Subgroups — C₂³×C₂₃×#108 #19207736

Order 5888.

2846 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂⁴×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×#108	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₂₃	—	2	14	no	—
6	C₂²×M₁₆×C₂₃	C₂²	1	28	yes	—
7	C₂³×C₈×C₂₃	—	2	2	no	—
8	C₂³×C₈×C₂₃	C₄	1	2	yes	—
9	C₂³×C₈×C₂₃	C₄	1	14	yes	—
10	C₂³×C₈×C₂₃	C₂²	1	14	yes	—
11	C₂×C₂₃×#108	C₂²	1	112	yes	direct product factor
12	C₂⁵×C₂₃	Q₈	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	C₂×C₄	1	7	yes	—
15	C₂³×C₄×C₂₃	D₄	1	7	yes	—
16	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₂₃	Q₈	1	7	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	7	yes	—
19	C₂×M₁₆×C₂₃	—	2	84	no	—
20	C₂×M₁₆×C₂₃	C₂³	1	56	yes	—
21	C₂²×C₈×C₂₃	—	2	28	no	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	84	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	28	yes	—
24	C₂²×C₈×C₂₃	C₂³	1	28	yes	—
25	C₂₃×#108	C₂³	1	64	yes	direct product factor
26	C₂⁴×C₂₃	—	2	8	no	—
27	C₂⁴×C₂₃	C₄⋊C₄	1	7	yes	—
28	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂⁴×C₂₃	C₂×Q₈	1	7	yes	—
30	C₂²×C₄×C₂₃	C₄⋊C₄	1	42	yes	—
31	C₂²×C₄×C₂₃	C₄⋊C₄	1	7	yes	—
32	C₂²×C₄×C₂₃	C₄⋊C₄	1	7	yes	—
33	C₂²×C₄×C₂₃	C₂²×C₄	1	21	yes	—
34	C₂²×C₄×C₂₃	C₂²×C₄	1	7	yes	—
35	C₂²×C₄×C₂₃	C₂×D₄	1	7	yes	—
36	C₂²×C₄×C₂₃	C₂×D₄	1	21	yes	—
37	C₂²×C₄×C₂₃	C₂×Q₈	1	21	yes	—
38	C₂²×C₄×C₂₃	C₂⁴	1	7	yes	—
39	M₁₆×C₂₃	—	2	56	no	—
40	M₁₆×C₂₃	C₂⁴	1	16	yes	—
41	C₂×C₈×C₂₃	—	2	56	no	—
42	C₂×C₈×C₂₃	C₂²×C₄	1	56	yes	—
43	C₂×C₈×C₂₃	C₂²×C₄	1	56	yes	—
44	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
45	C₂³×#108	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₂₃	—	2	28	no	—
47	C₂³×C₂₃	—	2	28	no	—
48	C₂³×C₂₃	#108	1	8	yes	direct product factor
49	C₂³×C₂₃	C₂×C₄⋊C₄	1	21	yes	—
50	C₂³×C₂₃	C₂×C₄⋊C₄	1	7	yes	—
51	C₂³×C₂₃	C₂²×Q₈	1	7	yes	—
52	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	42	yes	—
53	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	21	yes	—
54	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	21	yes	—
55	C₂×C₄×C₂₃	C₂³×C₄	1	7	yes	—
56	C₂×C₄×C₂₃	C₂³×C₄	1	7	yes	—
57	C₂×C₄×C₂₃	C₂²×D₄	1	7	yes	—
58	C₂×C₄×C₂₃	C₂²×D₄	1	7	yes	—
59	C₂×C₄×C₂₃	C₂²×Q₈	1	7	yes	—
60	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
61	C₈×C₂₃	—	2	16	no	—
62	C₈×C₂₃	C₂³×C₄	1	16	yes	—
63	C₂³×M₁₆	C₂×C₂₃	1	2	yes	—
64	C₂⁴×C₈	C₂×C₂₃	1	1	yes	characteristic
65	C₂²×#108	C₂×C₂₃	1	28	yes	direct product factor
66	C₂²×C₂₃	—	2	42	no	—
67	C₂²×C₂₃	—	2	14	no	—
68	C₂²×C₂₃	C₂×#108	1	28	yes	direct product factor
69	C₂²×C₂₃	C₂²×C₄⋊C₄	1	7	yes	—
70	C₂²×C₂₃	C₂²×C₄⋊C₄	1	7	yes	—
71	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
72	C₄×C₂₃	C₂²×C₄⋊C₄	1	7	yes	—
73	C₄×C₂₃	C₂²×C₄⋊C₄	1	7	yes	—
74	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
75	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
76	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
77	C₂²×M₁₆	—	2	14	no	—
78	C₂²×M₁₆	C₂²×C₂₃	1	28	yes	—
79	C₂³×C₈	—	2	2	no	—
80	C₂³×C₈	C₄×C₂₃	1	14	yes	—
81	C₂³×C₈	C₄×C₂₃	1	2	yes	—
82	C₂³×C₈	C₂²×C₂₃	1	14	yes	—
83	C₂×#108	C₂²×C₂₃	1	112	yes	direct product factor
84	C₂×C₂₃	—	2	7	no	—
85	C₂×C₂₃	—	2	1	no	—
86	C₂×C₂₃	C₂²×#108	1	14	yes	direct product factor
87	C₂×C₂₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
