Subgroups — C₂₃×#3498 #15671117

Order 2944.

1748 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₂₃×#550	C₂	1	4	yes	maximal, complemented
3	D₄²×C₂₃	C₂	1	12	yes	maximal, complemented
4	C₂₃×#535	C₂	1	6	yes	maximal, complemented
5	C₂₃×#532	C₂	1	3	yes	maximal, complemented
6	C₂×C₂₃×#127	C₂	1	3	yes	maximal, complemented
7	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×H(5,𝔽₂)	—	2	16	no	—
9	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×#41×C₂₃	C₂²	1	4	yes	complemented
11	C₂×#41×C₂₃	C₂²	1	6	yes	complemented
12	C₂²×D₄×C₂₃	C₂²	1	6	yes	complemented
13	C₂²×D₄×C₂₃	C₂²	1	12	yes	complemented
14	C₂²×D₄×C₂₃	C₂²	1	12	yes	complemented
15	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₂₃×#127	C₂²	1	12	yes	complemented
17	C₂₃×#127	C₂²	1	3	yes	complemented
18	C₂₃×#127	C₂²	1	6	yes	complemented
19	C₂₃×#124	C₂²	1	6	yes	complemented
20	C₂₃×#121	C₂²	1	24	yes	complemented
21	C₂₃×#121	C₂²	1	12	yes	complemented
22	C₂₃×#120	C₂²	1	24	yes	complemented
23	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
24	C₄×D₄×C₂₃	C₂²	1	12	yes	complemented
25	C₄×D₄×C₂₃	C₂²	1	6	yes	complemented
26	C₂₃×#117	C₂²	1	3	yes	complemented
27	C₂×C₄²×C₂₃	C₂²	1	3	yes	complemented
28	C₂⁴×C₂₃	C₂³	1	12	yes	—
29	#41×C₂₃	—	2	16	no	—
30	#41×C₂₃	—	2	24	no	—
31	#41×C₂₃	D₄	1	8	yes	complemented
32	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
33	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×D₄×C₂₃	—	2	48	no	—
35	C₂×D₄×C₂₃	—	2	24	no	—
36	C₂×D₄×C₂₃	—	4	24	no	—
37	C₂×D₄×C₂₃	C₂³	1	12	yes	—
38	C₂×D₄×C₂₃	C₂³	1	12	yes	—
39	C₂×D₄×C₂₃	C₂³	1	12	yes	—
40	C₂×D₄×C₂₃	C₂³	1	6	yes	—
41	C₂×D₄×C₂₃	C₂³	1	24	yes	—
42	C₂×D₄×C₂₃	C₂³	1	3	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	12	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
46	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
47	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
49	C₂²⋊C₄×C₂₃	C₂³	1	24	yes	—
50	C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₄²×C₂₃	C₂³	1	3	yes	—
52	C₄²×C₂₃	C₂³	1	6	yes	—
53	C₂³×C₂₃	—	2	24	no	—
54	C₂³×C₂₃	—	4	24	no	—
55	C₂³×C₂₃	C₂⁴	1	12	yes	—
56	C₂³×C₂₃	C₂⁴	1	3	yes	—
57	Q₈×C₂₃	—	2	4	no	—
58	Q₈×C₂₃	C₂×D₄	1	4	yes	complemented
59	D₄×C₂₃	—	2	24	no	—
60	D₄×C₂₃	—	2	24	no	—
61	D₄×C₂₃	—	2	12	no	—
62	D₄×C₂₃	—	2	24	no	—
63	D₄×C₂₃	—	4	24	no	—
64	D₄×C₂₃	C₂×D₄	1	12	yes	complemented
65	C₂×C₄×C₂₃	—	2	24	no	—
66	C₂×C₄×C₂₃	—	2	12	no	—
67	C₂×C₄×C₂₃	C₂×D₄	1	12	yes	complemented
68	C₂×C₄×C₂₃	C₂⁴	1	6	yes	—
69	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
70	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
72	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
73	#3498	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₂₃	—	2	6	no	—
75	C₂²×C₂₃	—	2	12	no	—
76	C₂²×C₂₃	—	2	24	no	—
77	C₂²×C₂₃	—	8	24	no	—
