Subgroups — C₂₃×#9607 #19157233

Order 5888.

306 conjugacy classes of subgroups.

The subgroup lattice has 216 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×#334	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1450	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	1	yes	characteristic
9	C₂×C₂₃×#108	C₄	1	1	yes	characteristic
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#341	C₄	1	2	yes	—
14	C₂₃×#335	—	2	2	no	—
15	C₂₃×#334	—	2	1	no	—
16	C₂₃×#334	—	2	1	no	—
17	C₂₃×#334	—	2	2	no	—
18	C₂₃×#322	—	2	1	no	—
19	C₈²×C₂₃	—	2	1	no	—
20	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×#108	—	2	2	no	—
27	C₂₃×#108	C₈	1	4	yes	complemented
28	C₂₃×C₄⋊C₈	—	2	2	no	—
29	C₂₃×C₄⋊C₈	—	2	2	no	—
30	C₂₃×C₄⋊C₈	—	2	1	no	—
31	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
32	C₂₃×C₂²⋊C₈	—	2	1	no	—
33	C₂₃×C₂²⋊C₈	Q₈	1	2	yes	—
34	C₂₃×#97	—	2	2	no	—
35	C₄×C₈×C₂₃	—	2	1	no	—
36	C₄×C₈×C₂₃	—	2	1	no	—
37	C₄×C₈×C₂₃	—	2	2	no	—
38	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
45	M₁₆×C₂₃	—	2	2	no	—
46	M₁₆×C₂₃	—	2	2	no	—
47	M₁₆×C₂₃	—	2	2	no	—
48	M₁₆×C₂₃	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	2	1	no	—
51	C₂×C₈×C₂₃	—	2	1	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	4	2	no	—
56	C₂×C₈×C₂₃	—	8	2	no	—
57	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
63	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₂₃	#41	1	1	yes	characteristic
67	#9607	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	4	1	no	—
71	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
81	C₈×C₂₃	—	2	1	no	—
82	C₈×C₂₃	—	2	1	no	—
83	C₈×C₂₃	—	2	4	no	—
84	C₈×C₂₃	—	4	2	no	—
85	C₈×C₂₃	—	4	2	no	—
86	C₈×C₂₃	—	4	2	no	—
87	C₈×C₂₃	—	8	2	no	—
88	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
89	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
90	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
91	#1565	C₂×C₂₃	1	2	yes	—
92	C₂×#334	C₂×C₂₃	1	1	yes	characteristic
93	#1450	C₂×C₂₃	1	1	yes	characteristic
94	#1283	C₂×C₂₃	1	2	yes	—
95	C₂²×C₂₃	—	2	1	no	—
96	C₂²×C₂₃	—	2	1	no	—
97	C₂²×C₂₃	#389	1	1	yes	characteristic
98	C₂²×C₂₃	C₈×Q₈	1	1	yes	characteristic
99	C₂²×C₂₃	#446	1	1	yes	characteristic
100	C₄×C₂₃	—	2	1	no	—
101	C₄×C₂₃	—	2	1	no	—
102	C₄×C₂₃	—	2	1	no	—
103	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₂₃	#424	1	1	yes	characteristic
105	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
106	C₄×C₂₃	#436	1	1	yes	characteristic
107	C₄×C₂₃	#471	1	2	yes	—
108	#431	C₂²×C₂₃	1	1	yes	characteristic
109	#431	C₂²×C₂₃	1	1	yes	characteristic
110	C₂×#108	C₄×C₂₃	1	1	yes	characteristic
111	C₂×#108	C₄×C₂₃	1	1	yes	characteristic
112	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
113	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
115	#341	C₄×C₂₃	1	2	yes	—
116	#335	—	2	2	no	—
117	#334	—	2	2	no	—
118	#334	—	2	1	no	—
119	#334	—	2	1	no	—
120	#322	—	2	1	no	—
121	C₈²	—	2	1	no	—
122	C₂×C₂₃	—	2	1	no	—
123	C₂×C₂₃	—	2	1	no	—
124	C₂×C₂₃	#1939	1	1	yes	characteristic
125	C₂×C₂₃	#1951	1	1	yes	characteristic
126	C₂×C₂₃	#1953	1	1	yes	characteristic
127	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
128	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
132	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
133	#108	—	2	2	no	—
134	#108	C₈×C₂₃	1	4	yes	complemented
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	1	no	—
138	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
139	C₂²⋊C₈	—	2	1	no	—
140	C₂²⋊C₈	Q₈×C₂₃	1	2	yes	—
141	#97	—	2	2	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
146	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
147	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
148	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
149	C₂₃	#9607	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
152	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	M₁₆	C₂×C₈×C₂₃	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	8	2	no	—
165	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
166	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
167	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
168	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
170	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
171	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
172	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
173	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
174	C₄²	#41×C₂₃	1	1	yes	characteristic
175	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
181	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
182	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
183	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
187	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	4	no	—
191	C₈	—	4	2	no	—
192	C₈	—	4	2	no	—
193	C₈	—	4	2	no	—
194	C₈	—	8	2	no	—
195	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
196	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₂₃×#389	1	1	yes	characteristic
200	C₂²	C₈×Q₈×C₂₃	1	1	yes	characteristic
201	C₂²	C₂₃×#446	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
206	C₄	C₂₃×#424	1	1	yes	characteristic
207	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
208	C₄	C₂₃×#436	1	1	yes	characteristic
209	C₄	C₂₃×#471	1	2	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	C₂₃×#1939	1	1	yes	characteristic
213	C₂	C₂₃×#1951	1	1	yes	characteristic
214	C₂	C₂₃×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#389
prime radical	C₂³×C₂₃	C₄×Q₈
LCS	G ▷ C₂×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
27. C₂₃×#108	C₈	semidirect
67. #9607	C₂₃	direct (N × H)
134. #108	C₈×C₂₃	semidirect
149. C₂₃	#9607	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×#334	2
C₂₃×#1450	2
C₂₃×#1283	2
#9607	23

Subgroups — C23×#9607 #19157233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9607 #19157233