Finite Group Databasebeta

← C23×#13401 #19161027

Subgroups — C23×#13401 #19161027

Order 5888.

390 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3067C211yesmaximal, characteristic
2C23×#2155C211yesmaximal, characteristic, complemented
3C23×#2145C211yesmaximal, characteristic, complemented
4C23×#2138C211yesmaximal, characteristic, complemented
5C2×C23×#350C211yesmaximal, characteristic, complemented
6C23×#1812C211yesmaximal, characteristic, complemented
7C23×#1382C211yesmaximal, characteristic, complemented
8C23×#575—21no—
9C2×C23×#135C2211yescharacteristic
10C2×C23×#131C2211yescharacteristic
11C2×C23×M32C2211yescharacteristic
12C22×C16×C23C2211yescharacteristic
13C23×#473—21no—
14C23×#472—21no—
15C23×#471—22no—
16C2×C23×#108C2211yescharacteristic
17C23×#427C2211yescharacteristic
18C23×#427C2211yescharacteristic
19C23×#413—21no—
20C23×#410—21no—
21C23×#365—22no—
22C23×#360—22no—
23C23×#358—21no—
24C23×#357—21no—
25C23×#350—22no—
26C23×H(3,ℤ/4)—21no—
27C2×#41×C23—21no—
28C2×#41×C23D411yescharacteristic
29C23×#137—22no—
30C23×#137—41no—
31C23×AGL(1,ℤ/8)—22no—
32C23×AGL(1,ℤ/8)—41no—
33C23×#135—21no—
34C23×#135—21no—
35C23×#135—21no—
36C23×#135—21no—
37C23×#135—42no—
38C2×Q16×C23D411yescharacteristic
39C2×SD16×C23—21no—
40C2×D8×C23D411yescharacteristic
41C23×#131—21no—
42C23×#131—21no—
43C23×#131—21no—
44C23×#131—21no—
45C2×M16×C23—21no—
46C2×M16×C23—21no—
47C2×M16×C23D411yescharacteristic
48C22×C8×C23—21no—
49C22×C8×C23C2311yescharacteristic
50C23×#117—21no—
51C23×#117—21no—
52C23×M32—21no—
53C23×M32D412yes—
54C2×C16×C23—21no—
55C2×C16×C23—21no—
56C2×C16×C23—21no—
57C2×C16×C23—21no—
58C23×#108—21no—
59C23×#108D412yes—
60C23×#107—21no—
61C23×#107—21no—
62C23×#106—21no—
63C23×#106—21no—
64C23×#101—22no—
65C23×#100—22no—
66C23×#99—42no—
67#41×C23—21no—
68#41×C23—21no—
69#41×C23—21no—
70#41×C23—21no—
71#41×C23—41no—
72#41×C23—41no—
73#41×C23—41no—
74#41×C23—41no—
75C2×Q8×C23—21no—
76C2×Q8×C23SD1611yescharacteristic
77C2×D4×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—41no—
80C2×D4×C23SD1611yescharacteristic
81C22×C4×C23—21no—
82C22×C4×C23—41no—
83C22×C4×C23C2×D411yescharacteristic
84Q16×C23—21no—
85Q16×C23—21no—
86Q16×C23—42no—
87SD16×C23—42no—
88SD16×C23—42no—
89SD16×C23—41no—
90SD16×C23—41no—
91D8×C23—21no—
92D8×C23—21no—
93D8×C23—42no—
94M16×C23—21no—
95M16×C23—22no—
96M16×C23—21no—
97M16×C23—41no—
98M16×C23—41no—
99M16×C23—41no—
100C2×C8×C23—21no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C2×C8×C23C2×D411yescharacteristic
105C2×C8×C23C2×D411yescharacteristic
106C2×C8×C23C2×D411yescharacteristic
107C2×C8×C23#4111yescharacteristic
108C2×C8×C23#4111yescharacteristic
109C2×C8×C23#4111yescharacteristic
110C4⋊C4×C23—41no—
111C4⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C42×C23—41no—
115C42×C23—41no—
116C16×C23—22no—
117C16×C23—41no—
118#13401C2311yesmaximal, characteristic, direct product factor, Sylow
119C23×C23—21no—
120C23×C23—41no—
121C23×C23AGL(1,ℤ/8)11yescharacteristic
122Q8×C23—21no—
123Q8×C23—21no—
124Q8×C23—41no—
125Q8×C23—41no—
126D4×C23—21no—
127D4×C23—21no—
128D4×C23—41no—
129D4×C23—41no—
130D4×C23—41no—
131D4×C23—41no—
132D4×C23—81no—
133D4×C23—81no—
134C2×C4×C23—21no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—81no—
139C2×C4×C23—81no—
140C2×C4×C23—81no—
141C2×C4×C23—81no—
142C2×C4×C23#12011yescharacteristic
143C2×C4×C23#12111yescharacteristic
144C2×C4×C23#12311yescharacteristic
145C2×C4×C23C2×SD1611yescharacteristic
146C2×C4×C23#13511yescharacteristic
147C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
148C8×C23—41no—
149C8×C23—42no—
150C8×C23#12111yescharacteristic
151C8×C23#12111yescharacteristic
152C8×C23#12311yescharacteristic
153C8×C23#12411yescharacteristic
154#3067C2×C2311yescharacteristic
155#2155C2×C2311yescharacteristic, complemented
156#2145C2×C2311yescharacteristic, complemented
157#2138C2×C2311yescharacteristic, complemented
158C2×#350C2×C2311yescharacteristic, complemented
159#1812C2×C2311yescharacteristic, complemented
160#1382C2×C2311yescharacteristic, complemented
161C22×C23—41no—
162C22×C23—41no—
163C22×C23—81no—
164C22×C23—81no—
165C22×C23—161no—
166C22×C23#44911yescharacteristic
167C22×C23#46911yescharacteristic
168C22×C23#48111yescharacteristic
