Finite Group Databasebeta

← C23×#13029 #19160655

Subgroups — C23×#13029 #19160655

Order 5888.

512 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#509C211yesmaximal, characteristic
2C2×C23×#443C211yesmaximal, characteristic, complemented
3C2×C23×#361C211yesmaximal, characteristic, complemented
4C23×#2139C211yesmaximal, characteristic, complemented
5C23×#1970C211yesmaximal, characteristic, complemented
6C23×#1387C211yesmaximal, characteristic, complemented
7C23×#1376C211yesmaximal, characteristic, complemented
8C2×C23×#135C2211yescharacteristic
9C22×D8×C23C2211yescharacteristic
10C2×C23×#131—21no—
11C23×#509—22no—
12C23×#509C414yescomplemented
13C2×C23×SD32—21no—
14C2×C23×D16—21no—
15C2×C23×M32C2211yescharacteristic
16C2×C23×M32C2211yescharacteristic
17C23×#443—22no—
18C23×#443—22no—
19C2×C23×#108C2211yescharacteristic
20C2×C23×#107C2211yescharacteristic
21C2×C23×#104—21no—
22C2×C23×#102—21no—
23C2×C4×C8×C23C2211yescharacteristic
24C23×#390—21no—
25C23×#361—22no—
26C23×#358—21no—
27C23×#357—21no—
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×C412yes—
42C2×D8×C23D411yescharacteristic
43C23×#131—24no—
44C23×#131—42no—
45C2×M16×C23—21no—
46C2×M16×C23—41no—
47C22×C8×C23—41no—
48C22×C8×C23D411yescharacteristic
49C22×C8×C23C2311yescharacteristic
50C2×C4⋊C4×C23—21no—
51C2×C22⋊C4×C23—42no—
52C2×C42×C23D411yescharacteristic
53C23×SD32—24no—
54C23×D16—24no—
55C23×M32—21no—
56C23×M32—21no—
57C23×M32—21no—
58C23×M32C2×C412yes—
59C2×C16×C23—21no—
60C2×C16×C23—21no—
61C23×#108—21no—
62C23×#108D412yes—
63C23×#107—21no—
64C23×#107—21no—
65C23×#104—24no—
66C23×#102—41no—
67C23×#102—41no—
68C4×C8×C23—21no—
69C4×C8×C23—21no—
70C24×C23—41no—
71#41×C23—22no—
72#41×C23—22no—
73#41×C23—41no—
74#41×C23—41no—
75C2×Q8×C23—21no—
76C2×D4×C23—22no—
77C2×D4×C23—21no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—41no—
84C22×C4×C23—21no—
85C22×C4×C23—41no—
86C22×C4×C23—41no—
87C22×C4×C23C2×D411yescharacteristic
88Q16×C23—21no—
89Q16×C23C22⋊C412yescomplemented
90SD16×C23—22no—
91SD16×C23—22no—
92D8×C23—22no—
93D8×C23—22no—
94D8×C23—21no—
95D8×C23—41no—
96D8×C23—41no—
97D8×C23C22⋊C412yescomplemented
98M16×C23—22no—
99M16×C23—44no—
100M16×C23—41no—
101C2×C8×C23—21no—
102C2×C8×C23—21no—
103C2×C8×C23—44no—
104C2×C8×C23—41no—
105C2×C8×C23D811yescharacteristic
106C2×C8×C23D811yescharacteristic
107C2×C8×C23C22×C411yescharacteristic
108C2×C8×C23C2×D411yescharacteristic
109C2×C8×C23C2×D411yescharacteristic
110C2×C8×C23C2×D411yescharacteristic
111C2×C8×C23#4111yescharacteristic
112C2×C8×C23#4111yescharacteristic
113C4⋊C4×C23—41no—
114C4⋊C4×C23—41no—
115C22⋊C4×C23—84no—
116C42×C23—21no—
117C42×C23—21no—
118C16×C23—22no—
119C16×C23—41no—
120#13029C2311yesmaximal, characteristic, direct product factor, Sylow
121C23×C23—41no—
122C23×C23—42no—
123C23×C23—82no—
124C23×C23—82no—
125C23×C23—81no—
126C23×C23—81no—
127C23×C23AGL(1,ℤ/8)11yescharacteristic
