Subgroups — C₂×C₂₃×#3415 #19210258

Order 5888.

2126 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#576	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3415	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#509	C₂	1	6	yes	maximal, complemented
4	C₂×C₂₃×#508	C₂	1	3	yes	maximal, complemented
5	C₂²×C₂₃×D₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×H(5,𝔽₂)	—	2	2	no	—
8	C₂₃×#576	—	2	8	no	—
9	C₂₃×#576	C₂²	1	16	yes	complemented
10	C₂×C₂₃×AGL(1,ℤ/8)	—	2	6	no	—
11	C₂×C₂₃×#135	C₂²	1	6	yes	complemented
12	C₂²×D₈×C₂₃	C₂²	1	6	yes	complemented
13	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
14	C₂₃×#509	C₂²	1	48	yes	complemented
15	C₂₃×#508	C₂²	1	24	yes	complemented
16	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₂₃×SD₃₂	C₂²	1	6	yes	complemented
18	C₂×C₂₃×D₁₆	C₂²	1	3	yes	complemented
19	C₂×C₂₃×D₁₆	C₂²	1	6	yes	complemented
20	C₂×C₂₃×D₁₆	C₂²	1	24	yes	complemented
21	C₂₃×#504	C₂²	1	8	yes	complemented
22	C₂×C₂₃×M₃₂	C₂²	1	3	yes	complemented
23	C₂²×C₁₆×C₂₃	C₂²	1	3	yes	complemented
24	C₂₃×H(5,𝔽₂)	—	2	16	no	—
25	C₂₃×H(5,𝔽₂)	—	4	8	no	—
26	C₂×#41×C₂₃	—	2	6	no	—
27	C₂×#41×C₂₃	—	4	2	no	—
28	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₂₃	—	2	6	no	—
30	C₂²×D₄×C₂₃	—	4	6	no	—
31	C₂₃×AGL(1,ℤ/8)	—	2	48	no	—
32	C₂₃×#135	—	2	12	no	—
33	C₂₃×#135	C₂³	1	24	yes	complemented
34	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₂₃	—	2	6	no	—
36	C₂×D₈×C₂₃	—	2	6	no	—
37	C₂×D₈×C₂₃	—	2	12	no	—
38	C₂×D₈×C₂₃	C₂³	1	6	yes	—
39	C₂×D₈×C₂₃	C₂³	1	24	yes	complemented
40	C₂₃×#131	D₄	1	4	yes	—
41	C₂₃×#131	C₂³	1	4	yes	—
42	C₂×M₁₆×C₂₃	D₄	1	3	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
44	C₂₃×Q₃₂	C₂³	1	4	yes	complemented
45	C₂₃×SD₃₂	C₂³	1	24	yes	complemented
46	C₂₃×D₁₆	C₂³	1	12	yes	complemented
47	C₂₃×D₁₆	C₂³	1	24	yes	complemented
48	C₂₃×M₃₂	C₂³	1	12	yes	complemented
49	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₂₃	C₂³	1	3	yes	—
51	C₂×C₁₆×C₂₃	C₂³	1	12	yes	complemented
52	C₂⁴×C₂₃	—	4	6	no	—
53	#41×C₂₃	—	2	24	no	—
54	#41×C₂₃	—	4	16	no	—
55	#41×C₂₃	—	4	12	no	—
56	#41×C₂₃	D₈	1	4	yes	—
57	#41×C₂₃	C₂×D₄	1	4	yes	—
58	C₂×Q₈×C₂₃	—	2	2	no	—
59	C₂×Q₈×C₂₃	D₈	1	1	yes	characteristic
60	C₂×D₄×C₂₃	—	2	6	no	—
61	C₂×D₄×C₂₃	—	2	24	no	—
62	C₂×D₄×C₂₃	—	4	6	no	—
63	C₂×D₄×C₂₃	—	4	48	no	—
64	C₂×D₄×C₂₃	—	4	6	no	—
65	C₂×D₄×C₂₃	—	4	12	no	—
66	C₂×D₄×C₂₃	D₈	1	3	yes	—
67	C₂²×C₄×C₂₃	—	4	6	no	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
69	Q₁₆×C₂₃	—	2	2	no	—
70	Q₁₆×C₂₃	C₂⁴	1	4	yes	—
71	SD₁₆×C₂₃	—	2	24	no	—
