Subgroups — C₂₃×#33494 #19181120

Order 5888.

556 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2168	C₂	1	8	yes	maximal, complemented
3	C₂₃×#2163	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2113	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#131	C₂²	1	2	yes	—
6	C₂₃×#532	C₄	1	1	yes	characteristic
7	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
8	C₂²×C₁₆×C₂₃	C₂²	1	4	yes	—
9	C₂₃×#436	C₄	1	4	yes	—
10	C₈×D₄×C₂₃	C₂²	1	4	yes	—
11	C₂₃×#423	C₄	1	1	yes	characteristic
12	C₂₃×#408	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊C₁₆	C₂²	1	4	yes	—
15	C₂₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
16	C₂₃×C₂²⋊C₁₆	C₂²	1	8	yes	—
17	C₂₃×#346	C₂²	1	4	yes	—
18	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
19	C₂₃×#131	—	4	2	no	—
20	C₂₃×#131	—	4	2	no	—
21	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
26	C₂₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
27	C₂₃×#127	C₈	1	1	yes	characteristic
28	C₂₃×#124	C₈	1	2	yes	—
29	C₂₃×#121	C₈	1	4	yes	—
30	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂₃×M₃₂	—	4	4	no	—
33	C₂×C₁₆×C₂₃	—	4	4	no	—
34	C₂×C₁₆×C₂₃	C₂³	1	8	yes	—
35	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₂₃	C₂³	1	2	yes	—
39	#41×C₂₃	—	4	2	no	—
40	#41×C₂₃	—	4	2	no	—
41	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
42	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
43	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
44	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
48	M₁₆×C₂₃	—	4	2	no	—
49	M₁₆×C₂₃	—	4	4	no	—
50	C₂×C₈×C₂₃	—	2	4	no	—
51	C₂×C₈×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	4	1	no	—
54	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
59	C₂²⋊C₄×C₂₃	C₂×C₈	1	8	yes	—
60	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
63	C₁₆×C₂₃	—	2	8	no	—
64	#33494	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₂₃	C₂²×C₈	1	4	yes	—
66	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
67	Q₈×C₂₃	—	4	2	no	—
68	D₄×C₂₃	—	4	4	no	—
69	D₄×C₂₃	—	4	2	no	—
70	D₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	4	2	no	—
76	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
80	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₂₃	—	2	4	no	—
82	C₈×C₂₃	—	2	1	no	—
83	C₈×C₂₃	—	2	2	no	—
84	C₈×C₂₃	H(5,𝔽₂)	1	2	yes	—
85	#2931	C₂×C₂₃	1	1	yes	characteristic
86	#2168	C₂×C₂₃	1	8	yes	complemented
87	#2163	C₂×C₂₃	1	2	yes	complemented
88	#2113	C₂×C₂₃	1	4	yes	complemented
89	C₂²×C₂₃	—	2	2	no	—
90	C₂²×C₂₃	—	2	8	no	—
91	C₂²×C₂₃	—	2	1	no	—
92	C₂²×C₂₃	—	2	4	no	—
93	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₂₃	—	2	2	no	—
95	C₄×C₂₃	—	2	2	no	—
96	C₄×C₂₃	—	2	1	no	—
97	C₄×C₂₃	—	2	4	no	—
98	C₄×C₂₃	#518	1	1	yes	characteristic
99	C₄×C₂₃	#568	1	1	yes	characteristic
100	C₂×#131	C₂²×C₂₃	1	2	yes	—
101	#532	C₄×C₂₃	1	1	yes	characteristic
102	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
103	C₂²×C₁₆	C₂²×C₂₃	1	4	yes	—
104	#436	C₄×C₂₃	1	4	yes	—
105	C₈×D₄	C₂²×C₂₃	1	4	yes	—
106	#423	C₄×C₂₃	1	1	yes	characteristic
107	#408	C₄×C₂₃	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
109	C₄⋊C₁₆	C₂²×C₂₃	1	4	yes	—
110	C₂²⋊C₁₆	C₂²×C₂₃	1	4	yes	—
111	C₂²⋊C₁₆	C₂²×C₂₃	1	8	yes	—
112	#346	C₂²×C₂₃	1	4	yes	—
113	C₂×C₂₃	—	4	1	no	—
114	C₂×C₂₃	—	4	4	no	—
115	C₂×C₂₃	#2972	1	1	yes	characteristic
116	C₂×C₂₃	#3407	1	2	yes	—
117	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
118	#131	—	4	2	no	—
119	#131	—	4	2	no	—
120	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
121	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
122	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
125	C₄⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
126	#127	C₈×C₂₃	1	1	yes	characteristic
127	#124	C₈×C₂₃	1	2	yes	—
128	#121	C₈×C₂₃	1	4	yes	—
129	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
130	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
131	M₃₂	—	4	4	no	—
132	C₂×C₁₆	—	4	4	no	—
133	C₂×C₁₆	C₂³×C₂₃	1	8	yes	—
134	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
135	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
136	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
137	C₄×C₈	C₂³×C₂₃	1	2	yes	—
138	C₂₃	#33494	1	1	yes	characteristic, direct product factor, Sylow
139	#41	—	4	2	no	—
140	#41	—	4	2	no	—
141	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
142	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
143	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
144	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
145	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
146	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
147	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
148	M₁₆	—	4	4	no	—
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
155	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
156	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
157	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
158	C₄⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
159	C₂²⋊C₄	C₂×C₈×C₂₃	1	8	yes	—
160	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
161	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
162	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
163	C₁₆	—	2	8	no	—
164	C₂³	C₂²×C₈×C₂₃	1	4	yes	—
165	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
177	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
178	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
179	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	1	no	—
182	C₈	—	2	4	no	—
183	C₈	C₂₃×H(5,𝔽₂)	1	2	yes	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	8	no	—
188	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	C₂₃×#518	1	1	yes	characteristic
194	C₄	C₂₃×#568	1	1	yes	characteristic
195	C₂	—	4	1	no	—
196	C₂	—	4	4	no	—
197	C₂	C₂₃×#2972	1	1	yes	characteristic
198	C₂	C₂₃×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₈
prime radical	C₂₃×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2168	C₂	semidirect
3. C₂₃×#2163	C₂	semidirect
4. C₂₃×#2113	C₂	semidirect
64. #33494	C₂₃	direct (N × H)
86. #2168	C₂×C₂₃	semidirect
87. #2163	C₂×C₂₃	semidirect
88. #2113	C₂×C₂₃	semidirect
138. C₂₃	#33494	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×#2931	2
C₂₃×#2168	2
C₂₃×#2163	2
C₂₃×#2113	2
#33494	23

Subgroups — C23×#33494 #19181120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33494 #19181120