Subgroups — C₂²×C₄×D₈×C₂₃ #19207774

Order 5888.

3694 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈×C₂₃	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂₃×#102	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₂₃	—	2	2	no	—
9	C₂²×D₈×C₂₃	C₄	1	16	yes	direct product factor
10	C₂²×D₈×C₂₃	C₂²	1	12	yes	—
11	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₂₃	—	2	12	no	—
14	C₂×C₄×D₄×C₂₃	C₂²	1	24	yes	complemented
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×D₈×C₂₃	C₂²	1	64	yes	direct product factor
19	C₂×C₂₃×#107	C₂²	1	12	yes	complemented
20	C₂×C₂₃×#102	C₂²	1	24	yes	complemented
21	C₂×C₄×C₈×C₂₃	C₂²	1	12	yes	complemented
22	C₂⁵×C₂₃	—	2	2	no	—
23	C₂²×D₄×C₂₃	—	2	6	no	—
24	C₂²×D₄×C₂₃	—	2	8	no	—
25	C₂²×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
26	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
27	C₂³×C₄×C₂₃	—	2	2	no	—
28	C₂³×C₄×C₂₃	—	2	24	no	—
29	C₂³×C₄×C₂₃	—	4	16	no	—
30	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×D₈×C₂₃	C₂×C₄	1	96	yes	direct product factor
34	C₂×D₈×C₂₃	C₂³	1	16	yes	—
35	C₂²×C₈×C₂₃	—	2	4	no	—
36	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
37	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
39	C₄×D₄×C₂₃	—	2	48	no	—
40	C₄×D₄×C₂₃	C₂³	1	32	yes	complemented
41	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
43	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
44	C₂×C₄²×C₂₃	D₄	1	6	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
46	C₂₃×#107	C₂³	1	16	yes	complemented
47	C₂₃×#102	C₂³	1	32	yes	complemented
48	C₄×C₈×C₂₃	C₂³	1	16	yes	complemented
49	C₂⁴×C₂₃	—	2	16	no	—
50	C₂⁴×C₂₃	—	2	12	no	—
51	C₂⁴×C₂₃	—	4	8	no	—
52	C₂⁴×C₂₃	—	4	8	no	—
53	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂×D₄×C₂₃	—	2	48	no	—
55	C₂×D₄×C₂₃	—	2	12	no	—
56	C₂×D₄×C₂₃	—	2	24	no	—
57	C₂×D₄×C₂₃	C₂²×C₄	1	48	yes	complemented
58	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₂₃	—	2	4	no	—
60	C₂²×C₄×C₂₃	—	2	12	no	—
61	C₂²×C₄×C₂₃	—	2	32	no	—
62	C₂²×C₄×C₂₃	—	4	8	no	—
63	C₂²×C₄×C₂₃	—	4	96	no	—
64	C₂²×C₄×C₂₃	D₈	1	8	yes	direct product factor
65	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₂₃	#41	1	4	yes	—
70	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
71	D₈×C₂₃	C₂²×C₄	1	64	yes	direct product factor
72	C₂×C₈×C₂₃	—	2	12	no	—
73	C₂×C₈×C₂₃	C₂²×C₄	1	24	yes	complemented
74	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
75	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
76	C₄⋊C₄×C₂₃	—	2	12	no	—
77	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
78	C₂²⋊C₄×C₂₃	—	2	32	no	—
79	C₄²×C₂₃	C₂×D₄	1	12	yes	—
80	C₄²×C₂₃	C₂⁴	1	4	yes	—
81	C₂²×C₄×D₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	—	2	48	no	—
83	C₂³×C₂₃	—	2	8	no	—
84	C₂³×C₂₃	—	4	24	no	—
85	C₂³×C₂₃	—	4	24	no	—
86	C₂³×C₂₃	—	4	64	no	—
87	C₂³×C₂₃	C₄×D₄	1	4	yes	—
88	C₂³×C₂₃	C₂×D₈	1	4	yes	—
89	C₂³×C₂₃	#135	1	4	yes	—
