Subgroups — C₂²×C₂₃×#334 #19165409

Order 5888.

942 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	C₂²×C₂₃×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#334	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₂₃	—	2	2	no	—
5	C₂³×C₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₄⋊C₈	—	2	6	no	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	12	yes	—
10	C₂×C₄×C₈×C₂₃	C₄	1	6	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	6	yes	—
12	C₂₃×#334	C₂²	1	16	yes	direct product factor
13	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
15	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	—	2	4	no	—
17	C₂²×C₈×C₂₃	—	2	12	no	—
18	C₂²×C₈×C₂₃	—	4	8	no	—
19	C₂²×C₈×C₂₃	C₈	1	8	yes	complemented
20	C₂²×C₈×C₂₃	C₂×C₄	1	6	yes	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	6	yes	—
22	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
23	C₂×C₄²×C₂₃	D₄	1	3	yes	—
24	C₂×C₄²×C₂₃	Q₈	1	3	yes	—
25	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
26	C₂₃×C₄⋊C₈	—	2	12	no	—
27	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
28	C₄×C₈×C₂₃	C₂×C₄	1	12	yes	—
29	C₄×C₈×C₂₃	C₂³	1	4	yes	—
30	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	—	2	4	no	—
32	C₂²×C₄×C₂₃	C₄⋊C₄	1	3	yes	—
33	C₂²×C₄×C₂₃	C₄⋊C₄	1	3	yes	—
34	C₂²×C₄×C₂₃	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₂₃	M₁₆	1	4	yes	—
37	C₂²×C₄×C₂₃	SD₁₆	1	8	yes	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₂₃	C₂×Q₈	1	3	yes	—
41	C₂×C₈×C₂₃	—	2	12	no	—
42	C₂×C₈×C₂₃	—	2	8	no	—
43	C₂×C₈×C₂₃	—	4	24	no	—
44	C₂×C₈×C₂₃	C₂×C₈	1	24	yes	complemented
45	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
47	C₄²×C₂₃	C₄⋊C₄	1	6	yes	—
48	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
49	C₄²×C₂₃	C₂×D₄	1	3	yes	—
50	C₄²×C₂₃	C₂×Q₈	1	3	yes	—
51	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂²×#334	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₂₃	C₄⋊C₈	1	4	yes	—
54	C₂³×C₂₃	#106	1	4	yes	—
55	C₂³×C₂₃	#108	1	4	yes	—
56	C₂³×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
57	C₂×C₄×C₂₃	—	2	6	no	—
58	C₂×C₄×C₂₃	—	2	6	no	—
59	C₂×C₄×C₂₃	C₄⋊C₈	1	12	yes	—
60	C₂×C₄×C₂₃	#106	1	12	yes	—
61	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
62	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
63	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
64	C₂×C₄×C₂₃	C₂²×C₈	1	6	yes	—
65	C₂×C₄×C₂₃	C₂×M₁₆	1	6	yes	—
66	C₂×C₄×C₂₃	C₂×SD₁₆	1	12	yes	—
67	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
70	C₈×C₂₃	—	2	4	no	—
71	C₈×C₂₃	—	4	8	no	—
72	C₈×C₂₃	C₂²×C₈	1	8	yes	complemented
73	C₂²×C₄⋊C₈	C₂×C₂₃	1	2	yes	—
74	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
75	C₂×#334	C₂×C₂₃	1	12	yes	direct product factor
76	C₂²×C₂₃	#334	1	16	yes	direct product factor
77	C₂²×C₂₃	C₂×C₄⋊C₈	1	6	yes	—
78	C₂²×C₂₃	C₂×#106	1	6	yes	—
79	C₂²×C₂₃	C₂×#108	1	6	yes	—
80	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₂₃	—	2	3	no	—
82	C₄×C₂₃	—	2	1	no	—
83	C₄×C₂₃	C₂×C₄⋊C₈	1	6	yes	—
84	C₄×C₂₃	C₂×#106	1	6	yes	—
85	C₄×C₂₃	C₂³×C₈	1	1	yes	characteristic
86	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
87	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
88	C₂³×C₈	—	2	2	no	—
89	C₂³×C₈	C₄×C₂₃	1	1	yes	characteristic
90	C₂³×C₈	C₄×C₂₃	1	1	yes	characteristic
91	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
92	C₂×C₄⋊C₈	—	2	6	no	—
93	C₂×C₄⋊C₈	C₂²×C₂₃	1	12	yes	—
94	C₂×C₄×C₈	C₄×C₂₃	1	6	yes	—
95	C₂×C₄×C₈	C₂²×C₂₃	1	6	yes	—
96	#334	C₂²×C₂₃	1	16	yes	direct product factor
97	C₂×C₂₃	C₂×#334	1	12	yes	direct product factor
98	C₂×C₂₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₂₃	C₂²×#106	1	1	yes	characteristic
100	C₂×C₂₃	C₂²×#108	1	1	yes	characteristic
101	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
