Subgroups — C₂₃×#7278 #19154904

Order 5888.

246 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×M₃₂	—	2	1	no	—
5	C₂₃×#431	—	2	1	no	—
6	C₂×C₂₃×#106	—	2	1	no	—
7	C₂×C₂₃×#106	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#384	—	2	1	no	—
10	C₂₃×C₄⋊C₁₆	—	2	1	no	—
11	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
12	C₂₃×#335	C₄	1	2	yes	complemented
13	C₂₃×#334	—	2	1	no	—
14	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
17	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
18	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
19	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₂₃×M₃₂	—	4	1	no	—
21	C₂₃×M₃₂	—	4	1	no	—
22	C₂×C₁₆×C₂₃	—	2	2	no	—
23	C₂₃×#106	—	4	1	no	—
24	C₂₃×#106	—	4	1	no	—
25	C₂₃×#106	—	4	2	no	—
26	C₂₃×C₄⋊C₈	—	2	2	no	—
27	C₂₃×C₂²⋊C₈	—	4	1	no	—
28	C₄×C₈×C₂₃	—	4	1	no	—
29	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	—	4	2	no	—
34	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	4	2	no	—
40	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
45	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
48	C₄⋊C₄×C₂₃	—	4	2	no	—
49	C₄⋊C₄×C₂₃	—	4	2	no	—
50	C₄⋊C₄×C₂₃	—	8	2	no	—
51	C₄⋊C₄×C₂₃	—	8	2	no	—
52	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
55	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
56	C₁₆×C₂₃	—	4	2	no	—
57	#7278	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₂₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	8	2	no	—
62	C₂×C₄×C₂₃	—	8	2	no	—
63	C₂×C₄×C₂₃	—	8	2	no	—
64	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#100	1	1	yes	characteristic
66	C₂×C₄×C₂₃	#101	1	1	yes	characteristic
67	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
68	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
69	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
70	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
71	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
72	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
73	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
74	C₈×C₂₃	—	2	1	no	—
75	C₈×C₂₃	—	2	1	no	—
76	C₈×C₂₃	—	8	2	no	—
77	C₈×C₂₃	#102	1	1	yes	characteristic
78	C₈×C₂₃	#103	1	1	yes	characteristic
79	C₈×C₂₃	#106	1	1	yes	characteristic
80	C₈×C₂₃	#107	1	1	yes	characteristic
81	#2150	C₂×C₂₃	1	1	yes	characteristic
82	#1844	C₂×C₂₃	1	1	yes	characteristic
83	#1564	C₂×C₂₃	1	1	yes	characteristic
84	C₂²×C₂₃	—	2	1	no	—
85	C₂²×C₂₃	—	2	1	no	—
86	C₂²×C₂₃	#331	1	1	yes	characteristic
87	C₂²×C₂₃	#332	1	1	yes	characteristic
88	C₂²×C₂₃	#340	1	1	yes	characteristic
89	C₄×C₂₃	—	2	1	no	—
90	C₄×C₂₃	—	2	1	no	—
91	C₄×C₂₃	—	16	2	no	—
92	C₄×C₂₃	#332	1	1	yes	characteristic
93	C₄×C₂₃	#333	1	1	yes	characteristic
94	C₄×C₂₃	#340	1	1	yes	characteristic
95	C₄×C₂₃	#343	1	1	yes	characteristic
96	C₄×C₂₃	#360	1	2	yes	—
97	C₄×C₂₃	#365	1	2	yes	—
98	C₂×M₃₂	—	2	1	no	—
99	#431	—	2	1	no	—
100	C₂×#106	—	2	1	no	—
101	C₂×#106	C₄×C₂₃	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
103	#384	—	2	1	no	—
104	C₄⋊C₁₆	—	2	1	no	—
105	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	complemented
106	#335	C₄×C₂₃	1	2	yes	complemented
107	#334	—	2	1	no	—
108	C₂×C₂₃	—	2	1	no	—
109	C₂×C₂₃	—	2	1	no	—
110	C₂×C₂₃	#1296	1	1	yes	characteristic
111	C₂×C₂₃	#1383	1	1	yes	characteristic
112	C₂×C₂₃	#1386	1	1	yes	characteristic
113	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	4	1	no	—
117	C₂×C₄⋊C₄	—	4	1	no	—
118	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
119	M₃₂	—	4	1	no	—
120	M₃₂	—	4	1	no	—
121	C₂×C₁₆	—	2	2	no	—
122	#106	—	4	1	no	—
123	#106	—	4	1	no	—
124	#106	—	4	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₂²⋊C₈	—	4	1	no	—
127	C₄×C₈	—	4	1	no	—
128	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
129	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
130	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
131	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
132	C₂₃	#7278	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
135	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
136	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
142	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
143	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
144	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
145	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
146	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
147	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	8	2	no	—
151	C₄⋊C₄	—	8	2	no	—
152	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
153	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
154	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
155	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
156	C₁₆	—	4	2	no	—
157	C₂³	C₂₃×#102	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
164	C₂×C₄	C₂₃×#100	1	1	yes	characteristic
165	C₂×C₄	C₂₃×#101	1	1	yes	characteristic
166	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
167	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
168	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
169	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
170	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
171	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
172	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	8	2	no	—
176	C₈	C₂₃×#102	1	1	yes	characteristic
177	C₈	C₂₃×#103	1	1	yes	characteristic
178	C₈	C₂₃×#106	1	1	yes	characteristic
179	C₈	C₂₃×#107	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₂₃×#331	1	1	yes	characteristic
183	C₂²	C₂₃×#332	1	1	yes	characteristic
184	C₂²	C₂₃×#340	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	16	2	no	—
188	C₄	C₂₃×#332	1	1	yes	characteristic
189	C₄	C₂₃×#333	1	1	yes	characteristic
190	C₄	C₂₃×#340	1	1	yes	characteristic
191	C₄	C₂₃×#343	1	1	yes	characteristic
192	C₄	C₂₃×#360	1	2	yes	—
193	C₄	C₂₃×#365	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	C₂₃×#1296	1	1	yes	characteristic
197	C₂	C₂₃×#1383	1	1	yes	characteristic
198	C₂	C₂₃×#1386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#340
commutator	C₂×C₈	C₄²×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
prime radical	C₂³×C₂₃	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
11. C₂₃×C₄⋊C₁₆	C₄	semidirect
12. C₂₃×#335	C₄	semidirect
57. #7278	C₂₃	direct (N × H)
105. C₄⋊C₁₆	C₄×C₂₃	semidirect
106. #335	C₄×C₂₃	semidirect
132. C₂₃	#7278	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2150	2
C₂₃×#1844	2
C₂₃×#1564	2
#7278	23

Subgroups — C23×#7278 #19154904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7278 #19154904