Subgroups — C₂×C₂₃×#1373 #19159636

Order 5888.

230 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1373	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₂₃×M₃₂	—	2	2	no	—
5	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#364	—	2	2	no	—
9	C₂₃×#364	C₂²	1	4	yes	—
10	C₄×C₁₆×C₂₃	C₄	1	2	yes	—
11	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
15	C₂₃×M₃₂	—	2	4	no	—
16	C₂₃×M₃₂	—	4	2	no	—
17	C₂×C₁₆×C₂₃	—	2	1	no	—
18	C₂×C₁₆×C₂₃	—	2	1	no	—
19	C₂×C₁₆×C₂₃	—	4	2	no	—
20	C₂×C₁₆×C₂₃	C₈	1	2	yes	—
21	C₂×C₁₆×C₂₃	C₈	1	2	yes	—
22	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	—	2	1	no	—
29	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
35	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
38	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₂₃	—	2	1	no	—
43	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₁₆×C₂₃	—	2	2	no	—
46	C₁₆×C₂₃	—	4	4	no	—
47	C₁₆×C₂₃	C₂×C₈	1	4	yes	—
48	C₂×#1373	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₂₃	#108	1	1	yes	characteristic
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	2	1	no	—
52	C₂×C₄×C₂₃	—	2	1	no	—
53	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
56	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
57	C₂×C₄×C₂₃	#108	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₈×C₂₃	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₂₃	C₄⋊C₈	1	1	yes	characteristic
61	C₈×C₂₃	#107	1	1	yes	characteristic
62	C₈×C₂₃	#107	1	1	yes	characteristic
63	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
64	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
66	C₈×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×#364	C₂×C₂₃	1	2	yes	—
68	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
69	#1373	C₂×C₂₃	1	4	yes	direct product factor
70	C₂²×C₂₃	—	2	2	no	—
71	C₂²×C₂₃	#335	1	1	yes	characteristic
72	C₂²×C₂₃	#335	1	1	yes	characteristic
73	C₂²×C₂₃	C₂×#108	1	1	yes	characteristic
74	C₄×C₂₃	—	2	2	no	—
75	C₄×C₂₃	—	2	2	no	—
76	C₄×C₂₃	#335	1	1	yes	characteristic
77	C₄×C₂₃	#335	1	1	yes	characteristic
78	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
80	C₂×M₃₂	—	2	2	no	—
81	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
82	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
84	#364	—	2	2	no	—
85	#364	C₂²×C₂₃	1	4	yes	—
86	C₄×C₁₆	C₄×C₂₃	1	2	yes	—
87	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
88	C₂×C₂₃	—	2	1	no	—
89	C₂×C₂₃	—	2	1	no	—
90	C₂×C₂₃	#1373	1	2	yes	direct product factor
91	C₂×C₂₃	C₂×#335	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
93	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
94	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	M₃₂	—	4	2	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	4	2	no	—
100	C₂×C₁₆	C₈×C₂₃	1	2	yes	—
101	C₂×C₁₆	C₈×C₂₃	1	2	yes	—
102	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
103	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
104	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
105	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
106	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
107	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
108	C₂₃	C₂×#1373	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
111	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
112	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
113	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
114	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
115	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
116	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
117	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
118	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
119	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
120	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
121	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
123	C₄²	—	2	1	no	—
124	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
125	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
126	C₁₆	—	2	2	no	—
127	C₁₆	—	4	4	no	—
128	C₁₆	C₂×C₈×C₂₃	1	4	yes	—
129	C₂³	C₂₃×#108	1	1	yes	characteristic
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
134	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
136	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
137	C₂×C₄	C₂₃×#108	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
139	C₈	C₂₃×C₄⋊C₈	1	1	yes	characteristic
140	C₈	C₂₃×C₄⋊C₈	1	1	yes	characteristic
141	C₈	C₂₃×#107	1	1	yes	characteristic
142	C₈	C₂₃×#107	1	1	yes	characteristic
143	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
144	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
145	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
146	C₈	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
147	C₂²	—	2	2	no	—
148	C₂²	C₂₃×#335	1	1	yes	characteristic
149	C₂²	C₂₃×#335	1	1	yes	characteristic
150	C₂²	C₂×C₂₃×#108	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	C₂₃×#335	1	1	yes	characteristic
154	C₄	C₂₃×#335	1	1	yes	characteristic
155	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
156	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	C₂₃×#1373	1	2	yes	direct product factor
160	C₂	C₂×C₂₃×#335	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	D₈
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	#335
prime radical	C₂³×C₂₃	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1373	C₂	direct (N × H)
48. C₂×#1373	C₂₃	direct (N × H)
69. #1373	C₂×C₂₃	direct (N × H)
90. C₂×C₂₃	#1373	direct (N × H)
108. C₂₃	C₂×#1373	direct (N × H)
159. C₂	C₂₃×#1373	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×#364	2
C₂×C₄×C₁₆×C₂₃	2
C₂₃×#1373	2
C₂×#1373	23

Subgroups — C2×C23×#1373 #19159636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1373 #19159636