128Q8×C23—22no—
129Q8×C23—41no—
130D4×C23—22no—
131D4×C23—42no—
132D4×C23—42no—
133D4×C23—42no—
134D4×C23—42no—
135D4×C23—41no—
136D4×C23—81no—
137D4×C23—81no—
138C2×C4×C23—22no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—42no—
143C2×C4×C23—42no—
144C2×C4×C23—82no—
145C2×C4×C23—81no—
146C2×C4×C23C4×D411yescharacteristic
147C2×C4×C23#12011yescharacteristic
148C2×C4×C23#12111yescharacteristic
149C2×C4×C23C2×D811yescharacteristic
150C2×C4×C23#13511yescharacteristic
151C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
152C8×C23—21no—
153C8×C23—21no—
154C8×C23—42no—
155C8×C23C2×C22⋊C411yescharacteristic
156C8×C23C4×D411yescharacteristic
157C8×C23#12111yescharacteristic
158C8×C23#12311yescharacteristic
159C2×#509C2×C2311yescharacteristic
160C2×#443C2×C2311yescharacteristic, complemented
161C2×#361C2×C2311yescharacteristic, complemented
162#2139C2×C2311yescharacteristic, complemented
163#1970C2×C2311yescharacteristic, complemented
164#1387C2×C2311yescharacteristic, complemented
165#1376C2×C2311yescharacteristic, complemented
166C22×C23—21no—
167C22×C23—21no—
168C22×C23—42no—
169C22×C23—84no—
170C22×C23—81no—
171C22×C23—81no—
172C22×C23—82no—
173C22×C23—162no—
174C22×C23—161no—
175C22×C23—161no—
176C22×C23#44211yescharacteristic
177C22×C23#44911yescharacteristic
178C22×C23#45911yescharacteristic
179C4×C23—42no—
180C4×C23—42no—
181C4×C23—161no—
182C4×C23#38611yescharacteristic
183C4×C23C4×D811yescharacteristic
184C4×C23#44711yescharacteristic
185C4×C23#46311yescharacteristic
186C2×#135C22×C2311yescharacteristic
187C22×D8C22×C2311yescharacteristic
188C2×#131—21no—
189#509—22no—
190#509C4×C2314yescomplemented
191C2×SD32—21no—
192C2×D16—21no—
193C2×M32C22×C2311yescharacteristic
194C2×M32C22×C2311yescharacteristic
195#443—22no—
196#443—22no—
197C2×#108C22×C2311yescharacteristic
198C2×#107C22×C2311yescharacteristic
199C2×#104—21no—
200C2×#102—21no—
201C2×C4×C8C22×C2311yescharacteristic
202#390—21no—
203#361—22no—
204#358—21no—
205#357—21no—
206C2×C23—21no—
207C2×C23—21no—
208C2×C23—82no—
209C2×C23—162no—
210C2×C23#187911yescharacteristic
211C2×C23#221211yescharacteristic
212C2×C23#221311yescharacteristic
213C2×#41—21no—
214C22×D4—21no—
215#135—21no—
216#135—21no—
217#135C2×C4×C2312yes—
218#135D4×C2312yes—
219C2×Q16D4×C2311yescharacteristic
220C2×SD16—21no—
221C2×D8—21no—
222C2×D8—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8C2×C4×C2312yes—
227C2×D8D4×C2311yescharacteristic
228#131—24no—
229#131—42no—
230C2×M16—21no—
231C2×M16—41no—
232C22×C8—41no—
233C22×C8D4×C2311yescharacteristic
234C22×C8C23×C2311yescharacteristic
235C2×C4⋊C4—21no—
236C2×C22⋊C4—42no—
237C2×C42D4×C2311yescharacteristic
238SD32—24no—
239D16—24no—
240M32—21no—
241M32—21no—
242M32—21no—
243M32C2×C4×C2312yes—
244C2×C16—21no—
245C2×C16—21no—
246#108—21no—
247#108D4×C2312yes—
248#107—21no—
249#107—21no—
250#104—24no—
251#102—41no—
252#102—41no—
