Subgroups — C₂×C₂₃×#473 #15670685

Order 2944.

448 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#137	C₂	1	2	yes	maximal
2	C₂²×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#473	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#101	C₂	1	2	yes	maximal, complemented
7	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
9	C₂₃×#137	—	2	4	no	—
10	C₂₃×#137	C₂²	1	8	yes	—
11	C₂×Q₁₆×C₂₃	—	2	2	no	—
12	C₂×Q₁₆×C₂₃	—	2	1	no	—
13	C₂×Q₁₆×C₂₃	—	2	1	no	—
14	C₂×Q₁₆×C₂₃	—	2	2	no	—
15	C₂×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
16	C₂×SD₁₆×C₂₃	—	2	2	no	—
17	C₂₃×#131	—	2	2	no	—
18	C₂₃×#131	C₂²	1	4	yes	—
19	C₂×M₁₆×C₂₃	—	2	1	no	—
20	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
22	C₂²×C₈×C₂₃	—	2	1	no	—
23	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#108	C₂²	1	4	yes	complemented
25	C₂₃×#101	C₂²	1	8	yes	complemented
26	#41×C₂₃	—	2	1	no	—
27	#41×C₂₃	—	2	1	no	—
28	#41×C₂₃	D₄	1	2	yes	—
29	#41×C₂₃	C₂³	1	2	yes	—
30	C₂×Q₈×C₂₃	—	2	2	no	—
31	C₂×Q₈×C₂₃	—	2	2	no	—
32	C₂×Q₈×C₂₃	—	2	2	no	—
33	C₂×Q₈×C₂₃	—	2	2	no	—
34	C₂×Q₈×C₂₃	—	2	2	no	—
35	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
37	C₂×D₄×C₂₃	—	2	1	no	—
38	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	—	2	1	no	—
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	Q₁₆×C₂₃	—	2	8	no	—
43	Q₁₆×C₂₃	—	2	4	no	—
44	Q₁₆×C₂₃	—	2	4	no	—
45	SD₁₆×C₂₃	—	2	8	no	—
46	M₁₆×C₂₃	—	2	1	no	—
47	M₁₆×C₂₃	—	2	2	no	—
48	M₁₆×C₂₃	—	2	4	no	—
49	M₁₆×C₂₃	C₂³	1	2	yes	—
50	M₁₆×C₂₃	C₂³	1	4	yes	—
51	C₂×C₈×C₂₃	—	2	1	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	4	no	—
54	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	D₄	1	2	yes	—
57	C₂×C₈×C₂₃	C₂³	1	2	yes	—
58	C₂³×C₂₃	—	2	1	no	—
59	C₂³×C₂₃	#41	1	1	yes	characteristic
60	Q₈×C₂₃	—	2	1	no	—
61	Q₈×C₂₃	—	2	4	no	—
62	Q₈×C₂₃	—	2	4	no	—
63	Q₈×C₂₃	—	4	2	no	—
64	Q₈×C₂₃	—	4	2	no	—
65	Q₈×C₂₃	C₂×D₄	1	2	yes	—
66	D₄×C₂₃	—	2	2	no	—
67	D₄×C₂₃	—	2	2	no	—
68	D₄×C₂₃	—	2	1	no	—
69	D₄×C₂₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	—	4	2	no	—
76	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₈×C₂₃	—	2	2	no	—
83	C₈×C₂₃	—	2	4	no	—
84	C₈×C₂₃	C₂×D₄	1	2	yes	—
85	C₈×C₂₃	C₂×D₄	1	2	yes	—
86	C₂×#473	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	—	2	2	no	—
89	C₂²×C₂₃	—	4	2	no	—
90	C₂²×C₂₃	#121	1	1	yes	characteristic
91	C₂²×C₂₃	#121	1	1	yes	characteristic
92	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
93	C₄×C₂₃	—	2	2	no	—
94	C₄×C₂₃	—	4	4	no	—
95	C₄×C₂₃	#121	1	1	yes	characteristic
96	C₄×C₂₃	#121	1	1	yes	characteristic
97	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×#137	C₂×C₂₃	1	2	yes	—
100	C₂²×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
101	C₂×#131	C₂×C₂₃	1	1	yes	characteristic
102	#473	C₂×C₂₃	1	8	yes	direct product factor
103	C₂×#108	C₂×C₂₃	1	1	yes	characteristic, complemented
104	C₂×#101	C₂×C₂₃	1	2	yes	complemented
105	C₂×C₂₃	—	2	1	no	—
106	C₂×C₂₃	—	2	1	no	—
107	C₂×C₂₃	—	4	2	no	—
108	C₂×C₂₃	#473	1	2	yes	direct product factor
109	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
