Finite Group Databasebeta

← C4×C23×#557 #19181795

Subgroups — C4×C23×#557 #19181795

Order 5888.

2220 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#557C211yesmaximal, characteristic
2C23×#2370C216yesmaximal
3C23×#2365C213yesmaximal
4C23×#2355C213yesmaximal
5C23×#2351C212yesmaximal
6Q8×C4⋊C4×C23C211yesmaximal, characteristic
7C4×C23×C4⋊Q8C213yesmaximal
8C4×C23×#125C216yesmaximal
9C42×Q8×C23C212yesmaximal
10C42×Q8×C23C213yesmaximal
11C42×Q8×C23C211yesmaximal, characteristic
12C23×#557C4116yesdirect product factor
13C2×C23×C4⋊Q8C2213yes—
14C2×C23×#125C2216yes—
15C2×C4×Q8×C23C2213yes—
16C2×C4×Q8×C23C2212yes—
17C2×C4×Q8×C23C2211yescharacteristic
18C2×C4×Q8×C23C2213yes—
19C2×C4×Q8×C23C2211yescharacteristic
20C2×C4×Q8×C23C2212yes—
21C2×C4×Q8×C23C2213yes—
22C2×C4×Q8×C23C2211yescharacteristic
23C23×#391C2216yes—
24C23×#389C2216yes—
25C23×#389C2216yes—
26C23×#389C2216yes—
27C23×#389C2216yes—
28C23×#387C22112yes—
29C23×#387C22112yes—
30C23×#384C2213yes—
31C23×#382C2216yes—
32C4×C4⋊C4×C23C2211yescharacteristic
33C4×C4⋊C4×C23C2216yes—
34C4×C4⋊C4×C23C2213yes—
35C4×C4⋊C4×C23C2216yes—
36C4×C4⋊C4×C23C2213yes—
37C4×C4⋊C4×C23C2216yes—
38C4×C4⋊C4×C23C2213yes—
39C4×C4⋊C4×C23C2213yes—
40C4×C4⋊C4×C23C2216yes—
41C4×C4⋊C4×C23C2216yes—
42C4×C4⋊C4×C23C2216yes—
43C4×C4⋊C4×C23C2213yes—
44C4×C4⋊C4×C23C2216yes—
45C43×C23C2216yes—
46C43×C23C2213yes—
47C22×Q8×C23C2313yes—
48C22×Q8×C23C2311yescharacteristic
49C23×C4⋊Q8C2×C4124yes—
50C23×#125C2×C4148yes—
51C4×Q8×C23—28no—
52C4×Q8×C23—24no—
53C4×Q8×C23—212no—
54C4×Q8×C23—212no—
55C4×Q8×C23C2×C4116yes—
56C4×Q8×C23C2×C4124yes—
57C4×Q8×C23C2×C418yes—
58C4×Q8×C23Q818yescomplemented
59C2×C4⋊C4×C23C2313yes—
60C2×C4⋊C4×C23C2316yes—
61C2×C4⋊C4×C23C2313yes—
62C2×C4⋊C4×C23C2316yes—
63C2×C4⋊C4×C23C2316yes—
64C2×C4⋊C4×C23C2316yes—
65C2×C4⋊C4×C23C2311yescharacteristic
66C2×C4⋊C4×C23C2316yes—
67C2×C4⋊C4×C23C2313yes—
68C2×C4⋊C4×C23C2316yes—
69C2×C4⋊C4×C23C2316yes—
70C2×C4⋊C4×C23C2313yes—
71C2×C4⋊C4×C23C2311yescharacteristic
72C2×C42×C23C2316yes—
73C2×C42×C23C2316yes—
74C2×C42×C23C2313yes—
75C2×C42×C23C2313yes—
76C2×C42×C23C2312yes—
77C2×C42×C23C2313yes—
78C2×C42×C23C2316yes—
79C2×C42×C23C2316yes—
80C2×C42×C23C2313yes—
81C2×C42×C23C2316yes—
82C2×C42×C23C2313yes—
83C2×C42×C23C2313yes—
84C2×C42×C23C2311yescharacteristic
85C23×#95C2312yes—
86C23×#95C2316yes—
87C23×#95C2316yes—
88C23×#95C2316yes—
89C23×#95C23112yes—
90C23×#95C23112yes—
91C2×Q8×C23—26no—
92C2×Q8×C23—26no—
93C2×Q8×C23C22×C4112yes—
94C2×Q8×C23C22×C414yes—
95C2×Q8×C23C2×Q814yes—
96C2×Q8×C23#4114yes—
97C22×C4×C23C2416yes—
98C22×C4×C23C2413yes—
