Finite Group Databasebeta

← C23×#7787 #19155413

Subgroups — C23×#7787 #19155413

Order 5888.

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

normal subgroups only

This group has 430 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×#2732C211yesmaximal, characteristic
2C23×#2099C211yesmaximal, characteristic, complemented
3C23×#2072C211yesmaximal, characteristic, complemented
4C23×#2071C211yesmaximal, characteristic
5C23×#1272C211yesmaximal, characteristic, complemented
6C23×#1271C211yesmaximal, characteristic, complemented
7C23×#1270C211yesmaximal, characteristic
8C2×C23×C22⋊Q8—21no—
9C2×C23×#103—21no—
10C2×C23×#102—21no—
11C23×#399—21no—
12C23×#398—21no—
13C23×#396—21no—
14C23×#391C2211yescharacteristic
15C23×#390C2211yescharacteristic
16C23×#389C2211yescharacteristic
17C23×#386—21no—
18C23×#385—21no—
19C23×#384C2211yescharacteristic
20C23×#340—21no—
21C23×#340—21no—
22C23×#340—21no—
23C23×#340—21no—
24C23×#336C2211yescharacteristic
25C23×#336C2211yescharacteristic
26C23×#336C2211yescharacteristic
27C22×Q8×C23D411yescharacteristic
28C22×D4×C23D411yescharacteristic
29C23×C4×C23—21no—
30C22×C8×C23—21no—
31C22×C8×C23—21no—
32C22×C8×C23—21no—
33C23×C22⋊Q8—42no—
34C23×C22⋊Q8—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×C23—21no—
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×C42×C23C2311yescharacteristic
52C23×#103—42no—
53C23×#102—42no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95—21no—
58C24×C23—21no—
59C2×Q8×C23—21no—
60C2×Q8×C23—41no—
61C2×Q8×C23—41no—
62C2×Q8×C23D812yes—
63C2×D4×C23—21no—
64C2×D4×C23—41no—
65C2×D4×C23—41no—
66C2×D4×C23SD1612yes—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—42no—
73C22×C4×C23—41no—
74C22×C4×C23—81no—
75C22×C4×C23—81no—
76C22×C4×C23C2×D411yescharacteristic
77C22×C4×C23C2×D411yescharacteristic
78C22×C4×C23C2×D411yescharacteristic
79C2×C8×C23—41no—
80C2×C8×C23—41no—
81C2×C8×C23—41no—
82C2×C8×C23—41no—
83C2×C8×C23—41no—
84C2×C8×C23—41no—
85C2×C8×C23—41no—
86C2×C8×C23—41no—
87C2×C8×C23—41no—
88C4⋊C4×C23—21no—
89C4⋊C4×C23—21no—
90C4⋊C4×C23—41no—
91C4⋊C4×C23—41no—
92C4⋊C4×C23—41no—
93C4⋊C4×C23—42no—
94C4⋊C4×C23—42no—
95C4⋊C4×C23—42no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—41no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23—41no—
101C4⋊C4×C23—41no—
102C4⋊C4×C23—81no—
103C22⋊C4×C23—42no—
104C22⋊C4×C23—41no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C22⋊C4×C23—41no—
108C22⋊C4×C23—81no—
109C22⋊C4×C23—81no—
110C42×C23—41no—
111#7787C2311yesmaximal, characteristic, direct product factor, Sylow
112C23×C23—22no—
113C23×C23—41no—
114C23×C23—41no—
115C23×C23—41no—
116C23×C23—41no—
117C23×C23—41no—
118C23×C23—41no—
119C23×C23#12011yescharacteristic
120Q8×C23—42no—
121Q8×C23—81no—
122D4×C23—42no—
123D4×C23—81no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—22no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
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—84no—
146C2×C4×C23—81no—
147C2×C4×C23—81no—
148C2×C4×C23—81no—
149C2×C4×C23C2×D811yescharacteristic
150C2×C4×C23C2×SD1611yescharacteristic
151C2×C4×C23#13511yescharacteristic
152C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C23#13711yescharacteristic
154C2×C4×C23#13711yescharacteristic
155C8×C23—81no—
156C8×C23—81no—
157C8×C23—81no—
158#2732C2×C2311yescharacteristic
159#2099C2×C2311yescharacteristic, complemented
160#2072C2×C2311yescharacteristic, complemented
161#2071C2×C2311yescharacteristic
162#1272C2×C2311yescharacteristic, complemented
163#1271C2×C2311yescharacteristic, complemented
164#1270C2×C2311yescharacteristic
165C22×C23—42no—
166C22×C23—42no—
167C22×C23—42no—
168C22×C23—81no—
169C22×C23—81no—