78	C₂²×C₂₃	C₂²×D₄	1	6	yes	—
79	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
80	C₄×C₂₃	—	2	4	no	—
81	C₄×C₂₃	—	2	6	no	—
82	C₄×C₂₃	C₂²×D₄	1	6	yes	—
83	C₄×C₂₃	C₂²×D₄	1	2	yes	—
84	C₄×C₂₃	H(5,𝔽₂)	1	4	yes	complemented
85	C₂×H(5,𝔽₂)	C₂×C₂₃	1	2	yes	complemented
86	#550	C₂×C₂₃	1	4	yes	complemented
87	D₄²	C₂×C₂₃	1	12	yes	complemented
88	#535	C₂×C₂₃	1	6	yes	complemented
89	#532	C₂×C₂₃	1	3	yes	complemented
90	C₂×#127	C₂×C₂₃	1	3	yes	complemented
91	C₄×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
92	C₂×C₂₃	—	2	6	no	—
93	C₂×C₂₃	—	4	12	no	—
94	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
95	C₂×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
96	H(5,𝔽₂)	—	2	16	no	—
97	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
98	C₂×#41	C₂²×C₂₃	1	6	yes	complemented
99	C₂×#41	C₂²×C₂₃	1	4	yes	complemented
100	C₂²×D₄	C₂²×C₂₃	1	6	yes	complemented
101	C₂²×D₄	C₂²×C₂₃	1	12	yes	complemented
102	C₂²×D₄	C₂²×C₂₃	1	12	yes	complemented
103	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
104	#127	C₂²×C₂₃	1	3	yes	complemented
105	#127	C₂²×C₂₃	1	6	yes	complemented
106	#127	C₂²×C₂₃	1	12	yes	complemented
107	#124	C₂²×C₂₃	1	6	yes	complemented
108	#121	C₂²×C₂₃	1	12	yes	complemented
109	#121	C₂²×C₂₃	1	24	yes	complemented
110	#120	C₂²×C₂₃	1	24	yes	complemented
111	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
112	C₄×D₄	C₂²×C₂₃	1	6	yes	complemented
113	C₄×D₄	C₂²×C₂₃	1	12	yes	complemented
114	#117	C₂²×C₂₃	1	3	yes	complemented
115	C₂×C₄²	C₂²×C₂₃	1	3	yes	complemented
116	C₂₃	#3498	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴	C₂³×C₂₃	1	12	yes	—
118	#41	—	2	16	no	—
119	#41	—	2	24	no	—
120	#41	D₄×C₂₃	1	8	yes	complemented
121	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
122	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
123	C₂×D₄	—	2	48	no	—
124	C₂×D₄	—	2	24	no	—
125	C₂×D₄	—	4	24	no	—
126	C₂×D₄	C₂³×C₂₃	1	12	yes	—
127	C₂×D₄	C₂³×C₂₃	1	12	yes	—
128	C₂×D₄	C₂³×C₂₃	1	12	yes	—
129	C₂×D₄	C₂³×C₂₃	1	6	yes	—
130	C₂×D₄	C₂³×C₂₃	1	24	yes	—
131	C₂×D₄	C₂³×C₂₃	1	3	yes	—
132	C₂²×C₄	C₂³×C₂₃	1	6	yes	—
133	C₂²×C₄	C₂³×C₂₃	1	12	yes	—
134	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
135	C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
136	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
137	C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
138	C₂²⋊C₄	C₂³×C₂₃	1	24	yes	—
139	C₄²	C₂³×C₂₃	1	1	yes	characteristic
140	C₄²	C₂³×C₂₃	1	3	yes	—
141	C₄²	C₂³×C₂₃	1	6	yes	—
142	C₂³	—	2	24	no	—
143	C₂³	—	4	24	no	—
144	C₂³	C₂⁴×C₂₃	1	12	yes	—
145	C₂³	C₂⁴×C₂₃	1	3	yes	—
146	Q₈	—	2	4	no	—
147	Q₈	C₂×D₄×C₂₃	1	4	yes	complemented
148	D₄	—	2	24	no	—
149	D₄	—	2	24	no	—
150	D₄	—	2	12	no	—
151	D₄	—	2	24	no	—
152	D₄	—	4	24	no	—
153	D₄	C₂×D₄×C₂₃	1	12	yes	complemented
154	C₂×C₄	—	2	24	no	—
