Finite Group Databasebeta

← C23×#13025 #19160651

Subgroups — C23×#13025 #19160651

Order 5888.

522 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#509C211yesmaximal, characteristic
2C23×#2943C211yesmaximal, characteristic, complemented
3C2×C23×#360C211yesmaximal, characteristic, complemented
4C23×#2140C211yesmaximal, characteristic, complemented
5C23×#1972C211yesmaximal, characteristic, complemented
6C23×#1386C211yesmaximal, characteristic, complemented
7C23×#1377C211yesmaximal, characteristic, complemented
8C2×C23×#135C2211yescharacteristic
9C22×D8×C23C2211yescharacteristic
10C4×#41×C23—21no—
11C23×#509—22no—
12C23×#509C414yescomplemented
13C2×C23×SD32—21no—
14C2×C23×D16—21no—
15C2×C23×M32C2211yescharacteristic
16C2×C23×M32C2211yescharacteristic
17C23×#442—21no—
18C23×#441—21no—
19C23×#440—22no—
20C23×#427C2211yescharacteristic
21C2×C23×#106C2211yescharacteristic
22C23×#416—21no—
23C2×C23×#102—21no—
24C2×C23×#97C2211yescharacteristic
25C23×#390—21no—
26C23×#360—22no—
27C23×#357—22no—
28C2×#41×C23—21no—
29C22×D4×C23—21no—
30C23×#135—21no—
31C23×#135—21no—
32C23×#135C2×C412yes—
33C23×#135D412yes—
34C2×Q16×C23D411yescharacteristic
35C2×SD16×C23—21no—
36C2×D8×C23—21no—
37C2×D8×C23—21no—
38C2×D8×C23—21no—
39C2×D8×C23—21no—
40C2×D8×C23—21no—
41C2×D8×C23C2×C412yescomplemented
42C2×D8×C23D411yescharacteristic
43C22×C8×C23D411yescharacteristic
44C22×C8×C23C2311yescharacteristic
45C4×Q8×C23—22no—
46C4×D4×C23—22no—
47C4×D4×C23—42no—
48C23×#117—21no—
49C23×#117—41no—
50C2×C4⋊C4×C23—21no—
51C2×C22⋊C4×C23—42no—
52C2×C42×C23—41no—
53C2×C42×C23D411yescharacteristic
54C23×SD32—24no—
55C23×D16—24no—
56C23×M32—21no—
57C23×M32—21no—
58C23×M32—21no—
59C23×M32C2×C412yescomplemented
60C2×C16×C23—21no—
61C2×C16×C23—21no—
62C23×#107D412yes—
63C23×#106—21no—
64C23×#106—21no—
65C23×#106—21no—
66C23×#103—22no—
67C23×#102—22no—
68C23×#102—42no—
69C23×#97—21no—
70C23×#97—21no—
71C24×C23—41no—
72#41×C23—22no—
73#41×C23—22no—
74#41×C23—41no—
75#41×C23—41no—
76C2×Q8×C23—21no—
77C2×D4×C23—22no—
78C2×D4×C23—21no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—41no—
84C2×D4×C23—41no—
85C22×C4×C23—42no—
86C22×C4×C23—41no—
87C22×C4×C23—41no—
88C22×C4×C23D812yes—
89C22×C4×C23C2×D411yescharacteristic
90Q16×C23—21no—
91Q16×C23C22⋊C412yescomplemented
92SD16×C23—22no—
93SD16×C23—22no—
94D8×C23—22no—
95D8×C23—22no—
96D8×C23—21no—
97D8×C23—41no—
98D8×C23—41no—
99D8×C23C22⋊C412yescomplemented
100C2×C8×C23—21no—
101C2×C8×C23—21no—
102C2×C8×C23—21no—
103C2×C8×C23C22×C411yescharacteristic
104C2×C8×C23C2×D411yescharacteristic
105C2×C8×C23C2×D411yescharacteristic
106C2×C8×C23C2×D411yescharacteristic
107C2×C8×C23#4111yescharacteristic
108C2×C8×C23#4111yescharacteristic
109C4⋊C4×C23—22no—
110C4⋊C4×C23—42no—
111C4⋊C4×C23—41no—
112C4⋊C4×C23—41no—
113C22⋊C4×C23—42no—
114C22⋊C4×C23—41no—
115C22⋊C4×C23—84no—
116C42×C23—21no—
117C42×C23—21no—
118C42×C23—42no—
119C42×C23—41no—
120C16×C23—22no—
121C16×C23—41no—
122#13025C2311yesmaximal, characteristic, direct product factor, Sylow
123C23×C23—41no—