170C22×C23—81no—
171C22×C23—81no—
172C22×C23#44711yescharacteristic
173C22×C23#44811yescharacteristic
174C22×C23#45211yescharacteristic
175C22×C23#45411yescharacteristic
176C22×C23#45511yescharacteristic
177C22×C23#45511yescharacteristic
178C22×C23#45811yescharacteristic
179C4×C23—21no—
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—21no—
183C4×C23—21no—
184C4×C23—21no—
185C4×C23—42no—
186C4×C23—81no—
187C4×C23—81no—
188C4×C23—81no—
189C4×C23—81no—
190C4×C23—161no—
191C2×C22⋊Q8—21no—
192C2×#103—21no—
193C2×#102—21no—
194#399—21no—
195#398—21no—
196#396—21no—
197#391C22×C2311yescharacteristic
198#390C22×C2311yescharacteristic
199#389C22×C2311yescharacteristic
200#386—21no—
201#385—21no—
202#384C22×C2311yescharacteristic
203#340—21no—
204#340—21no—
205#340—21no—
206#340—21no—
207#336C22×C2311yescharacteristic
208#336C22×C2311yescharacteristic
209#336C22×C2311yescharacteristic
210C2×C23—82no—
211C2×C23#160711yescharacteristic
212C2×C23#160811yescharacteristic
213C2×C23#161511yescharacteristic
214C2×C23#162711yescharacteristic
215C2×C23#164111yescharacteristic
216C2×C23#164311yescharacteristic
217C2×C23#166211yescharacteristic
218C22×Q8D4×C2311yescharacteristic
219C22×D4D4×C2311yescharacteristic
220C23×C4—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22⋊Q8—42no—
225C22⋊Q8—42no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4D4×C2311yescharacteristic
234C2×C4⋊C4D4×C2311yescharacteristic
235C2×C4⋊C4D4×C2311yescharacteristic
236C2×C4⋊C4D4×C2311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C42C23×C2311yescharacteristic
243#103—42no—
244#102—42no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249C23#778711yescharacteristic, direct product factor, Sylow
250C24—21no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8D8×C2312yes—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4SD16×C2312yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4C2×D4×C2311yescharacteristic
269C22×C4C2×D4×C2311yescharacteristic
270C22×C4C2×D4×C2311yescharacteristic
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—81no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C42—41no—
303C23—22no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C23×#12011yescharacteristic
311Q8—42no—
312Q8—81no—
313D4—42no—
314D4—81no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—84no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4C2×D8×C2311yescharacteristic
341C2×C4C2×SD16×C2311yescharacteristic
342C2×C4C23×#13511yescharacteristic
343C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
344C2×C4C23×#13711yescharacteristic
345C2×C4C23×#13711yescharacteristic
346C8—81no—
347C8—81no—
348C8—81no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22C23×#44711yescharacteristic
357C22C23×#44811yescharacteristic
358C22C23×#45211yescharacteristic
359C22C23×#45411yescharacteristic
360C22C23×#45511yescharacteristic
361C22C23×#45511yescharacteristic
362C22C23×#45811yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—161no—
375C2—82no—
376C2C23×#160711yescharacteristic
377C2C23×#160811yescharacteristic
378C2C23×#161511yescharacteristic
379C2C23×#162711yescharacteristic
380C2C23×#164111yescharacteristic
381C2C23×#164311yescharacteristic
382C2C23×#166211yescharacteristic
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×#2099C2semidirect
3. C23×#2072C2semidirect
5. C23×#1272C2semidirect
6. C23×#1271C2semidirect
111. #7787C23direct (N × H)
159. #2099C2×C23semidirect
160. #2072C2×C23semidirect
162. #1272C2×C23semidirect
163. #1271C2×C23semidirect
249. C23#7787direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#27322
C23×#20992
C23×#20722
C23×#20712
C23×#12722
C23×#12712
C23×#12702
#778723