Subgroups — C₂×C₂₃×#2231 #19181262

Order 5888.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2231	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#360	C₂	1	2	yes	maximal
7	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#131	—	2	1	no	—
9	C₂×C₂₃×#125	—	2	1	no	—
10	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
11	C₂₃×#499	—	2	2	no	—
12	C₂₃×#499	C₂²	1	4	yes	complemented
13	C₂₃×#443	—	2	2	no	—
14	C₂₃×#443	C₂²	1	4	yes	—
15	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#107	—	2	1	no	—
17	C₂×C₂₃×#106	C₂²	1	2	yes	—
18	C₂×C₂₃×#104	—	2	1	no	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#365	C₂²	1	8	yes	complemented
21	C₂₃×#364	C₂²	1	4	yes	complemented
22	C₂₃×#360	C₂²	1	8	yes	—
23	C₂×#41×C₂₃	—	2	1	no	—
24	C₂₃×#135	—	2	1	no	—
25	C₂₃×#135	—	2	1	no	—
26	C₂₃×#135	Q₈	1	2	yes	—
27	C₂₃×#135	C₂³	1	2	yes	—
28	C₂×Q₁₆×C₂₃	Q₈	1	1	yes	characteristic
29	C₂×SD₁₆×C₂₃	—	2	1	no	—
30	C₂×D₈×C₂₃	Q₈	1	1	yes	characteristic
31	C₂₃×#131	—	2	4	no	—
32	C₂₃×#131	—	4	2	no	—
33	C₂×M₁₆×C₂₃	—	2	1	no	—
34	C₂×M₁₆×C₂₃	—	4	1	no	—
35	C₂²×C₈×C₂₃	—	4	1	no	—
36	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×#125	—	2	4	no	—
39	C₂₃×#125	—	4	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
42	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
43	C₂₃×M₃₂	—	2	2	no	—
44	C₂₃×M₃₂	C₂³	1	4	yes	—
45	C₂×C₁₆×C₂₃	—	2	2	no	—
46	C₂₃×#108	—	2	1	no	—
47	C₂₃×#108	C₂³	1	2	yes	—
48	C₂₃×#107	—	2	4	no	—
49	C₂₃×#106	—	2	2	no	—
50	C₂₃×#106	C₂³	1	4	yes	—
51	C₂₃×#104	—	2	4	no	—
52	C₄×C₈×C₂₃	D₄	1	2	yes	—
53	C₄×C₈×C₂₃	C₂³	1	2	yes	—
54	#41×C₂₃	—	2	2	no	—
55	#41×C₂₃	—	2	2	no	—
56	#41×C₂₃	—	4	1	no	—
57	#41×C₂₃	—	4	1	no	—
58	C₂×Q₈×C₂₃	—	2	1	no	—
59	C₂×D₄×C₂₃	—	2	1	no	—
60	C₂×D₄×C₂₃	—	4	1	no	—
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₄×C₂₃	—	4	1	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₂₃	—	2	1	no	—
66	Q₁₆×C₂₃	C₂×Q₈	1	2	yes	—
67	SD₁₆×C₂₃	—	2	2	no	—
68	SD₁₆×C₂₃	—	2	2	no	—
69	D₈×C₂₃	—	2	1	no	—
70	D₈×C₂₃	C₂×Q₈	1	2	yes	—
71	M₁₆×C₂₃	—	2	2	no	—
72	M₁₆×C₂₃	—	4	1	no	—
73	M₁₆×C₂₃	—	4	4	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	4	no	—
76	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
77	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
78	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
79	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₂₃	—	2	4	no	—
87	C₄⋊C₄×C₂₃	—	4	8	no	—
88	C₄⋊C₄×C₂₃	—	4	2	no	—
89	C₄²×C₂₃	—	2	1	no	—
90	C₄²×C₂₃	C₂×D₄	1	2	yes	—
91	C₁₆×C₂₃	—	2	4	no	—
92	C₂×#2231	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	4	1	no	—
94	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	Q₈×C₂₃	—	2	2	no	—
96	Q₈×C₂₃	—	4	1	no	—
97	D₄×C₂₃	—	2	2	no	—
98	D₄×C₂₃	—	4	2	no	—
99	D₄×C₂₃	—	4	2	no	—
100	D₄×C₂₃	—	4	1	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	1	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	8	4	no	—
108	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₂₃	—	4	2	no	—
115	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
118	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
119	C₈×C₂₃	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
121	C₂×#499	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂×#443	C₂×C₂₃	1	1	yes	characteristic
123	#2231	C₂×C₂₃	1	8	yes	direct product factor
124	C₂×#365	C₂×C₂₃	1	2	yes	complemented
125	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂×#360	C₂×C₂₃	1	2	yes	—
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	4	2	no	—
129	C₂²×C₂₃	—	8	2	no	—
130	C₂²×C₂₃	#476	1	1	yes	characteristic
131	C₂²×C₂₃	#476	1	1	yes	characteristic
132	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₄×C₂₃	—	4	2	no	—
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	—	8	4	no	—
136	C₄×C₂₃	#476	1	1	yes	characteristic
137	C₄×C₂₃	#476	1	1	yes	characteristic
138	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
140	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×#131	—	2	1	no	—
142	C₂×#125	—	2	1	no	—
143	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
