Subgroups — C₂₃×#12881 #19160507

Order 5888.

298 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	2	yes	maximal
2	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1377	C₂	1	2	yes	maximal
4	C₂₃×#1375	C₂	1	2	yes	maximal
5	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
6	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#376	C₄	1	1	yes	characteristic
13	C₂₃×#364	—	2	4	no	—
14	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂₃×M₃₂	—	2	4	no	—
22	C₂₃×M₃₂	—	2	4	no	—
23	C₂×C₁₆×C₂₃	—	2	4	no	—
24	C₂₃×#97	—	2	2	no	—
25	C₂₃×#97	—	2	2	no	—
26	C₂₃×#97	—	2	2	no	—
27	C₄×C₈×C₂₃	—	2	2	no	—
28	C₄×C₈×C₂₃	C₈	1	4	yes	—
29	C₄×C₈×C₂₃	D₄	1	2	yes	—
30	C₄×C₈×C₂₃	Q₈	1	2	yes	—
31	C₂₃×#95	—	2	2	no	—
32	C₂²×C₄×C₂₃	—	2	1	no	—
33	C₂²×C₄×C₂₃	—	2	1	no	—
34	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₂₃	—	2	1	no	—
37	C₂×C₈×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	2	1	no	—
40	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₂₃	#41	1	2	yes	—
48	C₂×C₈×C₂₃	#41	1	2	yes	—
49	C₄²×C₂₃	—	2	1	no	—
50	C₄²×C₂₃	—	2	1	no	—
51	C₄²×C₂₃	—	2	1	no	—
52	C₄²×C₂₃	—	2	1	no	—
53	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₂₃	C₂×C₈	1	2	yes	—
55	C₁₆×C₂₃	—	4	4	no	—
56	#12881	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	4	2	no	—
62	C₂×C₄×C₂₃	—	4	2	no	—
63	C₂×C₄×C₂₃	—	4	2	no	—
64	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₂₃	—	2	2	no	—
71	C₈×C₂₃	—	2	2	no	—
72	C₈×C₂₃	C₄⋊C₈	1	4	yes	—
73	C₈×C₂₃	#124	1	2	yes	—
74	C₈×C₂₃	#125	1	2	yes	—
75	C₂×#364	C₂×C₂₃	1	2	yes	—
76	#1726	C₂×C₂₃	1	1	yes	characteristic
77	#1377	C₂×C₂₃	1	2	yes	—
78	#1375	C₂×C₂₃	1	2	yes	—
79	C₂²×C₂₃	—	2	2	no	—
80	C₂²×C₂₃	#423	1	1	yes	characteristic
81	C₂²×C₂₃	#431	1	1	yes	characteristic
82	C₂²×C₂₃	#432	1	1	yes	characteristic
83	C₄×C₂₃	—	4	2	no	—
84	C₄×C₂₃	—	4	2	no	—
85	C₄×C₂₃	—	4	2	no	—
86	C₄×C₂₃	#382	1	1	yes	characteristic
87	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₄×C₂₃	#431	1	1	yes	characteristic
89	C₄×C₂₃	#432	1	1	yes	characteristic
90	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
91	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
92	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
93	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
94	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
95	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
97	#376	C₄×C₂₃	1	1	yes	characteristic
98	#364	—	2	4	no	—
99	C₂×C₂₃	—	2	1	no	—
100	C₂×C₂₃	—	2	1	no	—
101	C₂×C₂₃	#1831	1	1	yes	characteristic
102	C₂×C₂₃	#2166	1	2	yes	—
103	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
104	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
105	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
109	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
110	M₃₂	—	2	4	no	—
111	M₃₂	—	2	4	no	—
112	C₂×C₁₆	—	2	4	no	—
113	#97	—	2	2	no	—
114	#97	—	2	2	no	—
115	#97	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	C₈×C₂₃	1	4	yes	—
118	C₄×C₈	D₄×C₂₃	1	2	yes	—
119	C₄×C₈	Q₈×C₂₃	1	2	yes	—
120	#95	—	2	2	no	—
121	C₂₃	#12881	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
125	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
131	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
132	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
133	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
134	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
135	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
136	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
137	C₂×C₈	#41×C₂₃	1	2	yes	—
138	C₂×C₈	#41×C₂₃	1	2	yes	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
144	C₄²	C₂×C₈×C₂₃	1	2	yes	—
145	C₁₆	—	4	4	no	—
146	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
155	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
158	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	C₂₃×C₄⋊C₈	1	4	yes	—
162	C₈	C₂₃×#124	1	2	yes	—
163	C₈	C₂₃×#125	1	2	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	C₂₃×#423	1	1	yes	characteristic
166	C₂²	C₂₃×#431	1	1	yes	characteristic
167	C₂²	C₂₃×#432	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	C₂₃×#382	1	1	yes	characteristic
172	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
173	C₄	C₂₃×#431	1	1	yes	characteristic
174	C₄	C₂₃×#432	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	C₂₃×#1831	1	1	yes	characteristic
178	C₂	C₂₃×#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#117
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	#432
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
56. #12881	C₂₃	direct (N × H)
121. C₂₃	#12881	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×#364	2
C₂₃×#1726	2
C₂₃×#1377	2
C₂₃×#1375	2
#12881	23

Subgroups — C23×#12881 #19160507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12881 #19160507