Finite Group Databasebeta

← C23×#12760 #19160386

Subgroups — C23×#12760 #19160386

Order 5888.

876 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#559C211yesmaximal, characteristic
2C2×C23×#353C211yesmaximal, characteristic
3C2×C23×#352C212yesmaximal, complemented
4C23×#1890C212yesmaximal, complemented
5C23×#1887C211yesmaximal, characteristic
6C23×#559—22no—
7C23×#559—24no—
8C23×#559C414yescomplemented
9C2×C23×#127C2211yescharacteristic
10C2×C23×#124—22no—
11C2×C23×#123—21no—
12C2×C23×#123C2212yes—
13C23×#518C2211yescharacteristic
14C2×C4×Q8×C23—21no—
15C2×C4×D4×C23—22no—
16C23×#418—21no—
17C2×C23×#100C2211yescharacteristic
18C2×C23×#99C2212yes—
19C23×#381—22no—
20C23×#353—22no—
21C23×#352—24no—
22C23×#342—22no—
23C23×#339—21no—
24C22×Q8×C23—21no—
25C22×D4×C23—22no—
26C22×D4×C23D411yescharacteristic
27C22×D4×C23C2311yescharacteristic
28C23×C4×C23—21no—
29C23×C4×C23D412yes—
30C2×M16×C23—21no—
31C23×#127—22no—
32C23×#127C2×C412yes—
33C23×#127D412yes—
34C23×#124—28no—
35C23×#124—28no—
36C23×#123—24no—
37C23×#123—24no—
38C23×#123—22no—
39C23×#123—22no—
40C23×#123C2×C414yes—
41C23×#123D414yes—
42C4×Q8×C23—24no—
43C4×Q8×C23—24no—
44C4×D4×C23—28no—
45C4×D4×C23—28no—
46C4×D4×C23—42no—
47C23×#117—21no—
48C2×C4⋊C4×C23—21no—
49C2×C4⋊C4×C23—22no—
50C2×C22⋊C4×C23—22no—
51C2×C22⋊C4×C23—21no—
52C2×C22⋊C4×C23—22no—
53C2×C22⋊C4×C23—22no—
54C2×C22⋊C4×C23—22no—
55C2×C22⋊C4×C23D412yes—
56C2×C42×C23—22no—
57C2×C42×C23D411yescharacteristic
58C23×#102—42no—
59C23×#100—22no—
60C23×#99—22no—
61C23×#99—22no—
62C23×#95—44no—
63C24×C23—21no—
64C24×C23C2×D412yes—
65C2×Q8×C23—24no—
66C2×Q8×C23—44no—
67C2×D4×C23—24no—
68C2×D4×C23—24no—
69C2×D4×C23—21no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C2×D4×C23—48no—
73C2×D4×C23—41no—
74C2×D4×C23—41no—
75C2×D4×C23C22⋊C412yes—
76C2×D4×C23C22×C411yescharacteristic
77C2×D4×C23C2×D411yescharacteristic
78C2×D4×C23C2×D412yes—
79C22×C4×C23—21no—
80C22×C4×C23—22no—
81C22×C4×C23—22no—
82C22×C4×C23—24no—
83C22×C4×C23—22no—
84C22×C4×C23—24no—
85C22×C4×C23—42no—
86C22×C4×C23—42no—
87C22×C4×C23—42no—
88C22×C4×C23—81no—
89C22×C4×C23—81no—
90C22×C4×C23C22⋊C414yes—
91C22×C4×C23C2×D411yescharacteristic
92M16×C23—41no—
93M16×C23—41no—
94C2×C8×C23—41no—
95C4⋊C4×C23—24no—
96C4⋊C4×C23—24no—
97C4⋊C4×C23—24no—
98C4⋊C4×C23—22no—
99C22⋊C4×C23—28no—
100C22⋊C4×C23—24no—
101C22⋊C4×C23—24no—
102C22⋊C4×C23—24no—
103C22⋊C4×C23—22no—
104C22⋊C4×C23—42no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—42no—
107C22⋊C4×C23—42no—
108C22⋊C4×C23C22⋊C414yes—
109C42×C23—28no—
110C42×C23—21no—
111C42×C23—41no—
112C42×C23C22⋊C412yes—
113#12760C2311yesmaximal, characteristic, direct product factor, Sylow
114C23×C23—22no—
115C23×C23—22no—
116C23×C23—22no—
117C23×C23—44no—
118C23×C23—44no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23C2×C22⋊C412yes—
122C23×C23#12012yes—
123C23×C23#12011yescharacteristic
124Q8×C23—48no—
125D4×C23—41no—
126D4×C23—42no—
127D4×C23—41no—
128D4×C23—42no—
129D4×C23—48no—
130D4×C23—48no—
131D4×C23—81no—
132C2×C4×C23—24no—
133C2×C4×C23—22no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—24no—
137C2×C4×C23—24no—
138C2×C4×C23—44no—
139C2×C4×C23—42no—
140C2×C4×C23—42no—
141C2×C4×C23—44no—
142C2×C4×C23—42no—
143C2×C4×C23—82no—
144C2×C4×C23—84no—
145C2×C4×C23—84no—
146C2×C4×C23—82no—
147C2×C4×C23—84no—
148C2×C4×C23C2×C22⋊C411yescharacteristic
149C2×C4×C23#12011yescharacteristic
150C8×C23—81no—
151C2×#559C2×C2311yescharacteristic
152C2×#353C2×C2311yescharacteristic
153C2×#352C2×C2312yescomplemented
154#1890C2×C2312yescomplemented
