Subgroups — C₂₃×#2961 #15670580

Order 2944.

348 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	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#446	C₂	1	8	yes	maximal, complemented
3	C₂₃×#424	C₂	1	2	yes	maximal
4	C₂₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#405	C₂	1	2	yes	maximal
6	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×C₄⋊Q₈	C₄	1	2	yes	—
11	C₂₃×#125	C₄	1	2	yes	—
12	C₂₃×C₂²⋊Q₈	C₄	1	4	yes	—
13	C₄×Q₈×C₂₃	C₂²	1	4	yes	—
14	C₂₃×#117	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
17	C₂₃×C₄⋊C₈	C₂²	1	8	yes	—
18	C₂₃×#97	C₂²	1	4	yes	—
19	C₂₃×#97	C₂²	1	4	yes	—
20	C₄×C₈×C₂₃	C₂²	1	4	yes	—
21	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	M₁₆×C₂₃	—	2	4	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	Q₈	1	4	yes	complemented
27	C₂×C₈×C₂₃	C₂³	1	2	yes	—
28	C₂×C₈×C₂₃	C₂³	1	4	yes	—
29	C₂×C₈×C₂₃	C₂³	1	2	yes	—
30	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
31	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
32	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
34	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₂₃	C₂³	1	2	yes	—
37	C₄²×C₂₃	C₂³	1	4	yes	—
38	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₂₃	—	4	4	no	—
40	C₂×C₄×C₂₃	—	4	2	no	—
41	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
48	C₈×C₂₃	—	2	2	no	—
49	C₈×C₂₃	—	2	4	no	—
50	C₈×C₂₃	C₂×Q₈	1	4	yes	—
51	#2961	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₂₃	—	2	2	no	—
53	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
54	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₂₃	—	2	2	no	—
56	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	—	2	8	no	—
58	C₄×C₂₃	C₄×Q₈	1	2	yes	—
59	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
60	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
61	#533	C₂×C₂₃	1	1	yes	characteristic
62	#446	C₂×C₂₃	1	8	yes	complemented
63	#424	C₂×C₂₃	1	2	yes	—
64	#423	C₂×C₂₃	1	1	yes	characteristic
65	#405	C₂×C₂₃	1	2	yes	—
66	C₂×#97	C₂×C₂₃	1	1	yes	characteristic
67	C₂×C₂₃	—	2	2	no	—
68	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
69	C₂×C₂₃	#568	1	2	yes	—
70	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
71	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
73	C₄⋊Q₈	C₄×C₂₃	1	2	yes	—
74	#125	C₄×C₂₃	1	2	yes	—
75	C₂²⋊Q₈	C₄×C₂₃	1	4	yes	—
76	C₄×Q₈	C₂²×C₂₃	1	4	yes	—
77	#117	C₂²×C₂₃	1	2	yes	—
78	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
79	C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
80	C₄⋊C₈	C₂²×C₂₃	1	8	yes	—
81	#97	C₂²×C₂₃	1	4	yes	—
82	#97	C₂²×C₂₃	1	4	yes	—
83	C₄×C₈	C₂²×C₂₃	1	4	yes	—
84	C₂₃	#2961	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	—
86	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
87	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Q₈×C₂₃	1	4	yes	complemented
91	C₂×C₈	C₂³×C₂₃	1	4	yes	—
92	C₂×C₈	C₂³×C₂₃	1	2	yes	—
93	C₂×C₈	C₂³×C₂₃	1	2	yes	—
94	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
95	C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
96	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
97	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
98	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
99	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
100	C₄²	C₂³×C₂₃	1	4	yes	—
101	C₄²	C₂³×C₂₃	1	2	yes	—
102	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
106	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
107	C₂×C₄	C₂²×C₄×C₂₃	1	8	yes	—
108	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
109	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
110	C₂×C₄	#41×C₂₃	1	2	yes	—
111	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	4	no	—
114	C₈	C₂×Q₈×C₂₃	1	4	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₄×Q₈×C₂₃	1	2	yes	—
117	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	C₄×Q₈×C₂₃	1	2	yes	—
122	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
123	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×C₄×Q₈×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	C₂³×C₂₃	C₂²×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
2. C₂₃×#446	C₂	semidirect
26. C₂×C₈×C₂₃	Q₈	semidirect
51. #2961	C₂₃	direct (N × H)
62. #446	C₂×C₂₃	semidirect
84. C₂₃	#2961	direct (N × H)
90. C₂×C₈	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#533	2
C₂₃×#446	2
C₂₃×#424	2
C₂₃×#423	2
C₂₃×#405	2
C₂×C₂₃×#97	2
#2961	23

Subgroups — C23×#2961 #15670580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2961 #15670580