72	D₈×C₂₃	—	2	6	no	—
73	D₈×C₂₃	—	2	24	no	—
74	D₈×C₂₃	C₂⁴	1	12	yes	—
75	M₁₆×C₂₃	C₂×D₄	1	12	yes	—
76	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
77	C₂×C₈×C₂₃	C₂×D₄	1	6	yes	—
78	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
80	C₁₆×C₂₃	C₂⁴	1	2	yes	—
81	C₁₆×C₂₃	C₂⁴	1	6	yes	—
82	C₂×#3415	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₂₃	—	4	6	no	—
84	C₂³×C₂₃	—	4	24	no	—
85	C₂³×C₂₃	—	16	12	no	—
86	C₂³×C₂₃	C₂×D₈	1	3	yes	—
87	Q₈×C₂₃	—	2	4	no	—
88	Q₈×C₂₃	—	4	2	no	—
89	Q₈×C₂₃	C₂×D₈	1	4	yes	—
90	D₄×C₂₃	—	2	12	no	—
91	D₄×C₂₃	—	4	24	no	—
92	D₄×C₂₃	—	4	24	no	—
93	D₄×C₂₃	—	4	6	no	—
94	D₄×C₂₃	C₂×D₈	1	12	yes	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	4	24	no	—
97	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
98	C₂×C₄×C₂₃	C₂×D₈	1	3	yes	—
99	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
100	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₂₃	C₂²×D₄	1	6	yes	—
103	C₈×C₂₃	C₂⁵	1	1	yes	characteristic
104	C₂×#576	C₂×C₂₃	1	2	yes	complemented
105	#3415	C₂×C₂₃	1	16	yes	direct product factor
106	C₂×#509	C₂×C₂₃	1	6	yes	complemented
107	C₂×#508	C₂×C₂₃	1	3	yes	complemented
108	C₂²×D₁₆	C₂×C₂₃	1	3	yes	complemented
109	C₂×#504	C₂×C₂₃	1	1	yes	characteristic, complemented
110	C₂²×C₂₃	—	2	6	no	—
111	C₂²×C₂₃	—	4	12	no	—
112	C₂²×C₂₃	—	8	12	no	—
113	C₂²×C₂₃	—	16	24	no	—
114	C₂²×C₂₃	C₂²×D₈	1	6	yes	—
115	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₂₃	—	4	4	no	—
117	C₄×C₂₃	C₂²×D₈	1	6	yes	—
118	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
120	C₂×H(5,𝔽₂)	—	2	2	no	—
121	#576	—	2	8	no	—
122	#576	C₂²×C₂₃	1	16	yes	complemented
123	C₂×AGL(1,ℤ/8)	—	2	6	no	—
124	C₂×#135	C₂²×C₂₃	1	6	yes	complemented
125	C₂²×D₈	C₂²×C₂₃	1	6	yes	complemented
126	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
127	#509	C₂²×C₂₃	1	48	yes	complemented
128	#508	C₂²×C₂₃	1	24	yes	complemented
129	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic, complemented
130	C₂×SD₃₂	C₂²×C₂₃	1	6	yes	complemented
131	C₂×D₁₆	C₂²×C₂₃	1	3	yes	complemented
132	C₂×D₁₆	C₂²×C₂₃	1	6	yes	complemented
133	C₂×D₁₆	C₂²×C₂₃	1	24	yes	complemented
134	#504	C₂²×C₂₃	1	8	yes	complemented
135	C₂×M₃₂	C₂²×C₂₃	1	3	yes	complemented
136	C₂²×C₁₆	C₂²×C₂₃	1	3	yes	complemented
137	C₂×C₂₃	—	2	6	no	—
138	C₂×C₂₃	—	8	12	no	—
139	C₂×C₂₃	#3415	1	2	yes	direct product factor
140	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
141	H(5,𝔽₂)	—	2	16	no	—
142	H(5,𝔽₂)	—	4	8	no	—
143	C₂×#41	—	2	6	no	—
144	C₂×#41	—	4	2	no	—
