Subgroups — C₂₃×#3054 #15670673

Order 2944.

500 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	C₂×C₂₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#525	C₂	1	2	yes	maximal, complemented
4	C₂₃×#470	C₂	1	2	yes	maximal, complemented
5	C₂₃×#469	C₂	1	2	yes	maximal, complemented
6	C₂₃×#468	C₂	1	4	yes	maximal, complemented
7	C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#416	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₃	C₂²	1	2	yes	—
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#135	—	2	4	no	—
12	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
14	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂×M₁₆×C₂₃	—	2	4	no	—
16	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#123	—	2	2	no	—
21	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
22	C₂₃×#121	C₂²	1	4	yes	—
23	C₂₃×#120	—	2	2	no	—
24	C₄×D₄×C₂₃	—	2	4	no	—
25	C₂₃×#117	C₂²	1	2	yes	—
26	C₂₃×#107	C₂²	1	2	yes	complemented
27	C₂₃×#106	C₂²	1	2	yes	complemented
28	C₂₃×#103	C₂²	1	4	yes	complemented
29	C₂₃×#102	C₂²	1	4	yes	complemented
30	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
31	#41×C₂₃	—	2	4	no	—
32	#41×C₂₃	—	2	4	no	—
33	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
34	C₂×D₄×C₂₃	—	2	4	no	—
35	C₂×D₄×C₂₃	—	2	2	no	—
36	C₂×D₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	—	2	2	no	—
38	C₂²×C₄×C₂₃	—	2	2	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	D₄	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	Q₁₆×C₂₃	—	2	2	no	—
45	SD₁₆×C₂₃	—	2	4	no	—
46	D₈×C₂₃	—	2	2	no	—
47	M₁₆×C₂₃	—	2	4	no	—
48	M₁₆×C₂₃	—	2	4	no	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
51	C₂×C₈×C₂₃	C₂³	1	2	yes	—
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	—	2	4	no	—
55	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
56	C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂²⋊C₄×C₂₃	—	2	4	no	—
58	C₂²⋊C₄×C₂₃	—	2	4	no	—
59	C₄²×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	—	2	2	no	—
62	C₂³×C₂₃	—	4	2	no	—
63	C₂³×C₂₃	C₂×D₄	1	2	yes	—
64	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₂₃	—	2	2	no	—
66	Q₈×C₂₃	—	2	2	no	—
67	D₄×C₂₃	—	2	2	no	—
68	D₄×C₂₃	—	2	2	no	—
69	D₄×C₂₃	—	4	8	no	—
70	D₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	4	4	no	—
78	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#41	1	2	yes	—
82	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
83	C₈×C₂₃	—	2	2	no	—
84	C₈×C₂₃	C₂×D₄	1	2	yes	—
85	C₈×C₂₃	C₂×D₄	1	2	yes	—
86	#3054	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	—	2	2	no	—
89	C₂²×C₂₃	—	2	4	no	—
90	C₂²×C₂₃	—	4	4	no	—
91	C₂²×C₂₃	—	4	2	no	—
92	C₂²×C₂₃	—	4	4	no	—
93	C₂²×C₂₃	#121	1	2	yes	—
94	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
95	C₄×C₂₃	—	2	2	no	—
96	C₄×C₂₃	—	4	2	no	—
97	C₄×C₂₃	—	4	4	no	—
98	C₄×C₂₃	#121	1	2	yes	—
99	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
101	C₂×#135	C₂×C₂₃	1	1	yes	characteristic, complemented
102	C₂²×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
103	#525	C₂×C₂₃	1	2	yes	complemented
104	#470	C₂×C₂₃	1	2	yes	complemented
105	#469	C₂×C₂₃	1	2	yes	complemented
106	#468	C₂×C₂₃	1	4	yes	complemented
107	#427	C₂×C₂₃	1	1	yes	characteristic, complemented
108	#416	C₂×C₂₃	1	2	yes	complemented
109	C₂×C₂₃	—	2	2	no	—
110	C₂×C₂₃	—	4	2	no	—
111	C₂×C₂₃	—	8	2	no	—
112	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
113	C₂×C₂₃	#575	1	2	yes	—
114	C₂×#41	C₂²×C₂₃	1	2	yes	—
115	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
116	#135	—	2	4	no	—
