Subgroups — C₂₃×#3060 #15670679

Order 2944.

484 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	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#537	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×#465	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#417	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₃	C₂²	1	2	yes	—
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#135	—	2	4	no	—
12	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#131	—	2	4	no	—
16	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
19	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×C₂²⋊Q₈	—	2	2	no	—
21	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
22	C₂₃×#121	—	2	2	no	—
23	C₂₃×#121	—	2	4	no	—
24	C₂₃×#121	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#106	C₂²	1	2	yes	—
27	C₂₃×#106	C₂²	1	2	yes	—
28	C₂₃×#103	C₂²	1	4	yes	—
29	C₂₃×#102	C₂²	1	4	yes	—
30	#41×C₂₃	—	2	2	no	—
31	#41×C₂₃	—	2	2	no	—
32	#41×C₂₃	—	2	4	no	—
33	#41×C₂₃	—	2	4	no	—
34	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
36	C₂×D₄×C₂₃	—	2	2	no	—
37	C₂×D₄×C₂₃	—	2	4	no	—
38	C₂×D₄×C₂₃	D₄	1	2	yes	—
39	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
40	C₂×D₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	—	2	2	no	—
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	Q₁₆×C₂₃	—	2	2	no	—
46	SD₁₆×C₂₃	—	2	4	no	—
47	D₈×C₂₃	—	2	2	no	—
48	M₁₆×C₂₃	—	2	2	no	—
49	M₁₆×C₂₃	—	2	4	no	—
50	C₂×C₈×C₂₃	—	2	4	no	—
51	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
52	C₂×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	4	no	—
57	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
58	C₂²⋊C₄×C₂₃	—	2	4	no	—
59	C₂²⋊C₄×C₂₃	—	2	4	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	C₂×D₄	1	2	yes	—
62	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
63	Q₈×C₂₃	—	2	2	no	—
64	Q₈×C₂₃	—	2	2	no	—
65	Q₈×C₂₃	—	2	2	no	—
66	D₄×C₂₃	—	2	4	no	—
67	D₄×C₂₃	—	2	2	no	—
68	D₄×C₂₃	—	2	2	no	—
69	D₄×C₂₃	—	2	2	no	—
70	D₄×C₂₃	—	4	4	no	—
71	D₄×C₂₃	—	8	4	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	4	4	no	—
77	C₂×C₄×C₂₃	—	8	2	no	—
78	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
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	#3060	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	2	no	—
91	C₂²×C₂₃	—	4	2	no	—
92	C₂²×C₂₃	—	4	2	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	C₂×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
103	#537	C₂×C₂₃	1	2	yes	complemented
104	#470	C₂×C₂₃	1	2	yes	complemented
105	#469	C₂×C₂₃	1	2	yes	complemented
106	#465	C₂×C₂₃	1	4	yes	complemented
107	C₂×#106	C₂×C₂₃	1	1	yes	characteristic, complemented
108	#417	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	#577	1	2	yes	—
114	C₂×#41	C₂²×C₂₃	1	2	yes	—
115	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
116	#135	—	2	4	no	—
117	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
118	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
119	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
120	#131	—	2	4	no	—
121	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
122	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
123	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
124	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₂²⋊Q₈	—	2	2	no	—
126	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
127	#121	—	2	2	no	—
128	#121	—	2	4	no	—
129	#121	C₂²×C₂₃	1	4	yes	—
130	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
131	#106	C₂²×C₂₃	1	2	yes	—
132	#106	C₂²×C₂₃	1	2	yes	—
133	#103	C₂²×C₂₃	1	4	yes	—
134	#102	C₂²×C₂₃	1	4	yes	—
135	C₂₃	#3060	1	1	yes	characteristic, direct product factor, Sylow
136	#41	—	2	2	no	—
137	#41	—	2	2	no	—
138	#41	—	2	4	no	—
139	#41	—	2	4	no	—
140	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
141	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	D₄×C₂₃	1	2	yes	—
145	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
146	C₂×D₄	C₂³×C₂₃	1	2	yes	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
151	Q₁₆	—	2	2	no	—
152	SD₁₆	—	2	4	no	—
153	D₈	—	2	2	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	D₄×C₂₃	1	4	yes	complemented
158	C₂×C₈	C₂³×C₂₃	1	2	yes	—
159	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
160	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₂²⋊C₄	—	2	4	no	—
166	C₂³	—	2	2	no	—
167	C₂³	C₂×D₄×C₂₃	1	2	yes	—
168	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	D₄	—	2	4	no	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	8	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	8	2	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	2	no	—
196	C₂²	—	4	2	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×#577	1	2	yes	—

	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 ◁ C₂×#41×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. C₂×C₂₃×#131	C₂	semidirect
3. C₂₃×#537	C₂	semidirect
4. C₂₃×#470	C₂	semidirect
5. C₂₃×#469	C₂	semidirect
6. C₂₃×#465	C₂	semidirect
7. C₂×C₂₃×#106	C₂	semidirect
8. C₂₃×#417	C₂	semidirect
51. C₂×C₈×C₂₃	D₄	semidirect
86. #3060	C₂₃	direct (N × H)
101. C₂×#135	C₂×C₂₃	semidirect
102. C₂×#131	C₂×C₂₃	semidirect
103. #537	C₂×C₂₃	semidirect
104. #470	C₂×C₂₃	semidirect
105. #469	C₂×C₂₃	semidirect
106. #465	C₂×C₂₃	semidirect
107. C₂×#106	C₂×C₂₃	semidirect
108. #417	C₂×C₂₃	semidirect
135. C₂₃	#3060	direct (N × H)
157. 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
C₂×C₂₃×#131	2
C₂₃×#537	2
C₂₃×#470	2
C₂₃×#469	2
C₂₃×#465	2
C₂×C₂₃×#106	2
C₂₃×#417	2
#3060	23

Subgroups — C23×#3060 #15670679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3060 #15670679