Finite Group Databasebeta

← C23×#13345 #19160971

Subgroups — C23×#13345 #19160971

Order 5888.

772 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 98 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×#458C212yesmaximal
2C23×#2724C211yesmaximal, characteristic, complemented
3C2×C23×#352C212yesmaximal, complemented
4C23×#1792C211yesmaximal, characteristic
5C23×#1393C211yesmaximal, characteristic, complemented
6C2×C23×H(5,𝔽2)C2211yescharacteristic
7C2×C23×#123—22no—
8C2×C23×#123C2212yes—
9C23×#458—24no—
10C23×#458—24no—
11C2×C23×#104—21no—
12C23×#417—21no—
13C2×C23×#100C2211yescharacteristic
14C2×C23×#99—22no—
15C2×C23×#99C2212yes—
16C23×#398—21no—
17C23×#397—22no—
18C23×#390C2211yescharacteristic
19C23×#389—21no—
20C23×#385—22no—
21C23×#352—24no—
22C23×H(5,𝔽2)—22no—
23C23×H(5,𝔽2)—21no—
24C23×H(5,𝔽2)—42no—
25C23×H(5,𝔽2)D412yes—
26C2×#41×C23—21no—
27C2×#41×C23—41no—
28C22×D4×C23—22no—
29C22×D4×C23—41no—
30C22×D4×C23C2311yescharacteristic
31C23×C4×C23D412yes—
32C2×M16×C23—21no—
33C23×#123—28no—
34C23×#123—22no—
35C23×#123—22no—
36C23×#123—22no—
37C23×#123—22no—
38C23×#123—44no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23—22no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—42no—
44C2×C4⋊C4×C23D411yescharacteristic
45C2×C22⋊C4×C23—22no—
46C2×C22⋊C4×C23—22no—
47C2×C22⋊C4×C23—22no—
48C2×C22⋊C4×C23—42no—
49C2×C22⋊C4×C23D412yes—
50C2×C42×C23D411yescharacteristic
51C23×#104—42no—
52C23×#103—41no—
53C23×#102—41no—
54C23×#100—22no—
55C23×#99—28no—
56C23×#99—22no—
57C23×#99D414yes—
58C23×#95—22no—
59C23×#95—22no—
60C23×#95—21no—
61C24×C23—22no—
62C24×C23C2×D412yes—
63#41×C23—22no—
64#41×C23—22no—
65#41×C23—42no—
66#41×C23—41no—
67#41×C23—44no—
68#41×C23—42no—
69#41×C23—41no—
70C2×Q8×C23—21no—
71C2×D4×C23—21no—
72C2×D4×C23—24no—
73C2×D4×C23—24no—
74C2×D4×C23—41no—
75C2×D4×C23—41no—
76C2×D4×C23—42no—
77C2×D4×C23—42no—
78C2×D4×C23—42no—
79C2×D4×C23—42no—
80C2×D4×C23—44no—
81C2×D4×C23—41no—
82C2×D4×C23—42no—
83C2×D4×C23C2×D411yescharacteristic
84C2×D4×C23C2×D412yes—
85C2×D4×C23#4111yescharacteristic
86C22×C4×C23—22no—
87C22×C4×C23—21no—
88C22×C4×C23—22no—
89C22×C4×C23—21no—
90C22×C4×C23—22no—
91C22×C4×C23—22no—
92C22×C4×C23—22no—
93C22×C4×C23—41no—
94C22×C4×C23—42no—
95C22×C4×C23—42no—
96C22×C4×C23—42no—
97C22×C4×C23C2×D411yescharacteristic
98M16×C23—41no—
99M16×C23—41no—
100C2×C8×C23—41no—
101C4⋊C4×C23—41no—
102C4⋊C4×C23—42no—
103C4⋊C4×C23—48no—
104C4⋊C4×C23—42no—
105C4⋊C4×C23—42no—
106C4⋊C4×C23—81no—
107C4⋊C4×C23—81no—
108C22⋊C4×C23—22no—
109C22⋊C4×C23—24no—
110C22⋊C4×C23—22no—
111C22⋊C4×C23—44no—
112C22⋊C4×C23—42no—
113C22⋊C4×C23—48no—
114C22⋊C4×C23—42no—
115C22⋊C4×C23—42no—
116C42×C23—41no—
117#13345C2311yesmaximal, characteristic, direct product factor, Sylow
118C23×C23—22no—
119C23×C23—22no—
120C23×C23—24no—
121C23×C23—42no—
122C23×C23—42no—
123C23×C23—42no—
124C23×C23—41no—
125C23×C23—82no—
126C23×C23—82no—
127C23×C23#12012yes—
128C23×C23#12011yescharacteristic
129C23×C23#12112yes—
130Q8×C23—22no—
131Q8×C23—41no—
132D4×C23—22no—
133D4×C23—42no—
134D4×C23—41no—
135D4×C23—41no—
136D4×C23—41no—
137D4×C23—44no—
138D4×C23—44no—
139D4×C23—42no—
140D4×C23—42no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—22no—
144C2×C4×C23—24no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23—41no—
148C2×C4×C23—42no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—44no—
152C2×C4×C23—44no—
153C2×C4×C23—42no—
154C2×C4×C23—42no—
155C2×C4×C23—42no—
156C2×C4×C23—42no—
157C2×C4×C23—42no—
158C2×C4×C23—42no—
159C2×C4×C23—41no—
160C2×C4×C23—42no—
161C2×C4×C23—82no—
162C2×C4×C23—82no—
163C2×C4×C23—82no—
164C2×C4×C23#12111yescharacteristic
165C2×C4×C23#12711yescharacteristic
166C8×C23—81no—
167C2×#458C2×C2312yes—
168#2724C2×C2311yescharacteristic, complemented
