Subgroups — C₂²×C₂₃×#433 #19207742

Order 5888.

2310 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#433	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₂₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂₃×C₂²⋊C₈	C₂	1	2	yes	maximal
5	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
9	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂×C₂₃×#117	C₄	1	12	yes	—
11	C₂×C₂₃×#117	C₂²	1	12	yes	—
12	C₂²×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
13	C₂²×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
14	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#433	C₂²	1	64	yes	direct product factor
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	24	yes	complemented
18	C₂×C₂₃×C₂²⋊C₈	C₂²	1	24	yes	—
19	C₂×C₂₃×#97	C₂²	1	12	yes	complemented
20	C₂×C₄×C₈×C₂₃	C₂²	1	12	yes	complemented
21	C₂⁵×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	—	2	8	no	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	C₂×C₄	1	6	yes	—
25	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
28	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	—	2	8	no	—
30	C₂²×C₈×C₂₃	—	2	8	no	—
31	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
33	C₂₃×#117	C₂×C₄	1	48	yes	—
34	C₂₃×#117	C₂³	1	16	yes	—
35	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	24	yes	—
36	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	24	yes	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	6	yes	—
38	C₂×C₄²×C₂₃	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
41	C₂₃×C₄⋊C₈	C₂³	1	32	yes	complemented
42	C₂₃×C₂²⋊C₈	C₂³	1	32	yes	—
43	C₂₃×#97	C₂³	1	16	yes	complemented
44	C₄×C₈×C₂₃	C₂³	1	16	yes	complemented
45	C₂⁴×C₂₃	—	2	4	no	—
46	C₂⁴×C₂₃	—	2	8	no	—
47	C₂⁴×C₂₃	C₂²×C₄	1	6	yes	—
48	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	4	no	—
50	C₂²×C₄×C₂₃	—	2	8	no	—
51	C₂²×C₄×C₂₃	—	2	8	no	—
52	C₂²×C₄×C₂₃	—	2	24	no	—
53	C₂²×C₄×C₂₃	—	2	24	no	—
54	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
57	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
59	C₂²×C₄×C₂₃	#41	1	8	yes	—
60	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
61	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
62	C₂×C₈×C₂₃	—	2	24	no	—
63	C₂×C₈×C₂₃	—	2	24	no	—
64	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
65	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
66	C₄⋊C₄×C₂₃	C₂²×C₄	1	32	yes	—
67	C₂²⋊C₄×C₂₃	C₂²×C₄	1	32	yes	—
68	C₄²×C₂₃	C₂²×C₄	1	12	yes	—
69	C₄²×C₂₃	C₂²×C₄	1	12	yes	—
70	C₄²×C₂₃	C₂⁴	1	4	yes	—
71	C₄²×C₂₃	C₂⁴	1	4	yes	—
72	C₂²×#433	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	—	2	12	no	—
74	C₂³×C₂₃	—	2	24	no	—
75	C₂³×C₂₃	—	4	16	no	—
76	C₂³×C₂₃	#117	1	4	yes	—
77	C₂³×C₂₃	#131	1	8	yes	—
78	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
79	C₂³×C₂₃	C₂³×C₄	1	3	yes	—
80	C₂×C₄×C₂₃	—	2	12	no	—
81	C₂×C₄×C₂₃	—	2	24	no	—
82	C₂×C₄×C₂₃	—	2	24	no	—
83	C₂×C₄×C₂₃	—	2	24	no	—
84	C₂×C₄×C₂₃	—	2	8	no	—
85	C₂×C₄×C₂₃	#117	1	12	yes	—
86	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
87	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
88	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
