Finite Group Databasebeta

← C23×#12987 #19160613

Subgroups — C23×#12987 #19160613

Order 5888.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#504C211yesmaximal, characteristic, complemented
2C23×#2955C211yesmaximal, characteristic
3C23×#2117C211yesmaximal, characteristic, complemented
4C2×C23×#350C211yesmaximal, characteristic, complemented
5C23×#2115C211yesmaximal, characteristic, complemented
6C23×#1379C211yesmaximal, characteristic, complemented
7C23×#1378C211yesmaximal, characteristic, complemented
8C23×#575C411yescharacteristic
9C2×C23×#131C2211yescharacteristic
10C2×C23×#131C2211yescharacteristic
11C23×#504—22no—
12C23×#504—22no—
13C2×C23×M32—21no—
14C2×C23×M32C2211yescharacteristic
15C2×C23×M32C2211yescharacteristic
16C2×C23×M32C2211yescharacteristic
17C22×C16×C23—21no—
18C22×C16×C23C2211yescharacteristic
19C23×#444—22no—
20C23×#443—22no—
21C23×#430C411yescharacteristic
22C23×#421C411yescharacteristic
23C23×#421C411yescharacteristic
24C23×#405C2211yescharacteristic
25C23×#350—22no—
26C23×#350—21no—
27C23×#350—21no—
28C23×#349—22no—
29C2×#41×C23C2×C411yescharacteristic
30C2×#41×C23C2×C411yescharacteristic
31C23×#137—21no—
32C23×#137C812yes—
33C23×AGL(1,ℤ/8)—21no—
34C23×AGL(1,ℤ/8)C812yes—
35C23×#135—22no—
36C23×#131—21no—
37C23×#131—21no—
38C23×#131—21no—
39C23×#131—21no—
40C23×#131—21no—
41C23×#131—21no—
42C23×#131D412yes—
43C23×#131D412yes—
44C2×M16×C23—21no—
45C2×M16×C23—21no—
46C2×M16×C23C2×C411yescharacteristic
47C2×M16×C23C2×C411yescharacteristic
48C2×M16×C23C2×C411yescharacteristic
49C22×C8×C23—21no—
50C22×C8×C23—21no—
51C22×C8×C23C2311yescharacteristic
52C23×#117C2×C411yescharacteristic
53C23×M32—24no—
54C23×M32—21no—
55C23×M32—21no—
56C23×M32—21no—
57C23×M32—21no—
58C23×M32—21no—
59C23×M32D412yes—
60C2×C16×C23—24no—
61C2×C16×C23—21no—
62C2×C16×C23—21no—
63C2×C16×C23—21no—
64C2×C16×C23—21no—
65C2×C16×C23—21no—
66C2×C16×C23D411yescharacteristic
67C2×C16×C23D411yescharacteristic
68C23×#108—22no—
69C23×#104—22no—
70C23×#104—22no—
71C23×#97—21no—
72C4×C8×C23—21no—
73#41×C23—21no—
74#41×C23—21no—
75#41×C23—21no—
76#41×C23—21no—
77#41×C23—21no—
78#41×C23—21no—
79#41×C23C22⋊C412yes—
80#41×C23C2×C812yes—
81C2×Q8×C23C2×C811yescharacteristic
82C2×Q8×C23M1611yescharacteristic
83C2×D4×C23—21no—
84C2×D4×C23—21no—
85C2×D4×C23C2×C811yescharacteristic
86C2×D4×C23M1611yescharacteristic
87C22×C4×C23—21no—
88C22×C4×C23—21no—
89C22×C4×C23C22×C411yescharacteristic
90Q16×C23—22no—
91SD16×C23—22no—
92SD16×C23—22no—
93D8×C23—22no—
94M16×C23—21no—
95M16×C23—21no—
96M16×C23—22no—
97M16×C23—22no—
98M16×C23—21no—
99M16×C23—21no—
100M16×C23—41no—
101M16×C23—41no—
102M16×C23C22⋊C412yes—
103M16×C23C2×C812yes—
104C2×C8×C23—21no—
105C2×C8×C23—21no—
106C2×C8×C23—22no—
107C2×C8×C23—22no—
108C2×C8×C23—21no—
109C2×C8×C23—41no—
110C2×C8×C23—41no—
111C2×C8×C23C2×D411yescharacteristic
112C2×C8×C23C2×D411yescharacteristic
113C2×C8×C23C2×D411yescharacteristic
114C2×C8×C23C2×D411yescharacteristic
115C2×C8×C23#4111yescharacteristic
116C2×C8×C23#4111yescharacteristic