99C22×C4×C23C2412yes—
100C22×C4×C23C2413yes—
101C22×C4×C23C2416yes—
102C22×C4×C23C2413yes—
103C22×C4×C23C2411yescharacteristic
104C22×C4×C23C2412yes—
105C22×C4×C23C2413yes—
106C22×C4×C23C2411yescharacteristic
107C22×C4×C23C2411yescharacteristic
108C4⋊C4×C23—212no—
109C4⋊C4×C23—212no—
110C4⋊C4×C23—26no—
111C4⋊C4×C23—212no—
112C4⋊C4×C23—26no—
113C4⋊C4×C23—46no—
114C4⋊C4×C23C22×C414yes—
115C4⋊C4×C23C22×C4124yes—
116C4⋊C4×C23C22×C4112yes—
117C4⋊C4×C23C22×C4124yes—
118C4⋊C4×C23C22×C4124yes—
119C4⋊C4×C23C2×Q8112yes—
120C4⋊C4×C23#4114yes—
121C42×C23—212no—
122C42×C23—26no—
123C42×C23—212no—
124C42×C23—26no—
125C42×C23—212no—
126C42×C23—24no—
127C42×C23—26no—
128C42×C23—43no—
129C42×C23—46no—
130C42×C23C22×C4124yes—
131C42×C23C22×C4112yes—
132C42×C23C2×Q8112yes—
133C42×C23#4114yes—
134C4×#557C2311yesmaximal, characteristic, direct product factor, Sylow
135C23×C23C2511yescharacteristic
136Q8×C23—224no—
137Q8×C23C4×Q8116yescomplemented
138C2×C4×C23—23no—
139C2×C4×C23—21no—
140C2×C4×C23—22no—
141C2×C4×C23—26no—
142C2×C4×C23—26no—
143C2×C4×C23—22no—
144C2×C4×C23—23no—
145C2×C4×C23—22no—
146C2×C4×C23—22no—
147C2×C4×C23—23no—
148C2×C4×C23—21no—
149C2×C4×C23—23no—
150C2×C4×C23—26no—
151C2×C4×C23—26no—
152C2×C4×C23—23no—
153C2×C4×C23—23no—
154C2×C4×C23—23no—
155C2×C4×C23—26no—
156C2×C4×C23C23×C414yes—
157C2×C4×C23C23×C416yes—
158C2×C4×C23C23×C4112yes—
159C2×C4×C23C23×C416yes—
160C2×C4×C23C23×C412yes—
161C2×C4×C23C22×Q816yes—
162C2×C4×C23C22×Q816yes—
163C2×C4×C23C22×Q812yes—
164C2×C4×C23C2×#4114yes—
165C2×C4×C23C2×#4112yes—
166C2×C4×C23C2×#4116yes—
167C2×C4×C23C2×#4112yes—
168C2×C4×C23C2×#4112yes—
169C2×C4×C23C2×#4112yes—
170C2×C4×C232−1+412yes—
171C2×#557C2×C2311yescharacteristic
172#2370C2×C2316yes—
173#2365C2×C2313yes—
174#2355C2×C2313yes—
175#2351C2×C2312yes—
176Q8×C4⋊C4C2×C2311yescharacteristic
177C4×C4⋊Q8C2×C2313yes—
178C4×#125C2×C2316yes—
179C42×Q8C2×C2313yes—
180C42×Q8C2×C2312yes—
181C42×Q8C2×C2311yescharacteristic
182C22×C23C24×C411yescharacteristic
183C22×C23C23×Q811yescharacteristic
184C22×C23C22×#4111yescharacteristic
185C22×C23C22×#4111yescharacteristic
186C22×C23C22×#4111yescharacteristic
187C22×C23C2×2−1+411yescharacteristic
188C22×C23#58511yescharacteristic
189C4×C23—212no—
190C4×C23—24no—
191C4×C23—26no—
192C4×C23—26no—
193C4×C23—22no—
194C4×C23—24no—
195C4×C23—46no—
196C4×C23—43no—
197C4×C23C2×C4×Q8112yes—
198C4×C23C4×#4114yes—
199C4×C23#55714yesdirect product factor
200#557C4×C23116yesdirect product factor
201C2×C4⋊Q8C22×C2313yes—
202C2×#125C22×C2316yes—
203C2×C4×Q8C22×C2313yes—
