Subgroups — C₂×C₂₃×#444 #15670573

Order 2944.

464 conjugacy classes of subgroups.

The subgroup lattice has 184 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
2	C₂×C₂₃×#131	C₂	1	2	yes	maximal, complemented
3	C₂₃×#444	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#104	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₂₃	C₂²	1	2	yes	—
8	C₂₃×#135	C₄	1	4	yes	complemented
9	C₂₃×#135	C₂²	1	4	yes	—
10	C₂×Q₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₁₆×C₂₃	C₄	1	2	yes	complemented
12	C₂×D₈×C₂₃	C₄	1	1	yes	characteristic, complemented
13	C₂₃×#131	—	2	4	no	—
14	C₂₃×#131	C₂²	1	8	yes	complemented
15	C₂×M₁₆×C₂₃	—	2	2	no	—
16	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂²×C₈×C₂₃	—	2	2	no	—
18	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#108	C₂²	1	4	yes	complemented
22	C₂₃×#104	C₂²	1	8	yes	complemented
23	C₂₃×#97	C₂²	1	4	yes	complemented
24	#41×C₂₃	—	2	2	no	—
25	#41×C₂₃	—	2	2	no	—
26	#41×C₂₃	C₂×C₄	1	4	yes	—
27	#41×C₂₃	C₂³	1	4	yes	—
28	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₂₃	—	2	2	no	—
30	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₃	—	2	2	no	—
32	C₂²×C₄×C₂₃	—	2	1	no	—
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	Q₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
35	SD₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
36	D₈×C₂₃	C₂×C₄	1	4	yes	complemented
37	M₁₆×C₂₃	—	2	2	no	—
38	M₁₆×C₂₃	—	2	8	no	—
39	M₁₆×C₂₃	C₂³	1	4	yes	—
40	C₂×C₈×C₂₃	—	2	2	no	—
41	C₂×C₈×C₂₃	—	2	8	no	—
42	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×C₂₃	D₄	1	2	yes	—
46	C₂×C₈×C₂₃	D₄	1	2	yes	—
47	C₂×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂×C₈×C₂₃	C₂³	1	2	yes	—
49	C₄²×C₂₃	—	2	1	no	—
50	C₄²×C₂₃	C₂³	1	2	yes	—
51	C₂³×C₂₃	—	2	2	no	—
52	C₂³×C₂₃	#41	1	1	yes	characteristic
53	Q₈×C₂₃	—	2	2	no	—
54	Q₈×C₂₃	C₂²×C₄	1	4	yes	—
55	D₄×C₂₃	—	2	4	no	—
56	D₄×C₂₃	—	2	4	no	—
57	D₄×C₂₃	—	2	2	no	—
58	D₄×C₂₃	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	4	no	—
61	C₂×C₄×C₂₃	—	2	4	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	4	2	no	—
64	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
70	C₈×C₂₃	—	2	4	no	—
71	C₈×C₂₃	C₂²×C₄	1	2	yes	—
72	C₈×C₂₃	C₂²×C₄	1	2	yes	—
73	C₈×C₂₃	C₂×D₄	1	4	yes	—
74	C₂×#444	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₂₃	—	2	2	no	—
76	C₂²×C₂₃	—	2	4	no	—
77	C₂²×C₂₃	—	4	4	no	—
78	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
79	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
80	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
81	C₄×C₂₃	—	2	4	no	—
82	C₄×C₂₃	—	4	2	no	—
83	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
84	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
85	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
86	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×#135	C₂×C₂₃	1	1	yes	characteristic
88	C₂×#131	C₂×C₂₃	1	2	yes	complemented
89	#444	C₂×C₂₃	1	8	yes	direct product factor
90	C₂×#108	C₂×C₂₃	1	1	yes	characteristic, complemented
91	C₂×#104	C₂×C₂₃	1	2	yes	complemented
92	C₂×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
93	C₂×C₂₃	—	2	1	no	—
94	C₂×C₂₃	—	2	1	no	—
95	C₂×C₂₃	—	4	4	no	—
96	C₂×C₂₃	#444	1	2	yes	direct product factor
97	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
98	C₂×#41	C₂²×C₂₃	1	2	yes	—
99	#135	C₄×C₂₃	1	4	yes	complemented
100	#135	C₂²×C₂₃	1	4	yes	—
101	C₂×Q₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
102	C₂×SD₁₆	C₄×C₂₃	1	2	yes	complemented
103	C₂×D₈	C₄×C₂₃	1	1	yes	characteristic, complemented
104	#131	—	2	4	no	—
105	#131	C₂²×C₂₃	1	8	yes	complemented
