Subgroups — C₂₃×#14354 #19161980

Order 5888.

1550 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#384	C₂	1	3	yes	maximal
2	C₂×C₂₃×#382	C₂	1	3	yes	maximal
3	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1444	C₂	1	8	yes	maximal
5	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
7	C₂²×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
9	C₂₃×#384	—	2	12	no	—
10	C₂₃×#382	—	2	12	no	—
11	C₂₃×#376	C₄	1	8	yes	complemented
12	C₂×C₂₃×#95	C₂²	1	12	yes	—
13	C₂×C₂₃×#95	C₂²	1	4	yes	—
14	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
16	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
19	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
20	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
22	C₂×C₄⋊C₄×C₂₃	—	4	24	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	12	yes	—
24	C₂×C₄²×C₂₃	D₄	1	12	yes	—
25	C₂×C₄²×C₂₃	Q₈	1	12	yes	—
26	C₂₃×#95	—	2	24	no	—
27	C₂₃×#95	C₂×C₄	1	16	yes	—
28	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	—	2	12	no	—
30	C₂²×C₄×C₂₃	—	2	12	no	—
31	C₂²×C₄×C₂₃	—	2	12	no	—
32	C₂²×C₄×C₂₃	—	2	6	no	—
33	C₂²×C₄×C₂₃	—	2	6	no	—
34	C₂²×C₄×C₂₃	—	2	24	no	—
35	C₂²×C₄×C₂₃	—	2	8	no	—
36	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
40	C₂²×C₄×C₂₃	C₂×Q₈	1	12	yes	—
41	C₂²×C₄×C₂₃	C₂×Q₈	1	6	yes	—
42	C₄⋊C₄×C₂₃	—	4	48	no	—
43	C₄⋊C₄×C₂₃	—	4	4	no	—
44	C₄⋊C₄×C₂₃	—	4	12	no	—
45	C₄²×C₂₃	C₄⋊C₄	1	48	yes	complemented
46	#14354	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
48	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
49	C₂³×C₂₃	C₂²×Q₈	1	3	yes	—
50	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
51	C₂³×C₂₃	2₋¹⁺⁴	1	4	yes	—
52	C₂×C₄×C₂₃	—	2	24	no	—
53	C₂×C₄×C₂₃	—	2	12	no	—
54	C₂×C₄×C₂₃	—	2	12	no	—
55	C₂×C₄×C₂₃	—	2	24	no	—
56	C₂×C₄×C₂₃	—	4	6	no	—
57	C₂×C₄×C₂₃	—	4	2	no	—
58	C₂×C₄×C₂₃	—	4	24	no	—
59	C₂×C₄×C₂₃	—	4	12	no	—
60	C₂×C₄×C₂₃	—	4	12	no	—
61	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	24	yes	—
62	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	12	yes	—
63	C₂×C₄×C₂₃	#127	1	4	yes	—
64	C₂×C₄×C₂₃	C₄⋊Q₈	1	12	yes	—
65	C₂×#384	C₂×C₂₃	1	3	yes	—
66	C₂×#382	C₂×C₂₃	1	3	yes	—
67	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
68	#1444	C₂×C₂₃	1	8	yes	—
69	C₂²×C₂₃	C₂²×C₄⋊C₄	1	3	yes	—
70	C₂²×C₂₃	#520	1	4	yes	—
71	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
72	C₂²×C₂₃	C₂×C₄⋊Q₈	1	3	yes	—
73	C₂²×C₂₃	#535	1	6	yes	—
74	C₂²×C₂₃	#536	1	6	yes	—
75	C₂²×C₂₃	#544	1	12	yes	—
76	C₄×C₂₃	—	2	24	no	—
77	C₄×C₂₃	—	8	6	no	—
78	C₄×C₂₃	—	8	2	no	—
79	C₄×C₂₃	#384	1	8	yes	—
80	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
81	C₂²×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
82	C₂²×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
83	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
84	#384	—	2	12	no	—
85	#382	—	2	12	no	—
86	#376	C₄×C₂₃	1	8	yes	complemented
87	C₂×#95	C₂²×C₂₃	1	12	yes	—
88	C₂×#95	C₂²×C₂₃	1	4	yes	—
