Subgroups — C₂₃×#1866 #15669485

Order 2944.

270 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#104	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#416	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#339	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#131	—	2	2	no	—
10	C₂₃×#131	—	2	2	no	—
11	C₂×M₁₆×C₂₃	—	2	1	no	—
12	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	—	2	1	no	—
15	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#117	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#108	—	2	2	no	—
20	C₂₃×C₄⋊C₈	—	2	1	no	—
21	C₂₃×C₄⋊C₈	—	2	1	no	—
22	C₂₃×#104	—	2	2	no	—
23	C₂₃×#103	C₄	1	1	yes	characteristic, complemented
24	C₂₃×#103	C₄	1	1	yes	characteristic, complemented
25	C₂₃×#102	C₄	1	1	yes	characteristic, complemented
26	C₂₃×#102	C₄	1	1	yes	characteristic, complemented
27	#41×C₂₃	—	2	1	no	—
28	#41×C₂₃	—	2	1	no	—
29	#41×C₂₃	D₄	1	2	yes	—
30	#41×C₂₃	Q₈	1	2	yes	—
31	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×D₄×C₂₃	—	2	1	no	—
33	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	—	2	1	no	—
35	C₂²×C₄×C₂₃	—	2	1	no	—
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	M₁₆×C₂₃	—	2	4	no	—
38	M₁₆×C₂₃	—	2	1	no	—
39	M₁₆×C₂₃	—	2	1	no	—
40	M₁₆×C₂₃	—	2	1	no	—
41	M₁₆×C₂₃	D₄	1	2	yes	—
42	C₂×C₈×C₂₃	—	2	4	no	—
43	C₂×C₈×C₂₃	—	2	1	no	—
44	C₂×C₈×C₂₃	—	2	1	no	—
45	C₂×C₈×C₂₃	—	2	1	no	—
46	C₂×C₈×C₂₃	—	2	1	no	—
47	C₂×C₈×C₂₃	—	2	1	no	—
48	C₂×C₈×C₂₃	—	2	1	no	—
49	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
52	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂²⋊C₄×C₂₃	—	2	1	no	—
56	C₄²×C₂₃	—	2	1	no	—
57	C₄²×C₂₃	—	2	1	no	—
58	C₄²×C₂₃	—	2	1	no	—
59	C₂³×C₂₃	—	2	1	no	—
60	C₂³×C₂₃	#41	1	1	yes	characteristic
61	Q₈×C₂₃	—	2	1	no	—
62	Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
63	D₄×C₂₃	—	2	2	no	—
64	D₄×C₂₃	—	2	2	no	—
65	D₄×C₂₃	—	2	1	no	—
66	D₄×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	4	1	no	—
74	C₂×C₄×C₂₃	—	4	1	no	—
75	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
81	C₈×C₂₃	—	2	1	no	—
82	C₈×C₂₃	—	2	1	no	—
83	C₈×C₂₃	—	2	1	no	—
84	C₈×C₂₃	—	2	2	no	—
85	C₈×C₂₃	—	2	1	no	—
86	C₈×C₂₃	—	4	1	no	—
87	#1866	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₂₃	—	2	1	no	—
89	C₂²×C₂₃	—	2	1	no	—
90	C₂²×C₂₃	—	2	2	no	—
91	C₂²×C₂₃	—	4	1	no	—
92	C₂²×C₂₃	—	4	1	no	—
93	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
94	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂²×C₂₃	#123	1	1	yes	characteristic
96	C₄×C₂₃	—	2	2	no	—
97	C₄×C₂₃	—	4	2	no	—
98	C₄×C₂₃	—	4	1	no	—
99	C₄×C₂₃	—	4	1	no	—
100	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
102	C₄×C₂₃	#120	1	1	yes	characteristic
103	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×#131	C₂×C₂₃	1	1	yes	characteristic
105	C₂×#108	C₂×C₂₃	1	1	yes	characteristic, complemented
106	#424	C₂×C₂₃	1	1	yes	characteristic, complemented
107	C₂×#104	C₂×C₂₃	1	1	yes	characteristic
108	#416	C₂×C₂₃	1	1	yes	characteristic
109	#339	C₂×C₂₃	1	1	yes	characteristic, complemented
110	#336	C₂×C₂₃	1	1	yes	characteristic, complemented
111	C₂×C₂₃	—	2	1	no	—
112	C₂×C₂₃	—	2	1	no	—