117C4⋊C4×C23—21no—
118C22⋊C4×C23—21no—
119C42×C23—21no—
120C16×C23—21no—
121C16×C23—22no—
122C16×C23—21no—
123C16×C23—41no—
124C16×C23—41no—
125#12987C2311yesmaximal, characteristic, direct product factor, Sylow
126C23×C23—21no—
127C23×C23—21no—
128C23×C23#13111yescharacteristic
129Q8×C23—21no—
130Q8×C23—21no—
131Q8×C23—21no—
132Q8×C23C22⋊C812yes—
133D4×C23—22no—
134D4×C23—22no—
135D4×C23—21no—
136D4×C23—21no—
137D4×C23—21no—
138D4×C23—41no—
139D4×C23—41no—
140D4×C23C22⋊C812yes—
141C2×C4×C23—21no—
142C2×C4×C23—22no—
143C2×C4×C23—22no—
144C2×C4×C23—21no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23C2×C22⋊C411yescharacteristic
149C2×C4×C23C4×D411yescharacteristic
150C2×C4×C23C4×D411yescharacteristic
151C2×C4×C23C22×C811yescharacteristic
152C2×C4×C23C2×M1611yescharacteristic
153C2×C4×C23#13111yescharacteristic
154C8×C23—22no—
155C8×C23—22no—
156C8×C23—41no—
157C8×C23#12011yescharacteristic
158C8×C23#12111yescharacteristic
159C8×C23#12111yescharacteristic
160C8×C23#12311yescharacteristic
161C2×#504C2×C2311yescharacteristic, complemented
162#2955C2×C2311yescharacteristic
163#2117C2×C2311yescharacteristic, complemented
164C2×#350C2×C2311yescharacteristic, complemented
165#2115C2×C2311yescharacteristic, complemented
166#1379C2×C2311yescharacteristic, complemented
167#1378C2×C2311yescharacteristic, complemented
168C22×C23—22no—
169C22×C23—41no—
170C22×C23—42no—
171C22×C23—41no—
172C22×C23—81no—
173C22×C23#40811yescharacteristic
174C22×C23C8×D411yescharacteristic
175C22×C23#43511yescharacteristic
176C4×C23—22no—
177C4×C23—41no—
178C4×C23—81no—
179C4×C23#38611yescharacteristic
180C4×C23C2×C22⋊C811yescharacteristic
181C4×C23C8×D411yescharacteristic
182C4×C23#43511yescharacteristic
183#575C4×C2311yescharacteristic
184C2×#131C22×C2311yescharacteristic
185C2×#131C22×C2311yescharacteristic
186#504—22no—
187#504—22no—
188C2×M32—21no—
189C2×M32C22×C2311yescharacteristic
190C2×M32C22×C2311yescharacteristic
191C2×M32C22×C2311yescharacteristic
192C22×C16—21no—
193C22×C16C22×C2311yescharacteristic
194#444—22no—
195#443—22no—
196#430C4×C2311yescharacteristic
197#421C4×C2311yescharacteristic
198#421C4×C2311yescharacteristic
199#405C22×C2311yescharacteristic
200#350—22no—
201#350—21no—
202#350—21no—
203#349—22no—
204C2×C23—21no—
205C2×C23—21no—
206C2×C23—21no—
207C2×C23—42no—
208C2×C23—81no—
209C2×C23#186911yescharacteristic
210C2×#41C2×C4×C2311yescharacteristic
211C2×#41C2×C4×C2311yescharacteristic
212#137—21no—
213#137C8×C2312yes—
214AGL(1,ℤ/8)—21no—
215AGL(1,ℤ/8)C8×C2312yes—
216#135—22no—
217#131—21no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—21no—
223#131D4×C2312yes—
224#131D4×C2312yes—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16C2×C4×C2311yescharacteristic
228C2×M16C2×C4×C2311yescharacteristic
229C2×M16C2×C4×C2311yescharacteristic
230C22×C8—21no—
231C22×C8—21no—
232C22×C8C23×C2311yescharacteristic
233#117C2×C4×C2311yescharacteristic
234M32—24no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32D4×C2312yes—