89	C₂×C₄×C₂₃	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₂₃	C₂×#41	1	12	yes	—
91	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
92	C₈×C₂₃	—	2	8	no	—
93	C₈×C₂₃	—	2	8	no	—
94	C₂²×#117	C₂×C₂₃	1	1	yes	characteristic
95	C₂×#433	C₂×C₂₃	1	24	yes	direct product factor
96	C₂²×C₄⋊C₈	C₂×C₂₃	1	2	yes	complemented
97	C₂²×C₂²⋊C₈	C₂×C₂₃	1	2	yes	—
98	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
99	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
100	C₂²×C₂₃	—	2	4	no	—
101	C₂²×C₂₃	—	2	8	no	—
102	C₂²×C₂₃	—	4	24	no	—
103	C₂²×C₂₃	#433	1	16	yes	direct product factor
104	C₂²×C₂₃	C₂×#117	1	6	yes	—
105	C₂²×C₂₃	C₂×#131	1	12	yes	—
106	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
107	C₄×C₂₃	—	2	4	no	—
108	C₄×C₂₃	—	2	8	no	—
109	C₄×C₂₃	—	2	8	no	—
110	C₄×C₂₃	C₂×#117	1	6	yes	—
111	C₄×C₂₃	C₂²×#41	1	2	yes	—
112	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
113	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
114	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
115	C₂×#117	C₄×C₂₃	1	12	yes	—
116	C₂×#117	C₂²×C₂₃	1	12	yes	—
117	C₂²×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
118	C₂²×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
119	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
120	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
121	#433	C₂²×C₂₃	1	64	yes	direct product factor
122	C₂×C₄⋊C₈	C₂²×C₂₃	1	24	yes	complemented
123	C₂×C₂²⋊C₈	C₂²×C₂₃	1	24	yes	—
124	C₂×#97	C₂²×C₂₃	1	12	yes	complemented
125	C₂×C₄×C₈	C₂²×C₂₃	1	12	yes	complemented
126	C₂×C₂₃	—	4	4	no	—
127	C₂×C₂₃	C₂×#433	1	12	yes	direct product factor
128	C₂×C₂₃	C₂²×#117	1	1	yes	characteristic
129	C₂×C₂₃	C₂²×#131	1	2	yes	—
130	C₂⁵	C₂×C₄×C₂₃	1	1	yes	characteristic
131	C₂³×C₄	—	2	8	no	—
132	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
133	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
134	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
135	C₂³×C₄	C₂×C₄×C₂₃	1	6	yes	—
136	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
137	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
138	C₂²×C₈	—	2	8	no	—
139	C₂²×C₈	—	2	8	no	—
140	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
141	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
142	#117	C₂×C₄×C₂₃	1	48	yes	—
143	#117	C₂³×C₂₃	1	16	yes	—
144	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	24	yes	—
145	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	24	yes	—
146	C₂×C₄²	C₂×C₄×C₂₃	1	6	yes	—
147	C₂×C₄²	C₂×C₄×C₂₃	1	6	yes	—
148	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
149	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
150	C₄⋊C₈	C₂³×C₂₃	1	32	yes	complemented
151	C₂²⋊C₈	C₂³×C₂₃	1	32	yes	—
152	#97	C₂³×C₂₃	1	16	yes	complemented
153	C₄×C₈	C₂³×C₂₃	1	16	yes	complemented
154	C₂₃	C₂²×#433	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	—	2	4	no	—
156	C₂⁴	—	2	8	no	—
157	C₂⁴	C₂²×C₄×C₂₃	1	6	yes	—
158	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	24	no	—
163	C₂²×C₄	—	2	24	no	—
164	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
165	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
