Finite Group Databasebeta

← C23×#7776 #19155402

Subgroups — C23×#7776 #19155402

Order 5888.

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

normal subgroups only

This group has 446 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
1C23×#2727C211yesmaximal, characteristic
2C23×#2058C211yesmaximal, characteristic, complemented
3C23×#2043C211yesmaximal, characteristic, complemented
4C23×#2023C211yesmaximal, characteristic
5C23×#1271C211yesmaximal, characteristic, complemented
6C23×#1271C211yesmaximal, characteristic, complemented
7C23×#1270C211yesmaximal, characteristic
8C2×C23×C4⋊Q8C2211yescharacteristic
9C2×C23×#126—21no—
10C2×C23×#106—21no—
11C2×C23×#103—21no—
12C2×C23×#103—21no—
13C2×C23×#102—21no—
14C23×#399—21no—
15C23×#396—21no—
16C23×#394—21no—
17C23×#390C2211yescharacteristic
18C23×#389—21no—
19C23×#389C2211yescharacteristic
20C23×#389C2211yescharacteristic
21C23×#388—21no—
22C23×#340—21no—
23C23×#340—21no—
24C23×#336C2211yescharacteristic
25C23×#336C2211yescharacteristic
26C23×#336C2211yescharacteristic
27C22×Q8×C23—21no—
28C22×D4×C23D411yescharacteristic
29C22×C8×C23—21no—
30C22×C8×C23—21no—
31C22×C8×C23—21no—
32C23×C4⋊Q8—42no—
33C23×#126—42no—
34C23×#126—42no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23D411yescharacteristic
42C2×C4⋊C4×C23D411yescharacteristic
43C2×C4⋊C4×C23D411yescharacteristic
44C2×C4⋊C4×C23D411yescharacteristic
45C2×C4⋊C4×C23D411yescharacteristic
46C2×C22⋊C4×C23—21no—
47C2×C22⋊C4×C23—21no—
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—21no—
50C2×C22⋊C4×C23—21no—
51C2×C22⋊C4×C23—21no—
52C2×C42×C23—21no—
53C2×C42×C23C2311yescharacteristic
54C23×#106—42no—
55C23×#103—42no—
56C23×#103—42no—
57C23×#102—42no—
58C23×#95—21no—
59C23×#95—21no—
60C23×#95—21no—
61C24×C23—21no—
62C2×Q8×C23—22no—
63C2×Q8×C23—41no—
64C2×Q8×C23—41no—
65C2×Q8×C23—41no—
66C2×Q8×C23—41no—
67C2×Q8×C23—41no—
68C2×D4×C23—21no—
69C2×D4×C23—21no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—41no—
78C22×C4×C23C2×D411yescharacteristic
79C22×C4×C23C2×D411yescharacteristic
80C22×C4×C23C2×D411yescharacteristic
81C2×C8×C23—22no—
82C2×C8×C23—41no—
83C2×C8×C23—41no—
84C2×C8×C23—41no—
85C2×C8×C23—41no—
86C2×C8×C23—41no—
87C2×C8×C23—41no—
88C2×C8×C23—41no—
89C2×C8×C23—41no—
90C4⋊C4×C23—21no—
91C4⋊C4×C23—22no—
92C4⋊C4×C23—22no—
93C4⋊C4×C23—21no—
94C4⋊C4×C23—21no—
95C4⋊C4×C23—41no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—42no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23—41no—
101C4⋊C4×C23—41no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23—41no—
105C4⋊C4×C23—81no—
106C4⋊C4×C23SD1612yes—
107C22⋊C4×C23—42no—
108C22⋊C4×C23—42no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—81no—
114C22⋊C4×C23—81no—
115C42×C23—41no—
116C42×C23—42no—
117#7776C2311yesmaximal, characteristic, direct product factor, Sylow
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23#12011yescharacteristic
126Q8×C23—42no—
127Q8×C23—42no—
128Q8×C23—81no—
129Q8×C23—81no—
130D4×C23—42no—
131D4×C23—81no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—22no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—42no—
154C2×C4×C23—81no—
155C2×C4×C23—81no—
156C2×C4×C23C2×SD1611yescharacteristic
157C2×C4×C23#13511yescharacteristic
158C2×C4×C23#13511yescharacteristic
159C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C23#13711yescharacteristic
162C8×C23—42no—
163C8×C23—81no—
164C8×C23—81no—
165#2727C2×C2311yescharacteristic
166#2058C2×C2311yescharacteristic, complemented
167#2043C2×C2311yescharacteristic, complemented
168#2023C2×C2311yescharacteristic
169#1271C2×C2311yescharacteristic, complemented
170#1271C2×C2311yescharacteristic, complemented
171#1270C2×C2311yescharacteristic
