Finite Group Databasebeta

← C23×#7795 #19155421

Subgroups — C23×#7795 #19155421

Order 5888.

500 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2682C211yesmaximal, characteristic
2C23×#2099C211yesmaximal, characteristic, complemented
3C23×#2071C211yesmaximal, characteristic, complemented
4C23×#2071C211yesmaximal, characteristic, complemented
5C23×#1271C211yesmaximal, characteristic, complemented
6C23×#1270C211yesmaximal, characteristic, complemented
7C23×#1270C211yesmaximal, characteristic, complemented
8C2×C23×#123—21no—
9C2×C23×#121—21no—
10C2×C23×#102—21no—
11C2×C23×#102—21no—
12C23×#396—21no—
13C23×#394—21no—
14C23×#392—21no—
15C23×#390C2211yescharacteristic
16C23×#390C2211yescharacteristic
17C23×#389C2211yescharacteristic
18C23×#385—21no—
19C23×#384C2211yescharacteristic
20C23×#340—21no—
21C23×#340—21no—
22C23×#340—21no—
23C23×#340—21no—
24C23×#336C2211yescharacteristic
25C23×#336C2211yescharacteristic
26C23×#336C2211yescharacteristic
27C22×D4×C23—21no—
28C22×D4×C23—21no—
29C22×D4×C23D411yescharacteristic
30C22×D4×C23D411yescharacteristic
31C23×C4×C23—21no—
32C22×C8×C23—21no—
33C22×C8×C23—21no—
34C22×C8×C23—21no—
35C23×#123—42no—
36C23×#123—42no—
37C23×#121—42no—
38C23×#121—42no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—21no—
44C2×C4⋊C4×C23D411yescharacteristic
45C2×C4⋊C4×C23D411yescharacteristic
46C2×C4⋊C4×C23D411yescharacteristic
47C2×C4⋊C4×C23D411yescharacteristic
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—21no—
50C2×C22⋊C4×C23—21no—
51C2×C22⋊C4×C23—21no—
52C2×C22⋊C4×C23—21no—
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—21no—
55C2×C42×C23C2311yescharacteristic
56C23×#102—42no—
57C23×#102—42no—
58C23×#95—21no—
59C23×#95—21no—
60C24×C23—21no—
61C24×C23—21no—
62C2×D4×C23—21no—
63C2×D4×C23—21no—
64C2×D4×C23—21no—
65C2×D4×C23—41no—
66C2×D4×C23—41no—
67C2×D4×C23—41no—
68C2×D4×C23—41no—
69C2×D4×C23—42no—
70C2×D4×C23—42no—
71C2×D4×C23—42no—
72C2×D4×C23—81no—
73C2×D4×C23—81no—
74C2×D4×C23—81no—
75C2×D4×C23D812yes—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—42no—
81C22×C4×C23—42no—
82C22×C4×C23—41no—
83C22×C4×C23—81no—
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23C2×D411yescharacteristic
86C22×C4×C23C2×D411yescharacteristic
87C2×C8×C23—41no—
88C2×C8×C23—41no—
89C2×C8×C23—41no—
90C2×C8×C23—41no—
91C2×C8×C23—41no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C4⋊C4×C23—21no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—41no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23—42no—
101C4⋊C4×C23—42no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23—41no—
105C4⋊C4×C23—41no—
106C4⋊C4×C23—41no—
107C4⋊C4×C23—41no—
108C4⋊C4×C23SD1612yes—
109C22⋊C4×C23—42no—
110C22⋊C4×C23—42no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C22⋊C4×C23—41no—
115C22⋊C4×C23—41no—
116C22⋊C4×C23—41no—
117C22⋊C4×C23—41no—
118C22⋊C4×C23—41no—
119C22⋊C4×C23—81no—
120C42×C23—41no—
121#7795C2311yesmaximal, characteristic, direct product factor, Sylow
122C23×C23—22no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23—41no—
128C23×C23—41no—
129C23×C23—41no—
130C23×C23—41no—
131C23×C23—41no—
132C23×C23—41no—
133C23×C23—41no—
134C23×C23—41no—
135C23×C23—41no—
136C23×C23#12011yescharacteristic
137D4×C23—42no—
138D4×C23—42no—
139D4×C23—81no—
140D4×C23—81no—
141D4×C23—84no—
142D4×C23—84no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—22no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—84no—
162C2×C4×C23—81no—
163C2×C4×C23—81no—
164C2×C4×C23—81no—
165C2×C4×C23C2×D811yescharacteristic
166C2×C4×C23C2×SD1611yescharacteristic
167C2×C4×C23#13511yescharacteristic
168C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
171C8×C23—81no—
172C8×C23—81no—
173C8×C23—81no—
174#2682C2×C2311yescharacteristic
175#2099C2×C2311yescharacteristic, complemented
176#2071C2×C2311yescharacteristic, complemented
177#2071C2×C2311yescharacteristic, complemented
178#1271C2×C2311yescharacteristic, complemented
179#1270C2×C2311yescharacteristic, complemented
180#1270C2×C2311yescharacteristic, complemented
181C22×C23—42no—
182C22×C23—42no—
