Subgroups — C₂₃×#10693 #19158319

Order 5888.

1222 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#97	C₂	1	2	yes	maximal
2	C₂×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#336	C₂	1	4	yes	maximal
4	C₂³×C₈×C₂₃	C₂²	1	4	yes	—
5	C₂²×C₄²×C₂₃	C₄	1	4	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#97	—	2	4	no	—
9	C₂×C₂₃×#97	—	2	4	no	—
10	C₂×C₂₃×#97	C₄	1	8	yes	complemented
11	C₄³×C₂₃	—	2	4	no	—
12	C₂₃×#336	C₄	1	16	yes	complemented
13	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
14	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	—	2	4	no	—
17	C₂²×C₈×C₂₃	—	2	4	no	—
18	C₂²×C₈×C₂₃	—	2	8	no	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	16	yes	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
21	C₂×C₄²×C₂₃	—	2	8	no	—
22	C₂×C₄²×C₂₃	—	2	16	no	—
23	C₂×C₄²×C₂₃	—	2	4	no	—
24	C₂×C₄²×C₂₃	—	2	4	no	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₂₃	D₄	1	2	yes	—
28	C₂×C₄²×C₂₃	D₄	1	4	yes	—
29	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
30	C₂₃×#97	—	2	16	no	—
31	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	—	2	4	no	—
33	C₂²×C₄×C₂₃	—	2	4	no	—
34	C₂²×C₄×C₂₃	—	2	8	no	—
35	C₂²×C₄×C₂₃	—	2	8	no	—
36	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
37	C₂²×C₄×C₂₃	C₄²	1	8	yes	—
38	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₂₃	M₁₆	1	16	yes	—
41	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	#41	1	4	yes	—
47	C₂×C₈×C₂₃	—	2	8	no	—
48	C₂×C₈×C₂₃	—	2	8	no	—
49	C₂×C₈×C₂₃	—	2	16	no	—
50	C₂×C₈×C₂₃	—	2	16	no	—
51	C₂×C₈×C₂₃	C₄²	1	16	yes	complemented
52	C₄²×C₂₃	—	2	32	no	—
53	C₄²×C₂₃	—	2	16	no	—
54	C₄²×C₂₃	C₂²⋊C₄	1	8	yes	—
55	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
56	#10693	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
58	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
59	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂³×C₂₃	#117	1	2	yes	—
62	C₂³×C₂₃	C₂×M₁₆	1	8	yes	—
63	C₂×C₄×C₂₃	—	2	8	no	—
64	C₂×C₄×C₂₃	—	2	8	no	—
65	C₂×C₄×C₂₃	—	2	8	no	—
66	C₂×C₄×C₂₃	—	2	8	no	—
67	C₂×C₄×C₂₃	—	2	16	no	—
68	C₂×C₄×C₂₃	—	2	8	no	—
69	C₂×C₄×C₂₃	—	2	16	no	—
70	C₂×C₄×C₂₃	#95	1	8	yes	—
71	C₂×C₄×C₂₃	#97	1	16	yes	complemented
72	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
73	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₂₃	#117	1	4	yes	—
76	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₂₃	C₄×D₄	1	8	yes	—
78	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
79	C₈×C₂₃	—	2	16	no	—
80	C₂²×#97	C₂×C₂₃	1	2	yes	—
81	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic
82	C₂×#336	C₂×C₂₃	1	4	yes	—
83	C₂²×C₂₃	C₄³	1	1	yes	characteristic
84	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
85	C₂²×C₂₃	#376	1	1	yes	characteristic
86	C₂²×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
87	C₂²×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
88	C₂²×C₂₃	C₂×#97	1	4	yes	—
89	C₂²×C₂₃	C₄×M₁₆	1	4	yes	—
90	C₂²×C₂₃	C₄×M₁₆	1	4	yes	—
91	C₂²×C₂₃	#407	1	4	yes	—
92	C₂²×C₂₃	#423	1	4	yes	—
93	C₂²×C₂₃	#432	1	8	yes	—
94	C₄×C₂₃	—	2	8	no	—
95	C₄×C₂₃	—	2	16	no	—
96	C₄×C₂₃	C₄×C₂²⋊C₄	1	4	yes	—
97	C₄×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
98	C₂³×C₈	C₂²×C₂₃	1	4	yes	—
99	C₂²×C₄²	C₄×C₂₃	1	4	yes	—
100	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
101	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
102	C₂×#97	—	2	4	no	—
103	C₂×#97	—	2	4	no	—
104	C₂×#97	C₄×C₂₃	1	8	yes	complemented
105	C₄³	—	2	4	no	—
106	#336	C₄×C₂₃	1	16	yes	complemented
107	C₂×C₂₃	C₄×#95	1	1	yes	characteristic
108	C₂×C₂₃	C₄×#97	1	2	yes	—
109	C₂×C₂₃	#1728	1	2	yes	—
110	C₂×C₂₃	#1745	1	2	yes	—
111	C₂×C₂₃	#1749	1	4	yes	—
