Subgroups — C₂₃×#33324 #19180950

Order 5888.

826 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2108	C₂	1	8	yes	maximal, complemented
4	C₄×C₂₃×M₃₂	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#117	C₄	1	1	yes	characteristic
8	C₂×C₂₃×M₃₂	C₄	1	16	yes	complemented
9	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
10	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₂₃	C₂²	1	4	yes	—
12	C₂₃×#405	C₄	1	4	yes	—
13	C₂₃×#405	C₂²	1	4	yes	—
14	C₂×C₂₃×#97	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#346	C₂²	1	8	yes	complemented
17	C₄×C₁₆×C₂₃	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	C₈	1	8	yes	—
20	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
27	C₂₃×#117	C₈	1	4	yes	—
28	C₂₃×#117	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₂₃	C₈	1	2	yes	—
30	C₂×C₂²⋊C₄×C₂₃	C₈	1	2	yes	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
32	C₂₃×M₃₂	C₂×C₄	1	16	yes	—
33	C₂₃×M₃₂	C₂×C₄	1	16	yes	complemented
34	C₂×C₁₆×C₂₃	C₂×C₄	1	16	yes	complemented
35	C₂×C₁₆×C₂₃	C₂³	1	8	yes	—
36	C₂₃×#97	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₂₃	C₂³	1	4	yes	—
39	C₂⁴×C₂₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
41	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
47	M₁₆×C₂₃	C₂×C₈	1	16	yes	—
48	C₂×C₈×C₂₃	C₄²	1	4	yes	—
49	C₂×C₈×C₂₃	C₄²	1	8	yes	—
50	C₂×C₈×C₂₃	C₂×C₈	1	8	yes	—
51	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	C₂×C₈	1	8	yes	—
59	C₂²⋊C₄×C₂₃	C₂×C₈	1	8	yes	—
60	C₄²×C₂₃	C₂×C₈	1	4	yes	—
61	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
62	C₁₆×C₂₃	C₂²×C₄	1	16	yes	—
63	#33324	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₂₃	—	4	2	no	—
65	C₂³×C₂₃	C₄×C₈	1	4	yes	—
66	C₂³×C₂₃	C₂²×C₈	1	2	yes	—
67	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₂₃	—	4	4	no	—
69	C₂×C₄×C₂₃	C₄×C₈	1	4	yes	—
70	C₂×C₄×C₂₃	C₄×C₈	1	8	yes	—
71	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
72	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
73	C₂×C₄×C₂₃	C₂²×C₈	1	8	yes	—
74	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
76	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₂₃	C₂×C₄²	1	2	yes	—
80	C₈×C₂₃	C₂×C₄²	1	4	yes	—
81	C₈×C₂₃	C₂²×C₈	1	8	yes	—
82	C₈×C₂₃	C₂³×C₄	1	2	yes	—
83	C₂²×M₃₂	C₂×C₂₃	1	2	yes	—
84	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
85	#2108	C₂×C₂₃	1	8	yes	complemented
86	C₄×M₃₂	C₂×C₂₃	1	4	yes	complemented
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	—	2	4	no	—
89	C₂²×C₂₃	—	4	4	no	—
90	C₂²×C₂₃	C₂×C₄×C₈	1	2	yes	—
91	C₂²×C₂₃	C₂×C₄×C₈	1	4	yes	—
92	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
93	C₄×C₂₃	—	2	8	no	—
94	C₄×C₂₃	C₂×C₄×C₈	1	2	yes	—
95	C₄×C₂₃	C₂×C₄×C₈	1	4	yes	—
96	C₄×C₂₃	C₂²×C₄²	1	1	yes	characteristic
97	C₄×C₂₃	C₂³×C₈	1	1	yes	characteristic
98	C₂²×M₁₆	C₄×C₂₃	1	1	yes	characteristic
99	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
100	C₂×#117	C₄×C₂₃	1	1	yes	characteristic
101	C₂×M₃₂	C₄×C₂₃	1	16	yes	complemented
102	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
103	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
104	C₂²×C₁₆	C₂²×C₂₃	1	4	yes	—
105	#405	C₄×C₂₃	1	4	yes	—
106	#405	C₂²×C₂₃	1	4	yes	—
107	C₂×#97	C₄×C₂₃	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
109	#346	C₂²×C₂₃	1	8	yes	complemented
110	C₄×C₁₆	C₂²×C₂₃	1	8	yes	complemented
111	C₂×C₂₃	—	2	2	no	—
112	C₂×C₂₃	—	2	4	no	—
113	C₂×C₂₃	C₂²×C₄×C₈	1	1	yes	characteristic
114	C₂×C₂₃	#3407	1	2	yes	—
115	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
116	C₂×M₁₆	C₈×C₂₃	1	8	yes	—
117	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
118	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
