Subgroups — C₂₃×#33340 #19180966

Order 5888.

1114 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3573	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2117	C₂	1	6	yes	maximal, complemented
5	C₂×C₂₃×#350	C₂	1	6	yes	maximal, complemented
6	C₂₃×#585	C₄	1	1	yes	characteristic
7	C₂₃×#568	—	2	4	no	—
8	C₂₃×#568	C₄	1	6	yes	—
9	C₂₃×#568	C₄	1	1	yes	characteristic
10	C₂×C₂₃×#131	—	2	4	no	—
11	C₂×C₂₃×#131	C₂²	1	6	yes	—
12	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
13	C₂₃×#504	—	2	4	no	—
14	C₂₃×#504	—	2	4	no	—
15	C₂×C₂₃×M₃₂	C₂²	1	6	yes	complemented
16	C₂×C₂₃×M₃₂	C₂²	1	3	yes	complemented
17	C₂²×C₁₆×C₂₃	C₂²	1	3	yes	complemented
18	C₂₃×#350	C₂²	1	4	yes	complemented
19	C₂₃×#350	C₂²	1	12	yes	complemented
20	C₂₃×2₋¹⁺⁴	—	2	1	no	—
21	C₂₃×2₋¹⁺⁴	C₈	1	4	yes	—
22	C₂₃×H(5,𝔽₂)	—	2	3	no	—
23	C₂₃×H(5,𝔽₂)	C₈	1	4	yes	—
24	C₂×#41×C₂₃	—	2	4	no	—
25	C₂×#41×C₂₃	C₂×C₄	1	6	yes	—
26	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂₃×#131	—	2	4	no	—
28	C₂₃×#131	—	2	12	no	—
29	C₂₃×#131	—	2	6	no	—
30	C₂₃×#131	—	2	6	no	—
31	C₂₃×#131	C₂×C₄	1	12	yes	—
32	C₂₃×#131	D₄	1	4	yes	—
33	C₂₃×#131	D₄	1	4	yes	—
34	C₂₃×#131	C₂³	1	4	yes	—
35	C₂×M₁₆×C₂₃	—	2	12	no	—
36	C₂×M₁₆×C₂₃	—	2	6	no	—
37	C₂×M₁₆×C₂₃	C₂×C₄	1	6	yes	—
38	C₂×M₁₆×C₂₃	C₂×C₄	1	3	yes	—
39	C₂²×C₈×C₂₃	—	2	6	no	—
40	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
41	C₂₃×M₃₂	—	2	6	no	—
42	C₂₃×M₃₂	—	2	6	no	—
43	C₂₃×M₃₂	C₂³	1	4	yes	—
44	C₂×C₁₆×C₂₃	—	2	6	no	—
45	C₂×C₁₆×C₂₃	—	2	6	no	—
46	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₂₃	C₂³	1	3	yes	—
48	#41×C₂₃	—	2	4	no	—
49	#41×C₂₃	—	2	12	no	—
50	#41×C₂₃	—	2	6	no	—
51	#41×C₂₃	—	2	6	no	—
52	#41×C₂₃	C₂²⋊C₄	1	4	yes	—
53	#41×C₂₃	C₂×C₈	1	12	yes	—
54	#41×C₂₃	C₂×C₈	1	4	yes	—
55	#41×C₂₃	C₂²×C₄	1	4	yes	—
56	C₂×Q₈×C₂₃	—	2	4	no	—
57	C₂×Q₈×C₂₃	C₂×C₈	1	6	yes	—
58	C₂×Q₈×C₂₃	M₁₆	1	1	yes	characteristic
59	C₂×D₄×C₂₃	—	2	6	no	—
60	C₂×D₄×C₂₃	—	2	12	no	—
61	C₂×D₄×C₂₃	C₂×C₈	1	6	yes	—
62	C₂×D₄×C₂₃	M₁₆	1	3	yes	—
63	C₂²×C₄×C₂₃	—	2	6	no	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
65	M₁₆×C₂₃	—	2	6	no	—
66	M₁₆×C₂₃	—	2	12	no	—
67	M₁₆×C₂₃	—	2	4	no	—
68	M₁₆×C₂₃	C₂²⋊C₄	1	12	yes	—
69	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₂₃	—	2	6	no	—
71	C₂×C₈×C₂₃	—	2	12	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
74	C₂×C₈×C₂₃	C₂×D₄	1	6	yes	—
75	C₂×C₈×C₂₃	C₂×D₄	1	6	yes	—
76	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
77	C₁₆×C₂₃	—	2	1	no	—
78	C₁₆×C₂₃	—	2	3	no	—
79	C₁₆×C₂₃	—	2	4	no	—
80	#33340	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₃	—	2	6	no	—
82	C₂³×C₂₃	C₂×M₁₆	1	3	yes	—
83	Q₈×C₂₃	—	2	6	no	—
84	Q₈×C₂₃	C₂²⋊C₈	1	4	yes	—
85	Q₈×C₂₃	C₂²×C₈	1	4	yes	—
86	D₄×C₂₃	—	2	4	no	—
87	D₄×C₂₃	—	2	12	no	—
88	D₄×C₂₃	—	2	6	no	—
89	D₄×C₂₃	C₂²⋊C₈	1	12	yes	—