253C4×C8—21no—
254C4×C8—21no—
255C23#1302911yescharacteristic, 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—21no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C2×D4×C2311yescharacteristic
274Q16—21no—
275Q16C22⋊C4×C2312yescomplemented
276SD16—22no—
277SD16—22no—
278D8—22no—
279D8—22no—
280D8—21no—
281D8—41no—
282D8—41no—
283D8C22⋊C4×C2312yescomplemented
284M16—22no—
285M16—44no—
286M16—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—44no—
290C2×C8—41no—
291C2×C8D8×C2311yescharacteristic
292C2×C8D8×C2311yescharacteristic
293C2×C8C22×C4×C2311yescharacteristic
294C2×C8C2×D4×C2311yescharacteristic
295C2×C8C2×D4×C2311yescharacteristic
296C2×C8C2×D4×C2311yescharacteristic
297C2×C8#41×C2311yescharacteristic
298C2×C8#41×C2311yescharacteristic
299C4⋊C4—41no—
300C4⋊C4—41no—
301C22⋊C4—84no—
302C42—21no—
303C42—21no—
304C16—22no—
305C16—41no—
306C23—41no—
307C23—42no—
308C23—82no—
309C23—82no—
310C23—81no—
311C23—81no—
312C23C23×AGL(1,ℤ/8)11yescharacteristic
313Q8—22no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—41no—
321D4—81no—
322D4—81no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4C4×D4×C2311yescharacteristic
332C2×C4C23×#12011yescharacteristic
333C2×C4C23×#12111yescharacteristic
334C2×C4C2×D8×C2311yescharacteristic
335C2×C4C23×#13511yescharacteristic
336C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
337C8—21no—
338C8—21no—
339C8—42no—
340C8C2×C22⋊C4×C2311yescharacteristic
341C8C4×D4×C2311yescharacteristic
342C8C23×#12111yescharacteristic
343C8C23×#12311yescharacteristic
344C22—21no—
345C22—21no—
346C22—42no—
347C22—84no—
348C22—81no—
349C22—81no—
350C22—82no—
351C22—162no—
352C22—161no—
353C22—161no—
354C22C23×#44211yescharacteristic
355C22C23×#44911yescharacteristic
356C22C23×#45911yescharacteristic
357C4—42no—
358C4—42no—
359C4—161no—
360C4C23×#38611yescharacteristic
361C4C4×D8×C2311yescharacteristic
362C4C23×#44711yescharacteristic
363C4C23×#46311yescharacteristic
364C2—21no—
365C2—21no—
366C2—82no—
367C2—162no—
368C2C23×#187911yescharacteristic
369C2C23×#221211yescharacteristic
370C2C23×#221311yescharacteristic
subgroupG/H
center, socleC22×C23#459
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×C4×C8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#443C2semidirect
3. C2×C23×#361C2semidirect
4. C23×#2139C2semidirect
5. C23×#1970C2semidirect
6. C23×#1387C2semidirect
7. C23×#1376C2semidirect
12. C23×#509C4semidirect
89. Q16×C23C22⋊C4semidirect
97. D8×C23C22⋊C4semidirect
120. #13029C23direct (N × H)
160. C2×#443C2×C23semidirect
161. C2×#361C2×C23semidirect
162. #2139C2×C23semidirect
163. #1970C2×C23semidirect
164. #1387C2×C23semidirect
165. #1376C2×C23semidirect
190. #509C4×C23semidirect
255. C23#13029direct (N × H)
275. Q16C22⋊C4×C23semidirect
283. 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
C2×C23×#4432
C2×C23×#3612
C23×#21392
C23×#19702
C23×#13872
C23×#13762
#1302923