Subgroups — C₂₃×#62652 #19210278

Order 5888.

11674 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×H(5,𝔽₂)	C₂	1	32	yes	maximal, complemented
3	C₂₃×#3428	C₂	1	12	yes	maximal, complemented
4	C₂×C₂₃×#518	C₂	1	18	yes	maximal, complemented
5	C₂×C₂₃×H(5,𝔽₂)	C₄	1	32	yes	complemented
6	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	16	yes	—
7	C₂²×#41×C₂₃	C₂²	1	6	yes	—
8	C₂³×D₄×C₂₃	C₂²	1	9	yes	—
9	C₂₃×#520	C₂²	1	96	yes	complemented
10	C₂₃×#519	C₂²	1	8	yes	complemented
11	C₂₃×#518	C₂²	1	72	yes	complemented
12	C₂₃×#518	C₂²	1	144	yes	complemented
13	C₄×#41×C₂₃	C₂²	1	96	yes	complemented
14	C₂×C₄×D₄×C₂₃	C₂²	1	144	yes	complemented
15	C₂×C₂₃×#117	C₂²	1	36	yes	complemented
16	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	24	yes	complemented
17	C₂⁵×C₂₃	C₂³	1	6	yes	—
18	C₂₃×H(5,𝔽₂)	C₂×C₄	1	256	yes	complemented
19	C₂×#41×C₂₃	C₂×C₄	1	96	yes	complemented
20	C₂×#41×C₂₃	C₂³	1	48	yes	—
21	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
22	C₂²×D₄×C₂₃	C₂×C₄	1	144	yes	complemented
23	C₂²×D₄×C₂₃	C₂³	1	18	yes	—
24	C₂²×D₄×C₂₃	C₂³	1	72	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	9	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	48	yes	complemented
27	C₄×Q₈×C₂₃	C₂³	1	64	yes	complemented
28	C₄×D₄×C₂₃	C₂³	1	576	yes	complemented
29	C₂₃×#117	C₂³	1	48	yes	—
30	C₂₃×#117	C₂³	1	144	yes	complemented
31	C₂×C₄⋊C₄×C₂₃	C₂³	1	72	yes	complemented
32	C₂×C₂²⋊C₄×C₂₃	C₂³	1	72	yes	—
33	C₂×C₂²⋊C₄×C₂₃	C₂³	1	144	yes	complemented
34	C₂×C₄²×C₂₃	C₂³	1	72	yes	complemented
35	C₂⁴×C₂₃	—	8	12	no	—
36	C₂⁴×C₂₃	C₂²×C₄	1	48	yes	complemented
37	C₂⁴×C₂₃	C₂⁴	1	9	yes	—
38	C₂⁴×C₂₃	C₂⁴	1	24	yes	—
39	#41×C₂₃	C₂²×C₄	1	384	yes	complemented
40	C₂×Q₈×C₂₃	C₂²×C₄	1	16	yes	complemented
41	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
42	C₂×D₄×C₂₃	C₂²×C₄	1	144	yes	complemented
43	C₂×D₄×C₂₃	C₂²×C₄	1	576	yes	complemented
44	C₂×D₄×C₂₃	C₂⁴	1	72	yes	—
45	C₂²×C₄×C₂₃	—	8	48	no	—
46	C₂²×C₄×C₂₃	C₂²×C₄	1	72	yes	complemented
47	C₂²×C₄×C₂₃	C₂⁴	1	144	yes	—
48	C₂²×C₄×C₂₃	C₂⁴	1	36	yes	—
49	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
50	C₄⋊C₄×C₂₃	C₂⁴	1	96	yes	—
51	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
52	C₂²⋊C₄×C₂₃	C₂⁴	1	144	yes	—
53	C₄²×C₂₃	C₂⁴	1	96	yes	—
54	#62652	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	—	4	48	no	—
56	C₂³×C₂₃	—	4	18	no	—
57	C₂³×C₂₃	—	8	48	no	—
58	C₂³×C₂₃	C₂³×C₄	1	36	yes	—
59	C₂³×C₂₃	C₂³×C₄	1	96	yes	—
60	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
61	C₂³×C₂₃	C₂⁵	1	18	yes	—
62	Q₈×C₂₃	C₂³×C₄	1	32	yes	—
63	D₄×C₂₃	C₂³×C₄	1	288	yes	—
64	C₂×C₄×C₂₃	—	4	144	no	—
65	C₂×C₄×C₂₃	C₂³×C₄	1	24	yes	—
66	C₂×C₄×C₂₃	C₂³×C₄	1	144	yes	—
67	C₂×C₄×C₂₃	C₂⁵	1	32	yes	—
68	C₂×C₄×C₂₃	C₂⁵	1	12	yes	—
69	C₂²×H(5,𝔽₂)	C₂×C₂₃	1	1	yes	characteristic
70	C₄×H(5,𝔽₂)	C₂×C₂₃	1	32	yes	complemented
