Subgroups — C₂₃×#20752 #19168378

Order 5888.

1158 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#423	C₂	1	2	yes	maximal
2	C₂×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1831	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1728	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₂₃	C₄	1	4	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#423	—	2	8	no	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	8	yes	—
11	C₄³×C₂₃	C₄	1	2	yes	—
12	C₄³×C₂₃	C₄	1	2	yes	—
13	C₄³×C₂₃	C₂²	1	4	yes	—
14	C₂₃×#336	C₂²	1	16	yes	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	—	2	8	no	—
19	C₂×M₁₆×C₂₃	—	2	16	no	—
20	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
21	C₂×C₄²×C₂₃	—	2	8	no	—
22	C₂×C₄²×C₂₃	—	2	16	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₂₃	D₄	1	4	yes	—
30	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
33	C₂₃×C₄⋊C₈	—	2	16	no	—
34	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	—	2	8	no	—
36	C₂²×C₄×C₂₃	—	2	16	no	—
37	C₂²×C₄×C₂₃	—	2	8	no	—
38	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₂₃	#41	1	8	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
47	M₁₆×C₂₃	—	4	16	no	—
48	C₂×C₈×C₂₃	—	2	8	no	—
49	C₂×C₈×C₂₃	—	2	16	no	—
50	C₄²×C₂₃	—	2	8	no	—
51	C₄²×C₂₃	—	2	16	no	—
52	C₄²×C₂₃	—	2	8	no	—
53	C₄²×C₂₃	—	2	8	no	—
54	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
55	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
56	C₄²×C₂₃	M₁₆	1	16	yes	complemented
57	C₄²×C₂₃	C₂×D₄	1	4	yes	—
58	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
59	#20752	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₂₃	—	2	4	no	—
61	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
62	C₂³×C₂₃	#117	1	4	yes	—
63	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	—	2	32	no	—
65	C₂×C₄×C₂₃	—	2	8	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	2	8	no	—
68	C₂×C₄×C₂₃	—	2	16	no	—
69	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
70	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₂₃	#117	1	4	yes	—
73	C₂×C₄×C₂₃	#124	1	8	yes	—
74	C₂×C₄×C₂₃	#125	1	8	yes	—
75	C₂×C₄×C₂₃	C₂×M₁₆	1	16	yes	—
76	C₂×C₄×C₂₃	C₂×M₁₆	1	8	yes	—
77	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
80	C₈×C₂₃	—	4	8	no	—
81	C₂×#423	C₂×C₂₃	1	2	yes	—
82	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic
83	#1831	C₂×C₂₃	1	8	yes	complemented
84	#1728	C₂×C₂₃	1	4	yes	—
85	C₂²×C₂₃	—	2	4	no	—
86	C₂²×C₂₃	—	2	8	no	—
87	C₂²×C₂₃	#382	1	4	yes	—
88	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₂₃	C₂×#117	1	2	yes	—
90	C₂²×C₂₃	C₂²×M₁₆	1	4	yes	—
91	C₄×C₂₃	—	2	4	no	—
92	C₄×C₂₃	—	2	16	no	—
93	C₄×C₂₃	#382	1	4	yes	—
94	C₄×C₂₃	#423	1	8	yes	—
95	C₄×C₂₃	C₂×#124	1	2	yes	—
96	C₄×C₂₃	C₂×#125	1	2	yes	—
97	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
98	C₂²×C₄²	C₄×C₂₃	1	4	yes	—
99	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
100	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
101	#423	—	2	8	no	—
102	C₂×C₄⋊C₈	C₂²×C₂₃	1	8	yes	—
103	C₄³	C₄×C₂₃	1	2	yes	—
104	C₄³	C₄×C₂₃	1	2	yes	—
105	C₄³	C₂²×C₂₃	1	4	yes	—
106	#336	C₂²×C₂₃	1	16	yes	—
107	C₂×C₂₃	—	2	4	no	—
108	C₂×C₂₃	C₂×#382	1	1	yes	characteristic
109	C₂×C₂₃	C₂×#423	1	2	yes	—
110	C₂×C₂₃	#2920	1	4	yes	—
111	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
112	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
113	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
114	C₂×M₁₆	—	2	8	no	—
115	C₂×M₁₆	—	2	16	no	—
116	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
117	C₂×C₄²	—	2	8	no	—
118	C₂×C₄²	—	2	16	no	—
119	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
120	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
121	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
122	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
123	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
125	C₂×C₄²	D₄×C₂₃	1	4	yes	—
126	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
127	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
128	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
129	C₄⋊C₈	—	2	16	no	—
130	C₂₃	#20752	1	1	yes	characteristic, direct product factor, Sylow
131	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	—	2	16	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
136	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
137	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
138	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
140	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
141	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
142	C₂²×C₄	#41×C₂₃	1	8	yes	—
143	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
144	M₁₆	—	4	16	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	2	16	no	—
147	C₄²	—	2	8	no	—
148	C₄²	—	2	16	no	—
149	C₄²	—	2	8	no	—
150	C₄²	—	2	8	no	—
151	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
152	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
153	C₄²	M₁₆×C₂₃	1	16	yes	complemented
154	C₄²	C₂×D₄×C₂₃	1	4	yes	—
155	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
156	C₂³	—	2	4	no	—
157	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
158	C₂³	C₂₃×#117	1	4	yes	—
159	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄	—	2	32	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	16	no	—
165	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
167	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
168	C₂×C₄	C₂₃×#117	1	4	yes	—
169	C₂×C₄	C₂₃×#124	1	8	yes	—
170	C₂×C₄	C₂₃×#125	1	8	yes	—
171	C₂×C₄	C₂×M₁₆×C₂₃	1	16	yes	—
172	C₂×C₄	C₂×M₁₆×C₂₃	1	8	yes	—
173	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
174	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
175	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
176	C₈	—	4	8	no	—
177	C₂²	—	2	8	no	—
178	C₂²	—	2	4	no	—
179	C₂²	C₂₃×#382	1	4	yes	—
180	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
181	C₂²	C₂×C₂₃×#117	1	2	yes	—
182	C₂²	C₂²×M₁₆×C₂₃	1	4	yes	—
183	C₄	—	2	16	no	—
184	C₄	—	2	4	no	—
185	C₄	C₂₃×#382	1	4	yes	—
186	C₄	C₂₃×#423	1	8	yes	—
187	C₄	C₂×C₂₃×#124	1	2	yes	—
188	C₄	C₂×C₂₃×#125	1	2	yes	—
189	C₂	—	2	4	no	—
190	C₂	C₂×C₂₃×#382	1	1	yes	characteristic
191	C₂	C₂×C₂₃×#423	1	2	yes	—
192	C₂	C₂₃×#2920	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1831	C₂	semidirect
56. C₄²×C₂₃	M₁₆	semidirect
59. #20752	C₂₃	direct (N × H)
83. #1831	C₂×C₂₃	semidirect
130. C₂₃	#20752	direct (N × H)
153. C₄²	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#423	2
C₂×C₄³×C₂₃	2
C₂₃×#1831	2
C₂₃×#1728	2
#20752	23

Subgroups — C23×#20752 #19168378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20752 #19168378