102	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
103	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
104	C₂²×C₈	—	2	4	no	—
105	C₂²×C₈	—	2	12	no	—
106	C₂²×C₈	—	4	8	no	—
107	C₂²×C₈	C₈×C₂₃	1	8	yes	complemented
108	C₂²×C₈	C₂×C₄×C₂₃	1	6	yes	—
109	C₂²×C₈	C₂×C₄×C₂₃	1	6	yes	—
110	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
111	C₂×C₄²	D₄×C₂₃	1	3	yes	—
112	C₂×C₄²	Q₈×C₂₃	1	3	yes	—
113	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
114	C₄⋊C₈	—	2	12	no	—
115	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
116	C₄×C₈	C₂×C₄×C₂₃	1	12	yes	—
117	C₄×C₈	C₂³×C₂₃	1	4	yes	—
118	C₂₃	C₂²×#334	1	1	yes	characteristic, direct product factor, Sylow
119	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	C₄⋊C₄×C₂₃	1	3	yes	—
122	C₂²×C₄	C₄⋊C₄×C₂₃	1	3	yes	—
123	C₂²×C₄	C₄⋊C₄×C₂₃	1	3	yes	—
124	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
125	C₂²×C₄	M₁₆×C₂₃	1	4	yes	—
126	C₂²×C₄	SD₁₆×C₂₃	1	8	yes	—
127	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
128	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
129	C₂²×C₄	C₂×Q₈×C₂₃	1	3	yes	—
130	C₂×C₈	—	2	12	no	—
131	C₂×C₈	—	2	8	no	—
132	C₂×C₈	—	4	24	no	—
133	C₂×C₈	C₂×C₈×C₂₃	1	24	yes	complemented
134	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
135	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
136	C₄²	C₄⋊C₄×C₂₃	1	6	yes	—
137	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
138	C₄²	C₂×D₄×C₂₃	1	3	yes	—
139	C₄²	C₂×Q₈×C₂₃	1	3	yes	—
140	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
141	C₂³	C₂₃×C₄⋊C₈	1	4	yes	—
142	C₂³	C₂₃×#106	1	4	yes	—
143	C₂³	C₂₃×#108	1	4	yes	—
144	C₂³	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
145	C₂×C₄	—	2	6	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	C₂₃×C₄⋊C₈	1	12	yes	—
148	C₂×C₄	C₂₃×#106	1	12	yes	—
149	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
150	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
151	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
152	C₂×C₄	C₂²×C₈×C₂₃	1	6	yes	—
153	C₂×C₄	C₂×M₁₆×C₂₃	1	6	yes	—
154	C₂×C₄	C₂×SD₁₆×C₂₃	1	12	yes	—
155	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	4	8	no	—
160	C₈	C₂²×C₈×C₂₃	1	8	yes	complemented
161	C₂²	C₂₃×#334	1	16	yes	direct product factor
162	C₂²	C₂×C₂₃×C₄⋊C₈	1	6	yes	—
163	C₂²	C₂×C₂₃×#106	1	6	yes	—
164	C₂²	C₂×C₂₃×#108	1	6	yes	—
165	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
166	C₄	—	2	3	no	—
167	C₄	—	2	1	no	—
168	C₄	C₂×C₂₃×C₄⋊C₈	1	6	yes	—
169	C₄	C₂×C₂₃×#106	1	6	yes	—
170	C₄	C₂³×C₈×C₂₃	1	1	yes	characteristic
171	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
172	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
173	C₂	C₂×C₂₃×#334	1	12	yes	direct product factor
174	C₂	C₂²×C₂₃×C₄⋊C₈	1	1	yes	characteristic
175	C₂	C₂²×C₂₃×#106	1	1	yes	characteristic
176	C₂	C₂²×C₂₃×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₂₃	D₄
commutator	C₄	C₂³×C₈×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle, prime radical	C₂⁴×C₂₃	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#334	C₂	direct (N × H)
12. C₂₃×#334	C₂²	direct (N × H)
19. C₂²×C₈×C₂₃	C₈	semidirect
44. C₂×C₈×C₂₃	C₂×C₈	semidirect
52. C₂²×#334	C₂₃	direct (N × H)
72. C₈×C₂₃	C₂²×C₈	semidirect
75. C₂×#334	C₂×C₂₃	direct (N × H)
76. C₂²×C₂₃	#334	direct (N × H)
96. #334	C₂²×C₂₃	direct (N × H)
97. C₂×C₂₃	C₂×#334	direct (N × H)
107. C₂²×C₈	C₈×C₂₃	semidirect
118. C₂₃	C₂²×#334	direct (N × H)
133. C₂×C₈	C₂×C₈×C₂₃	semidirect
160. C₈	C₂²×C₈×C₂₃	semidirect
161. C₂²	C₂₃×#334	direct (N × H)
173. C₂	C₂×C₂₃×#334	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×C₄⋊C₈	2
C₂²×C₄×C₈×C₂₃	2
C₂×C₂₃×#334	2
C₂²×#334	23

Subgroups — C22×C23×#334 #19165409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#334 #19165409