Finite Group Databasebeta

← C23×#2739 #15670358

Subgroups — C23×#2739 #15670358

Order 2944.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#400C211yesmaximal, characteristic, complemented
2C23×#400C211yesmaximal, characteristic, complemented
3C23×#400C211yesmaximal, characteristic, complemented
4C23×#398C211yesmaximal, characteristic, complemented
5C23×#397C211yesmaximal, characteristic
6C23×#396C211yesmaximal, characteristic
7C23×#389C211yesmaximal, characteristic, complemented
8C23×#388C211yesmaximal, characteristic
9C23×#388C211yesmaximal, characteristic
10C23×#387C211yesmaximal, characteristic, complemented
11C23×#387C211yesmaximal, characteristic, complemented
12C23×#387C211yesmaximal, characteristic, complemented
13C23×#385C211yesmaximal, characteristic
14C23×#381C211yesmaximal, characteristic
15C23×#380C211yesmaximal, characteristic
16C23×C4×C23C2211yescharacteristic
17C2×C4⋊C4×C23C2211yescharacteristic
18C2×C4⋊C4×C23C2211yescharacteristic
19C2×C4⋊C4×C23C2211yescharacteristic
20C2×C4⋊C4×C23C2211yescharacteristic
21C2×C4⋊C4×C23C2211yescharacteristic
22C2×C4⋊C4×C23C2211yescharacteristic
23C2×C4⋊C4×C23C2211yescharacteristic
24C2×C4⋊C4×C23C2211yescharacteristic
25C2×C4⋊C4×C23C2211yescharacteristic
26C2×C4⋊C4×C23C2211yescharacteristic
27C2×C22⋊C4×C23C2211yescharacteristic
28C2×C22⋊C4×C23C2211yescharacteristic
29C2×C22⋊C4×C23C2211yescharacteristic
30C2×C22⋊C4×C23C2211yescharacteristic
31C2×C22⋊C4×C23C2211yescharacteristic
32C2×C22⋊C4×C23C2211yescharacteristic
33C2×C42×C23C2211yescharacteristic
34C2×C42×C23C2211yescharacteristic
35C23×#95C2211yescharacteristic
36C23×#95C2211yescharacteristic
37C23×#95C2211yescharacteristic
38C23×#95C2211yescharacteristic
39C23×#95C2211yescharacteristic
40C23×#95C2211yescharacteristic
41C23×#95C2211yescharacteristic
42C23×#95C2211yescharacteristic
43C23×#95C2211yescharacteristic
44C23×#95C2211yescharacteristic
45C23×#95C2211yescharacteristic
46C23×#95C2211yescharacteristic
47C23×#95C2211yescharacteristic
48C23×#95C2211yescharacteristic
49C23×#95C2211yescharacteristic
50C23×#95C2211yescharacteristic
51C24×C23C2311yescharacteristic
52C22×C4×C23—22no—
53C22×C4×C23—22no—
54C22×C4×C23—41no—
55C22×C4×C23C2311yescharacteristic
56C22×C4×C23C2311yescharacteristic
57C22×C4×C23C2311yescharacteristic
58C22×C4×C23C2311yescharacteristic
59C22×C4×C23C2311yescharacteristic
60C22×C4×C23C2311yescharacteristic
61C22×C4×C23C2311yescharacteristic
62C22×C4×C23C2311yescharacteristic
63C22×C4×C23C2311yescharacteristic
64C22×C4×C23C2311yescharacteristic
65C22×C4×C23C2311yescharacteristic
66C22×C4×C23C2311yescharacteristic
67C22×C4×C23C2311yescharacteristic
68C22×C4×C23C2311yescharacteristic
69C4⋊C4×C23—22no—
70C4⋊C4×C23—22no—
71C4⋊C4×C23—22no—
72C4⋊C4×C23—22no—
73C4⋊C4×C23—41no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—41no—
79C22⋊C4×C23—22no—
80C22⋊C4×C23—22no—
81C22⋊C4×C23—22no—
82C22⋊C4×C23—41no—
83C22⋊C4×C23—41no—
84C22⋊C4×C23—41no—
85C42×C23—41no—
86C42×C23—41no—
87C23×C23—21no—
88C23×C23—21no—
89C23×C23—21no—
90C23×C23—21no—
91C23×C23—21no—
92C23×C23—21no—
93C23×C23#4112yes—
94C23×C23C2411yescharacteristic
95C2×C4×C23—21no—
96C2×C4×C23—21no—
97C2×C4×C23—21no—
98C2×C4×C23—21no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—21no—
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
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—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—44no—
136C2×C4×C23#4112yes—
137C2×C4×C23#4112yes—
138#2739C2311yesmaximal, characteristic, direct product factor, Sylow
139C22×C23—22no—
140C22×C23—22no—
141C22×C23—22no—
142C22×C23—41no—
143C22×C23—41no—
144C22×C23—41no—
145C22×C23—41no—
146C22×C23C2×#4111yescharacteristic
147C22×C23C2×#4111yescharacteristic
148C22×C23C2×#4111yescharacteristic
149C22×C23H(5,𝔽2)11yescharacteristic
150C22×C23H(5,𝔽2)11yescharacteristic
151C22×C232−1+411yescharacteristic
152C22×C232−1+411yescharacteristic
153C4×C23—22no—
154C4×C23—22no—
155C4×C23—41no—
156C4×C23—41no—
157C4×C23—41no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23—41no—
161C4×C23—41no—
162C4×C23—41no—
163C4×C23—41no—
164C4×C23—41no—
165C4×C23—41no—
166C4×C23—41no—
167#400C2×C2311yescharacteristic, complemented
168#400C2×C2311yescharacteristic, complemented
