Subgroups — C₂×C₂₃×#2657 #19182248

Order 5888.

5142 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#127	C₂	1	3	yes	maximal, complemented
2	C₂²×C₂₃×#124	C₂	1	3	yes	maximal, complemented
3	C₂₃×#2657	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#392	C₂	1	8	yes	maximal, complemented
5	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×D₄×C₂₃	C₂²	1	3	yes	—
8	C₂³×D₄×C₂₃	C₂²	1	12	yes	—
9	C₂×C₂₃×#127	—	2	24	no	—
10	C₂×C₂₃×#127	C₂²	1	24	yes	complemented
11	C₂×C₂₃×#124	—	2	24	no	—
12	C₂×C₂₃×#124	C₂²	1	24	yes	complemented
13	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	12	yes	—
14	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
15	C₂₃×#392	C₂²	1	64	yes	complemented
16	C₂₃×#376	C₂²	1	8	yes	complemented
17	C₂×C₂₃×#95	C₂²	1	4	yes	—
18	C₂⁵×C₂₃	C₂³	1	6	yes	—
19	C₂²×Q₈×C₂₃	—	2	12	no	—
20	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
21	C₂²×D₄×C₂₃	—	2	12	no	—
22	C₂²×D₄×C₂₃	—	2	48	no	—
23	C₂²×D₄×C₂₃	—	2	24	no	—
24	C₂²×D₄×C₂₃	—	4	48	no	—
25	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
26	C₂²×D₄×C₂₃	C₂³	1	48	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
28	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂₃×#127	—	2	96	no	—
31	C₂₃×#124	—	2	96	no	—
32	C₂×C₂²⋊C₄×C₂₃	—	2	48	no	—
33	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
34	C₂×C₄²×C₂₃	D₄	1	24	yes	complemented
35	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
36	C₂₃×#95	C₂³	1	16	yes	—
37	C₂⁴×C₂₃	—	2	12	no	—
38	C₂⁴×C₂₃	—	2	24	no	—
39	C₂⁴×C₂₃	—	2	48	no	—
40	C₂⁴×C₂₃	C₂⁴	1	12	yes	—
41	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
42	C₂×Q₈×C₂₃	—	2	24	no	—
43	C₂×Q₈×C₂₃	—	4	16	no	—
44	C₂×D₄×C₂₃	—	2	24	no	—
45	C₂×D₄×C₂₃	—	2	96	no	—
46	C₂×D₄×C₂₃	—	2	48	no	—
47	C₂×D₄×C₂₃	—	4	96	no	—
48	C₂×D₄×C₂₃	—	4	192	no	—
49	C₂×D₄×C₂₃	—	4	48	no	—
50	C₂²×C₄×C₂₃	—	2	24	no	—
51	C₂²×C₄×C₂₃	—	2	12	no	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
53	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
54	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
55	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
56	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
57	C₂²⋊C₄×C₂₃	—	2	96	no	—
58	C₄²×C₂₃	C₂×D₄	1	48	yes	complemented
59	C₂×#2657	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₂₃	—	2	12	no	—
61	C₂³×C₂₃	—	2	24	no	—
62	C₂³×C₂₃	—	2	48	no	—
63	C₂³×C₂₃	—	4	24	no	—
64	C₂³×C₂₃	—	4	96	no	—
65	C₂³×C₂₃	—	4	48	no	—
66	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
67	C₂³×C₂₃	H(5,𝔽₂)	1	8	yes	—
68	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
69	Q₈×C₂₃	—	4	16	no	—
70	D₄×C₂₃	—	4	192	no	—
71	D₄×C₂₃	—	4	48	no	—
72	C₂×C₄×C₂₃	—	2	48	no	—
73	C₂×C₄×C₂₃	—	2	24	no	—
74	C₂×C₄×C₂₃	—	2	12	no	—
75	C₂×C₄×C₂₃	—	4	8	no	—
76	C₂×C₄×C₂₃	#127	1	16	yes	—
77	C₂×C₄×C₂₃	C₂²×D₄	1	24	yes	—
78	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
79	C₂²×#127	C₂×C₂₃	1	3	yes	complemented
80	C₂²×#124	C₂×C₂₃	1	3	yes	complemented
81	#2657	C₂×C₂₃	1	16	yes	direct product factor
82	C₂×#392	C₂×C₂₃	1	8	yes	complemented
83	C₂×#376	C₂×C₂₃	1	1	yes	characteristic, complemented
84	C₂²×C₂₃	—	4	12	no	—
85	C₂²×C₂₃	—	4	48	no	—
86	C₂²×C₂₃	—	4	24	no	—
87	C₂²×C₂₃	—	8	48	no	—
88	C₂²×C₂₃	C₂×#127	1	4	yes	—
89	C₂²×C₂₃	#535	1	24	yes	—
90	C₂²×C₂₃	C₂³×D₄	1	3	yes	—
91	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	4	yes	—
92	C₄×C₂₃	—	2	24	no	—
93	C₄×C₂₃	—	4	4	no	—
94	C₄×C₂₃	C₂×#127	1	8	yes	—
95	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
96	C₂³×D₄	C₂²×C₂₃	1	3	yes	—
97	C₂³×D₄	C₂²×C₂₃	1	12	yes	—
98	C₂×#127	—	2	24	no	—
99	C₂×#127	C₂²×C₂₃	1	24	yes	complemented
100	C₂×#124	—	2	24	no	—
101	C₂×#124	C₂²×C₂₃	1	24	yes	complemented
102	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	12	yes	—
103	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
104	#392	C₂²×C₂₃	1	64	yes	complemented
