Subgroups — C₂₃×#2330 #15669949

Order 2944.

444 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#389	C₂	1	2	yes	maximal
3	C₂₃×#389	C₂	1	4	yes	maximal
4	C₂₃×#387	C₂	1	4	yes	maximal
5	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
7	C₂₃×#376	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#125	C₄	1	8	yes	complemented
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#95	C₂²	1	2	yes	—
19	C₂₃×#95	C₂²	1	2	yes	—
20	C₂₃×#95	C₂²	1	4	yes	—
21	C₂₃×#95	C₂²	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₃	—	2	4	no	—
30	C₄⋊C₄×C₂₃	—	4	4	no	—
31	C₄⋊C₄×C₂₃	—	4	4	no	—
32	C₄⋊C₄×C₂₃	C₂×C₄	1	16	yes	—
33	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
34	C₄²×C₂₃	—	2	2	no	—
35	C₄²×C₂₃	—	4	4	no	—
36	C₄²×C₂₃	C₂×C₄	1	4	yes	—
37	C₄²×C₂₃	Q₈	1	4	yes	—
38	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	4	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	—	2	1	no	—
46	C₂×C₄×C₂₃	—	2	1	no	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
54	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
56	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₂₃	#41	1	2	yes	—
58	#2330	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
62	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
63	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
64	C₄×C₂₃	—	2	8	no	—
65	C₄×C₂₃	—	2	4	no	—
66	C₄×C₂₃	—	2	4	no	—
67	C₄×C₂₃	—	4	1	no	—
68	C₄×C₂₃	—	4	4	no	—
69	C₄×C₂₃	—	4	1	no	—
70	C₄×C₂₃	C₄×Q₈	1	4	yes	—
71	C₂×#125	C₂×C₂₃	1	1	yes	characteristic
72	#389	C₂×C₂₃	1	4	yes	—
73	#389	C₂×C₂₃	1	2	yes	—
74	#387	C₂×C₂₃	1	4	yes	—
75	#384	C₂×C₂₃	1	1	yes	characteristic
76	C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
77	#376	C₂×C₂₃	1	1	yes	characteristic
78	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
79	C₂×C₂₃	#520	1	2	yes	—
80	C₂×C₂₃	#540	1	1	yes	characteristic
81	C₂×C₂₃	#541	1	1	yes	characteristic
82	C₂×C₂₃	#544	1	2	yes	—
83	#125	C₄×C₂₃	1	8	yes	complemented
84	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
86	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
87	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
89	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
90	C₂×C₄²	C₂²×C₂₃	1	4	yes	—
91	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
93	#95	C₂²×C₂₃	1	2	yes	—
94	#95	C₂²×C₂₃	1	2	yes	—
95	#95	C₂²×C₂₃	1	4	yes	—
96	#95	C₂²×C₂₃	1	4	yes	—
97	C₂₃	#2330	1	1	yes	characteristic, direct product factor, Sylow
98	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
99	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
100	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
101	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
102	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
103	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
104	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	4	no	—
108	C₄⋊C₄	C₂×C₄×C₂₃	1	16	yes	—
109	C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
110	C₄²	—	2	2	no	—
111	C₄²	—	4	4	no	—
112	C₄²	C₂×C₄×C₂₃	1	4	yes	—
113	C₄²	Q₈×C₂₃	1	4	yes	—
114	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
129	C₂×C₄	C₂²×C₄×C₂₃	1	8	yes	—
130	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
131	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
132	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
133	C₂×C₄	#41×C₂₃	1	2	yes	—
134	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
135	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
136	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
137	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
138	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
139	C₄	—	2	8	no	—
140	C₄	—	2	4	no	—
141	C₄	—	2	4	no	—
142	C₄	—	4	1	no	—
143	C₄	—	4	4	no	—
144	C₄	—	4	1	no	—
145	C₄	C₄×Q₈×C₂₃	1	4	yes	—
146	C₂	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
147	C₂	C₂₃×#520	1	2	yes	—
148	C₂	C₂₃×#540	1	1	yes	characteristic
149	C₂	C₂₃×#541	1	1	yes	characteristic
150	C₂	C₂₃×#544	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
8. C₂₃×#125	C₄	semidirect
58. #2330	C₂₃	direct (N × H)
83. #125	C₄×C₂₃	semidirect
97. C₂₃	#2330	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×#125	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#384	2
C₄×C₄⋊C₄×C₂₃	2
C₂₃×#376	2
#2330	23

Subgroups — C23×#2330 #15669949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2330 #15669949