Finite Group Databasebeta

← C23×#10182 #19157808

Subgroups — C23×#10182 #19157808

Order 5888.

508 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3209C211yesmaximal, characteristic, complemented
2C23×#3077C211yesmaximal, characteristic
3C2×C23×#476C211yesmaximal, characteristic
4C23×#1549C211yesmaximal, characteristic, complemented
5C23×#1481C211yesmaximal, characteristic, complemented
6C23×#1460C211yesmaximal, characteristic
7C23×#1301C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C2×C23×C4⋊Q8C2211yescharacteristic
10C2×C4×D4×C23C2211yescharacteristic
11C23×#494—21no—
12C23×#487—21no—
13C23×#478—22no—
14C23×#476—22no—
15C23×#476—21no—
16C23×#476—22no—
17C23×#474—21no—
18C23×#462—22no—
19C23×#451—22no—
20C23×#431C2211yescharacteristic
21C23×#431C2211yescharacteristic
22C23×#428—21no—
23C2×C23×#106—21no—
24C23×#423C2211yescharacteristic
25C2×C23×C4⋊C8C2211yescharacteristic
26C23×#418—21no—
27C2×C23×#102—21no—
28C23×#332—22no—
29C23×Q8⋊C8—22no—
30C23×#323—22no—
31C22×Q8×C23—21no—
32C22×D4×C23D411yescharacteristic
33C23×C4×C23—21no—
34C2×Q16×C23—24no—
35C2×M16×C23—21no—
36C22×C8×C23—21no—
37C23×C4⋊Q8—21no—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8—21no—
40C23×C4⋊Q8D412yes—
41C23×C4⋊Q8D411yescharacteristic
42C23×C4⋊Q8D411yescharacteristic
43C23×#125—21no—
44C23×C22⋊Q8—21no—
45C23×C22⋊Q8—21no—
46C4×Q8×C23—21no—
47C4×Q8×C23—21no—
48C4×D4×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C4×D4×C23—21no—
52C23×#117—21no—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23—21no—
55C2×C4⋊C4×C23D411yescharacteristic
56C2×C22⋊C4×C23—21no—
57C2×C42×C23C2311yescharacteristic
58C23×#107—22no—
59C23×#106—24no—
60C23×#106—22no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8D412yes—
66C23×C4⋊C8D412yes—
67C23×#103—24no—
68C23×#102—24no—
69C23×#102—22no—
70C23×#102—22no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8D412yes—
73C4×C8×C23—21no—
74C4×C8×C23—21no—
75C24×C23—21no—
76C2×Q8×C23—22no—
77C2×Q8×C23—21no—
78C2×Q8×C23—21no—
79C2×Q8×C23—41no—
80C2×Q8×C23—82no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—21no—
85C2×D4×C23—21no—
86C2×D4×C23Q1612yescomplemented
87C22×C4×C23—21no—
88C22×C4×C23—41no—
89C22×C4×C23—41no—
90C22×C4×C23—41no—
91C22×C4×C23—82no—
92C22×C4×C23C2×D411yescharacteristic
93C22×C4×C23C2×D411yescharacteristic
94C22×C4×C23#4111yescharacteristic
95Q16×C23—84no—
96M16×C23—41no—
97M16×C23—41no—
98C2×C8×C23—22no—
99C2×C8×C23—22no—
100C2×C8×C23—22no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C4⋊C4×C23—22no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—21no—
112C4⋊C4×C23—41no—
113C4⋊C4×C23—41no—
114C4⋊C4×C23—41no—
115C4⋊C4×C23—41no—
116C22⋊C4×C23—41no—
117C22⋊C4×C23—41no—
118C22⋊C4×C23—41no—
119C42×C23—41no—
120C42×C23C2×D411yescharacteristic
121C42×C23C2×D411yescharacteristic
122C42×C23C2×D411yescharacteristic
123C42×C23C2×D411yescharacteristic
124#10182C2311yesmaximal, characteristic, direct product factor, Sylow
125C23×C23—22no—
126C23×C23—41no—
127C23×C23—41no—
128C23×C23—81no—
129C23×C23—81no—
130C23×C23#12111yescharacteristic
131Q8×C23—42no—
132Q8×C23—44no—
133Q8×C23—42no—
134Q8×C23—82no—
135D4×C23—22no—
136D4×C23—22no—
137D4×C23—41no—
138D4×C23—41no—
139C2×C4×C23—22no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—41no—
144C2×C4×C23—42no—
145C2×C4×C23—82no—
146C2×C4×C23—82no—
147C2×C4×C23—82no—
148C2×C4×C23—81no—
149C2×C4×C23#12011yescharacteristic
150C2×C4×C23#12011yescharacteristic
151C2×C4×C23#12011yescharacteristic
152C2×C4×C23#12111yescharacteristic
153C2×C4×C23#12111yescharacteristic
154C2×C4×C23#12711yescharacteristic
155C2×C4×C23C2×Q1611yescharacteristic
156C2×C4×C23#13511yescharacteristic
157C2×C4×C23#13711yescharacteristic
158C2×C4×C23#13711yescharacteristic
159C8×C23—42no—
160C8×C23—42no—
161C8×C23—42no—
162C8×C23—81no—
163#3209C2×C2311yescharacteristic, complemented
164#3077C2×C2311yescharacteristic
165C2×#476C2×C2311yescharacteristic
166#1549C2×C2311yescharacteristic, complemented
167#1481C2×C2311yescharacteristic, complemented
168#1460C2×C2311yescharacteristic
169#1301C2×C2311yescharacteristic, complemented
170C22×C23—21no—
171C22×C23—21no—
172C22×C23—42no—
