Subgroups — C₂×#124×D₂₃ #15686642

Order 2944.

1880 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×D₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#10625118	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10625046	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10625107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625056	C₂	1	4	yes	maximal, complemented
7	#124×D₂₃	C₂	1	16	yes	maximal, direct product factor
8	C₂×C₂²⋊C₄×D₂₃	C₂	1	4	yes	maximal, complemented
9	C₂×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₂₃	C₂²	1	2	yes	—
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₂₃	C₂²	1	2	yes	—
15	C₂×Q₈×D₂₃	—	2	8	no	—
16	C₂×Q₈×D₂₃	C₂²	1	8	yes	complemented
17	C₂×D₄×D₂₃	—	2	8	no	—
18	C₂×D₄×D₂₃	C₂²	1	8	yes	—
19	C₂²×D₉₂	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
21	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
22	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₄₆	C₂²	1	2	yes	complemented
24	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
25	C₂₃×#124	C₂²	1	8	yes	complemented
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
27	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	complemented
29	#10625118	C₂²	1	8	yes	complemented
30	#10625107	C₂²	1	8	yes	complemented
31	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	complemented
32	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	complemented
33	C₂×C₄×Dic₂₃	C₂²	1	2	yes	complemented
34	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
35	#10625056	C₂²	1	32	yes	complemented
36	C₂²⋊C₄×D₂₃	C₂²	1	32	yes	complemented
37	#10625046	C₂²	1	8	yes	complemented
38	C₄²×D₂₃	C₂²	1	8	yes	complemented
39	C₂⁴×C₂₃	C₂³	1	2	yes	—
40	C₂³×D₂₃	—	2	8	no	—
41	C₂³×D₂₃	—	2	16	no	—
42	C₂³×D₂₃	C₂³	1	8	yes	—
43	C₂³×D₂₃	C₂³	1	2	yes	—
44	C₂³×D₂₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₂₃	—	2	4	no	—
46	C₂×Q₈×C₂₃	C₂³	1	4	yes	complemented
47	C₂×D₄×C₂₃	—	2	4	no	—
48	C₂×D₄×C₂₃	C₂³	1	4	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₂₃	—	2	8	no	—
53	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
54	C₂²×Dic₂₃	C₂³	1	2	yes	—
55	C₂²×Dic₂₃	C₂³	1	2	yes	—
56	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
57	Q₈×D₂₃	—	2	32	no	—
58	D₄×D₂₃	—	2	32	no	—
59	C₂×D₉₂	—	2	4	no	—
60	C₂×D₉₂	C₂³	1	4	yes	—
61	C₂×C₄×D₂₃	—	2	8	no	—
62	C₂×C₄×D₂₃	—	2	8	no	—
63	C₂×C₄×D₂₃	D₄	1	8	yes	—
64	C₂×C₄×D₂₃	C₂³	1	8	yes	—
65	C₂×C₄×D₂₃	C₂³	1	8	yes	—
66	C₂×C₄×D₂₃	C₂³	1	4	yes	—
67	C₂×Dic₄₆	—	2	4	no	—
68	C₂×Dic₄₆	—	2	8	no	—
69	C₂×Dic₄₆	C₂³	1	4	yes	complemented