105	#376	C₂²×C₂₃	1	8	yes	complemented
106	C₂×#95	C₂²×C₂₃	1	4	yes	—
107	C₂×C₂₃	—	8	12	no	—
108	C₂×C₂₃	#2657	1	8	yes	direct product factor
109	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
110	C₂×C₂₃	C₂×#535	1	6	yes	—
111	C₂⁵	C₂³×C₂₃	1	6	yes	—
112	C₂²×Q₈	—	2	12	no	—
113	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
114	C₂²×D₄	—	2	12	no	—
115	C₂²×D₄	—	2	48	no	—
116	C₂²×D₄	—	2	24	no	—
117	C₂²×D₄	—	4	48	no	—
118	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
119	C₂²×D₄	C₂³×C₂₃	1	48	yes	—
120	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
121	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
122	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
123	#127	—	2	96	no	—
124	#124	—	2	96	no	—
125	C₂×C₂²⋊C₄	—	2	48	no	—
126	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
127	C₂×C₄²	D₄×C₂₃	1	24	yes	complemented
128	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
129	#95	C₂³×C₂₃	1	16	yes	—
130	C₂₃	C₂×#2657	1	1	yes	characteristic, direct product factor, Sylow
131	C₂⁴	—	2	12	no	—
132	C₂⁴	—	2	24	no	—
133	C₂⁴	—	2	48	no	—
134	C₂⁴	C₂⁴×C₂₃	1	12	yes	—
135	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
136	C₂×Q₈	—	2	24	no	—
137	C₂×Q₈	—	4	16	no	—
138	C₂×D₄	—	2	24	no	—
139	C₂×D₄	—	2	96	no	—
140	C₂×D₄	—	2	48	no	—
141	C₂×D₄	—	4	96	no	—
142	C₂×D₄	—	4	192	no	—
143	C₂×D₄	—	4	48	no	—
144	C₂²×C₄	—	2	24	no	—
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
147	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
148	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
149	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
150	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
151	C₂²⋊C₄	—	2	96	no	—
152	C₄²	C₂×D₄×C₂₃	1	48	yes	complemented
153	C₂³	—	2	12	no	—
154	C₂³	—	2	24	no	—
155	C₂³	—	2	48	no	—
156	C₂³	—	4	24	no	—
157	C₂³	—	4	96	no	—
158	C₂³	—	4	48	no	—
159	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
160	C₂³	C₂₃×H(5,𝔽₂)	1	8	yes	—
161	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
162	Q₈	—	4	16	no	—
163	D₄	—	4	192	no	—
164	D₄	—	4	48	no	—
165	C₂×C₄	—	2	24	no	—
166	C₂×C₄	—	2	48	no	—
167	C₂×C₄	—	2	12	no	—
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	C₂₃×#127	1	16	yes	—
170	C₂×C₄	C₂²×D₄×C₂₃	1	24	yes	—
171	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
172	C₂²	—	4	12	no	—
173	C₂²	—	4	48	no	—
174	C₂²	—	4	24	no	—
175	C₂²	—	8	48	no	—
176	C₂²	C₂×C₂₃×#127	1	4	yes	—
177	C₂²	C₂₃×#535	1	24	yes	—
178	C₂²	C₂³×D₄×C₂₃	1	3	yes	—
179	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	4	yes	—
180	C₄	—	2	24	no	—
181	C₄	—	4	4	no	—
182	C₄	C₂×C₂₃×#127	1	8	yes	—
183	C₂	—	8	12	no	—
184	C₂	C₂₃×#2657	1	8	yes	direct product factor
185	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic
186	C₂	C₂×C₂₃×#535	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×#127	C₂	semidirect
2. C₂²×C₂₃×#124	C₂	semidirect
3. C₂₃×#2657	C₂	direct (N × H)
4. C₂×C₂₃×#392	C₂	semidirect
5. C₂×C₂₃×#376	C₂	semidirect
10. C₂×C₂₃×#127	C₂²	semidirect
12. C₂×C₂₃×#124	C₂²	semidirect
15. C₂₃×#392	C₂²	semidirect
16. C₂₃×#376	C₂²	semidirect
34. C₂×C₄²×C₂₃	D₄	semidirect
58. C₄²×C₂₃	C₂×D₄	semidirect
59. C₂×#2657	C₂₃	direct (N × H)
79. C₂²×#127	C₂×C₂₃	semidirect
80. C₂²×#124	C₂×C₂₃	semidirect
81. #2657	C₂×C₂₃	direct (N × H)
82. C₂×#392	C₂×C₂₃	semidirect
83. C₂×#376	C₂×C₂₃	semidirect
99. C₂×#127	C₂²×C₂₃	semidirect
101. C₂×#124	C₂²×C₂₃	semidirect
104. #392	C₂²×C₂₃	semidirect
105. #376	C₂²×C₂₃	semidirect
108. C₂×C₂₃	#2657	direct (N × H)
127. C₂×C₄²	D₄×C₂₃	semidirect
130. C₂₃	C₂×#2657	direct (N × H)
152. C₄²	C₂×D₄×C₂₃	semidirect
184. C₂	C₂₃×#2657	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#127	2
C₂²×C₂₃×#124	2
C₂₃×#2657	2
C₂×C₂₃×#392	2
C₂×C₂₃×#376	2
C₂×#2657	23

Subgroups — C2×C23×#2657 #19182248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2657 #19182248