117	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
118	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
119	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
122	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
123	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
124	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	#123	—	2	2	no	—
126	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
127	#121	C₂²×C₂₃	1	4	yes	—
128	#120	—	2	2	no	—
129	C₄×D₄	—	2	4	no	—
130	#117	C₂²×C₂₃	1	2	yes	—
131	#107	C₂²×C₂₃	1	2	yes	complemented
132	#106	C₂²×C₂₃	1	2	yes	complemented
133	#103	C₂²×C₂₃	1	4	yes	complemented
134	#102	C₂²×C₂₃	1	4	yes	complemented
135	C₂₃	#3054	1	1	yes	characteristic, direct product factor, Sylow
136	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
137	#41	—	2	4	no	—
138	#41	—	2	4	no	—
139	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	C₂³×C₂₃	1	2	yes	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	D₄×C₂₃	1	2	yes	—
147	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
149	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
150	Q₁₆	—	2	2	no	—
151	SD₁₆	—	2	4	no	—
152	D₈	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	2	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	D₄×C₂₃	1	4	yes	complemented
157	C₂×C₈	C₂³×C₂₃	1	2	yes	—
158	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	4	2	no	—
169	C₂³	C₂×D₄×C₂₃	1	2	yes	—
170	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
171	Q₈	—	2	2	no	—
172	Q₈	—	2	2	no	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	4	8	no	—
176	D₄	—	4	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
187	C₂×C₄	#41×C₂₃	1	2	yes	—
188	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	C₂×D₄×C₂₃	1	2	yes	—
191	C₈	C₂×D₄×C₂₃	1	2	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	4	no	—
196	C₂²	—	4	4	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂₃×#121	1	2	yes	—
199	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	4	no	—
202	C₄	—	4	2	no	—
203	C₄	C₂₃×#121	1	2	yes	—
204	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
205	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	4	2	no	—
208	C₂	—	8	2	no	—
209	C₂	C₂×C₂₃×#121	1	1	yes	characteristic
210	C₂	C₂₃×#575	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#135	C₂	semidirect
2. C₂²×M₁₆×C₂₃	C₂	semidirect
3. C₂₃×#525	C₂	semidirect
4. C₂₃×#470	C₂	semidirect
5. C₂₃×#469	C₂	semidirect
6. C₂₃×#468	C₂	semidirect
7. C₂₃×#427	C₂	semidirect
8. C₂₃×#416	C₂	semidirect
12. C₂×Q₁₆×C₂₃	C₂²	semidirect
13. C₂×SD₁₆×C₂₃	C₂²	semidirect
14. C₂×D₈×C₂₃	C₂²	semidirect
26. C₂₃×#107	C₂²	semidirect
27. C₂₃×#106	C₂²	semidirect
28. C₂₃×#103	C₂²	semidirect
29. C₂₃×#102	C₂²	semidirect
50. C₂×C₈×C₂₃	D₄	semidirect
86. #3054	C₂₃	direct (N × H)
101. C₂×#135	C₂×C₂₃	semidirect
102. C₂²×M₁₆	C₂×C₂₃	semidirect
103. #525	C₂×C₂₃	semidirect
104. #470	C₂×C₂₃	semidirect
105. #469	C₂×C₂₃	semidirect
106. #468	C₂×C₂₃	semidirect
107. #427	C₂×C₂₃	semidirect
108. #416	C₂×C₂₃	semidirect
117. C₂×Q₁₆	C₂²×C₂₃	semidirect
118. C₂×SD₁₆	C₂²×C₂₃	semidirect
119. C₂×D₈	C₂²×C₂₃	semidirect
131. #107	C₂²×C₂₃	semidirect
132. #106	C₂²×C₂₃	semidirect
133. #103	C₂²×C₂₃	semidirect
134. #102	C₂²×C₂₃	semidirect
135. C₂₃	#3054	direct (N × H)
156. C₂×C₈	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#135	2
C₂²×M₁₆×C₂₃	2
C₂₃×#525	2
C₂₃×#470	2
C₂₃×#469	2
C₂₃×#468	2
C₂₃×#427	2
C₂₃×#416	2
#3054	23

Subgroups — C23×#3054 #15670673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3054 #15670673