145	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
146	C₂²×D₄	—	2	6	no	—
147	C₂²×D₄	—	4	6	no	—
148	AGL(1,ℤ/8)	—	2	48	no	—
149	#135	—	2	12	no	—
150	#135	C₂³×C₂₃	1	24	yes	complemented
151	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
152	C₂×SD₁₆	—	2	6	no	—
153	C₂×D₈	—	2	6	no	—
154	C₂×D₈	—	2	12	no	—
155	C₂×D₈	C₂³×C₂₃	1	6	yes	—
156	C₂×D₈	C₂³×C₂₃	1	24	yes	complemented
157	#131	D₄×C₂₃	1	4	yes	—
158	#131	C₂³×C₂₃	1	4	yes	—
159	C₂×M₁₆	D₄×C₂₃	1	3	yes	—
160	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
161	Q₃₂	C₂³×C₂₃	1	4	yes	complemented
162	SD₃₂	C₂³×C₂₃	1	24	yes	complemented
163	D₁₆	C₂³×C₂₃	1	12	yes	complemented
164	D₁₆	C₂³×C₂₃	1	24	yes	complemented
165	M₃₂	C₂³×C₂₃	1	12	yes	complemented
166	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
167	C₂×C₁₆	C₂³×C₂₃	1	3	yes	—
168	C₂×C₁₆	C₂³×C₂₃	1	12	yes	complemented
169	C₂₃	C₂×#3415	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	—	4	6	no	—
171	#41	—	2	24	no	—
172	#41	—	4	16	no	—
173	#41	—	4	12	no	—
174	#41	D₈×C₂₃	1	4	yes	—
175	#41	C₂×D₄×C₂₃	1	4	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	D₈×C₂₃	1	1	yes	characteristic
178	C₂×D₄	—	2	6	no	—
179	C₂×D₄	—	2	24	no	—
180	C₂×D₄	—	4	6	no	—
181	C₂×D₄	—	4	48	no	—
182	C₂×D₄	—	4	6	no	—
183	C₂×D₄	—	4	12	no	—
184	C₂×D₄	D₈×C₂₃	1	3	yes	—
185	C₂²×C₄	—	4	6	no	—
186	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
187	Q₁₆	—	2	2	no	—
188	Q₁₆	C₂⁴×C₂₃	1	4	yes	—
189	SD₁₆	—	2	24	no	—
190	D₈	—	2	6	no	—
191	D₈	—	2	24	no	—
192	D₈	C₂⁴×C₂₃	1	12	yes	—
193	M₁₆	C₂×D₄×C₂₃	1	12	yes	—
194	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
195	C₂×C₈	C₂×D₄×C₂₃	1	6	yes	—
196	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
197	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
198	C₁₆	C₂⁴×C₂₃	1	2	yes	—
199	C₁₆	C₂⁴×C₂₃	1	6	yes	—
200	C₂³	—	4	6	no	—
201	C₂³	—	4	24	no	—
202	C₂³	—	16	12	no	—
203	C₂³	C₂×D₈×C₂₃	1	3	yes	—
204	Q₈	—	2	4	no	—
205	Q₈	—	4	2	no	—
206	Q₈	C₂×D₈×C₂₃	1	4	yes	—
207	D₄	—	2	12	no	—
208	D₄	—	4	24	no	—
209	D₄	—	4	24	no	—
210	D₄	—	4	6	no	—
211	D₄	C₂×D₈×C₂₃	1	12	yes	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	24	no	—
214	C₂×C₄	C₂×D₈×C₂₃	1	3	yes	—
215	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
216	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
218	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
219	C₈	C₂²×D₄×C₂₃	1	6	yes	—
220	C₈	C₂⁵×C₂₃	1	1	yes	characteristic
221	C₂²	—	2	6	no	—
222	C₂²	—	4	12	no	—
223	C₂²	—	8	12	no	—
224	C₂²	—	16	24	no	—
225	C₂²	C₂²×D₈×C₂₃	1	6	yes	—
