Subgroups — C₂×#13092491 #15682079

Order 2944.

564 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67267	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₂²⋊D₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092491	C₂	1	8	yes	maximal, direct product factor
5	C₂×Dic₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×(C₂×D₂₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10624997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₂₃	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Dic₂₃	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₂₃	C₄	1	4	yes	—
16	C₄×C₂²⋊D₂₃	C₂²	1	4	yes	—
17	C₂×(C₂×D₂₃)⋊C₄	C₄	1	1	yes	characteristic
18	C₂×Dic₂₃⋊C₄	C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
20	C₂×#67267	—	2	4	no	—
21	C₂×#67267	C₂²	1	4	yes	complemented
22	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₂₃	—	2	4	no	—
25	C₂×C₈×D₂₃	C₂²	1	4	yes	complemented
26	C₂₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
27	(C₂×D₂₃)⋊C₈	C₂²	1	4	yes	complemented
28	#10624997	C₂²	1	4	yes	complemented
29	Dic₂₃⋊C₈	C₂²	1	4	yes	complemented
30	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	—	2	2	no	—
33	C₂²×C₄×C₂₃	—	2	2	no	—
34	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₂₃	—	2	4	no	—
39	C₂×C₂²⋊D₂₃	C₂×C₄	1	4	yes	—
40	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₂₃	—	2	2	no	—
44	C₂×C₄×D₂₃	—	2	2	no	—
45	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₃	C₂³	1	2	yes	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	C₂³	1	2	yes	—
50	C₂×C₈×C₂₃	C₂³	1	2	yes	—
51	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
52	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
53	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
54	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₃	C₂³	1	2	yes	—
56	#67267	—	2	8	no	—
57	C₂×C₂₃⋊C₈	—	2	2	no	—
58	C₂×C₂₃⋊C₈	D₄	1	4	yes	complemented
59	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
60	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
61	C₈×D₂₃	—	2	8	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	2	2	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₂₃	—	2	2	no	—
68	C₂²×D₂₃	—	2	2	no	—
69	C₂²×D₂₃	M₁₆	1	4	yes	complemented
70	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₂₃	#41	1	2	yes	—
79	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₂₃	—	2	8	no	—
81	C₂×Dic₂₃	—	2	2	no	—
82	C₂×Dic₂₃	—	2	2	no	—
83	C₂×Dic₂₃	—	2	2	no	—
84	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
87	C₄×D₂₃	—	2	4	no	—
88	C₄×D₂₃	—	2	2	no	—
89	C₄×D₂₃	—	2	2	no	—
90	C₈×C₂₃	—	2	2	no	—
91	C₈×C₂₃	—	2	2	no	—
92	C₂₃⋊C₈	—	2	2	no	—
93	C₂₃⋊C₈	C₂×D₄	1	4	yes	complemented
94	C₂×#435	—	23	1	no	maximal, Sylow
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	2	2	no	—
97	C₂²×C₂₃	—	2	2	no	—
98	C₂²×C₂₃	—	4	4	no	—
99	C₂²×C₂₃	C₄×D₄	1	2	yes	—
100	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
101	C₂²×C₂₃	#131	1	2	yes	—
102	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₂₃	—	2	2	no	—
104	C₂×D₂₃	—	2	2	no	—
105	C₂×D₂₃	—	2	8	no	—
106	C₂×D₂₃	C₂×M₁₆	1	4	yes	complemented
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	C₄×D₄	1	2	yes	—
109	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
111	Dic₂₃	—	2	2	no	—
112	Dic₂₃	—	2	2	no	—
113	Dic₂₃	—	2	2	no	—
114	C₂²×M₁₆	—	23	1	no	—
115	C₂³×C₈	—	23	1	no	—
116	C₂×C₄×D₄	—	23	1	no	—
117	#435	—	23	8	no	—
118	C₂×C₄⋊C₈	—	23	1	no	—
119	C₂×C₂²⋊C₈	—	23	1	no	—
120	C₂×C₂²⋊C₈	D₂₃	1	1	yes	characteristic, complemented
121	C₂×#97	—	23	1	no	—
122	C₂×C₂₃	—	4	2	no	—
123	C₂×C₂₃	#435	1	4	yes	complemented
124	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
126	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
127	D₂₃	—	2	4	no	—
128	C₂²×D₄	—	23	1	no	—
129	C₂³×C₄	—	23	1	no	—
130	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
131	C₂×M₁₆	—	23	4	no	—
132	C₂×M₁₆	—	46	4	no	—
133	C₂²×C₈	—	23	1	no	—
134	C₂²×C₈	—	23	1	no	—