169C2×#352C2×C2312yescomplemented
170#1792C2×C2311yescharacteristic
171#1393C2×C2311yescharacteristic, complemented
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—22no—
175C22×C23—22no—
176C22×C23—42no—
177C22×C23—42no—
178C22×C23—42no—
179C22×C23—42no—
180C22×C23—42no—
181C22×C23—82no—
182C22×C23—84no—
183C22×C23—81no—
184C22×C23—81no—
185C22×C23—162no—
186C22×C23#39211yescharacteristic
187C22×C23B(4,𝔽2)11yescharacteristic
188C22×C23B(4,𝔽2)11yescharacteristic
189C4×C23—21no—
190C4×C23—21no—
191C4×C23—42no—
192C4×C23—42no—
193C4×C23—82no—
194C4×C23—84no—
195C4×C23—81no—
196C4×C23—82no—
197C2×H(5,𝔽2)C22×C2311yescharacteristic
198C2×#123—22no—
199C2×#123C22×C2312yes—
200#458—24no—
201#458—24no—
202C2×#104—21no—
203#417—21no—
204C2×#100C22×C2311yescharacteristic
205C2×#99—22no—
206C2×#99C22×C2312yes—
207#398—21no—
208#397—22no—
209#390C22×C2311yescharacteristic
210#389—21no—
211#385—22no—
212#352—24no—
213C2×C23—21no—
214C2×C23—21no—
215C2×C23—42no—
216C2×C23—42no—
217C2×C23—82no—
218C2×C23#202111yescharacteristic
219C2×C23#219811yescharacteristic
220C2×C23#220111yescharacteristic
221H(5,𝔽2)—22no—
222H(5,𝔽2)—21no—
223H(5,𝔽2)—42no—
224H(5,𝔽2)D4×C2312yes—
225C2×#41—21no—
226C2×#41—41no—
227C22×D4—22no—
228C22×D4—41no—
229C22×D4C23×C2311yescharacteristic
230C23×C4D4×C2312yes—
231C2×M16—21no—
232#123—28no—
233#123—22no—
234#123—22no—
235#123—22no—
236#123—22no—
237#123—44no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4D4×C2311yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4D4×C2312yes—
249C2×C42D4×C2311yescharacteristic
250#104—42no—
251#103—41no—
252#102—41no—
253#100—22no—
254#99—28no—
255#99—22no—
256#99D4×C2314yes—
257#95—22no—
258#95—22no—
259#95—21no—
260C23#1334511yescharacteristic, direct product factor, Sylow
261C24—22no—
262C24C2×D4×C2312yes—
263#41—22no—
264#41—22no—
265#41—42no—
266#41—41no—
267#41—44no—
268#41—42no—
269#41—41no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4C2×D4×C2311yescharacteristic
284C2×D4C2×D4×C2312yes—
285C2×D4#41×C2311yescharacteristic
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4C2×D4×C2311yescharacteristic
298M16—41no—
299M16—41no—
300C2×C8—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C22⋊C4—22no—
309C22⋊C4—24no—
310C22⋊C4—22no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—48no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C42—41no—
317C23—22no—
318C23—22no—
319C23—24no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—41no—
324C23—82no—
325C23—82no—
326C23C23×#12011yescharacteristic
327C23C23×#12012yes—
328C23C23×#12112yes—
329Q8—22no—
330Q8—41no—
331D4—22no—
332D4—42no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—44no—
337D4—44no—
338D4—42no—
339D4—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4C23×#12111yescharacteristic
364C2×C4C23×#12711yescharacteristic
365C8—81no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—82no—
376C22—84no—
377C22—81no—
378C22—81no—
379C22—162no—
380C22C23×#39211yescharacteristic
381C22C23×B(4,𝔽2)11yescharacteristic
382C22C23×B(4,𝔽2)11yescharacteristic
383C4—21no—
384C4—21no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—84no—
389C4—81no—
390C4—82no—
391C2—21no—
392C2—21no—
393C2—42no—
394C2—42no—
395C2—82no—
396C2C23×#202111yescharacteristic
397C2C23×#219811yescharacteristic
398C2C23×#220111yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×H(5,𝔽2)C22
LCSG ▷ C22×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
2. C23×#2724C2semidirect
3. C2×C23×#352C2semidirect
5. C23×#1393C2semidirect
117. #13345C23direct (N × H)
168. #2724C2×C23semidirect
169. C2×#352C2×C23semidirect
171. #1393C2×C23semidirect
260. C23#13345direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4582
C23×#27242
C2×C23×#3522
C23×#17922
C23×#13932
#1334523