173C22×C23—82no—
174C22×C23—82no—
175C22×C23—81no—
176C22×C23—81no—
177C22×C23#39211yescharacteristic
178C22×C23#46711yescharacteristic
179C22×C23#47011yescharacteristic
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—21no—
183C4×C23—82no—
184C4×C23—81no—
185C4×C23—81no—
186C4×C23—81no—
187C4×C23#45111yescharacteristic
188C4×C23#45111yescharacteristic
189C4×C23#45211yescharacteristic
190C4×C23#45211yescharacteristic
191C4×C23#47112yes—
192#533C22×C2311yescharacteristic
193C2×C4⋊Q8C22×C2311yescharacteristic
194C2×C4×D4C22×C2311yescharacteristic
195#494—21no—
196#487—21no—
197#478—22no—
198#476—21no—
199#476—22no—
200#476—22no—
201#474—21no—
202#462—22no—
203#451—22no—
204#431C22×C2311yescharacteristic
205#431C22×C2311yescharacteristic
206#428—21no—
207C2×#106—21no—
208#423C22×C2311yescharacteristic
209C2×C4⋊C8C22×C2311yescharacteristic
210#418—21no—
211C2×#102—21no—
212#332—22no—
213Q8⋊C8—22no—
214#323—22no—
215C2×C23—21no—
216C2×C23—21no—
217C2×C23—82no—
218C2×C23#200111yescharacteristic
219C2×C23#200811yescharacteristic
220C2×C23#201911yescharacteristic
221C22×Q8—21no—
222C22×D4D4×C2311yescharacteristic
223C23×C4—21no—
224C2×Q16—24no—
225C2×M16—21no—
226C22×C8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8D4×C2312yes—
231C4⋊Q8D4×C2311yescharacteristic
232C4⋊Q8D4×C2311yescharacteristic
233#125—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242#117—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4D4×C2311yescharacteristic
246C2×C22⋊C4—21no—
247C2×C42C23×C2311yescharacteristic
248#107—22no—
249#106—24no—
250#106—22no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8D4×C2312yes—
256C4⋊C8D4×C2312yes—
257#103—24no—
258#102—24no—
259#102—22no—
260#102—22no—
261C22⋊C8—21no—
262C22⋊C8D4×C2312yes—
263C4×C8—21no—
264C4×C8—21no—
265C23#1018211yescharacteristic, direct product factor, Sylow
266C24—21no—
267C2×Q8—22no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—41no—
271C2×Q8—82no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4Q16×C2312yescomplemented
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—82no—
283C22×C4C2×D4×C2311yescharacteristic
284C22×C4C2×D4×C2311yescharacteristic
285C22×C4#41×C2311yescharacteristic
286Q16—84no—
287M16—41no—
288M16—41no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C42—41no—
311C42C2×D4×C2311yescharacteristic
312C42C2×D4×C2311yescharacteristic
313C42C2×D4×C2311yescharacteristic
314C42C2×D4×C2311yescharacteristic
315C23—22no—
316C23—41no—
317C23—41no—
318C23—81no—
319C23—81no—
320C23C23×#12111yescharacteristic
321Q8—42no—
322Q8—44no—
323Q8—42no—
324Q8—82no—
325D4—22no—
326D4—22no—
327D4—41no—
328D4—41no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4C23×#12011yescharacteristic
340C2×C4C23×#12011yescharacteristic
341C2×C4C23×#12011yescharacteristic
342C2×C4C23×#12111yescharacteristic
343C2×C4C23×#12111yescharacteristic
344C2×C4C23×#12711yescharacteristic
345C2×C4C2×Q16×C2311yescharacteristic
346C2×C4C23×#13511yescharacteristic
347C2×C4C23×#13711yescharacteristic
348C2×C4C23×#13711yescharacteristic
349C8—42no—
350C8—42no—
351C8—42no—
352C8—81no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—82no—
357C22—82no—
358C22—81no—
359C22—81no—
360C22C23×#39211yescharacteristic
361C22C23×#46711yescharacteristic
362C22C23×#47011yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4C23×#45111yescharacteristic
371C4C23×#45111yescharacteristic
372C4C23×#45211yescharacteristic
373C4C23×#45211yescharacteristic
374C4C23×#47112yes—
375C2—21no—
376C2—21no—
377C2—82no—
378C2C23×#200111yescharacteristic
379C2C23×#200811yescharacteristic
380C2C23×#201911yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
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
1. C23×#3209C2semidirect
4. C23×#1549C2semidirect
5. C23×#1481C2semidirect
7. C23×#1301C2semidirect
86. C2×D4×C23Q16semidirect
124. #10182C23direct (N × H)
163. #3209C2×C23semidirect
166. #1549C2×C23semidirect
167. #1481C2×C23semidirect
169. #1301C2×C23semidirect
265. C23#10182direct (N × H)
277. C2×D4Q16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32092
C23×#30772
C2×C23×#4762
C23×#15492
C23×#14812
C23×#14602
C23×#13012
#1018223