Subgroups — C₂²×C₂₃×#423 #19207721

Order 5888.

3438 conjugacy classes of subgroups.

The subgroup lattice has 216 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	C₂³×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#423	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₂₃	C₄	1	2	yes	—
6	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₂₃	—	2	16	no	—
8	C₂²×M₁₆×C₂₃	C₂²	1	24	yes	—
9	C₂³×C₈×C₂₃	C₂²	1	4	yes	—
10	C₂²×C₄²×C₂₃	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₂₃	C₄	1	12	yes	—
12	C₂²×C₄²×C₂₃	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₂₃	C₂²	1	12	yes	—
14	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#423	C₂²	1	64	yes	direct product factor
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	48	yes	complemented
17	C₂⁵×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	—	2	16	no	—
19	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₂₃	C₂×C₄	1	24	yes	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	C₂×C₄	1	6	yes	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₂₃	D₄	1	8	yes	—
25	C₂³×C₄×C₂₃	Q₈	1	8	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₂₃	—	2	96	no	—
29	C₂×M₁₆×C₂₃	C₂³	1	32	yes	—
30	C₂²×C₈×C₂₃	—	2	16	no	—
31	C₂²×C₈×C₂₃	C₂³	1	24	yes	—
32	C₂×C₄²×C₂₃	C₂×C₄	1	12	yes	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	48	yes	—
34	C₂×C₄²×C₂₃	C₂×C₄	1	12	yes	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	12	yes	—
36	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
37	C₂×C₄²×C₂₃	C₂³	1	16	yes	—
38	C₂₃×C₄⋊C₈	C₂³	1	64	yes	complemented
39	C₂⁴×C₂₃	—	2	8	no	—
40	C₂⁴×C₂₃	C₄⋊C₄	1	8	yes	—
41	C₂⁴×C₂₃	C₂²×C₄	1	6	yes	—
42	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	—	2	8	no	—
44	C₂²×C₄×C₂₃	—	2	96	no	—
45	C₂²×C₄×C₂₃	C₄⋊C₄	1	48	yes	—
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
47	C₂²×C₄×C₂₃	M₁₆	1	16	yes	complemented
48	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
49	C₂²×C₄×C₂₃	C₂²×C₄	1	32	yes	—
50	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
56	C₂²×C₄×C₂₃	C₂×Q₈	1	24	yes	—
57	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
59	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
60	M₁₆×C₂₃	—	2	64	no	—
61	C₂×C₈×C₂₃	—	2	48	no	—
62	C₂×C₈×C₂₃	C₂⁴	1	16	yes	—
63	C₄²×C₂₃	C₂²×C₄	1	16	yes	—
64	C₄²×C₂₃	C₂²×C₄	1	24	yes	—
65	C₄²×C₂₃	C₂²×C₄	1	16	yes	—
66	C₄²×C₂₃	C₂⁴	1	8	yes	—
67	C₂²×#423	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₂₃	—	2	24	no	—
69	C₂³×C₂₃	—	2	32	no	—
70	C₂³×C₂₃	C₂×C₄⋊C₄	1	24	yes	—
71	C₂³×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
72	C₂³×C₂₃	C₂×M₁₆	1	8	yes	—
73	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
74	C₂³×C₂₃	C₂³×C₄	1	3	yes	—
75	C₂×C₄×C₂₃	—	2	24	no	—
76	C₂×C₄×C₂₃	—	2	64	no	—
77	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	48	yes	—
78	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	24	yes	—
79	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	12	yes	—
80	C₂×C₄×C₂₃	C₂×M₁₆	1	48	yes	complemented
81	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
82	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
83	C₂×C₄×C₂₃	C₂³×C₄	1	3	yes	—
84	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
