Subgroups — C₂×C₂₃×#1849 #19168431

Order 5888.

1006 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#108	C₂	1	4	yes	maximal
2	C₂×C₂₃×#423	C₂	1	2	yes	maximal
3	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1849	C₂	1	8	yes	maximal, direct product factor
5	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
6	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#108	—	2	16	no	—
9	C₂×C₂₃×#108	C₂²	1	16	yes	—
10	C₂₃×#423	—	2	4	no	—
11	C₂₃×#423	C₂²	1	8	yes	—
12	C₂×C₂₃×C₄⋊C₈	—	2	4	no	—
13	C₂×C₂₃×#97	C₄	1	4	yes	—
14	C₂×C₂₃×#97	C₄	1	4	yes	—
15	C₂×C₂₃×#97	C₂²	1	4	yes	—
16	C₂³×C₄×C₂₃	Q₈	1	2	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	—	2	4	no	—
19	C₂×M₁₆×C₂₃	—	2	16	no	—
20	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
21	C₂²×C₈×C₂₃	—	2	4	no	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₂₃	D₄	1	8	yes	—
25	C₂²×C₈×C₂₃	Q₈	1	8	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	D₄	1	2	yes	—
30	C₂×C₄²×C₂₃	D₄	1	2	yes	—
31	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
33	C₂₃×#108	—	2	32	no	—
34	C₂₃×C₄⋊C₈	—	2	16	no	—
35	C₂₃×#97	C₂×C₄	1	16	yes	—
36	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	—	4	4	no	—
38	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
46	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
48	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
49	M₁₆×C₂₃	—	2	16	no	—
50	M₁₆×C₂₃	—	4	8	no	—
51	C₂×C₈×C₂₃	—	2	8	no	—
52	C₂×C₈×C₂₃	—	4	8	no	—
53	C₂×C₈×C₂₃	C₄⋊C₄	1	16	yes	—
54	C₂×C₈×C₂₃	C₄⋊C₄	1	16	yes	—
55	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
56	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
57	C₂×C₈×C₂₃	C₂×Q₈	1	8	yes	—
58	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
60	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
61	C₄²×C₂₃	C₂×D₄	1	4	yes	—
62	C₂×#1849	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	—	2	4	no	—
64	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂³×C₂₃	C₄⋊Q₈	1	4	yes	—
67	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₂₃	—	2	8	no	—
69	C₂×C₄×C₂₃	—	4	4	no	—
70	C₂×C₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
72	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
73	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
74	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#127	1	2	yes	—
77	C₂×C₄×C₂₃	#127	1	2	yes	—
78	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
79	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
81	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
85	C₈×C₂₃	—	4	8	no	—
86	C₈×C₂₃	C₂×C₄⋊C₄	1	16	yes	—
87	C₂²×#108	C₂×C₂₃	1	4	yes	—
88	C₂×#423	C₂×C₂₃	1	2	yes	—
89	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic
90	#1849	C₂×C₂₃	1	8	yes	direct product factor
91	C₂²×C₂₃	—	2	8	no	—
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	—	2	2	no	—
94	C₂²×C₂₃	#384	1	2	yes	—
95	C₂²×C₂₃	#384	1	2	yes	—
96	C₂²×C₂₃	#430	1	4	yes	—
97	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
99	C₄×C₂₃	—	2	8	no	—
100	C₄×C₂₃	#384	1	2	yes	—
101	C₄×C₂₃	#384	1	2	yes	—
102	C₄×C₂₃	C₂²×C₄⋊C₄	1	2	yes	—
103	C₄×C₂₃	C₂×#127	1	1	yes	characteristic
104	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
106	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
107	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
108	C₂×#108	—	2	16	no	—
109	C₂×#108	C₂²×C₂₃	1	16	yes	—
110	#423	—	2	4	no	—
111	#423	C₂²×C₂₃	1	8	yes	—
112	C₂×C₄⋊C₈	—	2	4	no	—
113	C₂×#97	C₄×C₂₃	1	4	yes	—
114	C₂×#97	C₄×C₂₃	1	4	yes	—
115	C₂×#97	C₂²×C₂₃	1	4	yes	—
116	C₂×C₂₃	—	2	2	no	—
117	C₂×C₂₃	—	2	2	no	—
118	C₂×C₂₃	#1849	1	4	yes	direct product factor
119	C₂×C₂₃	C₂×#384	1	1	yes	characteristic
120	C₂×C₂₃	C₂×#430	1	2	yes	—
121	C₂³×C₄	Q₈×C₂₃	1	2	yes	—
122	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
123	C₂×M₁₆	—	2	4	no	—
124	C₂×M₁₆	—	2	16	no	—
125	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
128	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
129	C₂²×C₈	D₄×C₂₃	1	8	yes	—
130	C₂²×C₈	Q₈×C₂₃	1	8	yes	—
131	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
132	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
133	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
134	C₂×C₄²	D₄×C₂₃	1	2	yes	—
135	C₂×C₄²	D₄×C₂₃	1	2	yes	—
136	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
137	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
138	#108	—	2	32	no	—
139	C₄⋊C₈	—	2	16	no	—
140	#97	C₂×C₄×C₂₃	1	16	yes	—
141	C₂₃	C₂×#1849	1	1	yes	characteristic, direct product factor, Sylow
142	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
145	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
146	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
149	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
152	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
154	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
155	M₁₆	—	2	16	no	—
156	M₁₆	—	4	8	no	—
157	C₂×C₈	—	2	8	no	—
158	C₂×C₈	—	4	8	no	—
159	C₂×C₈	C₄⋊C₄×C₂₃	1	16	yes	—
160	C₂×C₈	C₄⋊C₄×C₂₃	1	16	yes	—
161	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
162	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
163	C₂×C₈	C₂×Q₈×C₂₃	1	8	yes	—
164	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
165	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
166	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
167	C₄²	C₂×D₄×C₂₃	1	4	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
170	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
171	C₂³	C₂₃×C₄⋊Q₈	1	4	yes	—
172	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
177	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
180	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
181	C₂×C₄	C₂₃×#127	1	2	yes	—
182	C₂×C₄	C₂₃×#127	1	2	yes	—
183	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
184	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
185	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
186	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
188	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
190	C₈	—	4	8	no	—
191	C₈	C₂×C₄⋊C₄×C₂₃	1	16	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	8	no	—
195	C₂²	C₂₃×#384	1	2	yes	—
196	C₂²	C₂₃×#384	1	2	yes	—
197	C₂²	C₂₃×#430	1	4	yes	—
198	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
199	C₂²	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	C₂₃×#384	1	2	yes	—
202	C₄	C₂₃×#384	1	2	yes	—
203	C₄	C₂²×C₄⋊C₄×C₂₃	1	2	yes	—
204	C₄	C₂×C₂₃×#127	1	1	yes	characteristic
205	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	C₂₃×#1849	1	4	yes	direct product factor
209	C₂	C₂×C₂₃×#384	1	1	yes	characteristic
210	C₂	C₂×C₂₃×#430	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1849	C₂	direct (N × H)
62. C₂×#1849	C₂₃	direct (N × H)
90. #1849	C₂×C₂₃	direct (N × H)
118. C₂×C₂₃	#1849	direct (N × H)
141. C₂₃	C₂×#1849	direct (N × H)
208. C₂	C₂₃×#1849	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×#108	2
C₂×C₂₃×#423	2
C₂²×C₂₃×#97	2
C₂₃×#1849	2
C₂×#1849	23

Subgroups — C2×C23×#1849 #19168431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1849 #19168431