110	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
111	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
112	#137	—	2	4	no	—
113	#137	C₂²×C₂₃	1	8	yes	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	2	1	no	—
116	C₂×Q₁₆	—	2	1	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
119	C₂×SD₁₆	—	2	2	no	—
120	#131	—	2	2	no	—
121	#131	C₂²×C₂₃	1	4	yes	—
122	C₂×M₁₆	—	2	1	no	—
123	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
124	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
127	#108	C₂²×C₂₃	1	4	yes	complemented
128	#101	C₂²×C₂₃	1	8	yes	complemented
129	C₂₃	C₂×#473	1	1	yes	characteristic, direct product factor, Sylow
130	#41	—	2	1	no	—
131	#41	—	2	1	no	—
132	#41	D₄×C₂₃	1	2	yes	—
133	#41	C₂³×C₂₃	1	2	yes	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	2	2	no	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
140	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
146	Q₁₆	—	2	8	no	—
147	Q₁₆	—	2	4	no	—
148	Q₁₆	—	2	4	no	—
149	SD₁₆	—	2	8	no	—
150	M₁₆	—	2	1	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	2	4	no	—
153	M₁₆	C₂³×C₂₃	1	2	yes	—
154	M₁₆	C₂³×C₂₃	1	4	yes	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
159	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
160	C₂×C₈	D₄×C₂₃	1	2	yes	—
161	C₂×C₈	C₂³×C₂₃	1	2	yes	—
162	C₂³	—	2	1	no	—
163	C₂³	#41×C₂₃	1	1	yes	characteristic
164	Q₈	—	2	1	no	—
165	Q₈	—	2	4	no	—
166	Q₈	—	2	4	no	—
167	Q₈	—	4	2	no	—
168	Q₈	—	4	2	no	—
169	Q₈	C₂×D₄×C₂₃	1	2	yes	—
170	D₄	—	2	2	no	—
171	D₄	—	2	2	no	—
172	D₄	—	2	1	no	—
173	D₄	C₂×D₄×C₂₃	1	2	yes	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	2	4	no	—
188	C₈	C₂×D₄×C₂₃	1	2	yes	—
189	C₈	C₂×D₄×C₂₃	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	C₂₃×#121	1	1	yes	characteristic
194	C₂²	C₂₃×#121	1	1	yes	characteristic
195	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	4	4	no	—
198	C₄	C₂₃×#121	1	1	yes	characteristic
199	C₄	C₂₃×#121	1	1	yes	characteristic
200	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
201	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	—	4	2	no	—
205	C₂	C₂₃×#473	1	2	yes	direct product factor
206	C₂	C₂×C₂₃×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₁₆×C₂₃	C₂	semidirect
4. C₂₃×#473	C₂	direct (N × H)
5. C₂×C₂₃×#108	C₂	semidirect
6. C₂×C₂₃×#101	C₂	semidirect
15. C₂×Q₁₆×C₂₃	C₂²	semidirect
24. C₂₃×#108	C₂²	semidirect
25. C₂₃×#101	C₂²	semidirect
86. C₂×#473	C₂₃	direct (N × H)
100. C₂²×Q₁₆	C₂×C₂₃	semidirect
102. #473	C₂×C₂₃	direct (N × H)
103. C₂×#108	C₂×C₂₃	semidirect
104. C₂×#101	C₂×C₂₃	semidirect
108. C₂×C₂₃	#473	direct (N × H)
118. C₂×Q₁₆	C₂²×C₂₃	semidirect
127. #108	C₂²×C₂₃	semidirect
128. #101	C₂²×C₂₃	semidirect
129. C₂₃	C₂×#473	direct (N × H)
205. C₂	C₂₃×#473	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×#137	2
C₂²×Q₁₆×C₂₃	2
C₂×C₂₃×#131	2
C₂₃×#473	2
C₂×C₂₃×#108	2
C₂×C₂₃×#101	2
C₂×#473	23

Subgroups — C2×C23×#473 #15670685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#473 #15670685