155#1887C2×C2311yescharacteristic
156C22×C23—22no—
157C22×C23—22no—
158C22×C23—22no—
159C22×C23—42no—
160C22×C23—42no—
161C22×C23—44no—
162C22×C23—44no—
163C22×C23—42no—
164C22×C23—82no—
165C22×C23—82no—
166C22×C23—82no—
167C22×C23#37911yescharacteristic
168C22×C23B(4,𝔽2)11yescharacteristic
169C22×C23#45811yescharacteristic
170C4×C23—24no—
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—44no—
174C4×C23—44no—
175C4×C23—82no—
176C4×C23—162no—
177#559—22no—
178#559—24no—
179#559C4×C2314yescomplemented
180C2×#127C22×C2311yescharacteristic
181C2×#124—22no—
182C2×#123—21no—
183C2×#123C22×C2312yes—
184#518C22×C2311yescharacteristic
185C2×C4×Q8—21no—
186C2×C4×D4—22no—
187#418—21no—
188C2×#100C22×C2311yescharacteristic
189C2×#99C22×C2312yes—
190#381—22no—
191#353—22no—
192#352—24no—
193#342—22no—
194#339—21no—
195C2×C23—21no—
196C2×C23—21no—
197C2×C23—42no—
198C2×C23—42no—
199C2×C23—82no—
200C2×C23#179011yescharacteristic
201C2×C23#219812yes—
202C22×Q8—21no—
203C22×D4—22no—
204C22×D4D4×C2311yescharacteristic
205C22×D4C23×C2311yescharacteristic
206C23×C4—21no—
207C23×C4D4×C2312yes—
208C2×M16—21no—
209#127—22no—
210#127C2×C4×C2312yes—
211#127D4×C2312yes—
212#124—28no—
213#124—28no—
214#123—24no—
215#123—24no—
216#123—22no—
217#123—22no—
218#123C2×C4×C2314yes—
219#123D4×C2314yes—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—42no—
225#117—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4D4×C2312yes—
234C2×C42—22no—
235C2×C42D4×C2311yescharacteristic
236#102—42no—
237#100—22no—
238#99—22no—
239#99—22no—
240#95—44no—
241C23#1276011yescharacteristic, direct product factor, Sylow
242C24—21no—
243C24C2×D4×C2312yes—
244C2×Q8—24no—
245C2×Q8—44no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—48no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4C22⋊C4×C2312yes—
255C2×D4C22×C4×C2311yescharacteristic
256C2×D4C2×D4×C2311yescharacteristic
257C2×D4C2×D4×C2312yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4C22⋊C4×C2314yes—
270C22×C4C2×D4×C2311yescharacteristic
271M16—41no—
272M16—41no—
273C2×C8—41no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—42no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4C22⋊C4×C2314yes—
288C42—28no—
289C42—21no—
290C42—41no—
291C42C22⋊C4×C2312yes—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—44no—
296C23—44no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊C4×C2312yes—
300C23C23×#12011yescharacteristic
301C23C23×#12012yes—
302Q8—48no—
303D4—41no—
304D4—42no—
305D4—41no—
306D4—42no—
307D4—48no—
308D4—48no—
309D4—81no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—84no—
326C2×C4C2×C22⋊C4×C2311yescharacteristic
327C2×C4C23×#12011yescharacteristic
328C8—81no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—44no—
335C22—44no—
336C22—42no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22C23×#37911yescharacteristic
341C22C23×B(4,𝔽2)11yescharacteristic
342C22C23×#45811yescharacteristic
343C4—24no—
344C4—21no—
345C4—21no—
346C4—44no—
347C4—44no—
348C4—82no—
349C4—162no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2—82no—
355C2C23×#179011yescharacteristic
356C2C23×#219812yes—
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×#127C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#352C2semidirect
4. C23×#1890C2semidirect
8. C23×#559C4semidirect
113. #12760C23direct (N × H)
153. C2×#352C2×C23semidirect
154. #1890C2×C23semidirect
179. #559C4×C23semidirect
241. C23#12760direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5592
C2×C23×#3532
C2×C23×#3522
C23×#18902
C23×#18872
#1276023