124C23×C23—42no—
125C23×C23—82no—
126C23×C23—82no—
127C23×C23—81no—
128C23×C23—81no—
129C23×C23C2×D811yescharacteristic
130Q8×C23—22no—
131Q8×C23—41no—
132D4×C23—22no—
133D4×C23—42no—
134D4×C23—42no—
135D4×C23—42no—
136D4×C23—42no—
137D4×C23—41no—
138D4×C23—81no—
139D4×C23—81no—
140C2×C4×C23—22no—
141C2×C4×C23—41no—
142C2×C4×C23—42no—
143C2×C4×C23—42no—
144C2×C4×C23—42no—
145C2×C4×C23—42no—
146C2×C4×C23—84no—
147C2×C4×C23—82no—
148C2×C4×C23—81no—
149C2×C4×C23C4×D411yescharacteristic
150C2×C4×C23#12011yescharacteristic
151C2×C4×C23#12111yescharacteristic
152C2×C4×C23#13511yescharacteristic
153C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
155C8×C23—41no—
156C8×C23C2×C22⋊C411yescharacteristic
157C8×C23C4×D411yescharacteristic
158C8×C23#12111yescharacteristic
159C8×C23#12311yescharacteristic
160C2×#509C2×C2311yescharacteristic
161#2943C2×C2311yescharacteristic, complemented
162C2×#360C2×C2311yescharacteristic, complemented
163#2140C2×C2311yescharacteristic, complemented
164#1972C2×C2311yescharacteristic, complemented
165#1386C2×C2311yescharacteristic, complemented
166#1377C2×C2311yescharacteristic, complemented
167C22×C23—22no—
168C22×C23—42no—
169C22×C23—84no—
170C22×C23—82no—
171C22×C23—82no—
172C22×C23—161no—
173C22×C23—161no—
174C22×C23—162no—
175C22×C23C4×D811yescharacteristic
176C22×C23#44711yescharacteristic
177C22×C23#45911yescharacteristic
178C4×C23—42no—
179C4×C23—42no—
180C4×C23—82no—
181C4×C23—161no—
182C4×C23#38611yescharacteristic
183C4×C23#44211yescharacteristic
184C4×C23#44911yescharacteristic
185C4×C23#46311yescharacteristic
186C2×#135C22×C2311yescharacteristic
187C22×D8C22×C2311yescharacteristic
188C4×#41—21no—
189#509—22no—
190#509C4×C2314yescomplemented
191C2×SD32—21no—
192C2×D16—21no—
193C2×M32C22×C2311yescharacteristic
194C2×M32C22×C2311yescharacteristic
195#442—21no—
196#441—21no—
197#440—22no—
198#427C22×C2311yescharacteristic
199C2×#106C22×C2311yescharacteristic
200#416—21no—
201C2×#102—21no—
202C2×#97C22×C2311yescharacteristic
203#390—21no—
204#360—22no—
205#357—22no—
206C2×C23—21no—
207C2×C23—21no—
208C2×C23—82no—
209C2×C23—162no—
210C2×C23#187911yescharacteristic
211C2×C23#219012yes—
212C2×#41—21no—
213C22×D4—21no—
214#135—21no—
215#135—21no—
216#135C2×C4×C2312yes—
217#135D4×C2312yes—
218C2×Q16D4×C2311yescharacteristic
219C2×SD16—21no—
220C2×D8—21no—
221C2×D8—21no—
222C2×D8—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8C2×C4×C2312yescomplemented
226C2×D8D4×C2311yescharacteristic
227C22×C8D4×C2311yescharacteristic
228C22×C8C23×C2311yescharacteristic
229C4×Q8—22no—
230C4×D4—22no—
231C4×D4—42no—
232#117—21no—
233#117—41no—
234C2×C4⋊C4—21no—
235C2×C22⋊C4—42no—
236C2×C42—41no—
237C2×C42D4×C2311yescharacteristic
238SD32—24no—
239D16—24no—
240M32—21no—
241M32—21no—
242M32—21no—
243M32C2×C4×C2312yescomplemented
244C2×C16—21no—
245C2×C16—21no—
246#107D4×C2312yes—
247#106—21no—
248#106—21no—
249#106—21no—
250#103—22no—
251#102—22no—
252#102—42no—
253#97—21no—
254#97—21no—
255C23#1302511yescharacteristic, direct product factor, Sylow