90	D₄×C₂₃	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₂₃	—	2	6	no	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	12	no	—
94	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
95	C₂×C₄×C₂₃	C₂²×C₈	1	6	yes	—
96	C₂×C₄×C₂₃	C₂×M₁₆	1	3	yes	—
97	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
98	C₈×C₂₃	—	2	4	no	—
99	C₈×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
100	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	#3573	C₂×C₂₃	1	1	yes	characteristic
103	#3407	C₂×C₂₃	1	1	yes	characteristic, complemented
104	C₂×#504	C₂×C₂₃	1	1	yes	characteristic, complemented
105	#2117	C₂×C₂₃	1	6	yes	complemented
106	C₂×#350	C₂×C₂₃	1	6	yes	complemented
107	C₂²×C₂₃	—	2	4	no	—
108	C₂²×C₂₃	—	4	3	no	—
109	C₂²×C₂₃	—	8	6	no	—
110	C₂²×C₂₃	C₂×C₂²⋊C₈	1	6	yes	—
111	C₂²×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
112	C₄×C₂₃	—	2	4	no	—
113	C₄×C₂₃	C₂×C₂²⋊C₈	1	6	yes	—
114	C₄×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
115	C₄×C₂₃	C₂³×C₈	1	1	yes	characteristic
116	#585	C₄×C₂₃	1	1	yes	characteristic
117	#568	—	2	4	no	—
118	#568	C₄×C₂₃	1	6	yes	—
119	#568	C₄×C₂₃	1	1	yes	characteristic
120	C₂×#131	—	2	4	no	—
121	C₂×#131	C₂²×C₂₃	1	6	yes	—
122	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
123	#504	—	2	4	no	—
124	#504	—	2	4	no	—
125	C₂×M₃₂	C₂²×C₂₃	1	6	yes	complemented
126	C₂×M₃₂	C₂²×C₂₃	1	3	yes	complemented
127	C₂²×C₁₆	C₂²×C₂₃	1	3	yes	complemented
128	#350	C₂²×C₂₃	1	4	yes	complemented
129	#350	C₂²×C₂₃	1	12	yes	complemented
130	C₂×C₂₃	—	2	1	no	—
131	C₂×C₂₃	—	2	6	no	—
132	C₂×C₂₃	—	4	4	no	—
133	C₂×C₂₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
134	2₋¹⁺⁴	—	2	1	no	—
135	2₋¹⁺⁴	C₈×C₂₃	1	4	yes	—
136	H(5,𝔽₂)	—	2	3	no	—
137	H(5,𝔽₂)	C₈×C₂₃	1	4	yes	—
138	C₂×#41	—	2	4	no	—
139	C₂×#41	C₂×C₄×C₂₃	1	6	yes	—
140	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
141	#131	—	2	4	no	—
142	#131	—	2	12	no	—
143	#131	—	2	6	no	—
144	#131	—	2	6	no	—
145	#131	C₂×C₄×C₂₃	1	12	yes	—
146	#131	D₄×C₂₃	1	4	yes	—
147	#131	D₄×C₂₃	1	4	yes	—
148	#131	C₂³×C₂₃	1	4	yes	—
149	C₂×M₁₆	—	2	12	no	—
150	C₂×M₁₆	—	2	6	no	—
151	C₂×M₁₆	C₂×C₄×C₂₃	1	6	yes	—
152	C₂×M₁₆	C₂×C₄×C₂₃	1	3	yes	—
153	C₂²×C₈	—	2	6	no	—
154	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
155	M₃₂	—	2	6	no	—
156	M₃₂	—	2	6	no	—
157	M₃₂	C₂³×C₂₃	1	4	yes	—
158	C₂×C₁₆	—	2	6	no	—
159	C₂×C₁₆	—	2	6	no	—
160	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
161	C₂×C₁₆	C₂³×C₂₃	1	3	yes	—
162	C₂₃	#33340	1	1	yes	characteristic, direct product factor, Sylow
163	#41	—	2	4	no	—
164	#41	—	2	12	no	—
165	#41	—	2	6	no	—
166	#41	—	2	6	no	—
167	#41	C₂²⋊C₄×C₂₃	1	4	yes	—
168	#41	C₂×C₈×C₂₃	1	12	yes	—
169	#41	C₂×C₈×C₂₃	1	4	yes	—
170	#41	C₂²×C₄×C₂₃	1	4	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	C₂×C₈×C₂₃	1	6	yes	—
