Subgroups — C₂×C₂₃×#3261 #19208538

Order 5888.

1754 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#541	C₂	1	2	yes	maximal
2	C₂₃×#3261	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#479	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#470	C₂	1	2	yes	maximal
5	C₂×C₂₃×#467	C₂	1	2	yes	maximal
6	C₂×C₂₃×#462	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#541	—	2	8	no	—
13	C₂₃×#541	C₂²	1	16	yes	—
14	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#126	—	2	4	no	—
16	C₂×C₂₃×#125	—	2	4	no	—
17	C₂×C₂₃×#125	C₂²	1	2	yes	—
18	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂₃×#479	C₂²	1	32	yes	complemented
22	C₂₃×#470	C₂²	1	16	yes	—
23	C₂₃×#467	C₂²	1	16	yes	—
24	C₂₃×#462	C₂²	1	32	yes	complemented
25	C₂×C₂₃×#107	C₂²	1	2	yes	—
26	C₂×C₂₃×#106	C₂²	1	2	yes	—
27	C₂₃×#424	C₂²	1	8	yes	—
28	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
29	C₂×C₂₃×#103	C₂²	1	4	yes	—
30	C₂×C₂₃×#103	C₂²	1	4	yes	—
31	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×Q₁₆×C₂₃	—	4	4	no	—
34	C₂×Q₁₆×C₂₃	—	4	4	no	—
35	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
36	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
37	C₂×M₁₆×C₂₃	—	4	2	no	—
38	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
39	C₂²×C₈×C₂₃	—	4	2	no	—
40	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
43	C₂₃×C₄⋊Q₈	—	2	4	no	—
44	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
45	C₂₃×#126	—	2	32	no	—
46	C₂₃×#125	—	2	32	no	—
47	C₂₃×#125	—	2	4	no	—
48	C₂₃×#125	C₂³	1	8	yes	—
49	C₂₃×C₂²⋊Q₈	—	2	8	no	—
50	C₂₃×C₂²⋊Q₈	C₂³	1	16	yes	—
51	C₄×Q₈×C₂₃	—	2	8	no	—
52	C₄×Q₈×C₂₃	C₂³	1	16	yes	—
53	C₂₃×#117	D₄	1	4	yes	—
54	C₂₃×#117	C₂³	1	4	yes	—
55	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
56	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
62	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
63	C₂×C₄²×C₂₃	—	2	4	no	—
64	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
65	C₂₃×#107	C₂³	1	8	yes	—
66	C₂₃×#106	C₂³	1	8	yes	—
67	C₂₃×C₄⋊C₈	C₂³	1	16	yes	—
68	C₂₃×#103	C₂³	1	16	yes	—
69	C₂₃×#103	C₂³	1	16	yes	—
70	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₂₃	—	2	4	no	—
72	C₂×Q₈×C₂₃	—	2	8	no	—
73	C₂×Q₈×C₂₃	—	2	8	no	—
74	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
75	C₂²×C₄×C₂₃	—	2	2	no	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	—	2	4	no	—
78	C₂²×C₄×C₂₃	—	2	4	no	—
79	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₂₃	—	4	8	no	—
85	Q₁₆×C₂₃	—	4	8	no	—
86	M₁₆×C₂₃	—	4	4	no	—
87	C₂×C₈×C₂₃	—	2	4	no	—
88	C₂×C₈×C₂₃	—	2	4	no	—
89	C₂×C₈×C₂₃	—	4	4	no	—
90	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
91	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
92	C₄⋊C₄×C₂₃	—	2	16	no	—
93	C₄⋊C₄×C₂₃	—	2	16	no	—
94	C₄⋊C₄×C₂₃	—	2	16	no	—
95	C₄⋊C₄×C₂₃	—	2	16	no	—
96	C₄⋊C₄×C₂₃	—	2	4	no	—
97	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
98	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
99	C₂²⋊C₄×C₂₃	—	2	16	no	—
100	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
101	C₄²×C₂₃	—	2	16	no	—
102	C₄²×C₂₃	C₂×D₄	1	4	yes	—
103	C₄²×C₂₃	C₂⁴	1	4	yes	—
104	C₂×#3261	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	—	2	2	no	—
106	C₂³×C₂₃	—	2	4	no	—
107	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
108	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₂₃	—	2	8	no	—
110	Q₈×C₂₃	—	2	8	no	—
111	Q₈×C₂₃	—	4	8	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	8	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	8	no	—
118	C₂×C₄×C₂₃	—	2	8	no	—
119	C₂×C₄×C₂₃	—	4	8	no	—
120	C₂×C₄×C₂₃	—	4	8	no	—
121	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
122	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
125	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
126	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
127	C₈×C₂₃	—	2	4	no	—
128	C₈×C₂₃	—	2	4	no	—
129	C₂×#541	C₂×C₂₃	1	2	yes	—
130	#3261	C₂×C₂₃	1	16	yes	direct product factor
131	C₂×#479	C₂×C₂₃	1	4	yes	complemented
132	C₂×#470	C₂×C₂₃	1	2	yes	—
133	C₂×#467	C₂×C₂₃	1	2	yes	—
134	C₂×#462	C₂×C₂₃	1	4	yes	complemented
135	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
136	C₂²×C₂₃	—	2	2	no	—
137	C₂²×C₂₃	—	2	4	no	—
138	C₂²×C₂₃	—	4	4	no	—
139	C₂²×C₂₃	#537	1	2	yes	—
140	C₂²×C₂₃	#578	1	4	yes	—
141	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	8	no	—
144	C₄×C₂₃	—	4	8	no	—
145	C₄×C₂₃	—	4	8	no	—
146	C₄×C₂₃	#537	1	2	yes	—
147	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
149	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
150	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
