Finite Group Databasebeta

← C23×#11867 #19159493

Subgroups — C23×#11867 #19159493

Order 5888.

376 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3060C211yesmaximal, characteristic
2C23×#2115C211yesmaximal, characteristic, complemented
3C23×#1812C211yesmaximal, characteristic, complemented
4C23×#1340C212yesmaximal, complemented
5C23×#1339C212yesmaximal, complemented
6C2×C23×#135C411yescharacteristic, complemented
7C2×C23×#131C2211yescharacteristic
8C23×#537—21no—
9C2×C23×M32—21no—
10C23×#470C411yescharacteristic, complemented
11C23×#470C2211yescharacteristic
12C23×#469C411yescharacteristic, complemented
13C23×#469C2211yescharacteristic
14C23×#465—21no—
15C23×#465—21no—
16C23×#427—21no—
17C2×C23×#106C411yescharacteristic, complemented
18C23×#417—21no—
19C23×#410—21no—
20C23×#349—21no—
21C23×#349C2212yes—
22C23×H(3,ℤ/4)C2212yes—
23C2×#41×C23—21no—
24C2×#41×C23D411yescharacteristic
25C23×#135—41no—
26C23×#135—41no—
27C2×Q16×C23C2×C411yescharacteristic
28C2×SD16×C23—21no—
29C2×D8×C23C2×C411yescharacteristic
30C23×#131—41no—
31C23×#131—41no—
32C2×M16×C23C2×C411yescharacteristic
33C2×M16×C23D411yescharacteristic
34C2×M16×C23C2311yescharacteristic
35C22×C8×C23C2×C411yescharacteristic
36C22×C8×C23D411yescharacteristic
37C22×C8×C23D411yescharacteristic
38C23×C22⋊Q8—21no—
39C23×C22⋊Q8—21no—
40C23×C22⋊Q8—41no—
41C23×#121—21no—
42C23×#121—21no—
43C23×#121—41no—
44C23×#121—41no—
45C23×#121—41no—
46C23×#117—22no—
47C2×C4⋊C4×C23—21no—
48C23×M32—22no—
49C23×M32—41no—
50C2×C16×C23—41no—
51C23×#107—41no—
52C23×#106—21no—
53C23×#106—41no—
54C23×#106C2×C411yescharacteristic
55C23×#106C2×C411yescharacteristic
56C23×#103—21no—
57C23×#103—21no—
58C23×#102—21no—
59C23×#102—21no—
60C23×#99—41no—
61C23×#99—41no—
62#41×C23—41no—
63#41×C23—41no—
64#41×C23—42no—
65#41×C23—42no—
66C2×Q8×C23—21no—
67C2×Q8×C23SD1611yescharacteristic
68C2×D4×C23—21no—
69C2×D4×C23—41no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C2×D4×C23D811yescharacteristic
73C2×D4×C23D811yescharacteristic
74C2×D4×C23SD1611yescharacteristic
75C22×C4×C23—41no—
76C22×C4×C23—41no—
77C22×C4×C23C22⋊C411yescharacteristic
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23C2×D411yescharacteristic
80Q16×C23—41no—
81SD16×C23—41no—
82SD16×C23—41no—
83D8×C23—41no—
84M16×C23—41no—
85M16×C23—41no—
86M16×C23—41no—
87C2×C8×C23—21no—
88C2×C8×C23—21no—
89C2×C8×C23—21no—
90C2×C8×C23—21no—
91C2×C8×C23—21no—
92C2×C8×C23—21no—
93C2×C8×C23C22⋊C411yescharacteristic
94C2×C8×C23C22⋊C411yescharacteristic
95C2×C8×C23C22×C411yescharacteristic
96C2×C8×C23C2×D411yescharacteristic
97C4⋊C4×C23—21no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—42no—
100C4⋊C4×C23—41no—
101C4⋊C4×C23—41no—
102C4⋊C4×C23—41no—
103C22⋊C4×C23—41no—
104C22⋊C4×C23—41no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C22⋊C4×C23—42no—
108C42×C23—42no—
109C16×C23—42no—
110#11867C2311yesmaximal, characteristic, direct product factor, Sylow
111C23×C23—41no—
112C23×C23#10211yescharacteristic
113C23×C23#10211yescharacteristic
114C23×C23C2×D811yescharacteristic
115Q8×C23—41no—
116Q8×C23—41no—
117Q8×C23—41no—
118D4×C23—41no—
119D4×C23—41no—
120D4×C23—41no—
121D4×C23—41no—
122D4×C23—41no—
123D4×C23—82no—
124D4×C23—161no—
125D4×C23—161no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—82no—
136C2×C4×C23—82no—
137C2×C4×C23—161no—
138C2×C4×C23#10211yescharacteristic
139C2×C4×C23#10211yescharacteristic
140C2×C4×C23C2×C22⋊C411yescharacteristic
141C2×C4×C23C2×SD1611yescharacteristic
142C8×C23—21no—
143C8×C23—21no—
144C8×C23—21no—
145C8×C23#9911yescharacteristic
146C8×C23#9911yescharacteristic
147#3060C2×C2311yescharacteristic
148#2115C2×C2311yescharacteristic, complemented
149#1812C2×C2311yescharacteristic, complemented
150#1340C2×C2312yescomplemented
151#1339C2×C2312yescomplemented
152C22×C23—21no—
153C22×C23—21no—
154C22×C23—21no—
155C22×C23—41no—
156C22×C23—41no—
157C22×C23—41no—
158C22×C23—81no—
159C22×C23—81no—
160C22×C23—81no—
161C22×C23C2×#10211yescharacteristic
162C4×C23—21no—
163C4×C23—21no—
164C4×C23—21no—