70	C₂×Dic₄₆	C₂³	1	8	yes	complemented
71	C₂²⋊C₄×C₂₃	C₂³	1	16	yes	complemented
72	C₄²×C₂₃	C₂³	1	4	yes	complemented
73	C₂²⋊Dic₂₃	C₂³	1	16	yes	complemented
74	(C₂×D₂₃)⋊C₄	C₂³	1	16	yes	complemented
75	(C₂×D₂₃)⋊C₄	C₂³	1	16	yes	complemented
76	C₄×Dic₂₃	C₂³	1	8	yes	complemented
77	C₄×Dic₂₃	C₂³	1	4	yes	complemented
78	C₂³×C₂₃	—	2	4	no	—
79	C₂³×C₂₃	—	2	8	no	—
80	C₂³×C₂₃	C₂⁴	1	4	yes	—
81	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₂²×D₂₃	—	2	4	no	—
83	C₂²×D₂₃	—	2	8	no	—
84	C₂²×D₂₃	—	2	16	no	—
85	C₂²×D₂₃	—	2	32	no	—
86	C₂²×D₂₃	—	4	32	no	—
87	C₂²×D₂₃	C₂×D₄	1	4	yes	—
88	C₂²×D₂₃	#41	1	8	yes	—
89	C₂²×D₂₃	C₂⁴	1	4	yes	—
90	C₂²×D₂₃	C₂⁴	1	2	yes	—
91	Q₈×C₂₃	—	2	8	no	—
92	D₄×C₂₃	—	2	8	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
97	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
98	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
99	C₂²⋊D₂₃	—	2	16	no	—
100	C₂×Dic₂₃	—	2	4	no	—
101	C₂×Dic₂₃	—	2	4	no	—
102	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
103	C₂×Dic₂₃	C₂⁴	1	4	yes	—
104	C₂×Dic₂₃	C₂⁴	1	4	yes	—
105	C₂×Dic₂₃	C₂⁴	1	2	yes	—
106	D₉₂	—	2	8	no	—
107	C₄×D₂₃	—	2	16	no	—
108	C₄×D₂₃	—	2	16	no	—
109	C₄×D₂₃	C₂×D₄	1	16	yes	—
110	Dic₄₆	—	2	8	no	—
111	Dic₄₆	—	2	16	no	—
112	C₂²×#124	—	23	1	no	maximal, Sylow
113	C₂²×C₂₃	—	2	4	no	—
114	C₂²×C₂₃	—	2	8	no	—
115	C₂²×C₂₃	—	4	8	no	—
116	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
117	C₂²×C₂₃	C₂×#41	1	4	yes	—
118	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
119	C₂×D₂₃	—	2	4	no	—
120	C₂×D₂₃	—	2	8	no	—
121	C₂×D₂₃	—	4	8	no	—
122	C₂×D₂₃	—	4	32	no	—
123	C₂×D₂₃	#124	1	16	yes	direct product factor
124	C₂×D₂₃	C₂²×D₄	1	4	yes	—
125	C₂×D₂₃	C₂×#41	1	8	yes	—
126	C₄×C₂₃	—	2	4	no	—
127	C₄×C₂₃	—	2	4	no	—
128	C₄×C₂₃	C₂²×D₄	1	4	yes	—
129	Dic₂₃	—	2	4	no	—
130	Dic₂₃	—	2	4	no	—
131	Dic₂₃	C₂²×D₄	1	4	yes	—
132	C₂³×Q₈	—	23	1	no	—
133	C₂³×D₄	—	23	1	no	—
134	C₂×#124	—	23	1	no	—
135	C₂×#124	—	23	1	no	—
136	C₂×#124	—	23	16	no	—
137	C₂×#124	—	23	4	no	—
138	C₂×#124	—	23	1	no	—
139	C₂×#124	D₂₃	1	1	yes	characteristic, direct product factor
140	C₂²×C₂²⋊C₄	—	23	4	no	—
141	C₂²×C₄²	—	23	1	no	—
142	C₂×C₂₃	—	4	4	no	—
143	C₂×C₂₃	C₂×#124	1	4	yes	complemented
144	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
145	C₂×C₂₃	C₂²×#41	1	2	yes	—
146	D₂₃	—	4	4	no	—
147	D₂₃	C₂×#124	1	8	yes	direct product factor
148	C₂⁵	—	23	2	no	—
149	C₂²×Q₈	—	23	8	no	—
150	C₂²×Q₈	—	23	2	no	—
151	C₂²×Q₈	—	23	1	no	—
152	C₂²×Q₈	—	46	8	no	—
153	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
154	C₂²×D₄	—	23	2	no	—
155	C₂²×D₄	—	23	8	no	—
156	C₂²×D₄	—	23	1	no	—
