Finite Group Databasebeta

← C23×#12232 #19159858

Subgroups — C23×#12232 #19159858

Order 5888.

360 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2834C211yesmaximal, characteristic
2C23×#2062C211yesmaximal, characteristic
3C23×#1614C211yesmaximal, characteristic
4C23×#1608C211yesmaximal, characteristic
5C23×#1352C211yesmaximal, characteristic
6C23×#1344C211yesmaximal, characteristic
7C23×#1326C211yesmaximal, characteristic
8C23×#543C2211yescharacteristic
9C23×#484—21no—
10C23×#483—21no—
11C23×#401—21no—
12C23×#399C2211yescharacteristic
13C23×#398—21no—
14C23×#398—21no—
15C23×#398—21no—
16C23×#397—21no—
17C23×#397C2211yescharacteristic
18C23×#385—21no—
19C23×#385—21no—
20C23×#385C2211yescharacteristic
21C23×#342—21no—
22C23×#342C2211yescharacteristic
23C23×#328—21no—
24C23×#328—21no—
25C23×#323C2211yescharacteristic
26C23×#323C2211yescharacteristic
27C23×C4×C23D411yescharacteristic
28C23×C22⋊Q8—21no—
29C23×C22⋊Q8—21no—
30C23×C22⋊Q8—41no—
31C23×C22⋊Q8—41no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23—41no—
38C2×C4⋊C4×C23D411yescharacteristic
39C2×C4⋊C4×C23D411yescharacteristic
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23D411yescharacteristic
45C2×C22⋊C4×C23C2311yescharacteristic
46C23×#107—41no—
47C23×#106—41no—
48C23×#103—41no—
49C23×#103—41no—
50C23×C22⋊C8—21no—
51C23×C22⋊C8—21no—
52C23×#95—21no—
53C23×#95—21no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95—21no—
58C23×#95—21no—
59C23×#95—41no—
60C23×#95D411yescharacteristic
61C23×#95D411yescharacteristic
62C24×C23C2×D411yescharacteristic
63C2×Q8×C23—21no—
64C2×Q8×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—21no—
73C22×C4×C23—41no—
74C22×C4×C23—41no—
75C22×C4×C23—81no—
76C22×C4×C23C2×D411yescharacteristic
77C22×C4×C23C2×D411yescharacteristic
78C2×C8×C23—41no—
79C2×C8×C23—41no—
80C4⋊C4×C23—21no—
81C4⋊C4×C23—21no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23—41no—
84C4⋊C4×C23—41no—
85C4⋊C4×C23—41no—
86C4⋊C4×C23—81no—
87C4⋊C4×C23—81no—
88C4⋊C4×C23—81no—
89C4⋊C4×C23—81no—
90C4⋊C4×C23—81no—
91C4⋊C4×C23—82no—
92C22⋊C4×C23—22no—
93C22⋊C4×C23—21no—
94C22⋊C4×C23—21no—
95C22⋊C4×C23—41no—
96C22⋊C4×C23—41no—
97C22⋊C4×C23—41no—
98C22⋊C4×C23—41no—
99C22⋊C4×C23—41no—
100C22⋊C4×C23—41no—
101C22⋊C4×C23—81no—
102#12232C2311yesmaximal, characteristic, direct product factor, Sylow
103C23×C23—21no—
104C23×C23—21no—
105C23×C23—41no—
106C23×C23—41no—
107C23×C23#12011yescharacteristic
108C23×C23#13511yescharacteristic
109C23×C23#13711yescharacteristic
110Q8×C23—81no—
111Q8×C23—81no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—42no—
117C2×C4×C23—41no—
118C2×C4×C23—41no—
119C2×C4×C23—41no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23—41no—
123C2×C4×C23—41no—
124C2×C4×C23—41no—
125C2×C4×C23—41no—
126C2×C4×C23—41no—
127C2×C4×C23—42no—
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—81no—
139C2×C4×C23—81no—
140C2×C4×C23—81no—
141C2×C4×C23—81no—
142C2×C4×C23—81no—
143C8×C23—81no—
144C8×C23—81no—
145#2834C2×C2311yescharacteristic
146#2062C2×C2311yescharacteristic
147#1614C2×C2311yescharacteristic
148#1608C2×C2311yescharacteristic
149#1352C2×C2311yescharacteristic
150#1344C2×C2311yescharacteristic
151#1326C2×C2311yescharacteristic
152C22×C23—21no—
153C22×C23—21no—
154C22×C23—21no—
155C22×C23—21no—
156C22×C23—41no—
157C22×C23—41no—
158C22×C23—41no—
159C22×C23—41no—
160C22×C23—81no—
161C22×C23#45211yescharacteristic
162C22×C23#45611yescharacteristic
163C22×C23B(4,𝔽2)11yescharacteristic
164C4×C23—41no—
165C4×C23—41no—
166C4×C23—81no—
167C4×C23—81no—
168C4×C23—81no—
169C4×C23—82no—
170C4×C23—81no—
171C4×C23—81no—
172C4×C23—81no—
173C4×C23—81no—
174C4×C23—81no—
175#543C22×C2311yescharacteristic
176#484—21no—
177#483—21no—
178#401—21no—
179#399C22×C2311yescharacteristic
180#398—21no—
181#398—21no—
182#398—21no—
183#397—21no—
184#397C22×C2311yescharacteristic
185#385—21no—
186#385—21no—
187#385C22×C2311yescharacteristic
188#342—21no—
189#342C22×C2311yescharacteristic
190#328—21no—
191#328—21no—
192#323C22×C2311yescharacteristic
193#323C22×C2311yescharacteristic
194C2×C23—21no—
195C2×C23—21no—
196C2×C23—41no—
197C2×C23—41no—
198C2×C23#160311yescharacteristic
199C2×C23#220011yescharacteristic
200C2×C23#220111yescharacteristic
201C23×C4D4×C2311yescharacteristic
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4D4×C2311yescharacteristic
213C2×C4⋊C4D4×C2311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4D4×C2311yescharacteristic
219C2×C22⋊C4C23×C2311yescharacteristic
220#107—41no—
221#106—41no—
222#103—41no—
223#103—41no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—41no—
234#95D4×C2311yescharacteristic
235#95D4×C2311yescharacteristic
236C23#1223211yescharacteristic, direct product factor, Sylow
237C24C2×D4×C2311yescharacteristic
238C2×Q8—21no—
239C2×Q8—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—81no—
251C22×C4C2×D4×C2311yescharacteristic
252C22×C4C2×D4×C2311yescharacteristic
253C2×C8—41no—
254C2×C8—41no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4—82no—
267C22⋊C4—22no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—81no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23C23×#12011yescharacteristic
282C23C23×#13511yescharacteristic
283C23C23×#13711yescharacteristic
284Q8—81no—
285Q8—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C8—81no—
318C8—81no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22C23×#45211yescharacteristic
329C22C23×#45611yescharacteristic
330C22C23×B(4,𝔽2)11yescharacteristic
331C4—41no—
332C4—41no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—82no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2C23×#160311yescharacteristic
347C2C23×#220011yescharacteristic
348C2C23×#220111yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
102. #12232C23direct (N × H)
236. C23#12232direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28342
C23×#20622
C23×#16142
C23×#16082
C23×#13522
C23×#13442
C23×#13262
#1223223