Finite Group Databasebeta

← C23×#7788 #19155414

Subgroups — C23×#7788 #19155414

Order 5888.

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

normal subgroups only

This group has 418 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×#2724C211yesmaximal, characteristic
2C23×#2099C211yesmaximal, characteristic, complemented
3C23×#2083C211yesmaximal, characteristic, complemented
4C23×#2071C211yesmaximal, characteristic
5C23×#1271C211yesmaximal, characteristic, complemented
6C23×#1271C211yesmaximal, characteristic, complemented
7C23×#1270C211yesmaximal, characteristic
8C2×C23×#123—21no—
9C2×C23×#107—21no—
10C2×C23×#102—21no—
11C23×#398—21no—
12C23×#397—21no—
13C23×#397—21no—
14C23×#390C2211yescharacteristic
15C23×#389C2211yescharacteristic
16C23×#389C2211yescharacteristic
17C23×#385—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×D4×C23D411yescharacteristic
28C23×C4×C23—21no—
29C22×C8×C23—21no—
30C22×C8×C23—21no—
31C22×C8×C23—21no—
32C23×#123—42no—
33C23×#123—42no—
34C2×C4⋊C4×C23—21no—
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×C23D411yescharacteristic
41C2×C4⋊C4×C23D411yescharacteristic
42C2×C4⋊C4×C23D411yescharacteristic
43C2×C4⋊C4×C23D411yescharacteristic
44C2×C4⋊C4×C23D411yescharacteristic
45C2×C22⋊C4×C23—21no—
46C2×C22⋊C4×C23—21no—
47C2×C22⋊C4×C23—21no—
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—21no—
50C2×C42×C23C2311yescharacteristic
51C23×#107—42no—
52C23×#102—42no—
53C23×#95—21no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95—21no—
58C24×C23—21no—
59C2×D4×C23—21no—
60C2×D4×C23—41no—
61C2×D4×C23—41no—
62C2×D4×C23D812yes—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—42no—
69C22×C4×C23—41no—
70C22×C4×C23—81no—
71C22×C4×C23—81no—
72C22×C4×C23C2×D411yescharacteristic
73C22×C4×C23C2×D411yescharacteristic
74C22×C4×C23C2×D411yescharacteristic
75C2×C8×C23—22no—
76C2×C8×C23—41no—
77C2×C8×C23—41no—
78C2×C8×C23—41no—
79C2×C8×C23—41no—
80C2×C8×C23—41no—
81C2×C8×C23—41no—
82C2×C8×C23—41no—
83C2×C8×C23—41no—
84C4⋊C4×C23—21no—
85C4⋊C4×C23—21no—
86C4⋊C4×C23—21no—
87C4⋊C4×C23—21no—
88C4⋊C4×C23—41no—
89C4⋊C4×C23—41no—
90C4⋊C4×C23—41no—
91C4⋊C4×C23—42no—
92C4⋊C4×C23—41no—
93C4⋊C4×C23—41no—
94C4⋊C4×C23—41no—
95C4⋊C4×C23—41no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—41no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23—81no—
101C22⋊C4×C23—42no—
102C22⋊C4×C23—42no—
103C22⋊C4×C23—42no—
104C22⋊C4×C23—41no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C22⋊C4×C23—41no—
108C42×C23—41no—
109#7788C2311yesmaximal, characteristic, direct product factor, Sylow
110C23×C23—22no—
111C23×C23—41no—
112C23×C23—41no—
113C23×C23—41no—
114C23×C23—41no—
115C23×C23—41no—
116C23×C23—41no—
117C23×C23#12011yescharacteristic
118D4×C23—42no—
119D4×C23—81no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—41no—
127C2×C4×C23—41no—
128C2×C4×C23—41no—
129C2×C4×C23—41no—
130C2×C4×C23—41no—
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—84no—
142C2×C4×C23—81no—
143C2×C4×C23—81no—
144C2×C4×C23—81no—
145C2×C4×C23C2×D811yescharacteristic
146C2×C4×C23#13511yescharacteristic
147C2×C4×C23#13511yescharacteristic
148C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C23#13711yescharacteristic
151C8×C23—42no—
152C8×C23—81no—
153C8×C23—81no—
154#2724C2×C2311yescharacteristic
155#2099C2×C2311yescharacteristic, complemented
156#2083C2×C2311yescharacteristic, complemented
157#2071C2×C2311yescharacteristic
158#1271C2×C2311yescharacteristic, complemented
159#1271C2×C2311yescharacteristic, complemented
160#1270C2×C2311yescharacteristic
161C22×C23—42no—
162C22×C23—42no—