113	C₂×C₂₃	—	4	2	no	—
114	C₂×C₂₃	#385	1	1	yes	characteristic
115	C₂×C₂₃	#443	1	1	yes	characteristic
116	C₂×C₂₃	#444	1	1	yes	characteristic
117	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	—	2	1	no	—
121	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
122	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
126	#117	C₂²×C₂₃	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
128	#108	—	2	2	no	—
129	C₄⋊C₈	—	2	1	no	—
130	C₄⋊C₈	—	2	1	no	—
131	#104	—	2	2	no	—
132	#103	C₄×C₂₃	1	1	yes	characteristic, complemented
133	#103	C₄×C₂₃	1	1	yes	characteristic, complemented
134	#102	C₄×C₂₃	1	1	yes	characteristic, complemented
135	#102	C₄×C₂₃	1	1	yes	characteristic, complemented
136	C₂₃	#1866	1	1	yes	characteristic, direct product factor, Sylow
137	#41	—	2	1	no	—
138	#41	—	2	1	no	—
139	#41	D₄×C₂₃	1	2	yes	—
140	#41	Q₈×C₂₃	1	2	yes	—
141	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
147	M₁₆	—	2	4	no	—
148	M₁₆	—	2	1	no	—
149	M₁₆	—	2	1	no	—
150	M₁₆	—	2	1	no	—
151	M₁₆	D₄×C₂₃	1	2	yes	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
161	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
162	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
163	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
164	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
165	C₂²⋊C₄	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	#41×C₂₃	1	1	yes	characteristic
171	Q₈	—	2	1	no	—
172	Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	1	no	—
176	D₄	C₄⋊C₄×C₂₃	1	2	yes	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
190	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	—	4	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	1	no	—
202	C₂²	C₄×D₄×C₂₃	1	1	yes	characteristic
203	C₂²	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
204	C₂²	C₂₃×#123	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	1	no	—
209	C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
210	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
211	C₄	C₂₃×#120	1	1	yes	characteristic
212	C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	—	4	2	no	—
216	C₂	C₂₃×#385	1	1	yes	characteristic
217	C₂	C₂₃×#443	1	1	yes	characteristic
218	C₂	C₂₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	C₄×D₄
prime radical	C₂×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#108	C₂	semidirect
3. C₂₃×#424	C₂	semidirect
6. C₂₃×#339	C₂	semidirect
7. C₂₃×#336	C₂	semidirect
23. C₂₃×#103	C₄	semidirect
24. C₂₃×#103	C₄	semidirect
25. C₂₃×#102	C₄	semidirect
26. C₂₃×#102	C₄	semidirect
87. #1866	C₂₃	direct (N × H)
105. C₂×#108	C₂×C₂₃	semidirect
106. #424	C₂×C₂₃	semidirect
109. #339	C₂×C₂₃	semidirect
110. #336	C₂×C₂₃	semidirect
132. #103	C₄×C₂₃	semidirect
133. #103	C₄×C₂₃	semidirect
134. #102	C₄×C₂₃	semidirect
135. #102	C₄×C₂₃	semidirect
136. C₂₃	#1866	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#131	2
C₂×C₂₃×#108	2
C₂₃×#424	2
C₂×C₂₃×#104	2
C₂₃×#416	2
C₂₃×#339	2
C₂₃×#336	2
#1866	23

Subgroups — C23×#1866 #15669485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1866 #15669485