Subgroups — C₂₃×#8277 #19155903

Order 5888.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#334	C₂	1	2	yes	maximal
5	C₂₃×#1277	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₂₃×#97	—	2	1	no	—
9	C₂×C₄×C₈×C₂₃	—	2	1	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₄³×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#336	—	2	1	no	—
14	C₂₃×#336	—	2	1	no	—
15	C₂₃×#336	C₄	1	4	yes	—
16	C₂₃×#334	C₄	1	8	yes	complemented
17	C₂²×C₈×C₂₃	—	2	2	no	—
18	C₂²×C₈×C₂₃	—	2	2	no	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×C₄⋊C₈	—	2	2	no	—
27	C₂₃×C₄⋊C₈	—	2	2	no	—
28	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
30	C₂₃×#97	—	4	2	no	—
31	C₄×C₈×C₂₃	—	2	2	no	—
32	C₄×C₈×C₂₃	—	4	2	no	—
33	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₂₃	—	2	4	no	—
40	C₂×C₈×C₂₃	—	2	2	no	—
41	C₂×C₈×C₂₃	—	2	4	no	—
42	C₂×C₈×C₂₃	—	2	4	no	—
43	C₂×C₈×C₂₃	—	2	4	no	—
44	C₂×C₈×C₂₃	—	4	2	no	—
45	C₂×C₈×C₂₃	—	4	2	no	—
46	C₂×C₈×C₂₃	—	4	2	no	—
47	C₂×C₈×C₂₃	—	4	2	no	—
48	C₂×C₈×C₂₃	C₄²	1	4	yes	—
49	C₄²×C₂₃	—	2	4	no	—
50	C₄²×C₂₃	—	2	2	no	—
51	C₄²×C₂₃	—	4	2	no	—
52	C₄²×C₂₃	SD₁₆	1	2	yes	—
53	C₄²×C₂₃	SD₁₆	1	2	yes	—
54	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
55	C₄²×C₂₃	#41	1	2	yes	—
56	#8277	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₂₃	C₂×C₄⋊C₄	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	—	2	2	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	#106	1	4	yes	—
65	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₂₃	#131	1	4	yes	—
70	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₂₃	#135	1	2	yes	—
73	C₈×C₂₃	—	2	4	no	—
74	C₈×C₂₃	—	4	2	no	—
75	C₈×C₂₃	—	4	4	no	—
76	C₈×C₂₃	—	4	4	no	—
77	#1767	C₂×C₂₃	1	1	yes	characteristic
78	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
79	#1744	C₂×C₂₃	1	1	yes	characteristic
80	C₂×#334	C₂×C₂₃	1	2	yes	—
81	#1277	C₂×C₂₃	1	2	yes	—
82	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₂₃	#424	1	2	yes	—
84	C₂²×C₂₃	C₂×#106	1	1	yes	characteristic
85	C₂²×C₂₃	#427	1	1	yes	characteristic
86	C₂²×C₂₃	#430	1	2	yes	—
87	C₄×C₂₃	—	2	4	no	—
88	C₄×C₂₃	—	2	2	no	—
89	C₄×C₂₃	—	4	2	no	—
90	C₄×C₂₃	#405	1	2	yes	—
91	C₄×C₂₃	#433	1	2	yes	—
92	C₄×C₂₃	C₄×SD₁₆	1	2	yes	—
93	C₄×C₂₃	C₄×SD₁₆	1	2	yes	—
94	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
95	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
96	C₂×#97	—	2	1	no	—
97	C₂×C₄×C₈	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
100	C₄³	C₂²×C₂₃	1	1	yes	characteristic
101	#336	—	2	1	no	—
102	#336	—	2	1	no	—
103	#336	C₄×C₂₃	1	4	yes	—
104	#334	C₄×C₂₃	1	8	yes	complemented
105	C₂×C₂₃	#1570	1	2	yes	—
106	C₂×C₂₃	#1572	1	2	yes	—
107	C₂×C₂₃	#1768	1	1	yes	characteristic
108	C₂×C₂₃	C₄×#106	1	1	yes	characteristic
109	C₂×C₂₃	#1778	1	1	yes	characteristic
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
113	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
115	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
116	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
117	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
118	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
122	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
123	#97	—	4	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	4	2	no	—
126	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
127	C₂₃	#8277	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
129	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
130	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	C₄²×C₂₃	1	4	yes	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	SD₁₆×C₂₃	1	2	yes	—
147	C₄²	SD₁₆×C₂₃	1	2	yes	—
148	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
149	C₄²	#41×C₂₃	1	2	yes	—
150	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	C₂₃×#106	1	4	yes	—
158	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
159	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
160	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
161	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
162	C₂×C₄	C₂₃×#131	1	4	yes	—
163	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
164	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
165	C₂×C₄	C₂₃×#135	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	4	2	no	—
168	C₈	—	4	4	no	—
169	C₈	—	4	4	no	—
170	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
171	C₂²	C₂₃×#424	1	2	yes	—
172	C₂²	C₂×C₂₃×#106	1	1	yes	characteristic
173	C₂²	C₂₃×#427	1	1	yes	characteristic
174	C₂²	C₂₃×#430	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	C₂₃×#405	1	2	yes	—
179	C₄	C₂₃×#433	1	2	yes	—
180	C₄	C₄×SD₁₆×C₂₃	1	2	yes	—
181	C₄	C₄×SD₁₆×C₂₃	1	2	yes	—
182	C₂	C₂₃×#1570	1	2	yes	—
183	C₂	C₂₃×#1572	1	2	yes	—
184	C₂	C₂₃×#1768	1	1	yes	characteristic
185	C₂	C₄×C₂₃×#106	1	1	yes	characteristic
186	C₂	C₂₃×#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
16. C₂₃×#334	C₄	semidirect
56. #8277	C₂₃	direct (N × H)
104. #334	C₄×C₂₃	semidirect
127. C₂₃	#8277	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×#1767	2
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1744	2
C₂×C₂₃×#334	2
C₂₃×#1277	2
#8277	23

Subgroups — C23×#8277 #19155903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8277 #19155903