Subgroups — C₂₃×#3383 #15671002

Order 2944.

324 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#557	C₂	1	2	yes	maximal
2	C₂₃×#500	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#499	C₂	1	2	yes	maximal
4	C₂₃×#479	C₂	1	4	yes	maximal
5	C₂₃×#475	C₂	1	2	yes	maximal
6	C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₁₆×C₂₃	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×C₄⋊Q₈	—	2	2	no	—
11	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂₃×#125	—	2	4	no	—
13	C₂₃×#125	C₂²	1	4	yes	—
14	C₄×Q₈×C₂₃	—	2	2	no	—
15	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
16	C₄×Q₈×C₂₃	C₂²	1	4	yes	—
17	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#107	C₂²	1	1	yes	characteristic
19	C₂₃×#107	C₂²	1	2	yes	—
20	C₂₃×#107	C₂²	1	2	yes	—
21	C₂₃×#106	C₂²	1	4	yes	—
22	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂₃×#103	C₂²	1	4	yes	—
25	C₂₃×#103	C₂²	1	2	yes	—
26	C₄×C₈×C₂₃	C₂²	1	2	yes	—
27	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₂₃	—	2	2	no	—
29	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
30	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
31	Q₁₆×C₂₃	Q₈	1	4	yes	complemented
32	C₂×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₂₃	—	2	4	no	—
36	C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₄⋊C₄×C₂₃	—	2	4	no	—
40	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
42	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
43	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
44	C₄²×C₂₃	—	2	4	no	—
45	C₄²×C₂₃	—	2	2	no	—
46	C₄²×C₂₃	C₂³	1	2	yes	—
47	C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	Q₈×C₂₃	—	2	2	no	—
49	Q₈×C₂₃	—	2	2	no	—
50	Q₈×C₂₃	—	4	4	no	—
51	Q₈×C₂₃	C₂×Q₈	1	4	yes	—
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₈×C₂₃	—	2	2	no	—
61	C₈×C₂₃	—	2	1	no	—
62	C₈×C₂₃	C₂×Q₈	1	2	yes	—
63	#3383	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₂₃	—	2	4	no	—
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	—	2	2	no	—
68	C₄×C₂₃	—	2	1	no	—
69	C₄×C₂₃	—	4	4	no	—
70	C₄×C₂₃	—	4	2	no	—
71	C₄×C₂₃	#135	1	2	yes	—
72	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
73	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
74	#557	C₂×C₂₃	1	2	yes	—
75	#500	C₂×C₂₃	1	1	yes	characteristic
76	#499	C₂×C₂₃	1	2	yes	—
77	#479	C₂×C₂₃	1	4	yes	—
78	#475	C₂×C₂₃	1	2	yes	—
79	C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic
80	C₄×Q₁₆	C₂×C₂₃	1	2	yes	—
81	C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
82	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
83	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
84	C₂×C₂₃	#578	1	1	yes	characteristic
85	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
88	#125	—	2	4	no	—
89	#125	C₂²×C₂₃	1	4	yes	—
90	C₄×Q₈	—	2	2	no	—
91	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
92	C₄×Q₈	C₂²×C₂₃	1	4	yes	—
93	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
94	#107	C₂²×C₂₃	1	1	yes	characteristic
95	#107	C₂²×C₂₃	1	2	yes	—
96	#107	C₂²×C₂₃	1	2	yes	—
97	#106	C₂²×C₂₃	1	4	yes	—
98	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
99	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
100	#103	C₂²×C₂₃	1	4	yes	—
101	#103	C₂²×C₂₃	1	2	yes	—
102	C₄×C₈	C₂²×C₂₃	1	2	yes	—
103	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
104	C₂₃	#3383	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
107	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
108	Q₁₆	Q₈×C₂₃	1	4	yes	complemented
109	C₂×C₈	C₂³×C₂₃	1	2	yes	—
110	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
111	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
118	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
119	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
120	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
121	C₄²	—	2	4	no	—
122	C₄²	—	2	2	no	—
123	C₄²	C₂³×C₂₃	1	2	yes	—
124	C₄²	C₂³×C₂₃	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	2	2	no	—
127	Q₈	—	4	4	no	—
128	Q₈	C₂×Q₈×C₂₃	1	4	yes	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
133	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
134	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
135	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
136	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	1	no	—
139	C₈	C₂×Q₈×C₂₃	1	2	yes	—
140	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	4	no	—
146	C₄	—	4	2	no	—
147	C₄	C₂₃×#135	1	2	yes	—
148	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
149	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
150	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
151	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
152	C₂	C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
31. Q₁₆×C₂₃	Q₈	semidirect
63. #3383	C₂₃	direct (N × H)
104. C₂₃	#3383	direct (N × H)
108. Q₁₆	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×#557	2
C₂₃×#500	2
C₂₃×#499	2
C₂₃×#479	2
C₂₃×#475	2
C₈×Q₈×C₂₃	2
C₄×Q₁₆×C₂₃	2
#3383	23

Subgroups — C23×#3383 #15671002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3383 #15671002