120	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
121	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
122	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
124	#117	C₈×C₂₃	1	4	yes	—
125	#117	C₂×C₄×C₂₃	1	4	yes	—
126	C₂×C₄⋊C₄	C₈×C₂₃	1	2	yes	—
127	C₂×C₂²⋊C₄	C₈×C₂₃	1	2	yes	—
128	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
129	M₃₂	C₂×C₄×C₂₃	1	16	yes	—
130	M₃₂	C₂×C₄×C₂₃	1	16	yes	complemented
131	C₂×C₁₆	C₂×C₄×C₂₃	1	16	yes	complemented
132	C₂×C₁₆	C₂³×C₂₃	1	8	yes	—
133	#97	C₂×C₄×C₂₃	1	8	yes	—
134	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
135	C₄×C₈	C₂³×C₂₃	1	4	yes	—
136	C₂₃	#33324	1	1	yes	characteristic, direct product factor, Sylow
137	C₂⁴	C₂×C₈×C₂₃	1	1	yes	characteristic
138	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
139	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
140	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
141	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
143	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
145	M₁₆	C₂×C₈×C₂₃	1	16	yes	—
146	C₂×C₈	C₄²×C₂₃	1	4	yes	—
147	C₂×C₈	C₄²×C₂₃	1	8	yes	—
148	C₂×C₈	C₂×C₈×C₂₃	1	8	yes	—
149	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
150	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
151	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
152	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
153	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
154	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
155	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
156	C₄⋊C₄	C₂×C₈×C₂₃	1	8	yes	—
157	C₂²⋊C₄	C₂×C₈×C₂₃	1	8	yes	—
158	C₄²	C₂×C₈×C₂₃	1	4	yes	—
159	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
160	C₁₆	C₂²×C₄×C₂₃	1	16	yes	—
161	C₂³	—	4	2	no	—
162	C₂³	C₄×C₈×C₂₃	1	4	yes	—
163	C₂³	C₂²×C₈×C₂₃	1	2	yes	—
164	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄×C₈×C₂₃	1	4	yes	—
167	C₂×C₄	C₄×C₈×C₂₃	1	8	yes	—
168	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
169	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
170	C₂×C₄	C₂²×C₈×C₂₃	1	8	yes	—
171	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
172	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
173	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
175	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
176	C₈	C₂×C₄²×C₂₃	1	2	yes	—
177	C₈	C₂×C₄²×C₂₃	1	4	yes	—
178	C₈	C₂²×C₈×C₂₃	1	8	yes	—
179	C₈	C₂³×C₄×C₂₃	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×C₄×C₈×C₂₃	1	2	yes	—
184	C₂²	C₂×C₄×C₈×C₂₃	1	4	yes	—
185	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	C₂×C₄×C₈×C₂₃	1	2	yes	—
188	C₄	C₂×C₄×C₈×C₂₃	1	4	yes	—
189	C₄	C₂²×C₄²×C₂₃	1	1	yes	characteristic
190	C₄	C₂³×C₈×C₂₃	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	4	no	—
193	C₂	C₂²×C₄×C₈×C₂₃	1	1	yes	characteristic
194	C₂	C₂₃×#3407	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2108	C₂	semidirect
4. C₄×C₂₃×M₃₂	C₂	semidirect
8. C₂×C₂₃×M₃₂	C₄	semidirect
16. C₂₃×#346	C₂²	semidirect
17. C₄×C₁₆×C₂₃	C₂²	semidirect
33. C₂₃×M₃₂	C₂×C₄	semidirect
34. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
63. #33324	C₂₃	direct (N × H)
85. #2108	C₂×C₂₃	semidirect
86. C₄×M₃₂	C₂×C₂₃	semidirect
101. C₂×M₃₂	C₄×C₂₃	semidirect
109. #346	C₂²×C₂₃	semidirect
110. C₄×C₁₆	C₂²×C₂₃	semidirect
130. M₃₂	C₂×C₄×C₂₃	semidirect
131. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
136. C₂₃	#33324	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×C₂₃×M₃₂	2
C₂×C₂₃×#405	2
C₂₃×#2108	2
C₄×C₂₃×M₃₂	2
#33324	23

Subgroups — C23×#33324 #19180950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33324 #19180950