Subgroups — C₂×C₂₃×#1706 #19167219

Order 5888.

1046 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#492	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×#486	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1706	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×SD₁₆×C₂₃	—	2	4	no	—
8	C₂²×D₈×C₂₃	—	2	2	no	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₂₃×#127	C₂²	1	2	yes	—
11	C₂₃×#500	—	2	2	no	—
12	C₂₃×#500	C₂²	1	4	yes	complemented
13	C₂₃×#493	—	2	2	no	—
14	C₂₃×#493	C₂²	1	4	yes	—
15	C₂₃×#492	—	2	4	no	—
16	C₂₃×#492	C₂²	1	8	yes	complemented
17	C₂₃×#486	—	2	4	no	—
18	C₂₃×#486	C₂²	1	8	yes	complemented
19	C₂×C₂₃×#107	—	2	2	no	—
20	C₂×C₂₃×#102	—	2	4	no	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₈²×C₂₃	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₂₃	—	2	2	no	—
26	C₂²×D₄×C₂₃	—	2	2	no	—
27	C₂²×D₄×C₂₃	—	2	2	no	—
28	C₂²×D₄×C₂₃	—	2	2	no	—
29	C₂×SD₁₆×C₂₃	—	2	16	no	—
30	C₂×SD₁₆×C₂₃	—	4	16	no	—
31	C₂×D₈×C₂₃	—	2	8	no	—
32	C₂×D₈×C₂₃	—	4	8	no	—
33	C₂²×C₈×C₂₃	D₄	1	2	yes	—
34	C₂²×C₈×C₂₃	D₄	1	2	yes	—
35	C₂²×C₈×C₂₃	D₄	1	2	yes	—
36	C₂₃×C₄⋊Q₈	—	2	2	no	—
37	C₂₃×C₄⋊Q₈	—	2	2	no	—
38	C₂₃×C₄⋊Q₈	—	2	2	no	—
39	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₂₃×#127	—	2	2	no	—
41	C₂₃×#127	—	2	2	no	—
42	C₂₃×#127	—	2	2	no	—
43	C₂₃×#127	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×#107	—	2	8	no	—
48	C₂₃×#102	—	2	16	no	—
49	C₄×C₈×C₂₃	D₄	1	2	yes	—
50	C₄×C₈×C₂₃	D₄	1	2	yes	—
51	C₄×C₈×C₂₃	D₄	1	2	yes	—
52	C₄×C₈×C₂₃	C₂³	1	2	yes	—
53	C₄×C₈×C₂₃	C₂³	1	2	yes	—
54	C₄×C₈×C₂₃	C₂³	1	2	yes	—
55	C₂⁴×C₂₃	—	4	2	no	—
56	C₂×Q₈×C₂₃	—	2	4	no	—
57	C₂×Q₈×C₂₃	—	4	2	no	—
58	C₂×Q₈×C₂₃	—	4	8	no	—
59	C₂×D₄×C₂₃	—	2	4	no	—
60	C₂×D₄×C₂₃	—	2	4	no	—
61	C₂×D₄×C₂₃	—	2	4	no	—
62	C₂×D₄×C₂₃	—	4	2	no	—
63	C₂×D₄×C₂₃	—	4	2	no	—
64	C₂×D₄×C₂₃	—	4	2	no	—
65	C₂×D₄×C₂₃	—	4	8	no	—
66	C₂×D₄×C₂₃	—	4	8	no	—
67	C₂×D₄×C₂₃	—	4	8	no	—
68	C₂²×C₄×C₂₃	—	4	2	no	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	SD₁₆×C₂₃	—	4	32	no	—
73	D₈×C₂₃	—	4	16	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	D₈	1	8	yes	complemented
76	C₂×C₈×C₂₃	SD₁₆	1	8	yes	complemented
77	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	4	2	no	—
83	C₄⋊C₄×C₂₃	—	4	2	no	—
84	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₂×#1706	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	—	4	4	no	—
90	C₂³×C₂₃	—	8	4	no	—
91	C₂³×C₂₃	—	8	4	no	—
92	C₂³×C₂₃	—	8	4	no	—
93	C₂³×C₂₃	#127	1	1	yes	characteristic
94	Q₈×C₂₃	—	4	8	no	—
95	Q₈×C₂₃	—	8	4	no	—
96	D₄×C₂₃	—	4	8	no	—
97	D₄×C₂₃	—	4	8	no	—
98	D₄×C₂₃	—	4	8	no	—
99	D₄×C₂₃	—	8	4	no	—
100	D₄×C₂₃	—	8	4	no	—
101	D₄×C₂₃	—	8	4	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	8	4	no	—
104	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
