Subgroups — C₂₃×#3403 #15671022

Order 2944.

380 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#559	C₂	1	2	yes	maximal, complemented
2	C₂₃×#496	C₂	1	2	yes	maximal, complemented
3	C₂₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#464	C₂	1	2	yes	maximal, complemented
5	C₂₃×#463	C₂	1	2	yes	maximal, complemented
6	C₂₃×#459	C₂	1	2	yes	maximal, complemented
7	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#440	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#127	C₂²	1	2	yes	—
17	C₂₃×#124	—	2	2	no	—
18	C₂₃×#124	C₂²	1	2	yes	—
19	C₂₃×#124	C₂²	1	2	yes	—
20	C₂₃×#123	—	2	2	no	—
21	C₂₃×#123	—	2	2	no	—
22	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
23	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₂₃	—	2	2	no	—
25	C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂₃×#103	C₂²	1	2	yes	—
31	C₂₃×#102	C₂²	1	2	yes	—
32	C₂₃×#102	C₂²	1	2	yes	—
33	C₂₃×#97	C₂²	1	2	yes	—
34	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
37	C₂×D₄×C₂₃	—	2	2	no	—
38	C₂×D₄×C₂₃	—	2	2	no	—
39	C₂×D₄×C₂₃	C₂³	1	2	yes	—
40	C₂×D₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	—	2	2	no	—
43	Q₁₆×C₂₃	—	4	1	no	—
44	SD₁₆×C₂₃	—	2	4	no	—
45	SD₁₆×C₂₃	—	4	2	no	—
46	D₈×C₂₃	—	2	2	no	—
47	D₈×C₂₃	—	4	2	no	—
48	D₈×C₂₃	—	4	1	no	—
49	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₂³	1	2	yes	—
51	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
54	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₂²⋊C₄×C₂₃	—	2	2	no	—
62	C₄²×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₄²×C₂₃	C₂³	1	2	yes	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	Q₈×C₂₃	—	2	2	no	—
68	Q₈×C₂₃	—	2	2	no	—
69	Q₈×C₂₃	—	4	1	no	—
70	D₄×C₂₃	—	2	2	no	—
71	D₄×C₂₃	—	2	2	no	—
72	D₄×C₂₃	—	2	2	no	—
73	D₄×C₂₃	—	2	2	no	—
74	D₄×C₂₃	—	4	4	no	—
75	D₄×C₂₃	—	8	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	8	2	no	—
79	C₂×C₄×C₂₃	—	8	2	no	—
80	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
85	C₈×C₂₃	—	2	2	no	—
86	C₈×C₂₃	—	2	1	no	—
87	C₈×C₂₃	#41	1	2	yes	—
88	#3403	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₂₃	—	4	2	no	—
90	C₂²×C₂₃	—	4	2	no	—
91	C₂²×C₂₃	—	4	2	no	—
92	C₂²×C₂₃	—	4	2	no	—
93	C₂²×C₂₃	—	4	2	no	—
94	C₂²×C₂₃	—	4	2	no	—
95	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₂₃	—	2	2	no	—
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	4	2	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
103	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
104	#559	C₂×C₂₃	1	2	yes	complemented
105	#496	C₂×C₂₃	1	2	yes	complemented
106	#495	C₂×C₂₃	1	1	yes	characteristic, complemented
107	#464	C₂×C₂₃	1	2	yes	complemented
108	#463	C₂×C₂₃	1	2	yes	complemented
109	#459	C₂×C₂₃	1	2	yes	complemented
110	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
111	#440	C₂×C₂₃	1	2	yes	complemented
112	C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
113	C₂×C₂₃	—	8	2	no	—
114	C₂×C₂₃	—	8	2	no	—
115	C₂×C₂₃	#550	1	1	yes	characteristic
116	C₂×C₂₃	#575	1	1	yes	characteristic
117	C₂×C₂₃	#576	1	1	yes	characteristic
118	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
119	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
120	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
121	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
122	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
123	C₂×D₈	C₂²×C₂₃	1	2	yes	—
124	#127	C₂²×C₂₃	1	2	yes	—
125	#124	—	2	2	no	—
126	#124	C₂²×C₂₃	1	2	yes	—
127	#124	C₂²×C₂₃	1	2	yes	—
128	#123	—	2	2	no	—
129	#123	—	2	2	no	—
130	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
131	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
132	C₄×D₄	—	2	2	no	—
133	C₄×D₄	C₂²×C₂₃	1	2	yes	—
134	C₄×D₄	C₂²×C₂₃	1	2	yes	—
135	#107	C₂²×C₂₃	1	1	yes	characteristic
136	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
137	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
138	#103	C₂²×C₂₃	1	2	yes	—
139	#102	C₂²×C₂₃	1	2	yes	—
140	#102	C₂²×C₂₃	1	2	yes	—
141	#97	C₂²×C₂₃	1	2	yes	—
142	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₂₃	#3403	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
145	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	C₂³×C₂₃	1	2	yes	—
149	C₂×D₄	C₂³×C₂₃	1	2	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	Q₁₆	—	4	1	no	—
153	SD₁₆	—	2	4	no	—
154	SD₁₆	—	4	2	no	—
155	D₈	—	2	2	no	—
156	D₈	—	4	2	no	—
157	D₈	—	4	1	no	—
158	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₂₃	1	2	yes	—
160	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
163	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
164	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	C₂³×C₂₃	1	1	yes	characteristic
173	C₄²	C₂³×C₂₃	1	2	yes	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	Q₈	—	2	2	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	4	1	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	2	2	no	—
183	D₄	—	4	4	no	—
184	D₄	—	8	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
191	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
193	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	#41×C₂₃	1	2	yes	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
211	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
212	C₂	—	8	2	no	—
213	C₂	—	8	2	no	—
214	C₂	C₂₃×#550	1	1	yes	characteristic
215	C₂	C₂₃×#575	1	1	yes	characteristic
216	C₂	C₂₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#559	C₂	semidirect
2. C₂₃×#496	C₂	semidirect
3. C₂₃×#495	C₂	semidirect
4. C₂₃×#464	C₂	semidirect
5. C₂₃×#463	C₂	semidirect
6. C₂₃×#459	C₂	semidirect
7. C₂₃×#446	C₂	semidirect
8. C₂₃×#440	C₂	semidirect
9. C₄×D₈×C₂₃	C₂	semidirect
88. #3403	C₂₃	direct (N × H)
104. #559	C₂×C₂₃	semidirect
105. #496	C₂×C₂₃	semidirect
106. #495	C₂×C₂₃	semidirect
107. #464	C₂×C₂₃	semidirect
108. #463	C₂×C₂₃	semidirect
109. #459	C₂×C₂₃	semidirect
110. #446	C₂×C₂₃	semidirect
111. #440	C₂×C₂₃	semidirect
112. C₄×D₈	C₂×C₂₃	semidirect
143. C₂₃	#3403	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×#559	2
C₂₃×#496	2
C₂₃×#495	2
C₂₃×#464	2
C₂₃×#463	2
C₂₃×#459	2
C₂₃×#446	2
C₂₃×#440	2
C₄×D₈×C₂₃	2
#3403	23

Subgroups — C23×#3403 #15671022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3403 #15671022