Subgroups — C₂₃×#8295 #19155921

Order 5888.

334 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	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1451	C₂	1	2	yes	maximal
4	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1278	C₂	1	2	yes	maximal
6	C₂₃×#431	C₂²	1	2	yes	—
7	C₂×C₂₃×#108	C₄	1	2	yes	—
8	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#341	C₄	1	2	yes	—
13	C₂₃×#335	C₄	1	2	yes	—
14	C₂₃×#334	C₄	1	2	yes	—
15	C₂₃×#322	—	2	4	no	—
16	C₂₃×#322	C₄	1	2	yes	—
17	C₂₃×#322	C₄	1	2	yes	—
18	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₂₃	—	2	1	no	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	C₂₃×#108	—	2	4	no	—
25	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
26	C₂₃×C₂²⋊C₈	—	2	2	no	—
27	C₂₃×#97	—	2	2	no	—
28	C₄×C₈×C₂₃	—	2	2	no	—
29	C₄×C₈×C₂₃	—	2	2	no	—
30	C₄×C₈×C₂₃	—	2	1	no	—
31	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₂₃	D₄	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
41	M₁₆×C₂₃	—	2	4	no	—
42	M₁₆×C₂₃	—	2	4	no	—
43	C₂×C₈×C₂₃	—	2	2	no	—
44	C₂×C₈×C₂₃	—	2	2	no	—
45	C₂×C₈×C₂₃	—	2	4	no	—
46	C₂×C₈×C₂₃	—	2	4	no	—
47	C₂×C₈×C₂₃	—	4	2	no	—
48	C₂×C₈×C₂₃	—	4	2	no	—
49	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
54	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
55	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₂₃	#41	1	1	yes	characteristic
59	#8295	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₂₃	#131	1	2	yes	—
71	C₈×C₂₃	—	2	1	no	—
72	C₈×C₂₃	—	2	1	no	—
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	C₈×C₂₃	#97	1	2	yes	—
78	C₈×C₂₃	#97	1	2	yes	—
79	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
80	#1564	C₂×C₂₃	1	1	yes	characteristic
81	#1451	C₂×C₂₃	1	2	yes	—
82	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
83	#1278	C₂×C₂₃	1	2	yes	—
84	C₂²×C₂₃	—	2	1	no	—
85	C₂²×C₂₃	—	2	1	no	—
86	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₂₃	#446	1	2	yes	—
88	C₄×C₂₃	—	2	1	no	—
89	C₄×C₂₃	—	2	1	no	—
90	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
91	C₄×C₂₃	#405	1	1	yes	characteristic
92	C₄×C₂₃	#430	1	2	yes	—
93	C₄×C₂₃	#436	1	2	yes	—
94	C₄×C₂₃	#444	1	2	yes	—
95	#431	C₂²×C₂₃	1	2	yes	—
96	C₂×#108	C₄×C₂₃	1	2	yes	—
97	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
101	#341	C₄×C₂₃	1	2	yes	—
102	#335	C₄×C₂₃	1	2	yes	—
103	#334	C₄×C₂₃	1	2	yes	—
104	#322	—	2	4	no	—
105	#322	C₄×C₂₃	1	2	yes	—
106	#322	C₄×C₂₃	1	2	yes	—
107	C₂×C₂₃	—	2	2	no	—
108	C₂×C₂₃	#1770	1	1	yes	characteristic
109	C₂×C₂₃	#1773	1	1	yes	characteristic
110	C₂×C₂₃	#1778	1	1	yes	characteristic
111	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
115	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
116	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
117	#108	—	2	4	no	—
118	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
119	C₂²⋊C₈	—	2	2	no	—
120	#97	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
125	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
126	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
127	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
128	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
129	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
130	C₄×C₈	D₄×C₂₃	1	2	yes	—
131	C₂₃	#8295	1	1	yes	characteristic, direct product factor, Sylow
132	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
134	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
135	M₁₆	—	2	4	no	—
136	M₁₆	—	2	4	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
144	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
145	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
146	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
147	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
148	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
149	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
150	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
151	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
152	C₄²	#41×C₂₃	1	1	yes	characteristic
153	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
158	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
159	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
161	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
163	C₂×C₄	C₂₃×#131	1	2	yes	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	4	no	—
167	C₈	—	4	2	no	—
168	C₈	—	4	4	no	—
169	C₈	—	4	4	no	—
170	C₈	C₂₃×#97	1	2	yes	—
171	C₈	C₂₃×#97	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂²	C₂₃×#446	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
179	C₄	C₂₃×#405	1	1	yes	characteristic
180	C₄	C₂₃×#430	1	2	yes	—
181	C₄	C₂₃×#436	1	2	yes	—
182	C₄	C₂₃×#444	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	C₂₃×#1770	1	1	yes	characteristic
185	C₂	C₂₃×#1773	1	1	yes	characteristic
186	C₂	C₂₃×#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	C₄×C₄⋊C₄
prime radical	C₂³×C₂₃	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
59. #8295	C₂₃	direct (N × H)
131. C₂₃	#8295	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#108	2
C₂₃×#1564	2
C₂₃×#1451	2
C₂×C₂₃×#322	2
C₂₃×#1278	2
#8295	23

Subgroups — C23×#8295 #19155921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8295 #19155921