204C2×C4×Q8C22×C2312yes—
205C2×C4×Q8C22×C2311yescharacteristic
206C2×C4×Q8C22×C2313yes—
207C2×C4×Q8C22×C2311yescharacteristic
208C2×C4×Q8C22×C2312yes—
209C2×C4×Q8C22×C2313yes—
210C2×C4×Q8C22×C2311yescharacteristic
211#391C22×C2316yes—
212#389C22×C2316yes—
213#389C22×C2316yes—
214#389C22×C2316yes—
215#389C22×C2316yes—
216#387C22×C23112yes—
217#387C22×C23112yes—
218#384C22×C2313yes—
219#382C22×C2316yes—
220C4×C4⋊C4C22×C2316yes—
221C4×C4⋊C4C22×C2316yes—
222C4×C4⋊C4C22×C2313yes—
223C4×C4⋊C4C22×C2316yes—
224C4×C4⋊C4C22×C2313yes—
225C4×C4⋊C4C22×C2316yes—
226C4×C4⋊C4C22×C2313yes—
227C4×C4⋊C4C22×C2313yes—
228C4×C4⋊C4C22×C2316yes—
229C4×C4⋊C4C22×C2316yes—
230C4×C4⋊C4C22×C2316yes—
231C4×C4⋊C4C22×C2313yes—
232C4×C4⋊C4C22×C2311yescharacteristic
233C43C22×C2316yes—
234C43C22×C2313yes—
235C2×C23C22×C4×Q811yescharacteristic
236C2×C23C2×C4×#4111yescharacteristic
237C2×C23C4×2−1+411yescharacteristic
238C2×C23C2×#55711yescharacteristic
239C2×C23Q8×#4111yescharacteristic
240C2×C23#348011yescharacteristic
241C2×C23#348211yescharacteristic
242C22×Q8C23×C2313yes—
243C22×Q8C23×C2311yescharacteristic
244C4⋊Q8C2×C4×C23124yes—
245#125C2×C4×C23148yes—
246C4×Q8—28no—
247C4×Q8—24no—
248C4×Q8—212no—
249C4×Q8—212no—
250C4×Q8C2×C4×C23116yes—
251C4×Q8C2×C4×C23124yes—
252C4×Q8C2×C4×C2318yes—
253C4×Q8Q8×C2318yescomplemented
254C2×C4⋊C4C23×C2313yes—
255C2×C4⋊C4C23×C2316yes—
256C2×C4⋊C4C23×C2313yes—
257C2×C4⋊C4C23×C2316yes—
258C2×C4⋊C4C23×C2316yes—
259C2×C4⋊C4C23×C2316yes—
260C2×C4⋊C4C23×C2311yescharacteristic
261C2×C4⋊C4C23×C2316yes—
262C2×C4⋊C4C23×C2313yes—
263C2×C4⋊C4C23×C2316yes—
264C2×C4⋊C4C23×C2316yes—
265C2×C4⋊C4C23×C2313yes—
266C2×C4⋊C4C23×C2311yescharacteristic
267C2×C42C23×C2316yes—
268C2×C42C23×C2316yes—
269C2×C42C23×C2313yes—
270C2×C42C23×C2316yes—
271C2×C42C23×C2313yes—
272C2×C42C23×C2312yes—
273C2×C42C23×C2313yes—
274C2×C42C23×C2316yes—
275C2×C42C23×C2313yes—
276C2×C42C23×C2316yes—
277C2×C42C23×C2313yes—
278C2×C42C23×C2313yes—
279C2×C42C23×C2311yescharacteristic
280#95C23×C2312yes—
281#95C23×C2316yes—
282#95C23×C2316yes—
283#95C23×C2316yes—
284#95C23×C23112yes—
285#95C23×C23112yes—
286C23C4×#55711yescharacteristic, direct product factor, Sylow
287C2×Q8—26no—
288C2×Q8—26no—
289C2×Q8C22×C4×C23112yes—
290C2×Q8C22×C4×C2314yes—
291C2×Q8C2×Q8×C2314yes—
292C2×Q8#41×C2314yes—
293C22×C4C24×C2313yes—
294C22×C4C24×C2316yes—
295C22×C4C24×C2312yes—
296C22×C4C24×C2313yes—
297C22×C4C24×C2316yes—
298C22×C4C24×C2313yes—
299C22×C4C24×C2312yes—
300C22×C4C24×C2311yescharacteristic
301C22×C4C24×C2313yes—
302C22×C4C24×C2311yescharacteristic