173	C₂×Q₈	M₁₆×C₂₃	1	1	yes	characteristic
174	C₂×D₄	—	2	6	no	—
175	C₂×D₄	—	2	12	no	—
176	C₂×D₄	C₂×C₈×C₂₃	1	6	yes	—
177	C₂×D₄	M₁₆×C₂₃	1	3	yes	—
178	C₂²×C₄	—	2	6	no	—
179	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
180	M₁₆	—	2	6	no	—
181	M₁₆	—	2	12	no	—
182	M₁₆	—	2	4	no	—
183	M₁₆	C₂²⋊C₄×C₂₃	1	12	yes	—
184	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
185	C₂×C₈	—	2	6	no	—
186	C₂×C₈	—	2	12	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
189	C₂×C₈	C₂×D₄×C₂₃	1	6	yes	—
190	C₂×C₈	C₂×D₄×C₂₃	1	6	yes	—
191	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
192	C₁₆	—	2	1	no	—
193	C₁₆	—	2	4	no	—
194	C₁₆	—	2	3	no	—
195	C₂³	—	2	6	no	—
196	C₂³	C₂×M₁₆×C₂₃	1	3	yes	—
197	Q₈	—	2	6	no	—
198	Q₈	C₂₃×C₂²⋊C₈	1	4	yes	—
199	Q₈	C₂²×C₈×C₂₃	1	4	yes	—
200	D₄	—	2	4	no	—
201	D₄	—	2	12	no	—
202	D₄	—	2	6	no	—
203	D₄	C₂₃×C₂²⋊C₈	1	12	yes	—
204	D₄	C₂²×C₈×C₂₃	1	4	yes	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	12	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
209	C₂×C₄	C₂²×C₈×C₂₃	1	6	yes	—
210	C₂×C₄	C₂×M₁₆×C₂₃	1	3	yes	—
211	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₈	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
214	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
215	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
216	C₂²	—	2	4	no	—
217	C₂²	—	4	3	no	—
218	C₂²	—	8	6	no	—
219	C₂²	C₂×C₂₃×C₂²⋊C₈	1	6	yes	—
220	C₂²	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	C₂×C₂₃×C₂²⋊C₈	1	6	yes	—
223	C₄	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
224	C₄	C₂³×C₈×C₂₃	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	6	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂²×C₂₃×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	C₂²×D₄
commutator	C₄	C₂³×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂²×C₂²⋊C₈
prime radical	C₂₃×#585	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3407	C₂	semidirect
3. C₂×C₂₃×#504	C₂	semidirect
4. C₂₃×#2117	C₂	semidirect
5. C₂×C₂₃×#350	C₂	semidirect
15. C₂×C₂₃×M₃₂	C₂²	semidirect
16. C₂×C₂₃×M₃₂	C₂²	semidirect
17. C₂²×C₁₆×C₂₃	C₂²	semidirect
18. C₂₃×#350	C₂²	semidirect
19. C₂₃×#350	C₂²	semidirect
80. #33340	C₂₃	direct (N × H)
103. #3407	C₂×C₂₃	semidirect
104. C₂×#504	C₂×C₂₃	semidirect
105. #2117	C₂×C₂₃	semidirect
106. C₂×#350	C₂×C₂₃	semidirect
125. C₂×M₃₂	C₂²×C₂₃	semidirect
126. C₂×M₃₂	C₂²×C₂₃	semidirect
127. C₂²×C₁₆	C₂²×C₂₃	semidirect
128. #350	C₂²×C₂₃	semidirect
129. #350	C₂²×C₂₃	semidirect
162. C₂₃	#33340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3573	2
C₂₃×#3407	2
C₂×C₂₃×#504	2
C₂₃×#2117	2
C₂×C₂₃×#350	2
#33340	23

Subgroups — C23×#33340 #19180966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33340 #19180966