71	#3428	C₂×C₂₃	1	12	yes	complemented
72	C₂×#518	C₂×C₂₃	1	18	yes	complemented
73	C₂²×C₂₃	—	2	36	no	—
74	C₂²×C₂₃	—	2	2	no	—
75	C₂²×C₂₃	—	4	36	no	—
76	C₂²×C₂₃	—	4	96	no	—
77	C₂²×C₂₃	C₂⁴×C₄	1	2	yes	—
78	C₂²×C₂₃	C₂⁴×C₄	1	36	yes	—
79	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
80	C₄×C₂₃	—	2	32	no	—
81	C₄×C₂₃	C₂⁴×C₄	1	24	yes	—
82	C₂×H(5,𝔽₂)	C₄×C₂₃	1	32	yes	complemented
83	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	16	yes	—
84	C₂²×#41	C₂²×C₂₃	1	6	yes	—
85	C₂³×D₄	C₂²×C₂₃	1	9	yes	—
86	#520	C₂²×C₂₃	1	96	yes	complemented
87	#519	C₂²×C₂₃	1	8	yes	complemented
88	#518	C₂²×C₂₃	1	72	yes	complemented
89	#518	C₂²×C₂₃	1	144	yes	complemented
90	C₄×#41	C₂²×C₂₃	1	96	yes	complemented
91	C₂×C₄×D₄	C₂²×C₂₃	1	144	yes	complemented
92	C₂×#117	C₂²×C₂₃	1	36	yes	complemented
93	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	24	yes	complemented
94	C₂×C₂₃	—	2	2	no	—
95	C₂×C₂₃	—	2	36	no	—
96	C₂×C₂₃	C₂⁵×C₄	1	1	yes	characteristic
97	C₂×C₂₃	H(7,𝔽₂)	1	2	yes	—
98	C₂⁵	C₂³×C₂₃	1	6	yes	—
99	H(5,𝔽₂)	C₂×C₄×C₂₃	1	256	yes	complemented
100	C₂×#41	C₂×C₄×C₂₃	1	96	yes	complemented
101	C₂×#41	C₂³×C₂₃	1	48	yes	—
102	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
103	C₂²×D₄	C₂×C₄×C₂₃	1	144	yes	complemented
104	C₂²×D₄	C₂³×C₂₃	1	72	yes	—
105	C₂²×D₄	C₂³×C₂₃	1	18	yes	—
106	C₂³×C₄	C₂³×C₂₃	1	9	yes	—
107	C₂³×C₄	C₂³×C₂₃	1	48	yes	complemented
108	C₄×Q₈	C₂³×C₂₃	1	64	yes	complemented
109	C₄×D₄	C₂³×C₂₃	1	576	yes	complemented
110	#117	C₂³×C₂₃	1	48	yes	—
111	#117	C₂³×C₂₃	1	144	yes	complemented
112	C₂×C₄⋊C₄	C₂³×C₂₃	1	72	yes	complemented
113	C₂×C₂²⋊C₄	C₂³×C₂₃	1	72	yes	—
114	C₂×C₂²⋊C₄	C₂³×C₂₃	1	144	yes	complemented
115	C₂×C₄²	C₂³×C₂₃	1	72	yes	complemented
116	C₂₃	#62652	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴	—	8	12	no	—
118	C₂⁴	C₂²×C₄×C₂₃	1	48	yes	complemented
119	C₂⁴	C₂⁴×C₂₃	1	9	yes	—
120	C₂⁴	C₂⁴×C₂₃	1	24	yes	—
121	#41	C₂²×C₄×C₂₃	1	384	yes	complemented
122	C₂×Q₈	C₂²×C₄×C₂₃	1	16	yes	complemented
123	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
124	C₂×D₄	C₂²×C₄×C₂₃	1	144	yes	complemented
125	C₂×D₄	C₂²×C₄×C₂₃	1	576	yes	complemented
126	C₂×D₄	C₂⁴×C₂₃	1	72	yes	—
127	C₂²×C₄	—	8	48	no	—
128	C₂²×C₄	C₂²×C₄×C₂₃	1	72	yes	complemented
129	C₂²×C₄	C₂⁴×C₂₃	1	144	yes	—
130	C₂²×C₄	C₂⁴×C₂₃	1	36	yes	—
131	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
132	C₄⋊C₄	C₂⁴×C₂₃	1	96	yes	—
133	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
134	C₂²⋊C₄	C₂⁴×C₂₃	1	144	yes	—
135	C₄²	C₂⁴×C₂₃	1	96	yes	—
136	C₂³	—	4	48	no	—
137	C₂³	—	4	18	no	—
138	C₂³	—	8	48	no	—
139	C₂³	C₂³×C₄×C₂₃	1	36	yes	—
140	C₂³	C₂³×C₄×C₂₃	1	96	yes	—
141	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
142	C₂³	C₂⁵×C₂₃	1	18	yes	—
143	Q₈	C₂³×C₄×C₂₃	1	32	yes	—
144	D₄	C₂³×C₄×C₂₃	1	288	yes	—
145	C₂×C₄	—	4	144	no	—
146	C₂×C₄	C₂³×C₄×C₂₃	1	24	yes	—
147	C₂×C₄	C₂³×C₄×C₂₃	1	144	yes	—