88	C₂⁵	Q₈×C₂₃	1	1	yes	characteristic
89	C₂³×C₄	C₂×C₄×C₂₃	1	7	yes	—
90	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
91	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
92	C₂³×C₄	D₄×C₂₃	1	7	yes	—
93	C₂³×C₄	Q₈×C₂₃	1	7	yes	—
94	C₂³×C₄	C₂³×C₂₃	1	7	yes	—
95	C₂×M₁₆	—	2	84	no	—
96	C₂×M₁₆	C₂³×C₂₃	1	56	yes	—
97	C₂²×C₈	—	2	28	no	—
98	C₂²×C₈	C₂×C₄×C₂₃	1	84	yes	—
99	C₂²×C₈	C₂×C₄×C₂₃	1	28	yes	—
100	C₂²×C₈	C₂³×C₂₃	1	28	yes	—
101	#108	C₂³×C₂₃	1	64	yes	direct product factor
102	C₂₃	C₂³×#108	1	1	yes	characteristic, direct product factor, Sylow
103	C₂⁴	—	2	8	no	—
104	C₂⁴	C₄⋊C₄×C₂₃	1	7	yes	—
105	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
106	C₂⁴	C₂×Q₈×C₂₃	1	7	yes	—
107	C₂²×C₄	C₄⋊C₄×C₂₃	1	42	yes	—
108	C₂²×C₄	C₄⋊C₄×C₂₃	1	7	yes	—
109	C₂²×C₄	C₄⋊C₄×C₂₃	1	7	yes	—
110	C₂²×C₄	C₂²×C₄×C₂₃	1	21	yes	—
111	C₂²×C₄	C₂²×C₄×C₂₃	1	7	yes	—
112	C₂²×C₄	C₂×D₄×C₂₃	1	7	yes	—
113	C₂²×C₄	C₂×D₄×C₂₃	1	21	yes	—
114	C₂²×C₄	C₂×Q₈×C₂₃	1	21	yes	—
115	C₂²×C₄	C₂⁴×C₂₃	1	7	yes	—
116	M₁₆	—	2	56	no	—
117	M₁₆	C₂⁴×C₂₃	1	16	yes	—
118	C₂×C₈	—	2	56	no	—
119	C₂×C₈	C₂²×C₄×C₂₃	1	56	yes	—
120	C₂×C₈	C₂²×C₄×C₂₃	1	56	yes	—
121	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
122	C₂³	—	2	28	no	—
123	C₂³	—	2	28	no	—
124	C₂³	C₂₃×#108	1	8	yes	direct product factor
125	C₂³	C₂×C₄⋊C₄×C₂₃	1	21	yes	—
126	C₂³	C₂×C₄⋊C₄×C₂₃	1	7	yes	—
127	C₂³	C₂²×Q₈×C₂₃	1	7	yes	—
128	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	42	yes	—
129	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	21	yes	—
130	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	21	yes	—
131	C₂×C₄	C₂³×C₄×C₂₃	1	7	yes	—
132	C₂×C₄	C₂³×C₄×C₂₃	1	7	yes	—
133	C₂×C₄	C₂²×D₄×C₂₃	1	7	yes	—
134	C₂×C₄	C₂²×D₄×C₂₃	1	7	yes	—
135	C₂×C₄	C₂²×Q₈×C₂₃	1	7	yes	—
136	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
137	C₈	—	2	16	no	—
138	C₈	C₂³×C₄×C₂₃	1	16	yes	—
139	C₂²	—	2	42	no	—
140	C₂²	—	2	14	no	—
141	C₂²	C₂×C₂₃×#108	1	28	yes	direct product factor
142	C₂²	C₂²×C₄⋊C₄×C₂₃	1	7	yes	—
143	C₂²	C₂²×C₄⋊C₄×C₂₃	1	7	yes	—
144	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
145	C₄	C₂²×C₄⋊C₄×C₂₃	1	7	yes	—
146	C₄	C₂²×C₄⋊C₄×C₂₃	1	7	yes	—
147	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
148	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	7	no	—
151	C₂	C₂²×C₂₃×#108	1	14	yes	direct product factor
152	C₂	C₂³×C₄⋊C₄×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₂₃	D₄
commutator	C₄	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂⁴×C₂₃	C₄⋊C₄
prime radical	C₂⁵×C₂₃	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₂₃×#108	C₂	direct (N × H)
11. C₂×C₂₃×#108	C₂²	direct (N × H)
25. C₂₃×#108	C₂³	direct (N × H)
45. C₂³×#108	C₂₃	direct (N × H)
48. C₂³×C₂₃	#108	direct (N × H)
65. C₂²×#108	C₂×C₂₃	direct (N × H)
68. C₂²×C₂₃	C₂×#108	direct (N × H)
83. C₂×#108	C₂²×C₂₃	direct (N × H)
86. C₂×C₂₃	C₂²×#108	direct (N × H)
101. #108	C₂³×C₂₃	direct (N × H)
102. C₂₃	C₂³×#108	direct (N × H)
124. C₂³	C₂₃×#108	direct (N × H)
141. C₂²	C₂×C₂₃×#108	direct (N × H)
151. C₂	C₂²×C₂₃×#108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₂₃	2
C₂⁴×C₈×C₂₃	2
C₂²×C₂₃×#108	2
C₂³×#108	23

Subgroups — C23×C23×#108 #19207736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×#108 #19207736