Subgroups — C₂₃×#12009 #19159635

Order 5888.

242 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1372	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1371	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
8	C₂₃×#431	—	2	1	no	—
9	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#346	C₄	1	2	yes	—
12	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#335	—	2	1	no	—
14	C₂₃×#335	C₂²	1	2	yes	—
15	C₂₃×#335	C₂²	1	1	yes	characteristic
16	C₂₃×#335	C₂²	1	1	yes	characteristic
17	C₂₃×#334	—	2	1	no	—
18	C₂²×C₈×C₂₃	—	4	1	no	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
22	C₂₃×M₃₂	—	2	2	no	—
23	C₂₃×M₃₂	C₈	1	4	yes	complemented
24	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
26	C₂₃×C₄⋊C₈	—	2	1	no	—
27	C₂₃×C₄⋊C₈	—	2	1	no	—
28	C₂₃×C₄⋊C₈	—	2	2	no	—
29	C₂₃×C₄⋊C₈	—	4	1	no	—
30	C₂₃×C₂²⋊C₈	—	4	1	no	—
31	C₄×C₈×C₂₃	—	4	1	no	—
32	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₂₃	—	2	2	no	—
40	C₂×C₈×C₂₃	—	4	2	no	—
41	C₂×C₈×C₂₃	—	4	1	no	—
42	C₂×C₈×C₂₃	—	4	1	no	—
43	C₂×C₈×C₂₃	—	8	2	no	—
44	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
48	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₂₃	D₈	1	1	yes	characteristic
52	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
53	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₁₆×C₂₃	—	2	2	no	—
55	C₁₆×C₂₃	C₂×C₈	1	4	yes	complemented
56	#12009	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₂₃	#107	1	1	yes	characteristic
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	—	2	1	no	—
60	C₂×C₄×C₂₃	—	4	1	no	—
61	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
64	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#108	1	2	yes	—
66	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₈×C₂₃	—	2	1	no	—
70	C₈×C₂₃	—	2	1	no	—
71	C₈×C₂₃	—	8	1	no	—
72	C₈×C₂₃	—	8	2	no	—
73	C₈×C₂₃	—	8	1	no	—
74	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
75	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
76	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
77	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
78	#1564	C₂×C₂₃	1	1	yes	characteristic
79	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
80	#1372	C₂×C₂₃	1	2	yes	complemented
81	#1371	C₂×C₂₃	1	2	yes	complemented
82	C₂²×C₂₃	—	2	1	no	—
83	C₂²×C₂₃	—	2	1	no	—
84	C₂²×C₂₃	#335	1	2	yes	—
85	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
86	C₄×C₂₃	—	2	1	no	—
87	C₄×C₂₃	—	2	1	no	—
88	C₄×C₂₃	—	2	1	no	—
89	C₄×C₂₃	#335	1	2	yes	—
90	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₂₃	C₂×#108	1	1	yes	characteristic
92	C₄×C₂₃	#509	1	1	yes	characteristic
93	C₄×C₂₃	#510	1	1	yes	characteristic
94	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
95	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
96	#431	—	2	1	no	—
97	C₂×C₄⋊C₈	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
99	#346	C₄×C₂₃	1	2	yes	—
100	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
101	#335	—	2	1	no	—
102	#335	C₂²×C₂₃	1	2	yes	—
103	#335	C₂²×C₂₃	1	1	yes	characteristic
104	#335	C₂²×C₂₃	1	1	yes	characteristic
105	#334	—	2	1	no	—
106	C₂×C₂₃	—	2	2	no	—
107	C₂×C₂₃	C₂×#335	1	1	yes	characteristic
108	C₂×C₂₃	#2159	1	1	yes	characteristic
109	C₂×C₂₃	#2161	1	1	yes	characteristic
110	C₂²×C₈	—	4	1	no	—
111	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
112	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
113	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
114	M₃₂	—	2	2	no	—
115	M₃₂	C₈×C₂₃	1	4	yes	complemented
116	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
117	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
118	C₄⋊C₈	—	2	1	no	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	1	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
126	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
127	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
128	C₂₃	#12009	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
130	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
131	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	8	2	no	—
137	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
138	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
139	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
140	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
141	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
142	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
143	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
144	C₄²	D₈×C₂₃	1	1	yes	characteristic
145	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
146	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
147	C₁₆	—	2	2	no	—
148	C₁₆	C₂×C₈×C₂₃	1	4	yes	complemented
149	C₂³	C₂₃×#107	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
154	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
155	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
156	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
157	C₂×C₄	C₂₃×#108	1	2	yes	—
158	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
159	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
160	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	2	no	—
164	C₈	—	8	1	no	—
165	C₈	—	8	1	no	—
166	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
167	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
168	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₂₃×#335	1	2	yes	—
172	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	C₂₃×#335	1	2	yes	—
177	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
178	C₄	C₂×C₂₃×#108	1	1	yes	characteristic
179	C₄	C₂₃×#509	1	1	yes	characteristic
180	C₄	C₂₃×#510	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×C₂₃×#335	1	1	yes	characteristic
183	C₂	C₂₃×#2159	1	1	yes	characteristic
184	C₂	C₂₃×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#107
prime radical	C₂³×C₂₃	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1372	C₂	semidirect
5. C₂₃×#1371	C₂	semidirect
23. C₂₃×M₃₂	C₈	semidirect
55. C₁₆×C₂₃	C₂×C₈	semidirect
56. #12009	C₂₃	direct (N × H)
80. #1372	C₂×C₂₃	semidirect
81. #1371	C₂×C₂₃	semidirect
115. M₃₂	C₈×C₂₃	semidirect
128. C₂₃	#12009	direct (N × H)
148. C₁₆	C₂×C₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂₃×M₃₂	2
C₂₃×#1564	2
C₂×C₂₃×#335	2
C₂₃×#1372	2
C₂₃×#1371	2
#12009	23

Subgroups — C23×#12009 #19159635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12009 #19159635