90	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
91	D₄×C₂₃	—	2	48	no	—
92	D₄×C₂₃	—	2	8	no	—
93	D₄×C₂₃	C₂³×C₄	1	16	yes	complemented
94	C₂×C₄×C₂₃	—	2	12	no	—
95	C₂×C₄×C₂₃	—	2	8	no	—
96	C₂×C₄×C₂₃	—	4	24	no	—
97	C₂×C₄×C₂₃	—	4	64	no	—
98	C₂×C₄×C₂₃	C₄×D₄	1	12	yes	—
99	C₂×C₄×C₂₃	C₂×D₈	1	24	yes	direct product factor
100	C₂×C₄×C₂₃	C₂³×C₄	1	6	yes	—
101	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
102	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
103	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
104	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
105	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
106	C₈×C₂₃	—	2	4	no	—
107	C₈×C₂₃	C₂³×C₄	1	8	yes	complemented
108	C₂³×D₈	C₂×C₂₃	1	1	yes	characteristic
109	C₂²×C₄×D₄	C₂×C₂₃	1	2	yes	complemented
110	C₂×C₄×D₈	C₂×C₂₃	1	24	yes	direct product factor
111	C₂²×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
112	C₂²×#102	C₂×C₂₃	1	2	yes	complemented
113	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
114	C₂²×C₂₃	—	2	16	no	—
115	C₂²×C₂₃	—	4	96	no	—
116	C₂²×C₂₃	—	4	8	no	—
117	C₂²×C₂₃	—	4	8	no	—
118	C₂²×C₂₃	C₄×D₈	1	16	yes	direct product factor
119	C₂²×C₂₃	C₂×C₄×D₄	1	6	yes	—
120	C₂²×C₂₃	C₂²×D₈	1	6	yes	—
121	C₂²×C₂₃	C₂×#135	1	6	yes	—
122	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
123	C₄×C₂₃	—	2	4	no	—
124	C₄×C₂₃	—	4	8	no	—
125	C₄×C₂₃	C₂×C₄×D₄	1	6	yes	—
126	C₄×C₂₃	C₂²×D₈	1	8	yes	direct product factor
127	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
128	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
129	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
130	C₂⁴×C₄	—	2	2	no	—
131	C₂²×D₈	C₄×C₂₃	1	16	yes	direct product factor
132	C₂²×D₈	C₂²×C₂₃	1	12	yes	—
133	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
135	C₂×C₄×D₄	—	2	12	no	—
136	C₂×C₄×D₄	C₂²×C₂₃	1	24	yes	complemented
137	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
138	C₂²×C₂²⋊C₄	—	2	2	no	—
139	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
140	C₄×D₈	C₂²×C₂₃	1	64	yes	direct product factor
141	C₂×#107	C₂²×C₂₃	1	12	yes	complemented
142	C₂×#102	C₂²×C₂₃	1	24	yes	complemented
143	C₂×C₄×C₈	C₂²×C₂₃	1	12	yes	complemented
144	C₂×C₂₃	—	4	16	no	—
145	C₂×C₂₃	C₂×C₄×D₈	1	12	yes	direct product factor
146	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
147	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
148	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
149	C₂⁵	—	2	2	no	—
150	C₂²×D₄	—	2	6	no	—
151	C₂²×D₄	—	2	8	no	—
152	C₂²×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
153	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
154	C₂³×C₄	—	2	2	no	—
155	C₂³×C₄	—	2	24	no	—
156	C₂³×C₄	—	4	16	no	—
157	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
158	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
159	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
