Subgroups — C₂₃×#13182 #19160808

Order 5888.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2161	C₂	1	2	yes	maximal
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1389	C₂	1	2	yes	maximal
6	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₂₃×#108	—	2	1	no	—
8	C₂×C₂₃×#108	C₂²	1	2	yes	—
9	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
10	C₂₃×#427	C₂²	1	1	yes	characteristic
11	C₂₃×#423	—	2	1	no	—
12	C₂₃×#405	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#368	—	2	4	no	—
15	C₂₃×#364	C₄	1	2	yes	complemented
16	C₂₃×#364	C₄	1	2	yes	complemented
17	C₂₃×#339	—	2	1	no	—
18	C₂×M₁₆×C₂₃	—	2	1	no	—
19	C₂×M₁₆×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
22	C₂₃×#117	—	2	1	no	—
23	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
24	C₂₃×M₃₂	—	2	2	no	—
25	C₂₃×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₂₃	—	2	2	no	—
27	C₂₃×#108	—	2	2	no	—
28	C₂₃×#108	—	2	1	no	—
29	C₂₃×#108	—	2	2	no	—
30	C₂₃×#108	—	4	1	no	—
31	C₂₃×#108	—	4	1	no	—
32	C₂₃×#108	D₄	1	2	yes	—
33	C₂₃×#107	—	2	1	no	—
34	C₂₃×#106	Q₈	1	2	yes	—
35	C₂₃×C₄⋊C₈	—	4	2	no	—
36	C₂₃×#97	—	4	1	no	—
37	C₄×C₈×C₂₃	—	4	1	no	—
38	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₂₃	C₂×C₄	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	—	2	1	no	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₂₃	—	2	2	no	—
45	M₁₆×C₂₃	—	4	1	no	—
46	M₁₆×C₂₃	—	4	2	no	—
47	M₁₆×C₂₃	—	4	2	no	—
48	C₂×C₈×C₂₃	—	4	1	no	—
49	C₂×C₈×C₂₃	—	4	2	no	—
50	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
53	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
54	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
55	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
61	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₂²⋊C₄×C₂₃	—	4	1	no	—
64	C₄²×C₂₃	—	2	1	no	—
65	C₄²×C₂₃	—	4	1	no	—
66	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
68	C₁₆×C₂₃	—	2	4	no	—
69	#13182	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	#137	1	1	yes	characteristic
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	4	2	no	—
74	C₂×C₄×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₈×C₂₃	—	4	2	no	—
82	C₈×C₂₃	—	8	2	no	—
83	C₈×C₂₃	#107	1	1	yes	characteristic
84	C₈×C₂₃	#107	1	1	yes	characteristic
85	C₈×C₂₃	#107	1	1	yes	characteristic
86	C₈×C₂₃	#107	1	1	yes	characteristic
87	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
88	C₈×C₂₃	C₄×Q₈	1	1	yes	characteristic
89	C₈×C₂₃	#125	1	1	yes	characteristic
90	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
91	#2161	C₂×C₂₃	1	2	yes	—
92	C₂×#364	C₂×C₂₃	1	1	yes	characteristic
93	#1848	C₂×C₂₃	1	1	yes	characteristic
94	#1772	C₂×C₂₃	1	1	yes	characteristic
95	#1389	C₂×C₂₃	1	2	yes	—
96	C₂²×C₂₃	—	4	1	no	—
97	C₂²×C₂₃	#428	1	1	yes	characteristic
98	C₂²×C₂₃	#462	1	1	yes	characteristic
99	C₂²×C₂₃	#474	1	1	yes	characteristic
100	C₄×C₂₃	—	2	2	no	—
101	C₄×C₂₃	—	4	1	no	—
102	C₄×C₂₃	—	8	2	no	—
103	C₄×C₂₃	#389	1	1	yes	characteristic
104	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
105	C₄×C₂₃	#459	1	1	yes	characteristic
106	C₄×C₂₃	#477	1	1	yes	characteristic
107	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
108	C₂×#108	—	2	1	no	—
109	C₂×#108	C₂²×C₂₃	1	2	yes	—
110	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
111	#427	C₂²×C₂₃	1	1	yes	characteristic
112	#423	—	2	1	no	—
113	#405	—	2	1	no	—
114	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
115	#368	—	2	4	no	—
116	#364	C₄×C₂₃	1	2	yes	complemented
117	#364	C₄×C₂₃	1	2	yes	complemented
118	#339	—	2	1	no	—
119	C₂×C₂₃	—	2	1	no	—
120	C₂×C₂₃	—	2	1	no	—
121	C₂×C₂₃	—	2	1	no	—
122	C₂×C₂₃	#1947	1	1	yes	characteristic
123	C₂×M₁₆	—	2	1	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
127	#117	—	2	1	no	—
128	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	C₂×C₄×C₂₃	1	4	yes	—
131	C₂×C₁₆	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	2	1	no	—
134	#108	—	2	2	no	—
135	#108	—	4	1	no	—
136	#108	—	4	1	no	—
137	#108	D₄×C₂₃	1	2	yes	—
138	#107	—	2	1	no	—
139	#106	Q₈×C₂₃	1	2	yes	—
140	C₄⋊C₈	—	4	2	no	—
141	#97	—	4	1	no	—
142	C₄×C₈	—	4	1	no	—
143	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
144	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
145	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
146	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
147	C₂₃	#13182	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	1	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
158	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
159	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
160	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
161	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
164	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
165	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
166	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
167	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
173	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₂³	C₂₃×#137	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
182	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₈	C₂₃×#107	1	1	yes	characteristic
189	C₈	C₂₃×#107	1	1	yes	characteristic
190	C₈	C₂₃×#107	1	1	yes	characteristic
191	C₈	C₂₃×#107	1	1	yes	characteristic
192	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
193	C₈	C₄×Q₈×C₂₃	1	1	yes	characteristic
194	C₈	C₂₃×#125	1	1	yes	characteristic
195	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	C₂₃×#428	1	1	yes	characteristic
198	C₂²	C₂₃×#462	1	1	yes	characteristic
199	C₂²	C₂₃×#474	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	C₂₃×#389	1	1	yes	characteristic
204	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
205	C₄	C₂₃×#459	1	1	yes	characteristic
206	C₄	C₂₃×#477	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	C₂₃×#1947	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#459
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1947
prime radical	C₂³×C₂₃	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂₃×#364	C₄	semidirect
16. C₂₃×#364	C₄	semidirect
69. #13182	C₂₃	direct (N × H)
116. #364	C₄×C₂₃	semidirect
117. #364	C₄×C₂₃	semidirect
147. C₂₃	#13182	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×#2161	2
C₂×C₂₃×#364	2
C₂₃×#1848	2
C₂₃×#1772	2
C₂₃×#1389	2
#13182	23

Subgroups — C23×#13182 #19160808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13182 #19160808