Subgroups — C₂₃×#9610 #19157236

Order 5888.

294 conjugacy classes of subgroups.

The subgroup lattice has 194 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×#1565	C₂	1	2	yes	maximal
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1283	C₂	1	2	yes	maximal
6	C₂₃×#431	C₂²	1	1	yes	characteristic
7	C₂₃×#431	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#108	C₄	1	2	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#341	C₄	1	1	yes	characteristic
13	C₂₃×#341	C₄	1	1	yes	characteristic
14	C₂₃×#335	—	2	2	no	—
15	C₂₃×#335	—	2	2	no	—
16	C₂₃×#334	—	2	2	no	—
17	C₂₃×#322	—	2	2	no	—
18	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	C₂₃×#108	—	2	4	no	—
25	C₂₃×C₄⋊C₈	—	2	2	no	—
26	C₂₃×C₄⋊C₈	—	2	2	no	—
27	C₂₃×C₄⋊C₈	—	2	1	no	—
28	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
29	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
30	C₂₃×C₂²⋊C₈	—	2	1	no	—
31	C₂₃×C₂²⋊C₈	Q₈	1	2	yes	—
32	C₂₃×#97	—	2	2	no	—
33	C₄×C₈×C₂₃	—	2	1	no	—
34	C₄×C₈×C₂₃	—	2	1	no	—
35	C₄×C₈×C₂₃	—	2	2	no	—
36	C₄×C₈×C₂₃	D₄	1	2	yes	—
37	C₄×C₈×C₂₃	Q₈	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	2	no	—
42	M₁₆×C₂₃	—	2	2	no	—
43	M₁₆×C₂₃	—	2	2	no	—
44	M₁₆×C₂₃	M₁₆	1	4	yes	—
45	C₂×C₈×C₂₃	—	2	2	no	—
46	C₂×C₈×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	—	2	1	no	—
50	C₂×C₈×C₂₃	—	2	1	no	—
51	C₂×C₈×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	4	2	no	—
53	C₂×C₈×C₂₃	—	8	2	no	—
54	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
56	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₂₃	#41	1	1	yes	characteristic
60	#9610	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₂₃	—	2	1	no	—
63	C₂×C₄×C₂₃	—	2	1	no	—
64	C₂×C₄×C₂₃	—	4	1	no	—
65	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₂₃	#131	1	2	yes	—
73	C₈×C₂₃	—	2	2	no	—
74	C₈×C₂₃	—	2	2	no	—
75	C₈×C₂₃	—	2	4	no	—
76	C₈×C₂₃	—	4	2	no	—
77	C₈×C₂₃	—	4	2	no	—
78	C₈×C₂₃	—	4	1	no	—
79	C₈×C₂₃	—	4	1	no	—
80	C₈×C₂₃	—	8	2	no	—
81	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
82	#1565	C₂×C₂₃	1	2	yes	—
83	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
84	#1452	C₂×C₂₃	1	1	yes	characteristic
85	#1283	C₂×C₂₃	1	2	yes	—
86	C₂²×C₂₃	—	2	1	no	—
87	C₂²×C₂₃	—	2	1	no	—
88	C₂²×C₂₃	#389	1	1	yes	characteristic
89	C₂²×C₂₃	#446	1	2	yes	—
90	C₄×C₂₃	—	2	1	no	—
91	C₄×C₂₃	—	2	1	no	—
92	C₄×C₂₃	—	2	1	no	—
93	C₄×C₂₃	#423	1	1	yes	characteristic
94	C₄×C₂₃	#424	1	1	yes	characteristic
95	C₄×C₂₃	#435	1	2	yes	—
96	C₄×C₂₃	#472	1	1	yes	characteristic
97	C₄×C₂₃	#473	1	1	yes	characteristic
98	#431	C₂²×C₂₃	1	1	yes	characteristic
99	#431	C₂²×C₂₃	1	1	yes	characteristic
100	C₂×#108	C₄×C₂₃	1	2	yes	—
101	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
102	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
103	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
104	#341	C₄×C₂₃	1	1	yes	characteristic
105	#341	C₄×C₂₃	1	1	yes	characteristic
106	#335	—	2	2	no	—
107	#335	—	2	2	no	—
108	#334	—	2	2	no	—
109	#322	—	2	2	no	—
110	C₂×C₂₃	—	2	2	no	—
111	C₂×C₂₃	#1940	1	1	yes	characteristic
112	C₂×C₂₃	#1952	1	1	yes	characteristic
113	C₂×C₂₃	#1953	1	1	yes	characteristic
114	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
115	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
118	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
119	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
120	#108	—	2	4	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	1	no	—
124	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
125	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
126	C₂²⋊C₈	—	2	1	no	—
127	C₂²⋊C₈	Q₈×C₂₃	1	2	yes	—
128	#97	—	2	2	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	D₄×C₂₃	1	2	yes	—
133	C₄×C₈	Q₈×C₂₃	1	2	yes	—
134	C₂₃	#9610	1	1	yes	characteristic, direct product factor, Sylow
135	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
137	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	M₁₆×C₂₃	1	4	yes	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	8	2	no	—
151	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
152	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
153	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
154	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
155	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
156	C₄²	#41×C₂₃	1	1	yes	characteristic
157	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
164	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
165	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
166	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
167	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
168	C₂×C₄	C₂₃×#131	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	4	no	—
172	C₈	—	4	2	no	—
173	C₈	—	4	2	no	—
174	C₈	—	4	1	no	—
175	C₈	—	4	1	no	—
176	C₈	—	8	2	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	C₂₃×#389	1	1	yes	characteristic
180	C₂²	C₂₃×#446	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₂₃×#423	1	1	yes	characteristic
185	C₄	C₂₃×#424	1	1	yes	characteristic
186	C₄	C₂₃×#435	1	2	yes	—
187	C₄	C₂₃×#472	1	1	yes	characteristic
188	C₄	C₂₃×#473	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	C₂₃×#1940	1	1	yes	characteristic
191	C₂	C₂₃×#1952	1	1	yes	characteristic
192	C₂	C₂₃×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#389
prime radical	C₂³×C₂₃	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
60. #9610	C₂₃	direct (N × H)
134. C₂₃	#9610	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×#1565	2
C₂×C₂₃×#335	2
C₂₃×#1452	2
C₂₃×#1283	2
#9610	23

Subgroups — C23×#9610 #19157236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9610 #19157236