Subgroups — C₂×C₂₃×#1832 #19168377

Order 5888.

1006 conjugacy classes of subgroups.

The subgroup lattice has 166 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	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1832	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#336	C₂	1	4	yes	maximal
5	C₂³×C₈×C₂₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₄⋊C₈	—	2	8	no	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	8	yes	—
10	C₂₃×#376	C₄	1	4	yes	—
11	C₂₃×#376	C₂²	1	4	yes	—
12	C₂×C₂₃×#95	C₄	1	4	yes	—
13	C₂₃×#336	C₂²	1	16	yes	—
14	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
16	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	—	2	8	no	—
18	C₂²×C₈×C₂₃	—	2	16	no	—
19	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
20	C₂×C₄²×C₂₃	—	2	8	no	—
21	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₂₃	D₄	1	4	yes	—
24	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
25	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
26	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
27	C₂₃×C₄⋊C₈	—	2	16	no	—
28	C₂₃×#95	C₂×C₄	1	16	yes	—
29	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	—	2	8	no	—
31	C₂²×C₄×C₂₃	—	2	8	no	—
32	C₂²×C₄×C₂₃	—	2	16	no	—
33	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
34	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
35	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
39	C₂²×C₄×C₂₃	#41	1	8	yes	—
40	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₂₃	—	2	8	no	—
42	C₂×C₈×C₂₃	—	2	16	no	—
43	C₂×C₈×C₂₃	—	4	16	no	—
44	C₄²×C₂₃	—	2	8	no	—
45	C₄²×C₂₃	—	2	8	no	—
46	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
47	C₄²×C₂₃	C₂×D₄	1	4	yes	—
48	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
49	C₂×#1832	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
51	C₂³×C₂₃	#117	1	4	yes	—
52	C₂³×C₂₃	#131	1	8	yes	—
53	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	—	2	8	no	—
55	C₂×C₄×C₂₃	—	2	16	no	—
56	C₂×C₄×C₂₃	—	2	8	no	—
57	C₂×C₄×C₂₃	—	2	16	no	—
58	C₂×C₄×C₂₃	—	4	16	no	—
59	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
60	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₂₃	#117	1	4	yes	—
62	C₂×C₄×C₂₃	#124	1	8	yes	—
63	C₂×C₄×C₂₃	#125	1	8	yes	—
64	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
67	C₈×C₂₃	—	4	8	no	—
68	C₂²×C₄⋊C₈	C₂×C₂₃	1	2	yes	—
69	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
70	#1832	C₂×C₂₃	1	8	yes	direct product factor
71	C₂×#336	C₂×C₂₃	1	4	yes	—
72	C₂²×C₂₃	#382	1	4	yes	—
73	C₂²×C₂₃	#424	1	8	yes	—
74	C₂²×C₂₃	#433	1	16	yes	—
75	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₂₃	C₂×#117	1	2	yes	—
77	C₂²×C₂₃	C₂×#131	1	4	yes	—
78	C₄×C₂₃	—	2	8	no	—
79	C₄×C₂₃	—	4	8	no	—
80	C₄×C₂₃	#382	1	4	yes	—
81	C₄×C₂₃	C₂×#124	1	2	yes	—
82	C₄×C₂₃	C₂×#125	1	2	yes	—
83	C₂³×C₈	C₂²×C₂₃	1	4	yes	—
84	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
85	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
86	C₂×C₄⋊C₈	—	2	8	no	—
87	C₂×C₄⋊C₈	C₂²×C₂₃	1	8	yes	—
88	#376	C₄×C₂₃	1	4	yes	—
89	#376	C₂²×C₂₃	1	4	yes	—
90	C₂×#95	C₄×C₂₃	1	4	yes	—
91	#336	C₂²×C₂₃	1	16	yes	—
92	C₂×C₂₃	#1832	1	8	yes	direct product factor
93	C₂×C₂₃	C₂×#382	1	1	yes	characteristic
94	C₂×C₂₃	C₂×#424	1	2	yes	—
95	C₂×C₂₃	C₂×#433	1	4	yes	—
96	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
97	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
98	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
99	C₂²×C₈	—	2	8	no	—
100	C₂²×C₈	—	2	16	no	—
101	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
102	C₂×C₄²	—	2	8	no	—
103	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
104	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
105	C₂×C₄²	D₄×C₂₃	1	4	yes	—
106	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
107	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
108	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
109	C₄⋊C₈	—	2	16	no	—
110	#95	C₂×C₄×C₂₃	1	16	yes	—
111	C₂₃	C₂×#1832	1	1	yes	characteristic, direct product factor, Sylow
112	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	16	no	—
116	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
117	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
118	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
119	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
120	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
121	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
122	C₂²×C₄	#41×C₂₃	1	8	yes	—
123	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
124	C₂×C₈	—	2	8	no	—
125	C₂×C₈	—	2	16	no	—
126	C₂×C₈	—	4	16	no	—
127	C₄²	—	2	8	no	—
128	C₄²	—	2	8	no	—
129	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
130	C₄²	C₂×D₄×C₂₃	1	4	yes	—
131	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
132	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
133	C₂³	C₂₃×#117	1	4	yes	—
134	C₂³	C₂₃×#131	1	8	yes	—
135	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	2	16	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	4	16	no	—
141	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
142	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
143	C₂×C₄	C₂₃×#117	1	4	yes	—
144	C₂×C₄	C₂₃×#124	1	8	yes	—
145	C₂×C₄	C₂₃×#125	1	8	yes	—
146	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
147	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
148	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
149	C₈	—	4	8	no	—
150	C₂²	C₂₃×#382	1	4	yes	—
151	C₂²	C₂₃×#424	1	8	yes	—
152	C₂²	C₂₃×#433	1	16	yes	—
153	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
154	C₂²	C₂×C₂₃×#117	1	2	yes	—
155	C₂²	C₂×C₂₃×#131	1	4	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	4	8	no	—
158	C₄	C₂₃×#382	1	4	yes	—
159	C₄	C₂×C₂₃×#124	1	2	yes	—
160	C₄	C₂×C₂₃×#125	1	2	yes	—
161	C₂	C₂₃×#1832	1	8	yes	direct product factor
162	C₂	C₂×C₂₃×#382	1	1	yes	characteristic
163	C₂	C₂×C₂₃×#424	1	2	yes	—
164	C₂	C₂×C₂₃×#433	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1832	C₂	direct (N × H)
49. C₂×#1832	C₂₃	direct (N × H)
70. #1832	C₂×C₂₃	direct (N × H)
92. C₂×C₂₃	#1832	direct (N × H)
111. C₂₃	C₂×#1832	direct (N × H)
161. C₂	C₂₃×#1832	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
C₂×C₂₃×#376	2
C₂₃×#1832	2
C₂×C₂₃×#336	2
C₂×#1832	23

Subgroups — C2×C23×#1832 #19168377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1832 #19168377