Subgroups — C₂×C₂₃×#2864 #19184260

Order 5888.

2078 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2864	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₂₃×#389	C₂	1	4	yes	maximal
3	C₂×C₂₃×#387	C₂	1	8	yes	maximal
4	C₂×C₂₃×#382	C₂	1	2	yes	maximal
5	C₂×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
7	C₂²×C₄⋊C₄×C₂₃	C₂²	1	8	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
9	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#389	C₂²	1	32	yes	—
12	C₂₃×#387	C₂²	1	64	yes	—
13	C₂₃×#382	C₂²	1	16	yes	—
14	C₂×C₂₃×#95	C₂²	1	8	yes	—
15	C₂×C₂₃×#95	C₂²	1	8	yes	—
16	C₄³×C₂₃	C₂²	1	8	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
19	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
20	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	—	2	32	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
23	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
24	C₂×C₄⋊C₄×C₂₃	C₂³	1	32	yes	—
25	C₂×C₄²×C₂₃	—	2	16	no	—
26	C₂×C₄²×C₂₃	—	2	8	no	—
27	C₂×C₄²×C₂₃	Q₈	1	8	yes	—
28	C₂×C₄²×C₂₃	C₂³	1	16	yes	—
29	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
31	C₂₃×#95	C₂³	1	32	yes	—
32	C₂₃×#95	C₂³	1	32	yes	—
33	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	—	2	8	no	—
35	C₂²×C₄×C₂₃	—	2	8	no	—
36	C₂²×C₄×C₂₃	—	2	8	no	—
37	C₂²×C₄×C₂₃	—	2	16	no	—
38	C₂²×C₄×C₂₃	—	2	32	no	—
39	C₂²×C₄×C₂₃	C₂×Q₈	1	8	yes	—
40	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₂₃	#41	1	16	yes	—
42	C₂²×C₄×C₂₃	#41	1	8	yes	—
43	C₂²×C₄×C₂₃	C₂⁴	1	16	yes	—
44	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
45	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
47	C₄⋊C₄×C₂₃	—	2	64	no	—
48	C₄⋊C₄×C₂₃	—	2	32	no	—
49	C₄²×C₂₃	—	2	32	no	—
50	C₄²×C₂₃	—	2	16	no	—
51	C₄²×C₂₃	C₂×Q₈	1	16	yes	—
52	C₂×#2864	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₂₃	C₂²×Q₈	1	2	yes	—
54	C₂³×C₂₃	C₂×#41	1	4	yes	—
55	C₂³×C₂₃	C₂×#41	1	8	yes	—
56	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
57	C₂×C₄×C₂₃	—	2	32	no	—
58	C₂×C₄×C₂₃	—	2	16	no	—
59	C₂×C₄×C₂₃	—	2	8	no	—
60	C₂×C₄×C₂₃	—	2	8	no	—
61	C₂×C₄×C₂₃	—	2	8	no	—
62	C₂×C₄×C₂₃	—	2	16	no	—
63	C₂×C₄×C₂₃	—	2	32	no	—
64	C₂×C₄×C₂₃	—	4	32	no	—
65	C₂×C₄×C₂₃	#125	1	16	yes	—
66	C₂×C₄×C₂₃	C₂²×Q₈	1	8	yes	—
67	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
68	C₂×C₄×C₂₃	C₂×#41	1	16	yes	—
69	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
70	#2864	C₂×C₂₃	1	16	yes	direct product factor
71	C₂×#389	C₂×C₂₃	1	4	yes	—
72	C₂×#387	C₂×C₂₃	1	8	yes	—
73	C₂×#382	C₂×C₂₃	1	2	yes	—
74	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic
75	C₂²×C₂₃	C₂×#125	1	4	yes	—
76	C₂²×C₂₃	#529	1	16	yes	—
77	C₂²×C₂₃	#533	1	8	yes	—
78	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
79	C₂²×C₂₃	C₂²×#41	1	2	yes	—
80	C₂²×C₂₃	C₂²×#41	1	4	yes	—
81	C₄×C₂₃	—	2	16	no	—
82	C₄×C₂₃	—	2	8	no	—
83	C₄×C₂₃	—	4	16	no	—
84	C₄×C₂₃	C₂×#125	1	8	yes	—
85	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
86	C₂²×C₄⋊C₄	C₂²×C₂₃	1	8	yes	—
87	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
88	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