108	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
109	C₂×C₄×C₂₃	#135	1	4	yes	—
110	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₂₃	—	2	4	no	—
114	C₈×C₂₃	C₂×D₈	1	8	yes	complemented
115	C₈×C₂₃	C₂×SD₁₆	1	8	yes	complemented
116	C₂×#500	C₂×C₂₃	1	1	yes	characteristic, complemented
117	C₂×#493	C₂×C₂₃	1	1	yes	characteristic
118	C₂×#492	C₂×C₂₃	1	2	yes	complemented
119	C₂×#486	C₂×C₂₃	1	2	yes	complemented
120	#1706	C₂×C₂₃	1	8	yes	direct product factor
121	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂²×C₂₃	—	8	4	no	—
123	C₂²×C₂₃	—	8	4	no	—
124	C₂²×C₂₃	—	8	4	no	—
125	C₂²×C₂₃	—	16	8	no	—
126	C₂²×C₂₃	#492	1	2	yes	—
127	C₂²×C₂₃	#493	1	2	yes	—
128	C₂²×C₂₃	#495	1	2	yes	—
129	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
130	C₄×C₂₃	—	8	4	no	—
131	C₄×C₂₃	#492	1	2	yes	—
132	C₄×C₂₃	#493	1	2	yes	—
133	C₄×C₂₃	#495	1	2	yes	—
134	C₄×C₂₃	C₂²×D₈	1	2	yes	—
135	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
136	C₄×C₂₃	C₂×#135	1	2	yes	—
137	C₂²×SD₁₆	—	2	4	no	—
138	C₂²×D₈	—	2	2	no	—
139	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
140	C₂×#127	C₂²×C₂₃	1	2	yes	—
141	#500	—	2	2	no	—
142	#500	C₂²×C₂₃	1	4	yes	complemented
143	#493	—	2	2	no	—
144	#493	C₂²×C₂₃	1	4	yes	—
145	#492	—	2	4	no	—
146	#492	C₂²×C₂₃	1	8	yes	complemented
147	#486	—	2	4	no	—
148	#486	C₂²×C₂₃	1	8	yes	complemented
149	C₂×#107	—	2	2	no	—
150	C₂×#102	—	2	4	no	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₈²	C₂²×C₂₃	1	4	yes	complemented
155	C₂×C₂₃	—	16	4	no	—
156	C₂×C₂₃	#1706	1	4	yes	direct product factor
157	C₂×C₂₃	C₂×#492	1	1	yes	characteristic
158	C₂×C₂₃	C₂×#493	1	1	yes	characteristic
159	C₂×C₂₃	C₂×#495	1	1	yes	characteristic
160	C₂²×Q₈	—	2	2	no	—
161	C₂²×D₄	—	2	2	no	—
162	C₂²×D₄	—	2	2	no	—
163	C₂²×D₄	—	2	2	no	—
164	C₂×SD₁₆	—	2	16	no	—
165	C₂×SD₁₆	—	4	16	no	—
166	C₂×D₈	—	2	8	no	—
167	C₂×D₈	—	4	8	no	—
168	C₂²×C₈	D₄×C₂₃	1	2	yes	—
169	C₂²×C₈	D₄×C₂₃	1	2	yes	—
170	C₂²×C₈	D₄×C₂₃	1	2	yes	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
175	#127	—	2	2	no	—
176	#127	—	2	2	no	—
177	#127	—	2	2	no	—
178	#127	C₂³×C₂₃	1	4	yes	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
182	#107	—	2	8	no	—
183	#102	—	2	16	no	—
184	C₄×C₈	D₄×C₂₃	1	2	yes	—
185	C₄×C₈	D₄×C₂₃	1	2	yes	—
186	C₄×C₈	D₄×C₂₃	1	2	yes	—
187	C₄×C₈	C₂³×C₂₃	1	2	yes	—
188	C₄×C₈	C₂³×C₂₃	1	2	yes	—
189	C₄×C₈	C₂³×C₂₃	1	2	yes	—
190	C₂₃	C₂×#1706	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	—	4	2	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	4	8	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	8	no	—
202	C₂×D₄	—	4	8	no	—
203	C₂×D₄	—	4	8	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	SD₁₆	—	4	32	no	—
209	D₈	—	4	16	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	D₈×C₂₃	1	8	yes	complemented
212	C₂×C₈	SD₁₆×C₂₃	1	8	yes	complemented
