Subgroups — C₂₃×#12879 #19160505

Order 5888.

366 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1374	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₂₃	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₄³×C₂₃	C₄	1	1	yes	characteristic
10	C₂₃×C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₈	1	2	yes	—
15	C₂×C₄²×C₂₃	C₈	1	1	yes	characteristic
16	C₂×C₄²×C₂₃	C₈	1	1	yes	characteristic
17	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₂₃	—	2	4	no	—
20	C₂×C₁₆×C₂₃	—	2	8	no	—
21	C₄×C₈×C₂₃	—	2	4	no	—
22	C₄×C₈×C₂₃	—	2	4	no	—
23	C₄×C₈×C₂₃	—	2	2	no	—
24	C₄×C₈×C₂₃	D₄	1	2	yes	—
25	C₄×C₈×C₂₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₃	—	2	2	no	—
32	C₂×C₈×C₂₃	—	2	2	no	—
33	C₂×C₈×C₂₃	—	2	2	no	—
34	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
36	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₂₃	#41	1	4	yes	—
39	C₄²×C₂₃	—	2	4	no	—
40	C₄²×C₂₃	—	2	2	no	—
41	C₄²×C₂₃	—	2	2	no	—
42	C₄²×C₂₃	—	2	2	no	—
43	C₄²×C₂₃	C₁₆	1	4	yes	complemented
44	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
45	C₄²×C₂₃	M₁₆	1	2	yes	—
46	C₁₆×C₂₃	—	4	4	no	—
47	#12879	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₂₃	C₂×C₁₆	1	4	yes	—
57	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₂₃	M₃₂	1	4	yes	—
59	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	#117	1	2	yes	—
62	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
63	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₂₃	—	2	2	no	—
65	C₈×C₂₃	—	2	4	no	—
66	C₈×C₂₃	#124	1	2	yes	—
67	C₈×C₂₃	#125	1	2	yes	—
68	C₂×C₄⋊C₁₆	C₂×C₂₃	1	2	yes	—
69	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
70	#1374	C₂×C₂₃	1	4	yes	—
71	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₂²×C₂₃	#431	1	2	yes	—
73	C₂²×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
74	C₂²×C₂₃	C₂×M₃₂	1	1	yes	characteristic
75	C₂²×C₂₃	C₂×M₃₂	1	2	yes	—
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	—	2	4	no	—
78	C₄×C₂₃	—	2	4	no	—
79	C₄×C₂₃	C₄⋊C₁₆	1	4	yes	—
80	C₄×C₂₃	#382	1	1	yes	characteristic
81	C₄×C₂₃	#423	1	1	yes	characteristic
82	C₄×C₂₃	#432	1	2	yes	—
83	C₂²×C₁₆	C₂²×C₂₃	1	4	yes	—
84	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
85	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
87	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
88	C₄³	C₄×C₂₃	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	4	no	—
90	C₂×C₂₃	#1831	1	1	yes	characteristic
91	C₂×C₂₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
92	C₂×C₂₃	#2150	1	1	yes	characteristic
93	C₂×C₂₃	#2162	1	2	yes	—
94	C₂×C₂₃	#2163	1	2	yes	—
95	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
96	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
97	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
98	C₂×C₄²	C₈×C₂₃	1	2	yes	—
99	C₂×C₄²	C₈×C₂₃	1	1	yes	characteristic
100	C₂×C₄²	C₈×C₂₃	1	1	yes	characteristic
101	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
102	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	—	2	8	no	—
105	C₄×C₈	—	2	4	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	D₄×C₂₃	1	2	yes	—
109	C₄×C₈	Q₈×C₂₃	1	2	yes	—
110	C₂₃	#12879	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
113	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
114	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
120	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
121	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
122	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
123	C₂×C₈	#41×C₂₃	1	4	yes	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	C₁₆×C₂₃	1	4	yes	complemented
129	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
130	C₄²	M₁₆×C₂₃	1	2	yes	—
131	C₁₆	—	4	4	no	—
132	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
139	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
140	C₂×C₄	C₂×C₁₆×C₂₃	1	4	yes	—
141	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
142	C₂×C₄	C₂₃×M₃₂	1	4	yes	—
143	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
145	C₂×C₄	C₂₃×#117	1	2	yes	—
146	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
147	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	4	no	—
150	C₈	C₂₃×#124	1	2	yes	—
151	C₈	C₂₃×#125	1	2	yes	—
152	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
153	C₂²	C₂₃×#431	1	2	yes	—
154	C₂²	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
155	C₂²	C₂×C₂₃×M₃₂	1	1	yes	characteristic
156	C₂²	C₂×C₂₃×M₃₂	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	2	4	no	—
160	C₄	C₂₃×C₄⋊C₁₆	1	4	yes	—
161	C₄	C₂₃×#382	1	1	yes	characteristic
162	C₄	C₂₃×#423	1	1	yes	characteristic
163	C₄	C₂₃×#432	1	2	yes	—
164	C₂	C₂₃×#1831	1	1	yes	characteristic
165	C₂	C₂×C₂₃×C₄⋊C₁₆	1	1	yes	characteristic
166	C₂	C₂₃×#2150	1	1	yes	characteristic
167	C₂	C₂₃×#2162	1	2	yes	—
168	C₂	C₂₃×#2163	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₂₃	C₂³
commutator	C₂²	C₂²×C₁₆×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
43. C₄²×C₂₃	C₁₆	semidirect
47. #12879	C₂₃	direct (N × H)
110. C₂₃	#12879	direct (N × H)
128. C₄²	C₁₆×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×C₄⋊C₁₆	2
C₄²×C₈×C₂₃	2
C₂₃×#1374	2
#12879	23

Subgroups — C23×#12879 #19160505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12879 #19160505