151	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
152	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	#541	—	2	8	no	—
154	#541	C₂²×C₂₃	1	16	yes	—
155	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
156	C₂×#126	—	2	4	no	—
157	C₂×#125	—	2	4	no	—
158	C₂×#125	C₂²×C₂₃	1	2	yes	—
159	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
160	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
161	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
162	#479	C₂²×C₂₃	1	32	yes	complemented
163	#470	C₂²×C₂₃	1	16	yes	—
164	#467	C₂²×C₂₃	1	16	yes	—
165	#462	C₂²×C₂₃	1	32	yes	complemented
166	C₂×#107	C₂²×C₂₃	1	2	yes	—
167	C₂×#106	C₂²×C₂₃	1	2	yes	—
168	#424	C₂²×C₂₃	1	8	yes	—
169	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
170	C₂×#103	C₂²×C₂₃	1	4	yes	—
171	C₂×#103	C₂²×C₂₃	1	4	yes	—
172	C₂×C₂₃	—	4	2	no	—
173	C₂×C₂₃	#3261	1	4	yes	direct product factor
174	C₂×C₂₃	C₂×#537	1	1	yes	characteristic
175	C₂×C₂₃	C₂×#578	1	2	yes	—
176	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
177	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	C₂×Q₁₆	—	4	4	no	—
179	C₂×Q₁₆	—	4	4	no	—
180	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
181	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
182	C₂×M₁₆	—	4	2	no	—
183	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
184	C₂²×C₈	—	4	2	no	—
185	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
186	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
187	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
188	C₄⋊Q₈	—	2	4	no	—
189	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
190	#126	—	2	32	no	—
191	#125	—	2	32	no	—
192	#125	—	2	4	no	—
193	#125	C₂³×C₂₃	1	8	yes	—
194	C₂²⋊Q₈	—	2	8	no	—
195	C₂²⋊Q₈	C₂³×C₂₃	1	16	yes	—
196	C₄×Q₈	—	2	8	no	—
197	C₄×Q₈	C₂³×C₂₃	1	16	yes	—
198	#117	D₄×C₂₃	1	4	yes	—
199	#117	C₂³×C₂₃	1	4	yes	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
206	C₂×C₂²⋊C₄	—	2	4	no	—
207	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
210	#107	C₂³×C₂₃	1	8	yes	—
211	#106	C₂³×C₂₃	1	8	yes	—
212	C₄⋊C₈	C₂³×C₂₃	1	16	yes	—
213	#103	C₂³×C₂₃	1	16	yes	—
214	#103	C₂³×C₂₃	1	16	yes	—
215	C₂₃	C₂×#3261	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	2	8	no	—
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
230	Q₁₆	—	4	8	no	—
231	Q₁₆	—	4	8	no	—
232	M₁₆	—	4	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	4	no	—
236	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
237	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
238	C₄⋊C₄	—	2	16	no	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	2	16	no	—
241	C₄⋊C₄	—	2	16	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
244	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
245	C₂²⋊C₄	—	2	16	no	—
246	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
247	C₄²	—	2	16	no	—
248	C₄²	C₂×D₄×C₂₃	1	4	yes	—
249	C₄²	C₂⁴×C₂₃	1	4	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	2	no	—
252	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
253	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
254	Q₈	—	2	8	no	—
255	Q₈	—	2	8	no	—
256	Q₈	—	4	8	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
267	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
270	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
271	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₈	—	2	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	C₂₃×#537	1	2	yes	—
278	C₂²	C₂₃×#578	1	4	yes	—
279	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	8	no	—
282	C₄	—	4	8	no	—
283	C₄	—	4	8	no	—
284	C₄	C₂₃×#537	1	2	yes	—
285	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
286	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
287	C₂	—	4	2	no	—
288	C₂	C₂₃×#3261	1	4	yes	direct product factor
289	C₂	C₂×C₂₃×#537	1	1	yes	characteristic
290	C₂	C₂×C₂₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3261	C₂	direct (N × H)
3. C₂×C₂₃×#479	C₂	semidirect
6. C₂×C₂₃×#462	C₂	semidirect
21. C₂₃×#479	C₂²	semidirect
24. C₂₃×#462	C₂²	semidirect
104. C₂×#3261	C₂₃	direct (N × H)
130. #3261	C₂×C₂₃	direct (N × H)
131. C₂×#479	C₂×C₂₃	semidirect
134. C₂×#462	C₂×C₂₃	semidirect
162. #479	C₂²×C₂₃	semidirect
165. #462	C₂²×C₂₃	semidirect
173. C₂×C₂₃	#3261	direct (N × H)
215. C₂₃	C₂×#3261	direct (N × H)
288. C₂	C₂₃×#3261	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×#541	2
C₂₃×#3261	2
C₂×C₂₃×#479	2
C₂×C₂₃×#470	2
C₂×C₂₃×#467	2
C₂×C₂₃×#462	2
C₂×C₂₃×#424	2
C₂×#3261	23

Subgroups — C2×C23×#3261 #19208538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3261 #19208538