163C22×C23—42no—
164C22×C23—81no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23#44711yescharacteristic
169C22×C23#44911yescharacteristic
170C22×C23#45211yescharacteristic
171C22×C23#45411yescharacteristic
172C22×C23#45511yescharacteristic
173C22×C23#45611yescharacteristic
174C22×C23#45811yescharacteristic
175C4×C23—21no—
176C4×C23—21no—
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—21no—
181C4×C23—81no—
182C4×C23—81no—
183C4×C23—81no—
184C4×C23—81no—
185C4×C23—81no—
186C4×C23—161no—
187C2×#123—21no—
188C2×#107—21no—
189C2×#102—21no—
190#398—21no—
191#397—21no—
192#397—21no—
193#390C22×C2311yescharacteristic
194#389C22×C2311yescharacteristic
195#389C22×C2311yescharacteristic
196#385—21no—
197#385—21no—
198#384C22×C2311yescharacteristic
199#340—21no—
200#340—21no—
201#340—21no—
202#340—21no—
203#336C22×C2311yescharacteristic
204#336C22×C2311yescharacteristic
205#336C22×C2311yescharacteristic
206C2×C23—82no—
207C2×C23#160711yescharacteristic
208C2×C23#160911yescharacteristic
209C2×C23#161511yescharacteristic
210C2×C23#162511yescharacteristic
211C2×C23#164011yescharacteristic
212C2×C23#164711yescharacteristic
213C2×C23#166411yescharacteristic
214C22×D4D4×C2311yescharacteristic
215C23×C4—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219#123—42no—
220#123—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4D4×C2311yescharacteristic
228C2×C4⋊C4D4×C2311yescharacteristic
229C2×C4⋊C4D4×C2311yescharacteristic
230C2×C4⋊C4D4×C2311yescharacteristic
231C2×C4⋊C4D4×C2311yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42C23×C2311yescharacteristic
238#107—42no—
239#102—42no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245C23#778811yescharacteristic, direct product factor, Sylow
246C24—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4D8×C2312yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4C2×D4×C2311yescharacteristic
261C22×C4C2×D4×C2311yescharacteristic
262C22×C4C2×D4×C2311yescharacteristic
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—81no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C42—41no—
297C23—22no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23C23×#12011yescharacteristic
305D4—42no—
306D4—81no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—84no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4C2×D8×C2311yescharacteristic
333C2×C4C23×#13511yescharacteristic
334C2×C4C23×#13511yescharacteristic
335C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
336C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
337C2×C4C23×#13711yescharacteristic
338C8—42no—
339C8—81no—
340C8—81no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22C23×#44711yescharacteristic
349C22C23×#44911yescharacteristic
350C22C23×#45211yescharacteristic
351C22C23×#45411yescharacteristic
352C22C23×#45511yescharacteristic
353C22C23×#45611yescharacteristic
354C22C23×#45811yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—161no—
367C2—82no—
368C2C23×#160711yescharacteristic
369C2C23×#160911yescharacteristic
370C2C23×#161511yescharacteristic
371C2C23×#162511yescharacteristic
372C2C23×#164011yescharacteristic
373C2C23×#164711yescharacteristic
374C2C23×#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×#2099C2semidirect
3. C23×#2083C2semidirect
5. C23×#1271C2semidirect
6. C23×#1271C2semidirect
109. #7788C23direct (N × H)
155. #2099C2×C23semidirect
156. #2083C2×C23semidirect
158. #1271C2×C23semidirect
159. #1271C2×C23semidirect
245. C23#7788direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#27242
C23×#20992
C23×#20832
C23×#20712
C23×#12712
C23×#12702
#778823