169#400C2×C2311yescharacteristic, complemented
170#398C2×C2311yescharacteristic, complemented
171#397C2×C2311yescharacteristic
172#396C2×C2311yescharacteristic
173#389C2×C2311yescharacteristic, complemented
174#388C2×C2311yescharacteristic
175#388C2×C2311yescharacteristic
176#387C2×C2311yescharacteristic, complemented
177#387C2×C2311yescharacteristic, complemented
178#387C2×C2311yescharacteristic, complemented
179#385C2×C2311yescharacteristic
180#381C2×C2311yescharacteristic
181#380C2×C2311yescharacteristic
182C2×C23—42no—
183C2×C23#53911yescharacteristic
184C2×C23#54111yescharacteristic
185C2×C23#54211yescharacteristic
186C2×C23#55111yescharacteristic
187C2×C23#55211yescharacteristic
188C2×C23#55311yescharacteristic
189C2×C23#56211yescharacteristic
190C23×C4C22×C2311yescharacteristic
191C2×C4⋊C4C22×C2311yescharacteristic
192C2×C4⋊C4C22×C2311yescharacteristic
193C2×C4⋊C4C22×C2311yescharacteristic
194C2×C4⋊C4C22×C2311yescharacteristic
195C2×C4⋊C4C22×C2311yescharacteristic
196C2×C4⋊C4C22×C2311yescharacteristic
197C2×C4⋊C4C22×C2311yescharacteristic
198C2×C4⋊C4C22×C2311yescharacteristic
199C2×C4⋊C4C22×C2311yescharacteristic
200C2×C4⋊C4C22×C2311yescharacteristic
201C2×C22⋊C4C22×C2311yescharacteristic
202C2×C22⋊C4C22×C2311yescharacteristic
203C2×C22⋊C4C22×C2311yescharacteristic
204C2×C22⋊C4C22×C2311yescharacteristic
205C2×C22⋊C4C22×C2311yescharacteristic
206C2×C22⋊C4C22×C2311yescharacteristic
207C2×C42C22×C2311yescharacteristic
208C2×C42C22×C2311yescharacteristic
209#95C22×C2311yescharacteristic
210#95C22×C2311yescharacteristic
211#95C22×C2311yescharacteristic
212#95C22×C2311yescharacteristic
213#95C22×C2311yescharacteristic
214#95C22×C2311yescharacteristic
215#95C22×C2311yescharacteristic
216#95C22×C2311yescharacteristic
217#95C22×C2311yescharacteristic
218#95C22×C2311yescharacteristic
219#95C22×C2311yescharacteristic
220#95C22×C2311yescharacteristic
221#95C22×C2311yescharacteristic
222#95C22×C2311yescharacteristic
223#95C22×C2311yescharacteristic
224#95C22×C2311yescharacteristic
225C23#273911yescharacteristic, direct product factor, Sylow
226C24C23×C2311yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4C23×C2311yescharacteristic
231C22×C4C23×C2311yescharacteristic
232C22×C4C23×C2311yescharacteristic
233C22×C4C23×C2311yescharacteristic
234C22×C4C23×C2311yescharacteristic
235C22×C4C23×C2311yescharacteristic
236C22×C4C23×C2311yescharacteristic
237C22×C4C23×C2311yescharacteristic
238C22×C4C23×C2311yescharacteristic
239C22×C4C23×C2311yescharacteristic
240C22×C4C23×C2311yescharacteristic
241C22×C4C23×C2311yescharacteristic
242C22×C4C23×C2311yescharacteristic
243C22×C4C23×C2311yescharacteristic
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C42—41no—
261C42—41no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#41×C2312yes—
269C23C24×C2311yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—44no—
311C2×C4#41×C2312yes—
312C2×C4#41×C2312yes—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22C2×#41×C2311yescharacteristic
321C22C2×#41×C2311yescharacteristic
322C22C2×#41×C2311yescharacteristic
323C22C23×H(5,𝔽2)11yescharacteristic
324C22C23×H(5,𝔽2)11yescharacteristic
325C22C23×2−1+411yescharacteristic
326C22C23×2−1+411yescharacteristic
327C4—22no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C2—42no—
342C2C23×#53911yescharacteristic
343C2C23×#54111yescharacteristic
344C2C23×#54211yescharacteristic
345C2C23×#55111yescharacteristic
346C2C23×#55211yescharacteristic
347C2C23×#55311yescharacteristic
348C2C23×#56211yescharacteristic
subgroupG/H
center, socleC23×C23C24
commutator, Frattini, cosocleC23C24×C23
prime radicalC24×C23C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#400C2semidirect
2. C23×#400C2semidirect
3. C23×#400C2semidirect
4. C23×#398C2semidirect
7. C23×#389C2semidirect
10. C23×#387C2semidirect
11. C23×#387C2semidirect
12. C23×#387C2semidirect
138. #2739C23direct (N × H)
167. #400C2×C23semidirect
168. #400C2×C23semidirect
169. #400C2×C23semidirect
170. #398C2×C23semidirect
173. #389C2×C23semidirect
176. #387C2×C23semidirect
177. #387C2×C23semidirect
178. #387C2×C23semidirect
225. C23#2739direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#4002
C23×#3982
C23×#3972
C23×#3962
C23×#3892
C23×#3882
C23×#3872
C23×#3852
C23×#3812
C23×#3802
#273923