303C22×C4C24×C2311yescharacteristic
304C4⋊C4—212no—
305C4⋊C4—212no—
306C4⋊C4—26no—
307C4⋊C4—212no—
308C4⋊C4—26no—
309C4⋊C4—46no—
310C4⋊C4C22×C4×C2314yes—
311C4⋊C4C22×C4×C23124yes—
312C4⋊C4C22×C4×C23112yes—
313C4⋊C4C22×C4×C23124yes—
314C4⋊C4C22×C4×C23124yes—
315C4⋊C4C2×Q8×C23112yes—
316C4⋊C4#41×C2314yes—
317C42—212no—
318C42—26no—
319C42—212no—
320C42—26no—
321C42—212no—
322C42—24no—
323C42—26no—
324C42—43no—
325C42—46no—
326C42C22×C4×C23124yes—
327C42C22×C4×C23112yes—
328C42C2×Q8×C23112yes—
329C42#41×C2314yes—
330C23C25×C2311yescharacteristic
331Q8—224no—
332Q8C4×Q8×C23116yescomplemented
333C2×C4—23no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—26no—
337C2×C4—26no—
338C2×C4—22no—
339C2×C4—23no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—23no—
343C2×C4—21no—
344C2×C4—23no—
345C2×C4—26no—
346C2×C4—26no—
347C2×C4—23no—
348C2×C4—23no—
349C2×C4—23no—
350C2×C4—26no—
351C2×C4C23×C4×C2314yes—
352C2×C4C23×C4×C2316yes—
353C2×C4C23×C4×C23112yes—
354C2×C4C23×C4×C2316yes—
355C2×C4C23×C4×C2312yes—
356C2×C4C22×Q8×C2316yes—
357C2×C4C22×Q8×C2312yes—
358C2×C4C22×Q8×C2316yes—
359C2×C4C2×#41×C2314yes—
360C2×C4C2×#41×C2312yes—
361C2×C4C2×#41×C2312yes—
362C2×C4C2×#41×C2316yes—
363C2×C4C2×#41×C2312yes—
364C2×C4C2×#41×C2312yes—
365C2×C4C23×2−1+412yes—
366C22C24×C4×C2311yescharacteristic
367C22C23×Q8×C2311yescharacteristic
368C22C22×#41×C2311yescharacteristic
369C22C22×#41×C2311yescharacteristic
370C22C22×#41×C2311yescharacteristic
371C22C2×C23×2−1+411yescharacteristic
372C22C23×#58511yescharacteristic
373C4—212no—
374C4—24no—
375C4—26no—
376C4—26no—
377C4—22no—
378C4—24no—
379C4—46no—
380C4—43no—
381C4C2×C4×Q8×C23112yes—
382C4C4×#41×C2314yes—
383C4C23×#55714yesdirect product factor
384C2C22×C4×Q8×C2311yescharacteristic
385C2C2×C4×#41×C2311yescharacteristic
386C2C4×C23×2−1+411yescharacteristic
387C2C2×C23×#55711yescharacteristic
388C2Q8×#41×C2311yescharacteristic
389C2C23×#348011yescharacteristic
390C2C23×#348211yescharacteristic
subgroupG/H
centerC22×C4×C23C24
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
socle, prime radicalC23×C23C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
12. C23×#557C4direct (N × H)
58. C4×Q8×C23Q8semidirect
134. C4×#557C23direct (N × H)
137. Q8×C23C4×Q8semidirect
199. C4×C23#557direct (N × H)
200. #557C4×C23direct (N × H)
253. C4×Q8Q8×C23semidirect
286. C23C4×#557direct (N × H)
332. Q8C4×Q8×C23semidirect
383. C4C23×#557direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5572
C23×#23702
C23×#23652
C23×#23552
C23×#23512
Q8×C4⋊C4×C232
C4×C23×C4⋊Q82
C4×C23×#1252
C42×Q8×C232
C4×#55723