135	C₂²×C₈	—	23	4	no	—
136	C₂²×C₈	—	46	4	no	—
137	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
138	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
139	C₄×D₄	—	23	4	no	—
140	C₄×D₄	—	23	4	no	—
141	C₂×C₄⋊C₄	—	23	1	no	—
142	C₂×C₂²⋊C₄	—	23	1	no	—
143	C₂×C₂²⋊C₄	—	23	1	no	—
144	C₂×C₄²	—	23	1	no	—
145	C₄⋊C₈	—	23	4	no	—
146	C₂²⋊C₈	—	23	4	no	—
147	C₂²⋊C₈	C₂×D₂₃	1	4	yes	complemented
148	#97	—	23	4	no	—
149	C₂₃	C₂×#435	1	1	yes	characteristic, complemented, Sylow
150	C₂⁴	—	23	1	no	—
151	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
152	C₂×D₄	—	23	4	no	—
153	C₂×D₄	—	46	4	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	23	1	no	—
157	C₂²×C₄	—	23	1	no	—
158	C₂²×C₄	—	23	1	no	—
159	C₂²×C₄	—	23	2	no	—
160	C₂²×C₄	—	23	2	no	—
161	C₂²×C₄	—	46	2	no	—
162	C₂²×C₄	—	46	2	no	—
163	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
164	C₂²×C₄	C₄×D₂₃	1	2	yes	—
165	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
166	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
167	M₁₆	—	46	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	23	4	no	—
171	C₂×C₈	—	23	2	no	—
172	C₂×C₈	—	23	2	no	—
173	C₂×C₈	—	46	8	no	—
174	C₂×C₈	—	46	2	no	—
175	C₂×C₈	C₂²×D₂₃	1	2	yes	—
176	C₂×C₈	C₂²×D₂₃	1	2	yes	—
177	C₄⋊C₄	—	23	4	no	—
178	C₂²⋊C₄	—	23	4	no	—
179	C₂²⋊C₄	—	23	4	no	—
180	C₄²	—	23	2	no	—
181	C₄²	—	23	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	23	4	no	—
186	C₂³	—	23	2	no	—
187	C₂³	—	46	2	no	—
188	C₂³	—	46	2	no	—
189	C₂³	C₂×C₄×D₂₃	1	2	yes	—
190	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
191	D₄	—	46	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	23	2	no	—
197	C₂×C₄	—	23	2	no	—
198	C₂×C₄	—	23	2	no	—
199	C₂×C₄	—	46	2	no	—
200	C₂×C₄	—	46	2	no	—
201	C₂×C₄	—	46	2	no	—
202	C₂×C₄	—	46	4	no	—
203	C₂×C₄	—	46	2	no	—
204	C₂×C₄	—	46	2	no	—
205	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
207	C₂×C₄	D₄×D₂₃	1	2	yes	—
208	C₂×C₄	#67290	1	2	yes	—
209	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	—	23	4	no	—
213	C₈	—	46	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	23	4	no	—
219	C₂²	—	46	2	no	—
220	C₂²	—	46	2	no	—
221	C₂²	—	46	8	no	—
222	C₂²	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
223	C₂²	#10625072	1	2	yes	—
224	C₂²	M₁₆×D₂₃	1	2	yes	—
225	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	46	2	no	—
228	C₄	—	46	2	no	—
229	C₄	—	46	2	no	—
230	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
231	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
232	C₄	C₂×#67290	1	1	yes	characteristic
233	C₂	—	4	2	no	—
234	C₂	—	46	4	no	—
235	C₂	#13092491	1	4	yes	direct product factor
236	C₂	C₂×(C₂²⋊D₂₃)⋊C₄	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₂₃	1	1	yes	characteristic
238	C₂	C₂×#10625072	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×C₂²⋊C₈	C₂
socle	C₂³×C₂₃	C₂²×C₄
prime radical	C₂²×C₂²⋊D₂₃	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₂₃
hypocenter	C₂₃	C₂×#435
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#67267	C₂	semidirect
2. C₂²×C₈×D₂₃	C₂	semidirect
4. #13092491	C₂	direct (N × H)
5. C₂×Dic₂₃⋊C₈	C₂	semidirect
6. C₂×(C₂×D₂₃)⋊C₈	C₂	semidirect
7. C₂×#10624997	C₂	semidirect
8. C₂×C₂₃×C₂²⋊C₈	C₂	semidirect
21. C₂×#67267	C₂²	semidirect
25. C₂×C₈×D₂₃	C₂²	semidirect
26. C₂₃×C₂²⋊C₈	C₂²	semidirect
27. (C₂×D₂₃)⋊C₈	C₂²	semidirect
28. #10624997	C₂²	semidirect
29. Dic₂₃⋊C₈	C₂²	semidirect
58. C₂×C₂₃⋊C₈	D₄	semidirect
69. C₂²×D₂₃	M₁₆	semidirect
93. C₂₃⋊C₈	C₂×D₄	semidirect
106. C₂×D₂₃	C₂×M₁₆	semidirect
120. C₂×C₂²⋊C₈	D₂₃	semidirect
123. C₂×C₂₃	#435	semidirect
147. C₂²⋊C₈	C₂×D₂₃	semidirect
149. C₂₃	C₂×#435	semidirect
235. C₂	#13092491	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×#67267	2
C₂²×C₈×D₂₃	2
C₂×C₄×C₂²⋊D₂₃	2
#13092491	2
C₂×Dic₂₃⋊C₈	2
C₂×(C₂×D₂₃)⋊C₈	2
C₂×#10624997	2
C₂×C₂₃×C₂²⋊C₈	2
C₂×#435	23

Subgroups — C2×#13092491 #15682079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092491 #15682079