148	C₂×C₄	C₂⁵×C₂₃	1	32	yes	—
149	C₂×C₄	C₂⁵×C₂₃	1	12	yes	—
150	C₂²	—	2	36	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	36	no	—
153	C₂²	—	4	96	no	—
154	C₂²	C₂⁴×C₄×C₂₃	1	2	yes	—
155	C₂²	C₂⁴×C₄×C₂₃	1	36	yes	—
156	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
157	C₄	—	2	32	no	—
158	C₄	C₂⁴×C₄×C₂₃	1	24	yes	—
159	C₂	—	2	2	no	—
160	C₂	—	2	36	no	—
161	C₂	C₂⁵×C₄×C₂₃	1	1	yes	characteristic
162	C₂	C₂₃×H(7,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂⁶
commutator	C₂	C₂⁵×C₄×C₂₃
Frattini, cosocle	C₂²	C₂⁶×C₂₃
prime radical	C₂²×C₂₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×H(5,𝔽₂)	C₂	semidirect
3. C₂₃×#3428	C₂	semidirect
4. C₂×C₂₃×#518	C₂	semidirect
5. C₂×C₂₃×H(5,𝔽₂)	C₄	semidirect
9. C₂₃×#520	C₂²	semidirect
10. C₂₃×#519	C₂²	semidirect
11. C₂₃×#518	C₂²	semidirect
12. C₂₃×#518	C₂²	semidirect
13. C₄×#41×C₂₃	C₂²	semidirect
14. C₂×C₄×D₄×C₂₃	C₂²	semidirect
15. C₂×C₂₃×#117	C₂²	semidirect
16. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
18. C₂₃×H(5,𝔽₂)	C₂×C₄	semidirect
19. C₂×#41×C₂₃	C₂×C₄	semidirect
22. C₂²×D₄×C₂₃	C₂×C₄	semidirect
26. C₂³×C₄×C₂₃	C₂³	semidirect
27. C₄×Q₈×C₂₃	C₂³	semidirect
28. C₄×D₄×C₂₃	C₂³	semidirect
30. C₂₃×#117	C₂³	semidirect
31. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
33. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
34. C₂×C₄²×C₂₃	C₂³	semidirect
36. C₂⁴×C₂₃	C₂²×C₄	semidirect
39. #41×C₂₃	C₂²×C₄	semidirect
40. C₂×Q₈×C₂₃	C₂²×C₄	semidirect
42. C₂×D₄×C₂₃	C₂²×C₄	semidirect
43. C₂×D₄×C₂₃	C₂²×C₄	semidirect
46. C₂²×C₄×C₂₃	C₂²×C₄	semidirect
54. #62652	C₂₃	direct (N × H)
70. C₄×H(5,𝔽₂)	C₂×C₂₃	semidirect
71. #3428	C₂×C₂₃	semidirect
72. C₂×#518	C₂×C₂₃	semidirect
82. C₂×H(5,𝔽₂)	C₄×C₂₃	semidirect
86. #520	C₂²×C₂₃	semidirect
87. #519	C₂²×C₂₃	semidirect
88. #518	C₂²×C₂₃	semidirect
89. #518	C₂²×C₂₃	semidirect
90. C₄×#41	C₂²×C₂₃	semidirect
91. C₂×C₄×D₄	C₂²×C₂₃	semidirect
92. C₂×#117	C₂²×C₂₃	semidirect
93. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
99. H(5,𝔽₂)	C₂×C₄×C₂₃	semidirect
100. C₂×#41	C₂×C₄×C₂₃	semidirect
103. C₂²×D₄	C₂×C₄×C₂₃	semidirect
107. C₂³×C₄	C₂³×C₂₃	semidirect
108. C₄×Q₈	C₂³×C₂₃	semidirect
109. C₄×D₄	C₂³×C₂₃	semidirect
111. #117	C₂³×C₂₃	semidirect
112. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
114. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
115. C₂×C₄²	C₂³×C₂₃	semidirect
116. C₂₃	#62652	direct (N × H)
118. C₂⁴	C₂²×C₄×C₂₃	semidirect
121. #41	C₂²×C₄×C₂₃	semidirect
122. C₂×Q₈	C₂²×C₄×C₂₃	semidirect
124. C₂×D₄	C₂²×C₄×C₂₃	semidirect
125. C₂×D₄	C₂²×C₄×C₂₃	semidirect
128. C₂²×C₄	C₂²×C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×H(5,𝔽₂)	2
C₄×C₂₃×H(5,𝔽₂)	2
C₂₃×#3428	2
C₂×C₂₃×#518	2
#62652	23

Subgroups — C23×#62652 #19210278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#62652 #19210278