Subgroups — C₂₃×#12660 #19160286

Order 5888.

330 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	2	yes	maximal
2	C₂×C₂₃×#347	C₂	1	2	yes	maximal
3	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1377	C₂	1	2	yes	maximal
5	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
6	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#376	C₄	1	1	yes	characteristic
13	C₂₃×#364	C₄	1	8	yes	complemented
14	C₂₃×#347	—	2	4	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂₃×M₃₂	—	2	4	no	—
23	C₂₃×M₃₂	C₂×C₄	1	8	yes	—
24	C₂×C₁₆×C₂₃	—	2	4	no	—
25	C₂₃×#97	—	2	2	no	—
26	C₂₃×#97	—	2	2	no	—
27	C₂₃×#97	D₄	1	2	yes	—
28	C₂₃×#97	Q₈	1	2	yes	—
29	C₄×C₈×C₂₃	—	2	2	no	—
30	C₄×C₈×C₂₃	—	2	2	no	—
31	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
32	C₂₃×#95	—	2	2	no	—
33	C₂²×C₄×C₂₃	—	2	1	no	—
34	C₂²×C₄×C₂₃	—	2	1	no	—
35	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	2	1	no	—
40	C₂×C₈×C₂₃	—	2	1	no	—
41	C₂×C₈×C₂₃	C₄²	1	2	yes	—
42	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₂₃	#41	1	2	yes	—
50	C₄²×C₂₃	—	2	1	no	—
51	C₄²×C₂₃	—	2	1	no	—
52	C₄²×C₂₃	—	2	1	no	—
53	C₄²×C₂₃	—	2	1	no	—
54	C₄²×C₂₃	C₄²	1	2	yes	—
55	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
56	C₁₆×C₂₃	—	2	8	no	—
57	#12660	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	4	2	no	—
63	C₂×C₄×C₂₃	—	4	2	no	—
64	C₂×C₄×C₂₃	—	4	2	no	—
65	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₂₃	—	2	2	no	—
72	C₈×C₂₃	—	2	2	no	—
73	C₈×C₂₃	#97	1	4	yes	—
74	C₈×C₂₃	C₄×D₄	1	2	yes	—
75	C₈×C₂₃	C₄×Q₈	1	2	yes	—
76	C₂×#364	C₂×C₂₃	1	2	yes	—
77	C₂×#347	C₂×C₂₃	1	2	yes	—
78	#1726	C₂×C₂₃	1	1	yes	characteristic
79	#1377	C₂×C₂₃	1	2	yes	—
80	C₂²×C₂₃	—	2	2	no	—
81	C₂²×C₂₃	C₄×M₁₆	1	1	yes	characteristic
82	C₂²×C₂₃	#423	1	1	yes	characteristic
83	C₂²×C₂₃	#432	1	1	yes	characteristic
84	C₄×C₂₃	—	4	2	no	—
85	C₄×C₂₃	—	4	2	no	—
86	C₄×C₂₃	—	4	2	no	—
87	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
89	C₄×C₂₃	#423	1	1	yes	characteristic
90	C₄×C₂₃	#432	1	1	yes	characteristic
91	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
92	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
93	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
94	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
95	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
96	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
98	#376	C₄×C₂₃	1	1	yes	characteristic
99	#364	C₄×C₂₃	1	8	yes	complemented
100	#347	—	2	4	no	—
101	C₂×C₂₃	—	2	1	no	—
102	C₂×C₂₃	—	2	1	no	—
103	C₂×C₂₃	#1767	1	1	yes	characteristic
104	C₂×C₂₃	#2166	1	2	yes	—
105	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
106	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
109	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
112	M₃₂	—	2	4	no	—
113	M₃₂	C₂×C₄×C₂₃	1	8	yes	—
114	C₂×C₁₆	—	2	4	no	—
115	#97	—	2	2	no	—
116	#97	—	2	2	no	—
117	#97	D₄×C₂₃	1	2	yes	—
118	#97	Q₈×C₂₃	1	2	yes	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
122	#95	—	2	2	no	—
123	C₂₃	#12660	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	C₄²×C₂₃	1	2	yes	—
133	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
134	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
135	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
136	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
137	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
138	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
140	C₂×C₈	#41×C₂₃	1	2	yes	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₄²×C₂₃	1	2	yes	—
146	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
147	C₁₆	—	2	8	no	—
148	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
158	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
159	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
160	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	C₂₃×#97	1	4	yes	—
164	C₈	C₄×D₄×C₂₃	1	2	yes	—
165	C₈	C₄×Q₈×C₂₃	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	C₄×M₁₆×C₂₃	1	1	yes	characteristic
168	C₂²	C₂₃×#423	1	1	yes	characteristic
169	C₂²	C₂₃×#432	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
174	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
175	C₄	C₂₃×#423	1	1	yes	characteristic
176	C₄	C₂₃×#432	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₂₃×#1767	1	1	yes	characteristic
180	C₂	C₂₃×#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#117
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	#432
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×#364	C₄	semidirect
57. #12660	C₂₃	direct (N × H)
99. #364	C₄×C₂₃	semidirect
123. C₂₃	#12660	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#364	2
C₂×C₂₃×#347	2
C₂₃×#1726	2
C₂₃×#1377	2
#12660	23

Subgroups — C23×#12660 #19160286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12660 #19160286