Finite Group Databasebeta

← C23×#10553 #19158179

Subgroups — C23×#10553 #19158179

Order 5888.

416 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3138C211yesmaximal, characteristic
2C23×#1950C211yesmaximal, characteristic, complemented
3C23×#1550C211yesmaximal, characteristic, complemented
4C23×#1545C211yesmaximal, characteristic
5C2×C23×#331C211yesmaximal, characteristic
6C23×#1481C211yesmaximal, characteristic, complemented
7C23×#1283C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C2×C23×#127C2211yescharacteristic
10C23×#490—22no—
11C23×#486—21no—
12C23×#486—21no—
13C23×#431C2211yescharacteristic
14C2×C23×#108—21no—
15C23×#423C2211yescharacteristic
16C2×C23×C4⋊C8C2211yescharacteristic
17C2×C23×C4⋊C8C2211yescharacteristic
18C23×#418—21no—
19C23×#418—21no—
20C4×M16×C23C2211yescharacteristic
21C23×#341—21no—
22C23×#333—21no—
23C23×#332—21no—
24C23×#331—22no—
25C23×#331—22no—
26C23×Q8⋊C8—22no—
27C22×D4×C23—21no—
28C22×D4×C23—21no—
29C22×D4×C23—21no—
30C2×M16×C23—21no—
31C2×M16×C23D411yescharacteristic
32C2×M16×C23D411yescharacteristic
33C22×C8×C23—21no—
34C22×C8×C23—21no—
35C23×C4⋊Q8D411yescharacteristic
36C23×C4⋊Q8D411yescharacteristic
37C23×#127—21no—
38C23×#127—21no—
39C23×#127—21no—
40C23×#127D412yes—
41C23×#125—21no—
42C23×C22⋊Q8—21no—
43C23×C22⋊Q8—21no—
44C4×Q8×C23—21no—
45C4×Q8×C23—21no—
46C23×#117—21no—
47C2×C42×C23C2311yescharacteristic
48C23×#108—42no—
49C23×C4⋊C8—21no—
50C23×C4⋊C8—21no—
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8D412yes—
56C23×#102—22no—
57C23×#102—22no—
58C23×#102—22no—
59C23×#102—22no—
60C23×C22⋊C8—21no—
61C23×#97—21no—
62C4×C8×C23—21no—
63C4×C8×C23—21no—
64C24×C23—41no—
65C2×Q8×C23—21no—
66C2×Q8×C23—21no—
67C2×D4×C23—22no—
68C2×D4×C23—22no—
69C2×D4×C23—22no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C2×D4×C23—41no—
73C2×D4×C23—41no—
74C2×D4×C23—41no—
75C2×D4×C23—41no—
76C2×D4×C23—41no—
77C2×D4×C23—81no—
78C2×D4×C23—81no—
79C2×D4×C23—82no—
80C22×C4×C23C2×D411yescharacteristic
81C22×C4×C23#4111yescharacteristic
82C22×C4×C23#4111yescharacteristic
83M16×C23—21no—
84M16×C23—21no—
85M16×C23—21no—
86M16×C23—21no—
87M16×C23—41no—
88M16×C23—41no—
89C2×C8×C23—21no—
90C2×C8×C23—21no—
91C2×C8×C23—22no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C2×C8×C23—41no—
97C2×C8×C23—81no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23—41no—
104C22⋊C4×C23—41no—
105C42×C23—41no—
106C42×C23C2×D411yescharacteristic
107C42×C23C2×D411yescharacteristic
108C42×C23C2×D411yescharacteristic
109C42×C23#4111yescharacteristic
110#10553C2311yesmaximal, characteristic, direct product factor, Sylow
111C23×C23—42no—
112C23×C23—82no—
113C23×C23—82no—
114C23×C23—81no—
115C23×C23—81no—
116C23×C23#12411yescharacteristic
117Q8×C23—41no—
118Q8×C23—41no—
119Q8×C23—81no—
120D4×C23—42no—
121D4×C23—42no—
122D4×C23—82no—
123D4×C23—82no—
124D4×C23—82no—
125D4×C23—81no—
126D4×C23—81no—
127D4×C23—81no—
128D4×C23—81no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23#12011yescharacteristic
135C2×C4×C23#12111yescharacteristic
136C2×C4×C23#12111yescharacteristic
137C2×C4×C23#12111yescharacteristic
138C2×C4×C23#12311yescharacteristic
139C2×C4×C23#12311yescharacteristic
140C2×C4×C23#13511yescharacteristic
141C2×C4×C23#13511yescharacteristic
142C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C23#13711yescharacteristic
144C8×C23—41no—
145C8×C23—41no—
146C8×C23—42no—
147C8×C23—81no—
148C8×C23—81no—
149C8×C23—81no—
150#3138C2×C2311yescharacteristic
151#1950C2×C2311yescharacteristic, complemented
152#1550C2×C2311yescharacteristic, complemented
153#1545C2×C2311yescharacteristic
154C2×#331C2×C2311yescharacteristic
155#1481C2×C2311yescharacteristic, complemented
156#1283C2×C2311yescharacteristic, complemented
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—82no—
160C22×C23—82no—
161C22×C23—82no—
162C22×C23—162no—
163C22×C23—162no—
164C22×C23#39411yescharacteristic
