Finite Group Databasebeta

← C23×#8916 #19156542

Subgroups — C23×#8916 #19156542

Order 5888.

354 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2995C211yesmaximal, characteristic
2C23×#1849C211yesmaximal, characteristic, complemented
3C23×#1569C211yesmaximal, characteristic, complemented
4C23×#1524C211yesmaximal, characteristic
5C23×#1495C211yesmaximal, characteristic
6C23×#1302C211yesmaximal, characteristic, complemented
7C23×#1299C211yesmaximal, characteristic, complemented
8C23×#529C2211yescharacteristic
9C23×#446—21no—
10C23×#436—21no—
11C23×#435—21no—
12C23×#435—21no—
13C23×#433—21no—
14C23×#432C2211yescharacteristic
15C23×#431C2211yescharacteristic
16C2×C23×#108—21no—
17C2×C23×#108—21no—
18C23×#424—21no—
19C23×#423C2211yescharacteristic
20C23×#423C2211yescharacteristic
21C4×M16×C23C2211yescharacteristic
22C2×C23×#97C2211yescharacteristic
23C23×#335—21no—
24C23×#334—21no—
25C23×#330C411yescharacteristic
26C23×#330C411yescharacteristic
27C23×#329C411yescharacteristic
28C23×#329C411yescharacteristic
29C23×Q8⋊C8—21no—
30C23×D4⋊C8—21no—
31C2×M16×C23—21no—
32C2×M16×C23—21no—
33C2×M16×C23—21no—
34C2×M16×C23D411yescharacteristic
35C2×M16×C23D411yescharacteristic
36C22×C8×C23—21no—
37C22×C8×C23—21no—
38C23×#126—21no—
39C23×#125C2×C411yescharacteristic
40C23×#124C2×C411yescharacteristic
41C23×#123—21no—
42C23×C22⋊Q8Q811yescharacteristic
43C23×#121Q811yescharacteristic
44C4×Q8×C23D411yescharacteristic
45C4×D4×C23—21no—
46C4×D4×C23D411yescharacteristic
47C23×#117—21no—
48C2×C42×C23C2311yescharacteristic
49C23×#108—42no—
50C23×#108—42no—
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×C4⋊C8D411yescharacteristic
59C23×C4⋊C8D411yescharacteristic
60C23×C22⋊C8—21no—
61C23×C22⋊C8—21no—
62C23×C22⋊C8—21no—
63C23×C22⋊C8—21no—
64C23×#97—21no—
65C23×#97—21no—
66C23×#97—21no—
67C23×#97C2×C411yescharacteristic
68C23×#97C2×C411yescharacteristic
69C23×#97C2×C411yescharacteristic
70C23×#97C2×C411yescharacteristic
71C4×C8×C23—21no—
72C4×C8×C23—21no—
73C4×C8×C23—21no—
74C2×Q8×C23C4⋊C411yescharacteristic
75C2×D4×C23—21no—
76C2×D4×C23C4⋊C411yescharacteristic
77C22×C4×C23—21no—
78C22×C4×C23C2×D411yescharacteristic
79C22×C4×C23C2×Q811yescharacteristic
80C22×C4×C23#4111yescharacteristic
81M16×C23—42no—
82M16×C23—41no—
83M16×C23—41no—
84M16×C23—41no—
85M16×C23—41no—
86M16×C23—81no—
87C2×C8×C23—22no—
88C2×C8×C23—22no—
89C2×C8×C23—21no—
90C2×C8×C23—21no—
91C2×C8×C23—21no—
92C2×C8×C23—21no—
93C2×C8×C23—21no—
94C2×C8×C23—21no—
95C2×C8×C23—21no—
96C2×C8×C23—21no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C2×C8×C23—81no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—21no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23C4⋊C411yescharacteristic
105C4⋊C4×C23C4⋊C411yescharacteristic
106C22⋊C4×C23—21no—
107C22⋊C4×C23—21no—
108C22⋊C4×C23—21no—
109C22⋊C4×C23—21no—
110C22⋊C4×C23—21no—
111C42×C23—21no—
112C42×C23C22×C411yescharacteristic
113C42×C23C2×D411yescharacteristic
114C42×C23C2×D411yescharacteristic
115C42×C23#4111yescharacteristic
116#8916C2311yesmaximal, characteristic, direct product factor, Sylow
117C23×C23—21no—
118C23×C23C22⋊Q811yescharacteristic
119Q8×C23—41no—
120D4×C23—41no—
121D4×C23—81no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—41no—
128C2×C4×C23—41no—
129C2×C4×C23—41no—
130C2×C4×C23—41no—
131C2×C4×C23C2×C4⋊C411yescharacteristic
132C2×C4×C23C4×D411yescharacteristic
133C2×C4×C23C4×D411yescharacteristic
134C2×C4×C23#12011yescharacteristic
135C2×C4×C23C22⋊Q811yescharacteristic
136C2×C4×C23#12311yescharacteristic
137C8×C23—22no—
138C8×C23—22no—
139C8×C23—22no—
140C8×C23—41no—
141C8×C23—41no—
142C8×C23—41no—
143C8×C23—41no—
144C8×C23—41no—
145C8×C23—81no—
146C8×C23—81no—
147#2995C2×C2311yescharacteristic
148#1849C2×C2311yescharacteristic, complemented
149#1569C2×C2311yescharacteristic, complemented
150#1524C2×C2311yescharacteristic
151#1495C2×C2311yescharacteristic
152#1302C2×C2311yescharacteristic, complemented
