4×D8×C2311yescharacteristic
328C4C4×SD16×C2311yescharacteristic
329C4C23×#44311yescharacteristic
330C4C23×#44411yescharacteristic
331C4C23×#46511yescharacteristic
332C4C23×#46611yescharacteristic
333C2—21no—
334C2—21no—
335C2—161no—
336C2C23×#192911yescharacteristic
337C2C23×#193211yescharacteristic
338C2C23×#193411yescharacteristic
subgroupG/H
centerC2×C4×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#388
prime radicalC23×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1950C2semidirect
5. C8×M16×C23C2semidirect
6. C23×#1303C2semidirect
7. C23×#1279C2semidirect
19. C23×#332C4semidirect
20. C23×#332C4semidirect
21. C23×#331C4semidirect
22. C23×#331C4semidirect
105. #9590C23direct (N × H)
146. #1950C2×C23semidirect
149. C8×M16C2×C23semidirect
150. #1303C2×C23semidirect
151. #1279C2×C23semidirect
182. #332C4×C23semidirect
183. #332C4×C23semidirect
184. #331C4×C23semidirect
185. #331C4×C23semidirect
230. C23#9590direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#19802
C23×#19502
C23×#15402
C23×#14922
C8×M16×C232
C23×#13032
C23×#12792
#959023