155	C₂×C₄	—	2	12	no	—
156	C₂×C₄	C₂×D₄×C₂₃	1	12	yes	complemented
157	C₂×C₄	C₂⁴×C₂₃	1	6	yes	—
158	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
159	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
160	C₂×C₄	C₂⁴×C₂₃	1	3	yes	—
161	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
162	C₂²	—	2	6	no	—
163	C₂²	—	2	12	no	—
164	C₂²	—	2	24	no	—
165	C₂²	—	8	24	no	—
166	C₂²	C₂²×D₄×C₂₃	1	6	yes	—
167	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	6	no	—
170	C₄	C₂²×D₄×C₂₃	1	6	yes	—
171	C₄	C₂²×D₄×C₂₃	1	2	yes	—
172	C₄	C₂₃×H(5,𝔽₂)	1	4	yes	complemented
173	C₂	—	2	6	no	—
174	C₂	—	4	12	no	—
175	C₂	C₂³×D₄×C₂₃	1	1	yes	characteristic
176	C₂	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×H(5,𝔽₂)	C₂	semidirect
2. C₂₃×#550	C₂	semidirect
3. D₄²×C₂₃	C₂	semidirect
4. C₂₃×#535	C₂	semidirect
5. C₂₃×#532	C₂	semidirect
6. C₂×C₂₃×#127	C₂	semidirect
7. C₄×#41×C₂₃	C₂	semidirect
10. C₂×#41×C₂₃	C₂²	semidirect
11. C₂×#41×C₂₃	C₂²	semidirect
12. C₂²×D₄×C₂₃	C₂²	semidirect
13. C₂²×D₄×C₂₃	C₂²	semidirect
14. C₂²×D₄×C₂₃	C₂²	semidirect
15. C₂₃×C₄⋊Q₈	C₂²	semidirect
16. C₂₃×#127	C₂²	semidirect
17. C₂₃×#127	C₂²	semidirect
18. C₂₃×#127	C₂²	semidirect
19. C₂₃×#124	C₂²	semidirect
20. C₂₃×#121	C₂²	semidirect
21. C₂₃×#121	C₂²	semidirect
22. C₂₃×#120	C₂²	semidirect
23. C₄×Q₈×C₂₃	C₂²	semidirect
24. C₄×D₄×C₂₃	C₂²	semidirect
25. C₄×D₄×C₂₃	C₂²	semidirect
26. C₂₃×#117	C₂²	semidirect
27. C₂×C₄²×C₂₃	C₂²	semidirect
31. #41×C₂₃	D₄	semidirect
58. Q₈×C₂₃	C₂×D₄	semidirect
64. D₄×C₂₃	C₂×D₄	semidirect
67. C₂×C₄×C₂₃	C₂×D₄	semidirect
73. #3498	C₂₃	direct (N × H)
84. C₄×C₂₃	H(5,𝔽₂)	semidirect
85. C₂×H(5,𝔽₂)	C₂×C₂₃	semidirect
86. #550	C₂×C₂₃	semidirect
87. D₄²	C₂×C₂₃	semidirect
88. #535	C₂×C₂₃	semidirect
89. #532	C₂×C₂₃	semidirect
90. C₂×#127	C₂×C₂₃	semidirect
91. C₄×#41	C₂×C₂₃	semidirect
98. C₂×#41	C₂²×C₂₃	semidirect
99. C₂×#41	C₂²×C₂₃	semidirect
100. C₂²×D₄	C₂²×C₂₃	semidirect
101. C₂²×D₄	C₂²×C₂₃	semidirect
102. C₂²×D₄	C₂²×C₂₃	semidirect
103. C₄⋊Q₈	C₂²×C₂₃	semidirect
104. #127	C₂²×C₂₃	semidirect
105. #127	C₂²×C₂₃	semidirect
106. #127	C₂²×C₂₃	semidirect
107. #124	C₂²×C₂₃	semidirect
108. #121	C₂²×C₂₃	semidirect
109. #121	C₂²×C₂₃	semidirect
110. #120	C₂²×C₂₃	semidirect
111. C₄×Q₈	C₂²×C₂₃	semidirect
112. C₄×D₄	C₂²×C₂₃	semidirect
113. C₄×D₄	C₂²×C₂₃	semidirect
114. #117	C₂²×C₂₃	semidirect
115. C₂×C₄²	C₂²×C₂₃	semidirect
116. C₂₃	#3498	direct (N × H)
120. #41	D₄×C₂₃	semidirect
147. Q₈	C₂×D₄×C₂₃	semidirect
153. D₄	C₂×D₄×C₂₃	semidirect
156. C₂×C₄	C₂×D₄×C₂₃	semidirect
172. C₄	C₂₃×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×H(5,𝔽₂)	2
C₂₃×#550	2
D₄²×C₂₃	2
C₂₃×#535	2
C₂₃×#532	2
C₂×C₂₃×#127	2
C₄×#41×C₂₃	2
#3498	23

Subgroups — C23×#3498 #15671117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3498 #15671117