157	C₂²×D₄	—	46	8	no	—
158	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
159	C₂³×C₄	—	23	1	no	—
160	C₂³×C₄	—	23	2	no	—
161	C₂³×C₄	—	23	2	no	—
162	#124	—	23	8	no	—
163	#124	—	23	8	no	—
164	#124	—	23	32	no	—
165	#124	—	23	8	no	—
166	#124	C₂×D₂₃	1	8	yes	direct product factor
167	C₂×C₂²⋊C₄	—	23	4	no	—
168	C₂×C₂²⋊C₄	—	23	4	no	—
169	C₂×C₂²⋊C₄	—	23	4	no	—
170	C₂×C₂²⋊C₄	—	23	32	no	—
171	C₂×C₂²⋊C₄	C₂×D₂₃	1	4	yes	complemented
172	C₂×C₄²	—	23	2	no	—
173	C₂×C₄²	—	23	1	no	—
174	C₂×C₄²	—	23	8	no	—
175	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic, complemented
176	C₂₃	C₂²×#124	1	1	yes	characteristic, complemented, Sylow
177	C₂⁴	—	23	1	no	—
178	C₂⁴	—	23	8	no	—
179	C₂⁴	—	23	2	no	—
180	C₂⁴	—	46	8	no	—
181	C₂⁴	—	46	16	no	—
182	C₂⁴	C₂²×D₂₃	1	2	yes	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	23	8	no	—
185	C₂×Q₈	—	23	4	no	—
186	C₂×Q₈	—	46	8	no	—
187	C₂×Q₈	—	46	32	no	—
188	C₂×Q₈	—	46	4	no	—
189	C₂×Q₈	C₂²×D₂₃	1	4	yes	complemented
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	23	8	no	—
192	C₂×D₄	—	23	4	no	—
193	C₂×D₄	—	46	32	no	—
194	C₂×D₄	—	46	8	no	—
195	C₂×D₄	—	46	4	no	—
196	C₂×D₄	C₂²×D₂₃	1	4	yes	—
197	C₂²×C₄	—	23	1	no	—
198	C₂²×C₄	—	23	8	no	—
199	C₂²×C₄	—	23	4	no	—
200	C₂²×C₄	—	23	2	no	—
201	C₂²×C₄	—	23	2	no	—
202	C₂²×C₄	—	23	8	no	—
203	C₂²×C₄	—	23	8	no	—
204	C₂²×C₄	—	46	8	no	—
205	C₂²×C₄	—	46	8	no	—
206	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
207	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
208	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
209	C₂²⋊C₄	—	23	16	no	—
210	C₂²⋊C₄	—	23	16	no	—
211	C₂²⋊C₄	—	23	16	no	—
212	C₂²⋊C₄	C₂²×D₂₃	1	16	yes	complemented
213	C₄²	—	23	8	no	—
214	C₄²	—	23	4	no	—
215	C₄²	C₂²×D₂₃	1	4	yes	complemented
216	C₂³	—	2	4	no	—
217	C₂³	—	2	8	no	—
218	C₂³	—	23	4	no	—
219	C₂³	—	23	8	no	—
220	C₂³	—	23	2	no	—
221	C₂³	—	23	4	no	—
222	C₂³	—	46	16	no	—
223	C₂³	—	46	32	no	—
224	C₂³	—	46	4	no	—
225	C₂³	—	46	8	no	—
226	C₂³	—	92	32	no	—
227	C₂³	C₂³×D₂₃	1	4	yes	—
228	C₂³	C₂³×D₂₃	1	1	yes	characteristic
229	Q₈	—	2	8	no	—
230	Q₈	—	46	16	no	—
231	Q₈	—	46	8	no	—
232	D₄	—	2	8	no	—
233	D₄	—	46	16	no	—
234	D₄	—	46	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	23	4	no	—
238	C₂×C₄	—	23	2	no	—
239	C₂×C₄	—	23	16	no	—
240	C₂×C₄	—	23	4	no	—
241	C₂×C₄	—	23	4	no	—
242	C₂×C₄	—	46	16	no	—
243	C₂×C₄	—	46	16	no	—
244	C₂×C₄	—	46	4	no	—
245	C₂×C₄	—	46	4	no	—
246	C₂×C₄	D₄×D₂₃	1	4	yes	—
247	C₂×C₄	C₂³×D₂₃	1	4	yes	—
248	C₂×C₄	C₂³×D₂₃	1	4	yes	—
249	C₂×C₄	C₂³×D₂₃	1	2	yes	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	23	16	no	—