112	C₂×C₂₃	#1767	1	4	yes	—
113	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
114	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
115	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	—	2	8	no	—
119	C₂²×C₈	C₂×C₄×C₂₃	1	16	yes	—
120	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
121	C₂×C₄²	—	2	8	no	—
122	C₂×C₄²	—	2	16	no	—
123	C₂×C₄²	—	2	4	no	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
126	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
127	C₂×C₄²	D₄×C₂₃	1	2	yes	—
128	C₂×C₄²	D₄×C₂₃	1	4	yes	—
129	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
130	#97	—	2	16	no	—
131	C₂₃	#10693	1	1	yes	characteristic, direct product factor, Sylow
132	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
138	C₂²×C₄	C₄²×C₂₃	1	8	yes	—
139	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
140	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
141	C₂²×C₄	M₁₆×C₂₃	1	16	yes	—
142	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
143	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
144	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
146	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	#41×C₂₃	1	4	yes	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	16	no	—
151	C₂×C₈	—	2	16	no	—
152	C₂×C₈	C₄²×C₂₃	1	16	yes	complemented
153	C₄²	—	2	32	no	—
154	C₄²	—	2	16	no	—
155	C₄²	C₂²⋊C₄×C₂₃	1	8	yes	—
156	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
157	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
158	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
159	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
160	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
161	C₂³	C₂₃×#117	1	2	yes	—
162	C₂³	C₂×M₁₆×C₂₃	1	8	yes	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	16	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	C₂₃×#95	1	8	yes	—
171	C₂×C₄	C₂₃×#97	1	16	yes	complemented
172	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
173	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
175	C₂×C₄	C₂₃×#117	1	4	yes	—
176	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
177	C₂×C₄	C₄×D₄×C₂₃	1	8	yes	—
178	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
179	C₈	—	2	16	no	—
180	C₂²	C₄³×C₂₃	1	1	yes	characteristic
181	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
182	C₂²	C₂₃×#376	1	1	yes	characteristic
183	C₂²	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
184	C₂²	C₄×C₄⋊C₄×C₂₃	1	2	yes	—
185	C₂²	C₂×C₂₃×#97	1	4	yes	—
186	C₂²	C₄×M₁₆×C₂₃	1	4	yes	—
187	C₂²	C₄×M₁₆×C₂₃	1	4	yes	—
188	C₂²	C₂₃×#407	1	4	yes	—
189	C₂²	C₂₃×#423	1	4	yes	—
190	C₂²	C₂₃×#432	1	8	yes	—
191	C₄	—	2	8	no	—
192	C₄	—	2	16	no	—
193	C₄	C₄×C₂²⋊C₄×C₂₃	1	4	yes	—
194	C₄	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
195	C₂	C₄×C₂₃×#95	1	1	yes	characteristic
196	C₂	C₄×C₂₃×#97	1	2	yes	—
197	C₂	C₂₃×#1728	1	2	yes	—
198	C₂	C₂₃×#1745	1	2	yes	—
199	C₂	C₂₃×#1749	1	4	yes	—
200	C₂	C₂₃×#1767	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₃	C₂³
commutator	C₂²	C₄³×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle, prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×C₂₃×#97	C₄	semidirect
12. C₂₃×#336	C₄	semidirect
51. C₂×C₈×C₂₃	C₄²	semidirect
56. #10693	C₂₃	direct (N × H)
71. C₂×C₄×C₂₃	#97	semidirect
104. C₂×#97	C₄×C₂₃	semidirect
106. #336	C₄×C₂₃	semidirect
131. C₂₃	#10693	direct (N × H)
152. C₂×C₈	C₄²×C₂₃	semidirect
171. C₂×C₄	C₂₃×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#97	2
C₂×C₄³×C₂₃	2
C₂×C₂₃×#336	2
#10693	23

Subgroups — C23×#10693 #19158319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10693 #19158319