241C2×C16—24no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16D4×C2311yescharacteristic
248C2×C16D4×C2311yescharacteristic
249#108—22no—
250#104—22no—
251#104—22no—
252#97—21no—
253C4×C8—21no—
254C23#1298711yescharacteristic, direct product factor, Sylow
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—21no—
260#41—21no—
261#41C22⋊C4×C2312yes—
262#41C2×C8×C2312yes—
263C2×Q8C2×C8×C2311yescharacteristic
264C2×Q8M16×C2311yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C2×C8×C2311yescharacteristic
268C2×D4M16×C2311yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C22×C4×C2311yescharacteristic
272Q16—22no—
273SD16—22no—
274SD16—22no—
275D8—22no—
276M16—21no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—21no—
281M16—21no—
282M16—41no—
283M16—41no—
284M16C22⋊C4×C2312yes—
285M16C2×C8×C2312yes—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8C2×D4×C2311yescharacteristic
294C2×C8C2×D4×C2311yescharacteristic
295C2×C8C2×D4×C2311yescharacteristic
296C2×C8C2×D4×C2311yescharacteristic
297C2×C8#41×C2311yescharacteristic
298C2×C8#41×C2311yescharacteristic
299C4⋊C4—21no—
300C22⋊C4—21no—
301C42—21no—
302C16—21no—
303C16—22no—
304C16—21no—
305C16—41no—
306C16—41no—
307C23—21no—
308C23—21no—
309C23C23×#13111yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8C23×C22⋊C812yes—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4C23×C22⋊C812yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4C2×C22⋊C4×C2311yescharacteristic
330C2×C4C4×D4×C2311yescharacteristic
331C2×C4C4×D4×C2311yescharacteristic
332C2×C4C22×C8×C2311yescharacteristic
333C2×C4C2×M16×C2311yescharacteristic
334C2×C4C23×#13111yescharacteristic
335C8—22no—
336C8—22no—
337C8—41no—
338C8C23×#12011yescharacteristic
339C8C23×#12111yescharacteristic
340C8C23×#12111yescharacteristic
341C8C23×#12311yescharacteristic
342C22—22no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—81no—
347C22C23×#40811yescharacteristic
348C22C8×D4×C2311yescharacteristic
349C22C23×#43511yescharacteristic
350C4—22no—
351C4—41no—
352C4—81no—
353C4C23×#38611yescharacteristic
354C4C2×C23×C22⋊C811yescharacteristic
355C4C8×D4×C2311yescharacteristic
356C4C23×#43511yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2C23×#186911yescharacteristic
subgroupG/H
centerC8×C23#121
commutatorC2×C4C22×C8×C23
Frattini, cosocleC22×C8C23×C23
socleC2×C23#1869
prime radicalC23×#575C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C8×C23 ◁ C22×C8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C23×#504C2semidirect
3. C23×#2117C2semidirect
4. C2×C23×#350C2semidirect
5. C23×#2115C2semidirect
6. C23×#1379C2semidirect
7. C23×#1378C2semidirect
125. #12987C23direct (N × H)
161. C2×#504C2×C23semidirect
163. #2117C2×C23semidirect
164. C2×#350C2×C23semidirect
165. #2115C2×C23semidirect
166. #1379C2×C23semidirect
167. #1378C2×C23semidirect
254. C23#12987direct (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
C23×#29552
C23×#21172
C2×C23×#3502
C23×#21152
C23×#13792
C23×#13782
#1298723