89	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
90	#389	C₂²×C₂₃	1	32	yes	—
91	#387	C₂²×C₂₃	1	64	yes	—
92	#382	C₂²×C₂₃	1	16	yes	—
93	C₂×#95	C₂²×C₂₃	1	8	yes	—
94	C₂×#95	C₂²×C₂₃	1	8	yes	—
95	C₄³	C₂²×C₂₃	1	8	yes	—
96	C₂×C₂₃	#2864	1	8	yes	direct product factor
97	C₂×C₂₃	C₂²×#125	1	1	yes	characteristic
98	C₂×C₂₃	C₂×#529	1	4	yes	—
99	C₂×C₂₃	C₂×#533	1	2	yes	—
100	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
101	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
102	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
103	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	2	32	no	—
105	C₂×C₄⋊C₄	—	2	16	no	—
106	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
107	C₂×C₄⋊C₄	C₂³×C₂₃	1	32	yes	—
108	C₂×C₄²	—	2	16	no	—
109	C₂×C₄²	—	2	8	no	—
110	C₂×C₄²	Q₈×C₂₃	1	8	yes	—
111	C₂×C₄²	C₂³×C₂₃	1	16	yes	—
112	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
113	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
114	#95	C₂³×C₂₃	1	32	yes	—
115	#95	C₂³×C₂₃	1	32	yes	—
116	C₂₃	C₂×#2864	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
118	C₂²×C₄	—	2	8	no	—
119	C₂²×C₄	—	2	8	no	—
120	C₂²×C₄	—	2	8	no	—
121	C₂²×C₄	—	2	16	no	—
122	C₂²×C₄	—	2	32	no	—
123	C₂²×C₄	C₂×Q₈×C₂₃	1	8	yes	—
124	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
125	C₂²×C₄	#41×C₂₃	1	16	yes	—
126	C₂²×C₄	#41×C₂₃	1	8	yes	—
127	C₂²×C₄	C₂⁴×C₂₃	1	16	yes	—
128	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
129	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
130	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
131	C₄⋊C₄	—	2	64	no	—
132	C₄⋊C₄	—	2	32	no	—
133	C₄²	—	2	32	no	—
134	C₄²	—	2	16	no	—
135	C₄²	C₂×Q₈×C₂₃	1	16	yes	—
136	C₂³	C₂²×Q₈×C₂₃	1	2	yes	—
137	C₂³	C₂×#41×C₂₃	1	4	yes	—
138	C₂³	C₂×#41×C₂₃	1	8	yes	—
139	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	32	no	—
144	C₂×C₄	—	2	16	no	—
145	C₂×C₄	—	2	16	no	—
146	C₂×C₄	—	2	32	no	—
147	C₂×C₄	—	4	32	no	—
148	C₂×C₄	C₂₃×#125	1	16	yes	—
149	C₂×C₄	C₂²×Q₈×C₂₃	1	8	yes	—
150	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
151	C₂×C₄	C₂×#41×C₂₃	1	16	yes	—
152	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
153	C₂²	C₂×C₂₃×#125	1	4	yes	—
154	C₂²	C₂₃×#529	1	16	yes	—
155	C₂²	C₂₃×#533	1	8	yes	—
156	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
157	C₂²	C₂²×#41×C₂₃	1	2	yes	—
158	C₂²	C₂²×#41×C₂₃	1	4	yes	—
159	C₄	—	2	16	no	—
160	C₄	—	2	8	no	—
161	C₄	—	4	16	no	—
162	C₄	C₂×C₂₃×#125	1	8	yes	—
163	C₂	C₂₃×#2864	1	8	yes	direct product factor
164	C₂	C₂²×C₂₃×#125	1	1	yes	characteristic
165	C₂	C₂×C₂₃×#529	1	4	yes	—
166	C₂	C₂×C₂₃×#533	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2864	C₂	direct (N × H)
52. C₂×#2864	C₂₃	direct (N × H)
70. #2864	C₂×C₂₃	direct (N × H)
96. C₂×C₂₃	#2864	direct (N × H)
116. C₂₃	C₂×#2864	direct (N × H)
163. C₂	C₂₃×#2864	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×#2864	2
C₂×C₂₃×#389	2
C₂×C₂₃×#387	2
C₂×C₂₃×#382	2
C₂×C₄³×C₂₃	2
C₂×#2864	23

Subgroups — C2×C23×#2864 #19184260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2864 #19184260