213	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
214	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
215	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
224	C₂³	—	4	4	no	—
225	C₂³	—	8	4	no	—
226	C₂³	—	8	4	no	—
227	C₂³	—	8	4	no	—
228	C₂³	C₂₃×#127	1	1	yes	characteristic
229	Q₈	—	4	8	no	—
230	Q₈	—	8	4	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	8	no	—
233	D₄	—	4	8	no	—
234	D₄	—	8	4	no	—
235	D₄	—	8	4	no	—
236	D₄	—	8	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
240	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
241	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
242	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
243	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
244	C₂×C₄	C₂₃×#135	1	4	yes	—
245	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
248	C₈	—	2	4	no	—
249	C₈	C₂×D₈×C₂₃	1	8	yes	complemented
250	C₈	C₂×SD₁₆×C₂₃	1	8	yes	complemented
251	C₂²	—	8	4	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	16	8	no	—
255	C₂²	C₂₃×#492	1	2	yes	—
256	C₂²	C₂₃×#493	1	2	yes	—
257	C₂²	C₂₃×#495	1	2	yes	—
258	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
259	C₄	—	8	4	no	—
260	C₄	C₂₃×#492	1	2	yes	—
261	C₄	C₂₃×#493	1	2	yes	—
262	C₄	C₂₃×#495	1	2	yes	—
263	C₄	C₂²×D₈×C₂₃	1	2	yes	—
264	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
265	C₄	C₂×C₂₃×#135	1	2	yes	—
266	C₂	—	16	4	no	—
267	C₂	C₂₃×#1706	1	4	yes	direct product factor
268	C₂	C₂×C₂₃×#492	1	1	yes	characteristic
269	C₂	C₂×C₂₃×#493	1	1	yes	characteristic
270	C₂	C₂×C₂₃×#495	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#500	C₂	semidirect
3. C₂×C₂₃×#492	C₂	semidirect
4. C₂×C₂₃×#486	C₂	semidirect
5. C₂₃×#1706	C₂	direct (N × H)
6. C₂×C₈²×C₂₃	C₂	semidirect
12. C₂₃×#500	C₂²	semidirect
16. C₂₃×#492	C₂²	semidirect
18. C₂₃×#486	C₂²	semidirect
24. C₈²×C₂₃	C₂²	semidirect
75. C₂×C₈×C₂₃	D₈	semidirect
76. C₂×C₈×C₂₃	SD₁₆	semidirect
88. C₂×#1706	C₂₃	direct (N × H)
114. C₈×C₂₃	C₂×D₈	semidirect
115. C₈×C₂₃	C₂×SD₁₆	semidirect
116. C₂×#500	C₂×C₂₃	semidirect
118. C₂×#492	C₂×C₂₃	semidirect
119. C₂×#486	C₂×C₂₃	semidirect
120. #1706	C₂×C₂₃	direct (N × H)
121. C₂×C₈²	C₂×C₂₃	semidirect
142. #500	C₂²×C₂₃	semidirect
146. #492	C₂²×C₂₃	semidirect
148. #486	C₂²×C₂₃	semidirect
154. C₈²	C₂²×C₂₃	semidirect
156. C₂×C₂₃	#1706	direct (N × H)
190. C₂₃	C₂×#1706	direct (N × H)
211. C₂×C₈	D₈×C₂₃	semidirect
212. C₂×C₈	SD₁₆×C₂₃	semidirect
249. C₈	C₂×D₈×C₂₃	semidirect
250. C₈	C₂×SD₁₆×C₂₃	semidirect
267. C₂	C₂₃×#1706	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×#500	2
C₂×C₂₃×#493	2
C₂×C₂₃×#492	2
C₂×C₂₃×#486	2
C₂₃×#1706	2
C₂×C₈²×C₂₃	2
C₂×#1706	23

Subgroups — C2×C23×#1706 #19167219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1706 #19167219