165C4×C23—81no—
166C4×C23—81no—
167C4×C23—81no—
168C4×C23—162no—
169C4×C23C2×#9911yescharacteristic
170C4×C23#42111yescharacteristic
171C2×#135C4×C2311yescharacteristic, complemented
172C2×#131C22×C2311yescharacteristic
173#537—21no—
174C2×M32—21no—
175#470C4×C2311yescharacteristic, complemented
176#470C22×C2311yescharacteristic
177#469C4×C2311yescharacteristic, complemented
178#469C22×C2311yescharacteristic
179#465—21no—
180#465—21no—
181#427—21no—
182C2×#106C4×C2311yescharacteristic, complemented
183#417—21no—
184#410—21no—
185#349—21no—
186#349C22×C2312yes—
187H(3,ℤ/4)C22×C2312yes—
188C2×C23—21no—
189C2×C23—41no—
190C2×C23—41no—
191C2×C23—41no—
192C2×C23—161no—
193C2×C23#151911yescharacteristic
194C2×#41—21no—
195C2×#41D4×C2311yescharacteristic
196#135—41no—
197#135—41no—
198C2×Q16C2×C4×C2311yescharacteristic
199C2×SD16—21no—
200C2×D8C2×C4×C2311yescharacteristic
201#131—41no—
202#131—41no—
203C2×M16C2×C4×C2311yescharacteristic
204C2×M16D4×C2311yescharacteristic
205C2×M16C23×C2311yescharacteristic
206C22×C8C2×C4×C2311yescharacteristic
207C22×C8D4×C2311yescharacteristic
208C22×C8D4×C2311yescharacteristic
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—41no—
212#121—21no—
213#121—21no—
214#121—41no—
215#121—41no—
216#121—41no—
217#117—22no—
218C2×C4⋊C4—21no—
219M32—22no—
220M32—41no—
221C2×C16—41no—
222#107—41no—
223#106—21no—
224#106—41no—
225#106C2×C4×C2311yescharacteristic
226#106C2×C4×C2311yescharacteristic
227#103—21no—
228#103—21no—
229#102—21no—
230#102—21no—
231#99—41no—
232#99—41no—
233C23#1186711yescharacteristic, direct product factor, Sylow
234#41—41no—
235#41—41no—
236#41—42no—
237#41—42no—
238C2×Q8—21no—
239C2×Q8SD16×C2311yescharacteristic
240C2×D4—21no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4D8×C2311yescharacteristic
245C2×D4D8×C2311yescharacteristic
246C2×D4SD16×C2311yescharacteristic
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C22⋊C4×C2311yescharacteristic
250C22×C4C22⋊C4×C2311yescharacteristic
251C22×C4C2×D4×C2311yescharacteristic
252Q16—41no—
253SD16—41no—
254SD16—41no—
255D8—41no—
256M16—41no—
257M16—41no—
258M16—41no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8C22⋊C4×C2311yescharacteristic
266C2×C8C22⋊C4×C2311yescharacteristic
267C2×C8C22×C4×C2311yescharacteristic
268C2×C8C2×D4×C2311yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—42no—
280C42—42no—
281C16—42no—
282C23—41no—
283C23C23×#10211yescharacteristic
284C23C23×#10211yescharacteristic
285C23C2×D8×C2311yescharacteristic
286Q8—41no—
287Q8—41no—
288Q8—41no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—82no—
295D4—161no—
296D4—161no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—161no—
309C2×C4C23×#10211yescharacteristic
310C2×C4C23×#10211yescharacteristic
311C2×C4C2×C22⋊C4×C2311yescharacteristic
312C2×C4C2×SD16×C2311yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8C23×#9911yescharacteristic
317C8C23×#9911yescharacteristic
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C2×C23×#10211yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—162no—
335C4C2×C23×#9911yescharacteristic
336C4C23×#42111yescharacteristic
337C2—21no—
338C2—41no—
339C2—41no—
340C2—41no—
341C2—161no—
342C2C23×#151911yescharacteristic
subgroupG/H
center, socleC2×C23#1519
commutatorC2×C8C22×C4×C23
Frattini, cosocleC2×M16C23×C23
prime radicalC23×#3060C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×C23×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2115C2semidirect
3. C23×#1812C2semidirect
4. C23×#1340C2semidirect
5. C23×#1339C2semidirect
6. C2×C23×#135C4semidirect
10. C23×#470C4semidirect
12. C23×#469C4semidirect
17. C2×C23×#106C4semidirect
110. #11867C23direct (N × H)
148. #2115C2×C23semidirect
149. #1812C2×C23semidirect
150. #1340C2×C23semidirect
151. #1339C2×C23semidirect
171. C2×#135C4×C23semidirect
175. #470C4×C23semidirect
177. #469C4×C23semidirect
182. C2×#106C4×C23semidirect
233. C23#11867direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30602
C23×#21152
C23×#18122
C23×#13402
C23×#13392
#1186723