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
110	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
111	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
112	#108	C₂²×C₂₃	1	4	yes	complemented
113	#104	C₂²×C₂₃	1	8	yes	complemented
114	#97	C₂²×C₂₃	1	4	yes	complemented
115	C₂₃	C₂×#444	1	1	yes	characteristic, direct product factor, Sylow
116	#41	—	2	2	no	—
117	#41	—	2	2	no	—
118	#41	C₂×C₄×C₂₃	1	4	yes	—
119	#41	C₂³×C₂₃	1	4	yes	—
120	C₂×Q₈	C₂×C₄×C₂₃	1	2	yes	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
126	Q₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
127	SD₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
128	D₈	C₂×C₄×C₂₃	1	4	yes	complemented
129	M₁₆	—	2	2	no	—
130	M₁₆	—	2	8	no	—
131	M₁₆	C₂³×C₂₃	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	8	no	—
134	C₂×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	—
137	C₂×C₈	D₄×C₂₃	1	2	yes	—
138	C₂×C₈	D₄×C₂₃	1	2	yes	—
139	C₂×C₈	C₂³×C₂₃	1	2	yes	—
140	C₂×C₈	C₂³×C₂₃	1	2	yes	—
141	C₄²	—	2	1	no	—
142	C₄²	C₂³×C₂₃	1	2	yes	—
143	C₂³	—	2	2	no	—
144	C₂³	#41×C₂₃	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	Q₈	C₂²×C₄×C₂₃	1	4	yes	—
147	D₄	—	2	4	no	—
148	D₄	—	2	4	no	—
149	D₄	—	2	2	no	—
150	D₄	C₂²×C₄×C₂₃	1	4	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
158	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
159	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
161	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
162	C₈	—	2	4	no	—
163	C₈	C₂²×C₄×C₂₃	1	2	yes	—
164	C₈	C₂²×C₄×C₂₃	1	2	yes	—
165	C₈	C₂×D₄×C₂₃	1	4	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	4	4	no	—
169	C₂²	C₄×D₄×C₂₃	1	1	yes	characteristic
170	C₂²	C₄×D₄×C₂₃	1	1	yes	characteristic
171	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	4	2	no	—
174	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
175	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
176	C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
177	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	4	4	no	—
181	C₂	C₂₃×#444	1	2	yes	direct product factor
182	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₄
commutator	C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₄×D₄
prime radical	C₂×C₂₃×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#131	C₂	semidirect
3. C₂₃×#444	C₂	direct (N × H)
4. C₂×C₂₃×#108	C₂	semidirect
5. C₂×C₂₃×#104	C₂	semidirect
6. C₂×C₂₃×#97	C₂	semidirect
8. C₂₃×#135	C₄	semidirect
10. C₂×Q₁₆×C₂₃	C₄	semidirect
11. C₂×SD₁₆×C₂₃	C₄	semidirect
12. C₂×D₈×C₂₃	C₄	semidirect
14. C₂₃×#131	C₂²	semidirect
21. C₂₃×#108	C₂²	semidirect
22. C₂₃×#104	C₂²	semidirect
23. C₂₃×#97	C₂²	semidirect
34. Q₁₆×C₂₃	C₂×C₄	semidirect
35. SD₁₆×C₂₃	C₂×C₄	semidirect
36. D₈×C₂₃	C₂×C₄	semidirect
74. C₂×#444	C₂₃	direct (N × H)
88. C₂×#131	C₂×C₂₃	semidirect
89. #444	C₂×C₂₃	direct (N × H)
90. C₂×#108	C₂×C₂₃	semidirect
91. C₂×#104	C₂×C₂₃	semidirect
92. C₂×#97	C₂×C₂₃	semidirect
96. C₂×C₂₃	#444	direct (N × H)
99. #135	C₄×C₂₃	semidirect
101. C₂×Q₁₆	C₄×C₂₃	semidirect
102. C₂×SD₁₆	C₄×C₂₃	semidirect
103. C₂×D₈	C₄×C₂₃	semidirect
105. #131	C₂²×C₂₃	semidirect
112. #108	C₂²×C₂₃	semidirect
113. #104	C₂²×C₂₃	semidirect
114. #97	C₂²×C₂₃	semidirect
115. C₂₃	C₂×#444	direct (N × H)
126. Q₁₆	C₂×C₄×C₂₃	semidirect
127. SD₁₆	C₂×C₄×C₂₃	semidirect
128. D₈	C₂×C₄×C₂₃	semidirect
181. C₂	C₂₃×#444	direct (N × H)

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×#444	2
C₂×C₂₃×#108	2
C₂×C₂₃×#104	2
C₂×C₂₃×#97	2
C₂×#444	23

Subgroups — C2×C23×#444 #15670573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#444 #15670573