165C22×C23#48811yescharacteristic
166C22×C23#48911yescharacteristic
167C4×C23—21no—
168C4×C23—21no—
169C4×C23—21no—
170C4×C23—81no—
171C4×C23—81no—
172C4×C23#44911yescharacteristic
173C4×C23#45211yescharacteristic
174C4×C23#46311yescharacteristic
175C4×C23#46411yescharacteristic
176C4×C23#47111yescharacteristic
177C4×C23#47211yescharacteristic
178#533C22×C2311yescharacteristic
179C2×#127C22×C2311yescharacteristic
180#490—22no—
181#486—21no—
182#486—21no—
183#431C22×C2311yescharacteristic
184C2×#108—21no—
185#423C22×C2311yescharacteristic
186C2×C4⋊C8C22×C2311yescharacteristic
187C2×C4⋊C8C22×C2311yescharacteristic
188#418—21no—
189#418—21no—
190C4×M16C22×C2311yescharacteristic
191#341—21no—
192#333—21no—
193#332—21no—
194#331—22no—
195#331—22no—
196Q8⋊C8—22no—
197C2×C23—21no—
198C2×C23—21no—
199C2×C23—162no—
200C2×C23#204211yescharacteristic
201C2×C23#204811yescharacteristic
202C2×C23#205111yescharacteristic
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—21no—
206C2×M16—21no—
207C2×M16D4×C2311yescharacteristic
208C2×M16D4×C2311yescharacteristic
209C22×C8—21no—
210C22×C8—21no—
211C4⋊Q8D4×C2311yescharacteristic
212C4⋊Q8D4×C2311yescharacteristic
213#127—21no—
214#127—21no—
215#127—21no—
216#127D4×C2312yes—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222#117—21no—
223C2×C42C23×C2311yescharacteristic
224#108—42no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8D4×C2312yes—
232#102—22no—
233#102—22no—
234#102—22no—
235#102—22no—
236C22⋊C8—21no—
237#97—21no—
238C4×C8—21no—
239C4×C8—21no—
240C23#1055311yescharacteristic, direct product factor, Sylow
241C24—41no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—81no—
255C2×D4—81no—
256C2×D4—82no—
257C22×C4C2×D4×C2311yescharacteristic
258C22×C4#41×C2311yescharacteristic
259C22×C4#41×C2311yescharacteristic
260M16—21no—
261M16—21no—
262M16—21no—
263M16—21no—
264M16—41no—
265M16—41no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—81no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C22⋊C4—41no—
282C42—41no—
283C42C2×D4×C2311yescharacteristic
284C42C2×D4×C2311yescharacteristic
285C42C2×D4×C2311yescharacteristic
286C42#41×C2311yescharacteristic
287C23—42no—
288C23—82no—
289C23—82no—
290C23—81no—
291C23—81no—
292C23C23×#12411yescharacteristic
293Q8—41no—
294Q8—41no—
295Q8—81no—
296D4—42no—
297D4—42no—
298D4—82no—
299D4—82no—
300D4—82no—
301D4—81no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4C23×#12011yescharacteristic
311C2×C4C23×#12111yescharacteristic
312C2×C4C23×#12111yescharacteristic
313C2×C4C23×#12111yescharacteristic
314C2×C4C23×#12311yescharacteristic
315C2×C4C23×#12311yescharacteristic
316C2×C4C23×#13511yescharacteristic
317C2×C4C23×#13511yescharacteristic
318C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
319C2×C4C23×#13711yescharacteristic
320C8—41no—
321C8—41no—
322C8—42no—
323C8—81no—
324C8—81no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22C23×#39411yescharacteristic
334C22C23×#48811yescharacteristic
335C22C23×#48911yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—81no—
340C4—81no—
341C4C23×#44911yescharacteristic
342C4C23×#45211yescharacteristic
343C4C23×#46311yescharacteristic
344C4C23×#46411yescharacteristic
345C4C23×#47111yescharacteristic
346C4C23×#47211yescharacteristic
347C2—21no—
348C2—21no—
349C2—162no—
350C2C23×#204211yescharacteristic
351C2C23×#204811yescharacteristic
352C2C23×#205111yescharacteristic
subgroupG/H
center, socleC22×C23#394
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1950C2semidirect
3. C23×#1550C2semidirect
6. C23×#1481C2semidirect
7. C23×#1283C2semidirect
110. #10553C23direct (N × H)
151. #1950C2×C23semidirect
152. #1550C2×C23semidirect
155. #1481C2×C23semidirect
156. #1283C2×C23semidirect
240. C23#10553direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31382
C23×#19502
C23×#15502
C23×#15452
C2×C23×#3312
C23×#14812
C23×#12832
#1055323