89	C₂×C₂₃	C₂×#384	1	1	yes	characteristic
90	C₂×C₂₃	#2317	1	6	yes	—
91	C₂×C₂₃	#2657	1	1	yes	characteristic
92	C₂×C₂₃	#2658	1	1	yes	characteristic
93	C₂×C₂₃	#2660	1	6	yes	—
94	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
95	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
96	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
97	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	6	no	—
99	C₂×C₄⋊C₄	—	2	6	no	—
100	C₂×C₄⋊C₄	—	2	24	no	—
101	C₂×C₄⋊C₄	—	2	12	no	—
102	C₂×C₄⋊C₄	—	4	24	no	—
103	C₂×C₄²	C₂×C₄×C₂₃	1	12	yes	—
104	C₂×C₄²	D₄×C₂₃	1	12	yes	—
105	C₂×C₄²	Q₈×C₂₃	1	12	yes	—
106	#95	—	2	24	no	—
107	#95	C₂×C₄×C₂₃	1	16	yes	—
108	C₂₃	#14354	1	1	yes	characteristic, direct product factor, Sylow
109	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
110	C₂²×C₄	—	2	12	no	—
111	C₂²×C₄	—	2	12	no	—
112	C₂²×C₄	—	2	6	no	—
113	C₂²×C₄	—	2	6	no	—
114	C₂²×C₄	—	2	12	no	—
115	C₂²×C₄	—	2	24	no	—
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
118	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
119	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
120	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
121	C₂²×C₄	C₂×Q₈×C₂₃	1	12	yes	—
122	C₂²×C₄	C₂×Q₈×C₂₃	1	6	yes	—
123	C₄⋊C₄	—	4	48	no	—
124	C₄⋊C₄	—	4	4	no	—
125	C₄⋊C₄	—	4	12	no	—
126	C₄²	C₄⋊C₄×C₂₃	1	48	yes	complemented
127	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
128	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
129	C₂³	C₂²×Q₈×C₂₃	1	3	yes	—
130	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
131	C₂³	C₂₃×2₋¹⁺⁴	1	4	yes	—
132	C₂×C₄	—	2	24	no	—
133	C₂×C₄	—	2	12	no	—
134	C₂×C₄	—	2	12	no	—
135	C₂×C₄	—	2	24	no	—
136	C₂×C₄	—	4	6	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	24	no	—
139	C₂×C₄	—	4	12	no	—
140	C₂×C₄	—	4	12	no	—
141	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	24	yes	—
142	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	12	yes	—
143	C₂×C₄	C₂₃×#127	1	4	yes	—
144	C₂×C₄	C₂₃×C₄⋊Q₈	1	12	yes	—
145	C₂²	C₂²×C₄⋊C₄×C₂₃	1	3	yes	—
146	C₂²	C₂₃×#520	1	4	yes	—
147	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
148	C₂²	C₂×C₂₃×C₄⋊Q₈	1	3	yes	—
149	C₂²	C₂₃×#535	1	6	yes	—
150	C₂²	C₂₃×#536	1	6	yes	—
151	C₂²	C₂₃×#544	1	12	yes	—
152	C₄	—	2	24	no	—
153	C₄	—	8	6	no	—
154	C₄	—	8	2	no	—
155	C₄	C₂₃×#384	1	8	yes	—
156	C₂	C₂×C₂₃×#384	1	1	yes	characteristic
157	C₂	C₂₃×#2317	1	6	yes	—
158	C₂	C₂₃×#2657	1	1	yes	characteristic
159	C₂	C₂₃×#2658	1	1	yes	characteristic
160	C₂	C₂₃×#2660	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₂₃×#376	C₄	semidirect
45. C₄²×C₂₃	C₄⋊C₄	semidirect
46. #14354	C₂₃	direct (N × H)
86. #376	C₄×C₂₃	semidirect
108. C₂₃	#14354	direct (N × H)
126. C₄²	C₄⋊C₄×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×#384	2
C₂×C₂₃×#382	2
C₂×C₂₃×#376	2
C₂₃×#1444	2
#14354	23

Subgroups — C23×#14354 #19161980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#14354 #19161980