153#1299C2×C2311yescharacteristic, complemented
154C22×C23—21no—
155C22×C23—21no—
156C22×C23—21no—
157C22×C23—41no—
158C22×C23—41no—
159C22×C23—41no—
160C22×C23#38511yescharacteristic
161C4×C23—21no—
162C4×C23—21no—
163C4×C23—21no—
164C4×C23—21no—
165C4×C23—21no—
166C4×C23—41no—
167C4×C23—41no—
168C4×C23—41no—
169C4×C23#45311yescharacteristic
170C4×C23#45611yescharacteristic
171#529C22×C2311yescharacteristic
172#446—21no—
173#436—21no—
174#435—21no—
175#435—21no—
176#433—21no—
177#432C22×C2311yescharacteristic
178#431C22×C2311yescharacteristic
179C2×#108—21no—
180C2×#108—21no—
181#424—21no—
182#423C22×C2311yescharacteristic
183#423C22×C2311yescharacteristic
184C4×M16C22×C2311yescharacteristic
185C2×#97C22×C2311yescharacteristic
186#335—21no—
187#334—21no—
188#330C4×C2311yescharacteristic
189#330C4×C2311yescharacteristic
190#329C4×C2311yescharacteristic
191#329C4×C2311yescharacteristic
192Q8⋊C8—21no—
193D4⋊C8—21no—
194C2×C23—41no—
195C2×C23—81no—
196C2×C23#185811yescharacteristic
197C2×C23#185911yescharacteristic
198C2×C23#186111yescharacteristic
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16D4×C2311yescharacteristic
203C2×M16D4×C2311yescharacteristic
204C22×C8—21no—
205C22×C8—21no—
206#126—21no—
207#125C2×C4×C2311yescharacteristic
208#124C2×C4×C2311yescharacteristic
209#123—21no—
210C22⋊Q8Q8×C2311yescharacteristic
211#121Q8×C2311yescharacteristic
212C4×Q8D4×C2311yescharacteristic
213C4×D4—21no—
214C4×D4D4×C2311yescharacteristic
215#117—21no—
216C2×C42C23×C2311yescharacteristic
217#108—42no—
218#108—42no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8D4×C2311yescharacteristic
227C4⋊C8D4×C2311yescharacteristic
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232#97—21no—
233#97—21no—
234#97—21no—
235#97C2×C4×C2311yescharacteristic
236#97C2×C4×C2311yescharacteristic
237#97C2×C4×C2311yescharacteristic
238#97C2×C4×C2311yescharacteristic
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C23#891611yescharacteristic, direct product factor, Sylow
243C2×Q8C4⋊C4×C2311yescharacteristic
244C2×D4—21no—
245C2×D4C4⋊C4×C2311yescharacteristic
246C22×C4—21no—
247C22×C4C2×D4×C2311yescharacteristic
248C22×C4C2×Q8×C2311yescharacteristic
249C22×C4#41×C2311yescharacteristic
250M16—42no—
251M16—41no—
252M16—41no—
253M16—41no—
254M16—41no—
255M16—81no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—81no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4C4⋊C4×C2311yescharacteristic
274C4⋊C4C4⋊C4×C2311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C42—21no—
281C42C22×C4×C2311yescharacteristic
282C42C2×D4×C2311yescharacteristic
283C42C2×D4×C2311yescharacteristic
284C42#41×C2311yescharacteristic
285C23—21no—
286C23C23×C22⋊Q811yescharacteristic
287Q8—41no—
288D4—41no—
289D4—81no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4C2×C4⋊C4×C2311yescharacteristic
300C2×C4C4×D4×C2311yescharacteristic
301C2×C4C4×D4×C2311yescharacteristic
302C2×C4C23×#12011yescharacteristic
303C2×C4C23×C22⋊Q811yescharacteristic
304C2×C4C23×#12311yescharacteristic
305C8—22no—
306C8—22no—
307C8—22no—
308C8—41no—
309C8—41no—
310C8—41no—
311C8—41no—
312C8—41no—
313C8—81no—
314C8—81no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C23×#38511yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4C23×#45311yescharacteristic
331C4C23×#45611yescharacteristic
332C2—41no—
333C2—81no—
334C2C23×#185811yescharacteristic
335C2C23×#185911yescharacteristic
336C2C23×#186111yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#385
prime radicalC23×#121Q8
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×#1849C2semidirect
3. C23×#1569C2semidirect
6. C23×#1302C2semidirect
7. C23×#1299C2semidirect
116. #8916C23direct (N × H)
148. #1849C2×C23semidirect
149. #1569C2×C23semidirect
152. #1302C2×C23semidirect
153. #1299C2×C23semidirect
242. C23#8916direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29952
C23×#18492
C23×#15692
C23×#15242
C23×#14952
C23×#13022
C23×#12992
#891623