160	C₂×D₈	C₂×C₄×C₂₃	1	96	yes	direct product factor
161	C₂×D₈	C₂³×C₂₃	1	16	yes	—
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	complemented
164	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
165	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
166	C₄×D₄	—	2	48	no	—
167	C₄×D₄	C₂³×C₂₃	1	32	yes	complemented
168	C₂×C₄⋊C₄	—	2	6	no	—
169	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
170	C₂×C₂²⋊C₄	—	2	24	no	—
171	C₂×C₄²	D₄×C₂₃	1	6	yes	—
172	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
173	#107	C₂³×C₂₃	1	16	yes	complemented
174	#102	C₂³×C₂₃	1	32	yes	complemented
175	C₄×C₈	C₂³×C₂₃	1	16	yes	complemented
176	C₂₃	C₂²×C₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
177	C₂⁴	—	2	16	no	—
178	C₂⁴	—	2	12	no	—
179	C₂⁴	—	4	8	no	—
180	C₂⁴	—	4	8	no	—
181	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
182	C₂×D₄	—	2	48	no	—
183	C₂×D₄	—	2	12	no	—
184	C₂×D₄	—	2	24	no	—
185	C₂×D₄	C₂²×C₄×C₂₃	1	48	yes	complemented
186	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	12	no	—
189	C₂²×C₄	—	2	32	no	—
190	C₂²×C₄	—	4	8	no	—
191	C₂²×C₄	—	4	96	no	—
192	C₂²×C₄	D₈×C₂₃	1	8	yes	direct product factor
193	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
194	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
195	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
196	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
197	C₂²×C₄	#41×C₂₃	1	4	yes	—
198	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
199	D₈	C₂²×C₄×C₂₃	1	64	yes	direct product factor
200	C₂×C₈	—	2	12	no	—
201	C₂×C₈	C₂²×C₄×C₂₃	1	24	yes	complemented
202	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
203	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
204	C₄⋊C₄	—	2	12	no	—
205	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
206	C₂²⋊C₄	—	2	32	no	—
207	C₄²	C₂×D₄×C₂₃	1	12	yes	—
208	C₄²	C₂⁴×C₂₃	1	4	yes	—
209	C₂³	—	2	8	no	—
210	C₂³	—	2	48	no	—
211	C₂³	—	4	24	no	—
212	C₂³	—	4	24	no	—
213	C₂³	—	4	64	no	—
214	C₂³	C₄×D₄×C₂₃	1	4	yes	—
215	C₂³	C₂×D₈×C₂₃	1	4	yes	—
216	C₂³	C₂₃×#135	1	4	yes	—
217	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
218	D₄	—	2	8	no	—
219	D₄	—	2	48	no	—
220	D₄	C₂³×C₄×C₂₃	1	16	yes	complemented
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	4	24	no	—
224	C₂×C₄	—	4	64	no	—
225	C₂×C₄	C₄×D₄×C₂₃	1	12	yes	—
226	C₂×C₄	C₂×D₈×C₂₃	1	24	yes	direct product factor
227	C₂×C₄	C₂³×C₄×C₂₃	1	6	yes	—
228	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
229	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
231	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
232	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	C₂³×C₄×C₂₃	1	8	yes	complemented
235	C₂²	—	2	16	no	—
236	C₂²	—	4	96	no	—
237	C₂²	—	4	8	no	—
238	C₂²	—	4	8	no	—
239	C₂²	C₄×D₈×C₂₃	1	16	yes	direct product factor
240	C₂²	C₂×C₄×D₄×C₂₃	1	6	yes	—
241	C₂²	C₂²×D₈×C₂₃	1	6	yes	—
242	C₂²	C₂×C₂₃×#135	1	6	yes	—