254	C₂²	—	23	4	no	—
255	C₂²	—	23	8	no	—
256	C₂²	—	46	4	no	—
257	C₂²	—	46	8	no	—
258	C₂²	—	92	32	no	—
259	C₂²	—	92	8	no	—
260	C₂²	C₂×D₄×D₂₃	1	2	yes	—
261	C₂²	#41×D₂₃	1	4	yes	—
262	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	23	4	no	—
266	C₄	—	46	4	no	—
267	C₄	—	46	4	no	—
268	C₄	C₂×D₄×D₂₃	1	4	yes	—
269	C₂	—	4	4	no	—
270	C₂	—	23	8	no	—
271	C₂	—	92	4	no	—
272	C₂	#124×D₂₃	1	4	yes	direct product factor
273	C₂	C₂²×D₄×D₂₃	1	1	yes	characteristic
274	C₂	C₂×#41×D₂₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×#124	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂²×D₄×D₂₃	C₂
hypercenter	C₂×#124	D₂₃
hypocenter	C₂₃	C₂²×#124
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₂₃	C₂	semidirect
3. C₂×#10625118	C₂	semidirect
4. C₂×#10625046	C₂	semidirect
5. C₂×#10625107	C₂	semidirect
6. C₂×#10625056	C₂	semidirect
7. #124×D₂₃	C₂	direct (N × H)
8. C₂×C₂²⋊C₄×D₂₃	C₂	semidirect
9. C₂×C₄²×D₂₃	C₂	semidirect
10. C₂×C₂₃×#124	C₂	semidirect
12. C₂²×Q₈×C₂₃	C₂²	semidirect
16. C₂×Q₈×D₂₃	C₂²	semidirect
23. C₂²×Dic₄₆	C₂²	semidirect
24. C₂²×Dic₄₆	C₂²	semidirect
25. C₂₃×#124	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
27. C₂×C₄²×C₂₃	C₂²	semidirect
28. C₂×C₂²⋊Dic₂₃	C₂²	semidirect
29. #10625118	C₂²	semidirect
30. #10625107	C₂²	semidirect
31. C₂×(C₂×D₂₃)⋊C₄	C₂²	semidirect
32. C₂×(C₂×D₂₃)⋊C₄	C₂²	semidirect
33. C₂×C₄×Dic₂₃	C₂²	semidirect
34. C₂×C₄×Dic₂₃	C₂²	semidirect
35. #10625056	C₂²	semidirect
36. C₂²⋊C₄×D₂₃	C₂²	semidirect
37. #10625046	C₂²	semidirect
38. C₄²×D₂₃	C₂²	semidirect
46. C₂×Q₈×C₂₃	C₂³	semidirect
69. C₂×Dic₄₆	C₂³	semidirect
70. C₂×Dic₄₆	C₂³	semidirect
71. C₂²⋊C₄×C₂₃	C₂³	semidirect
72. C₄²×C₂₃	C₂³	semidirect
73. C₂²⋊Dic₂₃	C₂³	semidirect
74. (C₂×D₂₃)⋊C₄	C₂³	semidirect
75. (C₂×D₂₃)⋊C₄	C₂³	semidirect
76. C₄×Dic₂₃	C₂³	semidirect
77. C₄×Dic₂₃	C₂³	semidirect
123. C₂×D₂₃	#124	direct (N × H)
139. C₂×#124	D₂₃	direct (N × H)
143. C₂×C₂₃	C₂×#124	semidirect
147. D₂₃	C₂×#124	direct (N × H)
153. C₂²×Q₈	C₂×D₂₃	semidirect
166. #124	C₂×D₂₃	direct (N × H)
171. C₂×C₂²⋊C₄	C₂×D₂₃	semidirect
175. C₂×C₄²	C₂×D₂₃	semidirect
176. C₂₃	C₂²×#124	semidirect
189. C₂×Q₈	C₂²×D₂₃	semidirect
212. C₂²⋊C₄	C₂²×D₂₃	semidirect
215. C₄²	C₂²×D₂₃	semidirect
272. C₂	#124×D₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₂₃	2
C₂²×D₄×D₂₃	2
C₂×#10625118	2
C₂×#10625046	2
C₂×#10625107	2
C₂×#10625056	2
#124×D₂₃	2
C₂×C₂²⋊C₄×D₂₃	2
C₂×C₄²×D₂₃	2
C₂×C₂₃×#124	2
C₂²×#124	23

Subgroups — C2×#124×D23 #15686642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#124×D₂₃ #15686642