169C4×C23—41no—
170C4×C23—81no—
171C4×C23—161no—
172C4×C23—161no—
173C4×C23#39411yescharacteristic
174C4×C23#45011yescharacteristic
175C4×C23#46511yescharacteristic
176C4×C23#48211yescharacteristic
177#575—21no—
178C2×#135C22×C2311yescharacteristic
179C2×#131C22×C2311yescharacteristic
180C2×M32C22×C2311yescharacteristic
181C22×C16C22×C2311yescharacteristic
182#473—21no—
183#472—21no—
184#471—22no—
185C2×#108C22×C2311yescharacteristic
186#427C22×C2311yescharacteristic
187#427C22×C2311yescharacteristic
188#413—21no—
189#410—21no—
190#365—22no—
191#360—22no—
192#358—21no—
193#357—21no—
194#350—22no—
195H(3,ℤ/4)—21no—
196C2×C23—21no—
197C2×C23—21no—
198C2×C23—21no—
199C2×C23—81no—
200C2×C23—161no—
201C2×C23#203411yescharacteristic
202C2×#41—21no—
203C2×#41D4×C2311yescharacteristic
204#137—22no—
205#137—41no—
206AGL(1,ℤ/8)—22no—
207AGL(1,ℤ/8)—41no—
208#135—21no—
209#135—21no—
210#135—21no—
211#135—21no—
212#135—42no—
213C2×Q16D4×C2311yescharacteristic
214C2×SD16—21no—
215C2×D8D4×C2311yescharacteristic
216#131—21no—
217#131—21no—
218#131—21no—
219#131—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16D4×C2311yescharacteristic
223C22×C8—21no—
224C22×C8C23×C2311yescharacteristic
225#117—21no—
226#117—21no—
227M32—21no—
228M32D4×C2312yes—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233#108—21no—
234#108D4×C2312yes—
235#107—21no—
236#107—21no—
237#106—21no—
238#106—21no—
239#101—22no—
240#100—22no—
241#99—42no—
242C23#1340111yescharacteristic, direct product factor, Sylow
243#41—21no—
244#41—21no—
245#41—21no—
246#41—21no—
247#41—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×Q8SD16×C2311yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4SD16×C2311yescharacteristic
257C22×C4—21no—
258C22×C4—41no—
259C22×C4C2×D4×C2311yescharacteristic
260Q16—21no—
261Q16—21no—
262Q16—42no—
263SD16—42no—
264SD16—42no—
265SD16—41no—
266SD16—41no—
267D8—21no—
268D8—21no—
269D8—42no—
270M16—21no—
271M16—22no—
272M16—21no—
273M16—41no—
274M16—41no—
275M16—41no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8C2×D4×C2311yescharacteristic
281C2×C8C2×D4×C2311yescharacteristic
282C2×C8C2×D4×C2311yescharacteristic
283C2×C8#41×C2311yescharacteristic
284C2×C8#41×C2311yescharacteristic
285C2×C8#41×C2311yescharacteristic
286C4⋊C4—41no—
287C4⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C42—41no—
291C42—41no—
292C16—22no—
293C16—41no—
294C23—21no—
295C23—41no—
296C23C23×AGL(1,ℤ/8)11yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—41no—
300Q8—41no—
301D4—21no—
302D4—21no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—81no—
308D4—81no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4C23×#12011yescharacteristic
318C2×C4C23×#12111yescharacteristic
319C2×C4C23×#12311yescharacteristic
320C2×C4C2×SD16×C2311yescharacteristic
321C2×C4C23×#13511yescharacteristic
322C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
323C8—41no—
324C8—42no—
325C8C23×#12111yescharacteristic
326C8C23×#12111yescharacteristic
327C8C23×#12311yescharacteristic
328C8C23×#12411yescharacteristic
329C22—41no—
330C22—41no—
331C22—81no—
332C22—81no—
333C22—161no—
334C22C23×#44911yescharacteristic
335C22C23×#46911yescharacteristic
336C22C23×#48111yescharacteristic
337C4—41no—
338C4—81no—
339C4—161no—
340C4—161no—
341C4C23×#39411yescharacteristic
342C4C23×#45011yescharacteristic
343C4C23×#46511yescharacteristic
344C4C23×#48211yescharacteristic
345C2—21no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—161no—
350C2C23×#203411yescharacteristic
subgroupG/H
centerC4×C23#482
commutator, Frattini, cosocleC22×C8C23×C23
socleC2×C23#2034
prime radicalC23×#3067C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C22×C4×C23 ◁ C2×C23×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2155C2semidirect
3. C23×#2145C2semidirect
4. C23×#2138C2semidirect
5. C2×C23×#350C2semidirect
6. C23×#1812C2semidirect
7. C23×#1382C2semidirect
118. #13401C23direct (N × H)
155. #2155C2×C23semidirect
156. #2145C2×C23semidirect
157. #2138C2×C23semidirect
158. C2×#350C2×C23semidirect
159. #1812C2×C23semidirect
160. #1382C2×C23semidirect
242. C23#13401direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30672
C23×#21552
C23×#21452
C23×#21382
C2×C23×#3502
C23×#18122
C23×#13822
#1340123