243	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
244	C₄	—	2	4	no	—
245	C₄	—	4	8	no	—
246	C₄	C₂×C₄×D₄×C₂₃	1	6	yes	—
247	C₄	C₂²×D₈×C₂₃	1	8	yes	direct product factor
248	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
249	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
250	C₂	—	4	16	no	—
251	C₂	C₂×C₄×D₈×C₂₃	1	12	yes	direct product factor
252	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
253	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic
254	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₂₃	D₄
commutator	C₄	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂⁴×C₂₃	C₂×D₄
prime radical	C₂³×D₈×C₂₃	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₄×C₂₃	C₂	semidirect
3. C₂×C₄×D₈×C₂₃	C₂	direct (N × H)
4. C₂²×C₂₃×#107	C₂	semidirect
5. C₂²×C₂₃×#102	C₂	semidirect
6. C₂²×C₄×C₈×C₂₃	C₂	semidirect
9. C₂²×D₈×C₂₃	C₄	direct (N × H)
14. C₂×C₄×D₄×C₂₃	C₂²	semidirect
18. C₄×D₈×C₂₃	C₂²	direct (N × H)
19. C₂×C₂₃×#107	C₂²	semidirect
20. C₂×C₂₃×#102	C₂²	semidirect
21. C₂×C₄×C₈×C₂₃	C₂²	semidirect
25. C₂²×D₄×C₂₃	C₂×C₄	semidirect
33. C₂×D₈×C₂₃	C₂×C₄	direct (N × H)
36. C₂²×C₈×C₂₃	C₂×C₄	semidirect
40. C₄×D₄×C₂₃	C₂³	semidirect
46. C₂₃×#107	C₂³	semidirect
47. C₂₃×#102	C₂³	semidirect
48. C₄×C₈×C₂₃	C₂³	semidirect
57. C₂×D₄×C₂₃	C₂²×C₄	semidirect
64. C₂²×C₄×C₂₃	D₈	direct (N × H)
71. D₈×C₂₃	C₂²×C₄	direct (N × H)
73. C₂×C₈×C₂₃	C₂²×C₄	semidirect
81. C₂²×C₄×D₈	C₂₃	direct (N × H)
93. D₄×C₂₃	C₂³×C₄	semidirect
99. C₂×C₄×C₂₃	C₂×D₈	direct (N × H)
107. C₈×C₂₃	C₂³×C₄	semidirect
109. C₂²×C₄×D₄	C₂×C₂₃	semidirect
110. C₂×C₄×D₈	C₂×C₂₃	direct (N × H)
111. C₂²×#107	C₂×C₂₃	semidirect
112. C₂²×#102	C₂×C₂₃	semidirect
113. C₂²×C₄×C₈	C₂×C₂₃	semidirect
118. C₂²×C₂₃	C₄×D₈	direct (N × H)
126. C₄×C₂₃	C₂²×D₈	direct (N × H)
131. C₂²×D₈	C₄×C₂₃	direct (N × H)
136. C₂×C₄×D₄	C₂²×C₂₃	semidirect
140. C₄×D₈	C₂²×C₂₃	direct (N × H)
141. C₂×#107	C₂²×C₂₃	semidirect
142. C₂×#102	C₂²×C₂₃	semidirect
143. C₂×C₄×C₈	C₂²×C₂₃	semidirect
145. C₂×C₂₃	C₂×C₄×D₈	direct (N × H)
152. C₂²×D₄	C₂×C₄×C₂₃	semidirect
160. C₂×D₈	C₂×C₄×C₂₃	direct (N × H)
163. C₂²×C₈	C₂×C₄×C₂₃	semidirect
167. C₄×D₄	C₂³×C₂₃	semidirect
173. #107	C₂³×C₂₃	semidirect
174. #102	C₂³×C₂₃	semidirect
175. C₄×C₈	C₂³×C₂₃	semidirect
176. C₂₃	C₂²×C₄×D₈	direct (N × H)
185. C₂×D₄	C₂²×C₄×C₂₃	semidirect
192. C₂²×C₄	D₈×C₂₃	direct (N × H)
199. D₈	C₂²×C₄×C₂₃	direct (N × H)
201. C₂×C₈	C₂²×C₄×C₂₃	semidirect
220. D₄	C₂³×C₄×C₂₃	semidirect
226. C₂×C₄	C₂×D₈×C₂₃	direct (N × H)
234. C₈	C₂³×C₄×C₂₃	semidirect
239. C₂²	C₄×D₈×C₂₃	direct (N × H)
247. C₄	C₂²×D₈×C₂₃	direct (N × H)
251. C₂	C₂×C₄×D₈×C₂₃	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×D₈×C₂₃	2
C₂²×C₄×D₄×C₂₃	2
C₂×C₄×D₈×C₂₃	2
C₂²×C₂₃×#107	2
C₂²×C₂₃×#102	2
C₂²×C₄×C₈×C₂₃	2
C₂²×C₄×D₈	23

Subgroups — C22×C4×D8×C23 #19207774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×D₈×C₂₃ #19207774