183C22×C23—42no—
184C22×C23—81no—
185C22×C23—81no—
186C22×C23—81no—
187C22×C23—81no—
188C22×C23—81no—
189C22×C23—81no—
190C22×C23—81no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23—81no—
194C22×C23—81no—
195C22×C23#44711yescharacteristic
196C22×C23#44911yescharacteristic
197C22×C23#45011yescharacteristic
198C22×C23#45411yescharacteristic
199C22×C23#45511yescharacteristic
200C22×C23#45511yescharacteristic
201C22×C23#45811yescharacteristic
202C4×C23—21no—
203C4×C23—21no—
204C4×C23—21no—
205C4×C23—21no—
206C4×C23—21no—
207C4×C23—21no—
208C4×C23—42no—
209C4×C23—81no—
210C4×C23—81no—
211C4×C23—81no—
212C4×C23—161no—
213C2×#123—21no—
214C2×#121—21no—
215C2×#102—21no—
216C2×#102—21no—
217#396—21no—
218#394—21no—
219#392—21no—
220#390C22×C2311yescharacteristic
221#390C22×C2311yescharacteristic
222#389C22×C2311yescharacteristic
223#385—21no—
224#384C22×C2311yescharacteristic
225#340—21no—
226#340—21no—
227#340—21no—
228#340—21no—
229#336C22×C2311yescharacteristic
230#336C22×C2311yescharacteristic
231#336C22×C2311yescharacteristic
232C2×C23—82no—
233C2×C23—161no—
234C2×C23#160711yescharacteristic
235C2×C23#161211yescharacteristic
236C2×C23#161611yescharacteristic
237C2×C23#162511yescharacteristic
238C2×C23#163911yescharacteristic
239C2×C23#164211yescharacteristic
240C2×C23#166211yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4D4×C2311yescharacteristic
244C22×D4D4×C2311yescharacteristic
245C23×C4—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249#123—42no—
250#123—42no—
251#121—42no—
252#121—42no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4D4×C2311yescharacteristic
259C2×C4⋊C4D4×C2311yescharacteristic
260C2×C4⋊C4D4×C2311yescharacteristic
261C2×C4⋊C4D4×C2311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42C23×C2311yescharacteristic
270#102—42no—
271#102—42no—
272#95—21no—
273#95—21no—
274C23#779511yescharacteristic, direct product factor, Sylow
275C24—21no—
276C24—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—81no—
288C2×D4—81no—
289C2×D4—81no—
290C2×D4D8×C2312yes—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—81no—
299C22×C4C2×D4×C2311yescharacteristic
300C22×C4C2×D4×C2311yescharacteristic
301C22×C4C2×D4×C2311yescharacteristic
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4SD16×C2312yes—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—81no—
335C42—41no—
336C23—22no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23C23×#12011yescharacteristic
351D4—42no—
352D4—42no—
353D4—81no—
354D4—81no—
355D4—84no—
356D4—84no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—84no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4C2×D8×C2311yescharacteristic
380C2×C4C2×SD16×C2311yescharacteristic
381C2×C4C23×#13511yescharacteristic
382C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
384C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
385C8—81no—
386C8—81no—
387C8—81no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22C23×#44711yescharacteristic
403C22C23×#44911yescharacteristic
404C22C23×#45011yescharacteristic
405C22C23×#45411yescharacteristic
406C22C23×#45511yescharacteristic
407C22C23×#45511yescharacteristic
408C22C23×#45811yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—161no—
420C2—82no—
421C2—161no—
422C2C23×#160711yescharacteristic
423C2C23×#161211yescharacteristic
424C2C23×#161611yescharacteristic
425C2C23×#162511yescharacteristic
426C2C23×#163911yescharacteristic
427C2C23×#164211yescharacteristic
428C2C23×#166211yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#2682C2
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×#2099C2semidirect
3. C23×#2071C2semidirect
4. C23×#2071C2semidirect
5. C23×#1271C2semidirect
6. C23×#1270C2semidirect
7. C23×#1270C2semidirect
121. #7795C23direct (N × H)
175. #2099C2×C23semidirect
176. #2071C2×C23semidirect
177. #2071C2×C23semidirect
178. #1271C2×C23semidirect
179. #1270C2×C23semidirect
180. #1270C2×C23semidirect
274. C23#7795direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26822
C23×#20992
C23×#20712
C23×#12712
C23×#12702
#779523