144	#499	—	2	2	no	—
145	#499	C₂²×C₂₃	1	4	yes	complemented
146	#443	—	2	2	no	—
147	#443	C₂²×C₂₃	1	4	yes	—
148	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×#107	—	2	1	no	—
150	C₂×#106	C₂²×C₂₃	1	2	yes	—
151	C₂×#104	—	2	1	no	—
152	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	#365	C₂²×C₂₃	1	8	yes	complemented
154	#364	C₂²×C₂₃	1	4	yes	complemented
155	#360	C₂²×C₂₃	1	8	yes	—
156	C₂×C₂₃	—	2	1	no	—
157	C₂×C₂₃	—	2	1	no	—
158	C₂×C₂₃	—	8	2	no	—
159	C₂×C₂₃	#2231	1	2	yes	direct product factor
160	C₂×C₂₃	C₂×#476	1	1	yes	characteristic
161	C₂×#41	—	2	1	no	—
162	#135	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	Q₈×C₂₃	1	2	yes	—
165	#135	C₂³×C₂₃	1	2	yes	—
166	C₂×Q₁₆	Q₈×C₂₃	1	1	yes	characteristic
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×D₈	Q₈×C₂₃	1	1	yes	characteristic
169	#131	—	2	4	no	—
170	#131	—	4	2	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	4	1	no	—
173	C₂²×C₈	—	4	1	no	—
174	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
176	#125	—	2	4	no	—
177	#125	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	4	2	no	—
180	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	C₂³×C₂₃	1	4	yes	—
183	C₂×C₁₆	—	2	2	no	—
184	#108	—	2	1	no	—
185	#108	C₂³×C₂₃	1	2	yes	—
186	#107	—	2	4	no	—
187	#106	—	2	2	no	—
188	#106	C₂³×C₂₃	1	4	yes	—
189	#104	—	2	4	no	—
190	C₄×C₈	D₄×C₂₃	1	2	yes	—
191	C₄×C₈	C₂³×C₂₃	1	2	yes	—
192	C₂₃	C₂×#2231	1	1	yes	characteristic, direct product factor, Sylow
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	Q₁₆	—	2	1	no	—
205	Q₁₆	C₂×Q₈×C₂₃	1	2	yes	—
206	SD₁₆	—	2	2	no	—
207	SD₁₆	—	2	2	no	—
208	D₈	—	2	1	no	—
209	D₈	C₂×Q₈×C₂₃	1	2	yes	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	4	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
216	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
217	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
220	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
221	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
222	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
223	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
224	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	8	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×D₄×C₂₃	1	2	yes	—
230	C₁₆	—	2	4	no	—
231	C₂³	—	4	1	no	—
232	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
233	Q₈	—	2	2	no	—
234	Q₈	—	4	1	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
248	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
254	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
255	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
256	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
257	C₈	C₂²×Q₈×C₂₃	1	1	yes	characteristic
258	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₂₃×#476	1	1	yes	characteristic
263	C₂²	C₂₃×#476	1	1	yes	characteristic
264	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	8	4	no	—
268	C₄	C₂₃×#476	1	1	yes	characteristic
269	C₄	C₂₃×#476	1	1	yes	characteristic
270	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
271	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	2	no	—
275	C₂	C₂₃×#2231	1	2	yes	direct product factor
276	C₂	C₂×C₂₃×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#499	C₂	semidirect
3. C₂₃×#2231	C₂	direct (N × H)
4. C₂×C₂₃×#365	C₂	semidirect
5. C₂×C₂₃×#364	C₂	semidirect
12. C₂₃×#499	C₂²	semidirect
20. C₂₃×#365	C₂²	semidirect
21. C₂₃×#364	C₂²	semidirect
92. C₂×#2231	C₂₃	direct (N × H)
121. C₂×#499	C₂×C₂₃	semidirect
123. #2231	C₂×C₂₃	direct (N × H)
124. C₂×#365	C₂×C₂₃	semidirect
125. C₂×#364	C₂×C₂₃	semidirect
145. #499	C₂²×C₂₃	semidirect
153. #365	C₂²×C₂₃	semidirect
154. #364	C₂²×C₂₃	semidirect
159. C₂×C₂₃	#2231	direct (N × H)
192. C₂₃	C₂×#2231	direct (N × H)
275. C₂	C₂₃×#2231	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#499	2
C₂×C₂₃×#443	2
C₂₃×#2231	2
C₂×C₂₃×#365	2
C₂×C₂₃×#364	2
C₂×C₂₃×#360	2
C₂×#2231	23

Subgroups — C2×C23×#2231 #19181262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2231 #19181262