Subgroups — C₂₃×#33716 #19181342

Order 5888.

826 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×#508	C₂	1	4	yes	maximal, complemented
2	C₂₃×#3147	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2251	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2250	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#135	—	2	2	no	—
7	C₂×C₂₃×#135	C₂²	1	4	yes	—
8	C₂₃×#532	—	2	2	no	—
9	C₂₃×#508	—	2	16	no	—
10	C₂×C₂₃×Q₃₂	C₂²	1	4	yes	—
11	C₂×C₂₃×SD₃₂	C₂²	1	8	yes	—
12	C₂×C₂₃×D₁₆	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#495	—	2	2	no	—
15	C₂₃×#495	C₂²	1	4	yes	—
16	C₂₃×#494	C₂²	1	2	yes	—
17	C₂₃×#493	C₂²	1	2	yes	—
18	C₂₃×#492	—	2	2	no	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#346	C₂²	1	4	yes	—
21	C₂×#41×C₂₃	—	2	4	no	—
22	C₂₃×#135	—	2	16	no	—
23	C₂₃×#135	—	4	8	no	—
24	C₂×Q₁₆×C₂₃	—	2	2	no	—
25	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
26	C₂×SD₁₆×C₂₃	—	2	4	no	—
27	C₂×SD₁₆×C₂₃	—	2	4	no	—
28	C₂×D₈×C₂₃	—	2	2	no	—
29	C₂×D₈×C₂₃	C₂³	1	4	yes	—
30	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
32	C₂₃×C₄⋊Q₈	—	2	2	no	—
33	C₂₃×#127	—	2	2	no	—
34	C₂₃×#124	—	2	4	no	—
35	C₂₃×#121	—	4	4	no	—
36	C₄×D₄×C₂₃	—	4	4	no	—
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂₃×Q₃₂	—	2	8	no	—
39	C₂₃×SD₃₂	—	2	16	no	—
40	C₂₃×D₁₆	—	2	8	no	—
41	C₂×C₁₆×C₂₃	D₄	1	8	yes	complemented
42	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
43	C₄×C₈×C₂₃	D₄	1	2	yes	—
44	C₄×C₈×C₂₃	C₂³	1	2	yes	—
45	#41×C₂₃	—	4	8	no	—
46	#41×C₂₃	—	4	8	no	—
47	C₂×Q₈×C₂₃	—	2	4	no	—
48	C₂×D₄×C₂₃	—	2	4	no	—
49	C₂×D₄×C₂₃	—	4	4	no	—
50	C₂×D₄×C₂₃	—	4	4	no	—
51	C₂²×C₄×C₂₃	—	4	4	no	—
52	C₂²×C₄×C₂₃	D₈	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	Q₁₆×C₂₃	—	2	8	no	—
55	Q₁₆×C₂₃	—	4	4	no	—
56	SD₁₆×C₂₃	—	4	8	no	—
57	SD₁₆×C₂₃	—	4	8	no	—
58	D₈×C₂₃	—	2	8	no	—
59	D₈×C₂₃	—	4	4	no	—
60	C₂×C₈×C₂₃	D₈	1	4	yes	—
61	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	—	4	4	no	—
66	C₂²⋊C₄×C₂₃	—	4	8	no	—
67	C₄²×C₂₃	D₈	1	2	yes	—
68	C₄²×C₂₃	C₂×D₄	1	2	yes	—
69	C₁₆×C₂₃	C₂×D₄	1	8	yes	—
70	#33716	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₂₃	—	4	4	no	—
72	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₂₃	—	4	4	no	—
74	Q₈×C₂₃	—	4	4	no	—
75	D₄×C₂₃	—	4	4	no	—
76	D₄×C₂₃	—	4	4	no	—
77	D₄×C₂₃	—	8	8	no	—
78	D₄×C₂₃	—	8	8	no	—
79	C₂×C₄×C₂₃	—	4	2	no	—
80	C₂×C₄×C₂₃	—	4	4	no	—
81	C₂×C₄×C₂₃	—	8	8	no	—
82	C₂×C₄×C₂₃	#127	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
84	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
87	C₈×C₂₃	#127	1	2	yes	—
88	C₈×C₂₃	C₂×D₈	1	4	yes	—
89	C₈×C₂₃	C₂²×D₄	1	2	yes	—
90	C₂×#508	C₂×C₂₃	1	4	yes	complemented
91	#3147	C₂×C₂₃	1	2	yes	complemented
92	#2251	C₂×C₂₃	1	4	yes	complemented
93	#2250	C₂×C₂₃	1	4	yes	complemented
94	C₂×#346	C₂×C₂₃	1	1	yes	characteristic, complemented
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	8	4	no	—
97	C₂²×C₂₃	—	8	8	no	—
98	C₂²×C₂₃	#493	1	2	yes	—
99	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
100	C₄×C₂₃	—	2	4	no	—
101	C₄×C₂₃	—	8	4	no	—
102	C₄×C₂₃	#493	1	2	yes	—
103	C₄×C₂₃	C₂×#127	1	1	yes	characteristic
104	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
105	C₂×#135	—	2	2	no	—
106	C₂×#135	C₂²×C₂₃	1	4	yes	—
107	#532	—	2	2	no	—
108	#508	—	2	16	no	—
109	C₂×Q₃₂	C₂²×C₂₃	1	4	yes	—
110	C₂×SD₃₂	C₂²×C₂₃	1	8	yes	—
111	C₂×D₁₆	C₂²×C₂₃	1	4	yes	—
112	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