85	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
86	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
87	C₂×C₄×C₂₃	C₂²×Q₈	1	8	yes	—
88	C₂×C₄×C₂₃	C₂²×Q₈	1	6	yes	—
89	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
90	C₈×C₂₃	—	2	16	no	—
91	C₂³×M₁₆	C₂×C₂₃	1	2	yes	—
92	C₂³×C₄²	C₂×C₂₃	1	1	yes	characteristic
93	C₂×#423	C₂×C₂₃	1	24	yes	direct product factor
94	C₂²×C₄⋊C₈	C₂×C₂₃	1	4	yes	complemented
95	C₂²×C₂₃	—	2	48	no	—
96	C₂²×C₂₃	—	2	8	no	—
97	C₂²×C₂₃	#423	1	16	yes	direct product factor
98	C₂²×C₂₃	C₂²×C₄⋊C₄	1	8	yes	—
99	C₂²×C₂₃	C₂²×C₄⋊C₄	1	6	yes	—
100	C₂²×C₂₃	C₂²×M₁₆	1	12	yes	—
101	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
102	C₄×C₂₃	—	2	8	no	—
103	C₄×C₂₃	C₂²×C₄⋊C₄	1	6	yes	—
104	C₄×C₂₃	C₂²×C₄⋊C₄	1	8	yes	—
105	C₄×C₂₃	C₂²×M₁₆	1	16	yes	complemented
106	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
107	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
108	C₂⁴×C₄	C₄×C₂₃	1	2	yes	—
109	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
110	C₂²×M₁₆	—	2	16	no	—
111	C₂²×M₁₆	C₂²×C₂₃	1	24	yes	—
112	C₂³×C₈	C₂²×C₂₃	1	4	yes	—
113	C₂²×C₄²	C₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₄²	C₄×C₂₃	1	12	yes	—
115	C₂²×C₄²	C₄×C₂₃	1	1	yes	characteristic
116	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
117	C₂²×C₄²	C₂²×C₂₃	1	12	yes	—
118	#423	C₂²×C₂₃	1	64	yes	direct product factor
119	C₂×C₄⋊C₈	C₂²×C₂₃	1	48	yes	complemented
120	C₂×C₂₃	—	2	8	no	—
121	C₂×C₂₃	C₂×#423	1	12	yes	direct product factor
122	C₂×C₂₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₂₃	C₂³×M₁₆	1	2	yes	—
124	C₂⁵	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₂³×C₄	—	2	16	no	—
126	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
127	C₂³×C₄	C₂×C₄×C₂₃	1	24	yes	—
128	C₂³×C₄	C₂×C₄×C₂₃	1	6	yes	—
129	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
130	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
131	C₂³×C₄	D₄×C₂₃	1	8	yes	—
132	C₂³×C₄	Q₈×C₂₃	1	8	yes	—
133	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
134	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
135	C₂×M₁₆	—	2	96	no	—
136	C₂×M₁₆	C₂³×C₂₃	1	32	yes	—
137	C₂²×C₈	—	2	16	no	—
138	C₂²×C₈	C₂³×C₂₃	1	24	yes	—
139	C₂×C₄²	C₂×C₄×C₂₃	1	12	yes	—
140	C₂×C₄²	C₂×C₄×C₂₃	1	48	yes	—
141	C₂×C₄²	C₂×C₄×C₂₃	1	12	yes	—
142	C₂×C₄²	C₂×C₄×C₂₃	1	12	yes	—
143	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
144	C₂×C₄²	C₂³×C₂₃	1	16	yes	—
145	C₄⋊C₈	C₂³×C₂₃	1	64	yes	complemented
146	C₂₃	C₂²×#423	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁴	—	2	8	no	—
148	C₂⁴	C₄⋊C₄×C₂₃	1	8	yes	—
149	C₂⁴	C₂²×C₄×C₂₃	1	6	yes	—
150	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	96	no	—
153	C₂²×C₄	C₄⋊C₄×C₂₃	1	48	yes	—
154	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
155	C₂²×C₄	M₁₆×C₂₃	1	16	yes	complemented
156	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
157	C₂²×C₄	C₂²×C₄×C₂₃	1	32	yes	—
158	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
159	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
160	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
161	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
162	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
163	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