256C24—41no—
257#41—22no—
258#41—22no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×D4—22no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4D8×C2312yes—
274C22×C4C2×D4×C2311yescharacteristic
275Q16—21no—
276Q16C22⋊C4×C2312yescomplemented
277SD16—22no—
278SD16—22no—
279D8—22no—
280D8—22no—
281D8—21no—
282D8—41no—
283D8—41no—
284D8C22⋊C4×C2312yescomplemented
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C22×C4×C2311yescharacteristic
289C2×C8C2×D4×C2311yescharacteristic
290C2×C8C2×D4×C2311yescharacteristic
291C2×C8C2×D4×C2311yescharacteristic
292C2×C8#41×C2311yescharacteristic
293C2×C8#41×C2311yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—84no—
301C42—21no—
302C42—21no—
303C42—42no—
304C42—41no—
305C16—22no—
306C16—41no—
307C23—41no—
308C23—42no—
309C23—82no—
310C23—82no—
311C23—81no—
312C23—81no—
313C23C2×D8×C2311yescharacteristic
314Q8—22no—
315Q8—41no—
316D4—22no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—42no—
321D4—41no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—84no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4C4×D4×C2311yescharacteristic
334C2×C4C23×#12011yescharacteristic
335C2×C4C23×#12111yescharacteristic
336C2×C4C23×#13511yescharacteristic
337C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
338C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
339C8—41no—
340C8C2×C22⋊C4×C2311yescharacteristic
341C8C4×D4×C2311yescharacteristic
342C8C23×#12111yescharacteristic
343C8C23×#12311yescharacteristic
344C22—22no—
345C22—42no—
346C22—84no—
347C22—82no—
348C22—82no—
349C22—161no—
350C22—161no—
351C22—162no—
352C22C4×D8×C2311yescharacteristic
353C22C23×#44711yescharacteristic
354C22C23×#45911yescharacteristic
355C4—42no—
356C4—42no—
357C4—82no—
358C4—161no—
359C4C23×#38611yescharacteristic
360C4C23×#44211yescharacteristic
361C4C23×#44911yescharacteristic
362C4C23×#46311yescharacteristic
363C2—21no—
364C2—21no—
365C2—82no—
366C2—162no—
367C2C23×#187911yescharacteristic
368C2C23×#219012yes—
subgroupG/H
center, socleC22×C23#447
commutatorC2×C8C22×C4×C23
Frattini, cosocleC22×C8C23×C23
prime radicalC2×C23×#509C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×#97 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2943C2semidirect
3. C2×C23×#360C2semidirect
4. C23×#2140C2semidirect
5. C23×#1972C2semidirect
6. C23×#1386C2semidirect
7. C23×#1377C2semidirect
12. C23×#509C4semidirect
41. C2×D8×C23C2×C4semidirect
59. C23×M32C2×C4semidirect
91. Q16×C23C22⋊C4semidirect
99. D8×C23C22⋊C4semidirect
122. #13025C23direct (N × H)
161. #2943C2×C23semidirect
162. C2×#360C2×C23semidirect
163. #2140C2×C23semidirect
164. #1972C2×C23semidirect
165. #1386C2×C23semidirect
166. #1377C2×C23semidirect
190. #509C4×C23semidirect
225. C2×D8C2×C4×C23semidirect
243. M32C2×C4×C23semidirect
255. C23#13025direct (N × H)
276. Q16C22⋊C4×C23semidirect
284. D8C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#5092
C23×#29432
C2×C23×#3602
C23×#21402
C23×#19722
C23×#13862
C23×#13772
#1302523