172C22×C23—81no—
173C22×C23—81no—
174C22×C23—81no—
175C22×C23—81no—
176C22×C23—81no—
177C22×C23—81no—
178C22×C23—81no—
179C22×C23#44911yescharacteristic
180C22×C23#45011yescharacteristic
181C22×C23#45211yescharacteristic
182C22×C23#45411yescharacteristic
183C22×C23#45511yescharacteristic
184C22×C23#45611yescharacteristic
185C22×C23B(4,𝔽2)11yescharacteristic
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—21no—
189C4×C23—21no—
190C4×C23—21no—
191C4×C23—21no—
192C4×C23—42no—
193C4×C23—82no—
194C4×C23—81no—
195C4×C23—81no—
196C4×C23—81no—
197C4×C23—81no—
198C2×C4⋊Q8C22×C2311yescharacteristic
199C2×#126—21no—
200C2×#106—21no—
201C2×#103—21no—
202C2×#103—21no—
203C2×#102—21no—
204#399—21no—
205#396—21no—
206#394—21no—
207#390C22×C2311yescharacteristic
208#389—21no—
209#389C22×C2311yescharacteristic
210#389C22×C2311yescharacteristic
211#388—21no—
212#340—21no—
213#340—21no—
214#336C22×C2311yescharacteristic
215#336C22×C2311yescharacteristic
216#336C22×C2311yescharacteristic
217C2×C23—161no—
218C2×C23#159711yescharacteristic
219C2×C23#160411yescharacteristic
220C2×C23#160611yescharacteristic
221C2×C23#163511yescharacteristic
222C2×C23#164211yescharacteristic
223C2×C23#165211yescharacteristic
224C2×C23#166411yescharacteristic
225C22×Q8—21no—
226C22×D4D4×C2311yescharacteristic
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C4⋊Q8—42no—
231#126—42no—
232#126—42no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4D4×C2311yescharacteristic
240C2×C4⋊C4D4×C2311yescharacteristic
241C2×C4⋊C4D4×C2311yescharacteristic
242C2×C4⋊C4D4×C2311yescharacteristic
243C2×C4⋊C4D4×C2311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C42—21no—
251C2×C42C23×C2311yescharacteristic
252#106—42no—
253#103—42no—
254#103—42no—
255#102—42no—
256#95—21no—
257#95—21no—
258#95—21no—
259C23#777611yescharacteristic, direct product factor, Sylow
260C24—21no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4C2×D4×C2311yescharacteristic
278C22×C4C2×D4×C2311yescharacteristic
279C22×C4C2×D4×C2311yescharacteristic
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C4⋊C4—21no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C4⋊C4SD16×C2312yes—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—41no—
315C42—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23C23×#12011yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—81no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4C2×SD16×C2311yescharacteristic
355C2×C4C23×#13511yescharacteristic
356C2×C4C23×#13511yescharacteristic
357C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
358C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
359C2×C4C23×#13711yescharacteristic
360C8—42no—
361C8—81no—
362C8—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22C23×#44911yescharacteristic
371C22C23×#45011yescharacteristic
372C22C23×#45211yescharacteristic
373C22C23×#45411yescharacteristic
374C22C23×#45511yescharacteristic
375C22C23×#45611yescharacteristic
376C22C23×B(4,𝔽2)11yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—81no—
389C2—161no—
390C2C23×#159711yescharacteristic
391C2C23×#160411yescharacteristic
392C2C23×#160611yescharacteristic
393C2C23×#163511yescharacteristic
394C2C23×#164211yescharacteristic
395C2C23×#165211yescharacteristic
396C2C23×#166411yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2058C2semidirect
3. C23×#2043C2semidirect
5. C23×#1271C2semidirect
6. C23×#1271C2semidirect
117. #7776C23direct (N × H)
166. #2058C2×C23semidirect
167. #2043C2×C23semidirect
169. #1271C2×C23semidirect
170. #1271C2×C23semidirect
259. C23#7776direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#27272
C23×#20582
C23×#20432
C23×#20232
C23×#12712
C23×#12702
#777623