113	#495	—	2	2	no	—
114	#495	C₂²×C₂₃	1	4	yes	—
115	#494	C₂²×C₂₃	1	2	yes	—
116	#493	C₂²×C₂₃	1	2	yes	—
117	#492	—	2	2	no	—
118	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
119	#346	C₂²×C₂₃	1	4	yes	—
120	C₂×C₂₃	—	2	2	no	—
121	C₂×C₂₃	—	16	4	no	—
122	C₂×C₂₃	C₂×#493	1	1	yes	characteristic
123	C₂×C₂₃	#3414	1	2	yes	—
124	C₂×#41	—	2	4	no	—
125	#135	—	2	16	no	—
126	#135	—	4	8	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
129	C₂×SD₁₆	—	2	4	no	—
130	C₂×SD₁₆	—	2	4	no	—
131	C₂×D₈	—	2	2	no	—
132	C₂×D₈	C₂³×C₂₃	1	4	yes	—
133	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	2	no	—
136	#127	—	2	2	no	—
137	#124	—	2	4	no	—
138	#121	—	4	4	no	—
139	C₄×D₄	—	4	4	no	—
140	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
141	Q₃₂	—	2	8	no	—
142	SD₃₂	—	2	16	no	—
143	D₁₆	—	2	8	no	—
144	C₂×C₁₆	D₄×C₂₃	1	8	yes	complemented
145	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
146	C₄×C₈	D₄×C₂₃	1	2	yes	—
147	C₄×C₈	C₂³×C₂₃	1	2	yes	—
148	C₂₃	#33716	1	1	yes	characteristic, direct product factor, Sylow
149	#41	—	4	8	no	—
150	#41	—	4	8	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	4	4	no	—
154	C₂×D₄	—	4	4	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	D₈×C₂₃	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
158	Q₁₆	—	2	8	no	—
159	Q₁₆	—	4	4	no	—
160	SD₁₆	—	4	8	no	—
161	SD₁₆	—	4	8	no	—
162	D₈	—	2	8	no	—
163	D₈	—	4	4	no	—
164	C₂×C₈	D₈×C₂₃	1	4	yes	—
165	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
166	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
167	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
168	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
169	C₄⋊C₄	—	4	4	no	—
170	C₂²⋊C₄	—	4	8	no	—
171	C₄²	D₈×C₂₃	1	2	yes	—
172	C₄²	C₂×D₄×C₂₃	1	2	yes	—
173	C₁₆	C₂×D₄×C₂₃	1	8	yes	—
174	C₂³	—	4	4	no	—
175	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
176	Q₈	—	4	4	no	—
177	Q₈	—	4	4	no	—
178	D₄	—	4	4	no	—
179	D₄	—	4	4	no	—
180	D₄	—	8	8	no	—
181	D₄	—	8	8	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	8	no	—
185	C₂×C₄	C₂₃×#127	1	2	yes	—
186	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
187	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
188	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
190	C₈	C₂₃×#127	1	2	yes	—
191	C₈	C₂×D₈×C₂₃	1	4	yes	—
192	C₈	C₂²×D₄×C₂₃	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	8	4	no	—
195	C₂²	—	8	8	no	—
196	C₂²	C₂₃×#493	1	2	yes	—
197	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	4	no	—
200	C₄	C₂₃×#493	1	2	yes	—
201	C₄	C₂×C₂₃×#127	1	1	yes	characteristic
202	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	16	4	no	—
205	C₂	C₂×C₂₃×#493	1	1	yes	characteristic
206	C₂	C₂₃×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#508	C₂	semidirect
2. C₂₃×#3147	C₂	semidirect
3. C₂₃×#2251	C₂	semidirect
4. C₂₃×#2250	C₂	semidirect
5. C₂×C₂₃×#346	C₂	semidirect
41. C₂×C₁₆×C₂₃	D₄	semidirect
70. #33716	C₂₃	direct (N × H)
90. C₂×#508	C₂×C₂₃	semidirect
91. #3147	C₂×C₂₃	semidirect
92. #2251	C₂×C₂₃	semidirect
93. #2250	C₂×C₂₃	semidirect
94. C₂×#346	C₂×C₂₃	semidirect
144. C₂×C₁₆	D₄×C₂₃	semidirect
148. C₂₃	#33716	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×#508	2
C₂₃×#3147	2
C₂₃×#2251	2
C₂₃×#2250	2
C₂×C₂₃×#346	2
#33716	23

Subgroups — C23×#33716 #19181342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33716 #19181342