226	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
227	C₄	—	4	4	no	—
228	C₄	C₂²×D₈×C₂₃	1	6	yes	—
229	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
230	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
231	C₂	—	2	6	no	—
232	C₂	—	8	12	no	—
233	C₂	C₂₃×#3415	1	2	yes	direct product factor
234	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₂₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×#41×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#576	C₂	semidirect
2. C₂₃×#3415	C₂	direct (N × H)
3. C₂×C₂₃×#509	C₂	semidirect
4. C₂×C₂₃×#508	C₂	semidirect
5. C₂²×C₂₃×D₁₆	C₂	semidirect
6. C₂×C₂₃×#504	C₂	semidirect
9. C₂₃×#576	C₂²	semidirect
11. C₂×C₂₃×#135	C₂²	semidirect
12. C₂²×D₈×C₂₃	C₂²	semidirect
14. C₂₃×#509	C₂²	semidirect
15. C₂₃×#508	C₂²	semidirect
16. C₂×C₂₃×Q₃₂	C₂²	semidirect
17. C₂×C₂₃×SD₃₂	C₂²	semidirect
18. C₂×C₂₃×D₁₆	C₂²	semidirect
19. C₂×C₂₃×D₁₆	C₂²	semidirect
20. C₂×C₂₃×D₁₆	C₂²	semidirect
21. C₂₃×#504	C₂²	semidirect
22. C₂×C₂₃×M₃₂	C₂²	semidirect
23. C₂²×C₁₆×C₂₃	C₂²	semidirect
33. C₂₃×#135	C₂³	semidirect
39. C₂×D₈×C₂₃	C₂³	semidirect
44. C₂₃×Q₃₂	C₂³	semidirect
45. C₂₃×SD₃₂	C₂³	semidirect
46. C₂₃×D₁₆	C₂³	semidirect
47. C₂₃×D₁₆	C₂³	semidirect
48. C₂₃×M₃₂	C₂³	semidirect
51. C₂×C₁₆×C₂₃	C₂³	semidirect
82. C₂×#3415	C₂₃	direct (N × H)
104. C₂×#576	C₂×C₂₃	semidirect
105. #3415	C₂×C₂₃	direct (N × H)
106. C₂×#509	C₂×C₂₃	semidirect
107. C₂×#508	C₂×C₂₃	semidirect
108. C₂²×D₁₆	C₂×C₂₃	semidirect
109. C₂×#504	C₂×C₂₃	semidirect
122. #576	C₂²×C₂₃	semidirect
124. C₂×#135	C₂²×C₂₃	semidirect
125. C₂²×D₈	C₂²×C₂₃	semidirect
127. #509	C₂²×C₂₃	semidirect
128. #508	C₂²×C₂₃	semidirect
129. C₂×Q₃₂	C₂²×C₂₃	semidirect
130. C₂×SD₃₂	C₂²×C₂₃	semidirect
131. C₂×D₁₆	C₂²×C₂₃	semidirect
132. C₂×D₁₆	C₂²×C₂₃	semidirect
133. C₂×D₁₆	C₂²×C₂₃	semidirect
134. #504	C₂²×C₂₃	semidirect
135. C₂×M₃₂	C₂²×C₂₃	semidirect
136. C₂²×C₁₆	C₂²×C₂₃	semidirect
139. C₂×C₂₃	#3415	direct (N × H)
150. #135	C₂³×C₂₃	semidirect
156. C₂×D₈	C₂³×C₂₃	semidirect
161. Q₃₂	C₂³×C₂₃	semidirect
162. SD₃₂	C₂³×C₂₃	semidirect
163. D₁₆	C₂³×C₂₃	semidirect
164. D₁₆	C₂³×C₂₃	semidirect
165. M₃₂	C₂³×C₂₃	semidirect
168. C₂×C₁₆	C₂³×C₂₃	semidirect
169. C₂₃	C₂×#3415	direct (N × H)
233. C₂	C₂₃×#3415	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#576	2
C₂₃×#3415	2
C₂×C₂₃×#509	2
C₂×C₂₃×#508	2
C₂²×C₂₃×D₁₆	2
C₂×C₂₃×#504	2
C₂×#3415	23

Subgroups — C2×C23×#3415 #19210258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3415 #19210258