164	C₂²×C₄	C₂×Q₈×C₂₃	1	24	yes	—
165	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
166	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
167	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
168	M₁₆	—	2	64	no	—
169	C₂×C₈	—	2	48	no	—
170	C₂×C₈	C₂⁴×C₂₃	1	16	yes	—
171	C₄²	C₂²×C₄×C₂₃	1	16	yes	—
172	C₄²	C₂²×C₄×C₂₃	1	24	yes	—
173	C₄²	C₂²×C₄×C₂₃	1	16	yes	—
174	C₄²	C₂⁴×C₂₃	1	8	yes	—
175	C₂³	—	2	24	no	—
176	C₂³	—	2	32	no	—
177	C₂³	C₂×C₄⋊C₄×C₂₃	1	24	yes	—
178	C₂³	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
179	C₂³	C₂×M₁₆×C₂₃	1	8	yes	—
180	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
181	C₂³	C₂³×C₄×C₂₃	1	3	yes	—
182	C₂×C₄	—	2	24	no	—
183	C₂×C₄	—	2	64	no	—
184	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	48	yes	—
185	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	12	yes	—
186	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	24	yes	—
187	C₂×C₄	C₂×M₁₆×C₂₃	1	48	yes	complemented
188	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
189	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
190	C₂×C₄	C₂³×C₄×C₂₃	1	3	yes	—
191	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
192	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
193	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
194	C₂×C₄	C₂²×Q₈×C₂₃	1	8	yes	—
195	C₂×C₄	C₂²×Q₈×C₂₃	1	6	yes	—
196	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
197	C₈	—	2	16	no	—
198	C₂²	—	2	8	no	—
199	C₂²	—	2	48	no	—
200	C₂²	C₂₃×#423	1	16	yes	direct product factor
201	C₂²	C₂²×C₄⋊C₄×C₂₃	1	8	yes	—
202	C₂²	C₂²×C₄⋊C₄×C₂₃	1	6	yes	—
203	C₂²	C₂²×M₁₆×C₂₃	1	12	yes	—
204	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
205	C₄	—	2	8	no	—
206	C₄	C₂²×C₄⋊C₄×C₂₃	1	6	yes	—
207	C₄	C₂²×C₄⋊C₄×C₂₃	1	8	yes	—
208	C₄	C₂²×M₁₆×C₂₃	1	16	yes	complemented
209	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
210	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×C₂₃×#423	1	12	yes	direct product factor
213	C₂	C₂³×C₄⋊C₄×C₂₃	1	1	yes	characteristic
214	C₂	C₂³×M₁₆×C₂₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₃	C₂³
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂⁴×C₂₃	C₂²×C₄
prime radical	C₂⁵×C₂₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#423	C₂	direct (N × H)
4. C₂²×C₂₃×C₄⋊C₈	C₂	semidirect
15. C₂₃×#423	C₂²	direct (N × H)
16. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
38. C₂₃×C₄⋊C₈	C₂³	semidirect
47. C₂²×C₄×C₂₃	M₁₆	semidirect
67. C₂²×#423	C₂₃	direct (N × H)
80. C₂×C₄×C₂₃	C₂×M₁₆	semidirect
93. C₂×#423	C₂×C₂₃	direct (N × H)
94. C₂²×C₄⋊C₈	C₂×C₂₃	semidirect
97. C₂²×C₂₃	#423	direct (N × H)
105. C₄×C₂₃	C₂²×M₁₆	semidirect
118. #423	C₂²×C₂₃	direct (N × H)
119. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
121. C₂×C₂₃	C₂×#423	direct (N × H)
145. C₄⋊C₈	C₂³×C₂₃	semidirect
146. C₂₃	C₂²×#423	direct (N × H)
155. C₂²×C₄	M₁₆×C₂₃	semidirect
187. C₂×C₄	C₂×M₁₆×C₂₃	semidirect
200. C₂²	C₂₃×#423	direct (N × H)
208. C₄	C₂²×M₁₆×C₂₃	semidirect
212. C₂	C₂×C₂₃×#423	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
C₂³×C₄²×C₂₃	2
C₂×C₂₃×#423	2
C₂²×C₂₃×C₄⋊C₈	2
C₂²×#423	23

Subgroups — C22×C23×#423 #19207721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#423 #19207721