Subgroups — C₂₃×#2905 #15670524

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#424	C₂	1	8	yes	maximal, complemented
4	C₂₃×#423	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₂₃	—	2	8	no	—
7	C₂×M₁₆×C₂₃	C₂²	1	4	yes	—
8	C₂×M₁₆×C₂₃	C₂²	1	4	yes	—
9	C₂²×C₈×C₂₃	C₂²	1	4	yes	—
10	C₂₃×#117	C₄	1	4	yes	—
11	C₂₃×#117	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
13	C₂×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
15	C₂₃×C₄⋊C₈	C₂²	1	16	yes	complemented
16	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₃	D₄	1	4	yes	—
21	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
24	M₁₆×C₂₃	—	2	8	no	—
25	M₁₆×C₂₃	—	2	8	no	—
26	C₂×C₈×C₂₃	—	2	8	no	—
27	C₂×C₈×C₂₃	C₂³	1	8	yes	—
28	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
30	C₄²×C₂₃	C₂×C₄	1	4	yes	—
31	C₄²×C₂₃	C₂³	1	4	yes	—
32	C₂³×C₂₃	—	4	2	no	—
33	C₂³×C₂₃	C₄⋊C₄	1	4	yes	—
34	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
35	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	—	4	4	no	—
37	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
39	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
47	C₈×C₂₃	—	2	8	no	—
48	#2905	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂²×C₂₃	—	2	2	no	—
50	C₂²×C₂₃	—	2	4	no	—
51	C₂²×C₂₃	—	4	4	no	—
52	C₂²×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
53	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
54	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₂₃	—	2	8	no	—
56	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
58	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
60	C₂²×M₁₆	C₂×C₂₃	1	2	yes	—
61	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
62	#424	C₂×C₂₃	1	8	yes	complemented
63	#423	C₂×C₂₃	1	4	yes	complemented
64	C₂×C₂₃	—	2	2	no	—
65	C₂×C₂₃	—	2	4	no	—
66	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₂₃	#568	1	2	yes	—
68	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
69	C₂×M₁₆	—	2	8	no	—
70	C₂×M₁₆	C₂²×C₂₃	1	4	yes	—
71	C₂×M₁₆	C₂²×C₂₃	1	4	yes	—
72	C₂²×C₈	C₂²×C₂₃	1	4	yes	—
73	#117	C₄×C₂₃	1	4	yes	—
74	#117	C₂²×C₂₃	1	4	yes	—
75	C₂×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
76	C₂×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
77	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
78	C₄⋊C₈	C₂²×C₂₃	1	16	yes	complemented
79	C₂₃	#2905	1	1	yes	characteristic, direct product factor, Sylow
80	C₂⁴	C₂×C₄×C₂₃	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
82	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
83	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
84	C₂²×C₄	D₄×C₂₃	1	4	yes	—
85	C₂²×C₄	Q₈×C₂₃	1	4	yes	—
86	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
87	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
88	M₁₆	—	2	8	no	—
89	M₁₆	—	2	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	C₂³×C₂₃	1	8	yes	—
92	C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
93	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
94	C₄²	C₂×C₄×C₂₃	1	4	yes	—
95	C₄²	C₂³×C₂₃	1	4	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₄⋊C₄×C₂₃	1	4	yes	—
98	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
99	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
102	C₂×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
103	C₂×C₄	C₂²×C₄×C₂₃	1	8	yes	—
104	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
105	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
106	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
107	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
108	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
109	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
110	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
111	C₈	—	2	8	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
116	C₂²	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
117	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
120	C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
121	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
122	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
126	C₂	C₂₃×#568	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#424	C₂	semidirect
4. C₂₃×#423	C₂	semidirect
15. C₂₃×C₄⋊C₈	C₂²	semidirect
48. #2905	C₂₃	direct (N × H)
62. #424	C₂×C₂₃	semidirect
63. #423	C₂×C₂₃	semidirect
78. C₄⋊C₈	C₂²×C₂₃	semidirect
79. C₂₃	#2905	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₂₃	2
C₂×C₂₃×#117	2
C₂₃×#424	2
C₂₃×#423	2
#2905	23

Subgroups — C23×#2905 #15670524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2905 #15670524