166	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
167	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
168	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
169	C₂²×C₄	#41×C₂₃	1	8	yes	—
170	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
171	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
172	C₂×C₈	—	2	24	no	—
173	C₂×C₈	—	2	24	no	—
174	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
175	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
176	C₄⋊C₄	C₂²×C₄×C₂₃	1	32	yes	—
177	C₂²⋊C₄	C₂²×C₄×C₂₃	1	32	yes	—
178	C₄²	C₂²×C₄×C₂₃	1	12	yes	—
179	C₄²	C₂²×C₄×C₂₃	1	12	yes	—
180	C₄²	C₂⁴×C₂₃	1	4	yes	—
181	C₄²	C₂⁴×C₂₃	1	4	yes	—
182	C₂³	—	2	24	no	—
183	C₂³	—	2	12	no	—
184	C₂³	—	4	16	no	—
185	C₂³	C₂₃×#117	1	4	yes	—
186	C₂³	C₂₃×#131	1	8	yes	—
187	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
188	C₂³	C₂³×C₄×C₂₃	1	3	yes	—
189	C₂×C₄	—	2	24	no	—
190	C₂×C₄	—	2	24	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	24	no	—
194	C₂×C₄	C₂₃×#117	1	12	yes	—
195	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
196	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
197	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
198	C₂×C₄	C₂³×C₄×C₂₃	1	3	yes	—
199	C₂×C₄	C₂×#41×C₂₃	1	12	yes	—
200	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
201	C₈	—	2	8	no	—
202	C₈	—	2	8	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	8	no	—
205	C₂²	—	4	24	no	—
206	C₂²	C₂₃×#433	1	16	yes	direct product factor
207	C₂²	C₂×C₂₃×#117	1	6	yes	—
208	C₂²	C₂×C₂₃×#131	1	12	yes	—
209	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
210	C₄	—	2	4	no	—
211	C₄	—	2	8	no	—
212	C₄	—	2	8	no	—
213	C₄	C₂×C₂₃×#117	1	6	yes	—
214	C₄	C₂²×#41×C₂₃	1	2	yes	—
215	C₂	—	4	4	no	—
216	C₂	C₂×C₂₃×#433	1	12	yes	direct product factor
217	C₂	C₂²×C₂₃×#117	1	1	yes	characteristic
218	C₂	C₂²×C₂₃×#131	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
2. C₂×C₂₃×#433	C₂	direct (N × H)
3. C₂²×C₂₃×C₄⋊C₈	C₂	semidirect
5. C₂²×C₂₃×#97	C₂	semidirect
6. C₂²×C₄×C₈×C₂₃	C₂	semidirect
16. C₂₃×#433	C₂²	direct (N × H)
17. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
19. C₂×C₂₃×#97	C₂²	semidirect
20. C₂×C₄×C₈×C₂₃	C₂²	semidirect
41. C₂₃×C₄⋊C₈	C₂³	semidirect
43. C₂₃×#97	C₂³	semidirect
44. C₄×C₈×C₂₃	C₂³	semidirect
72. C₂²×#433	C₂₃	direct (N × H)
95. C₂×#433	C₂×C₂₃	direct (N × H)
96. C₂²×C₄⋊C₈	C₂×C₂₃	semidirect
98. C₂²×#97	C₂×C₂₃	semidirect
99. C₂²×C₄×C₈	C₂×C₂₃	semidirect
103. C₂²×C₂₃	#433	direct (N × H)
121. #433	C₂²×C₂₃	direct (N × H)
122. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
124. C₂×#97	C₂²×C₂₃	semidirect
125. C₂×C₄×C₈	C₂²×C₂₃	semidirect
127. C₂×C₂₃	C₂×#433	direct (N × H)
150. C₄⋊C₈	C₂³×C₂₃	semidirect
152. #97	C₂³×C₂₃	semidirect
153. C₄×C₈	C₂³×C₂₃	semidirect
154. C₂₃	C₂²×#433	direct (N × H)
206. C₂²	C₂₃×#433	direct (N × H)
216. C₂	C₂×C₂₃×#433	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×#117	2
C₂×C₂₃×#433	2
C₂²×C₂₃×C₄⋊C₈	2
C₂²×C₂₃×C₂²⋊C₈	2
C₂²×C₂₃×#97	2
C₂²×C₄×C₈×C₂₃	2
C₂²×#433	23

Subgroups — C22×C23×#433 #19207742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#433 #19207742