Subgroups — C₂×C₂₃×#2110 #19180954

Order 5888.

738 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2110	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#347	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#117	C₄	1	1	yes	characteristic
8	C₂×C₂₃×M₃₂	C₄	1	4	yes	complemented
9	C₂×C₂₃×M₃₂	C₄	1	8	yes	complemented
10	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
11	C₂×C₂₃×M₃₂	C₂²	1	8	yes	—
12	C₂²×C₁₆×C₂₃	C₄	1	4	yes	complemented
13	C₂₃×#405	C₄	1	4	yes	—
14	C₂₃×#405	C₂²	1	4	yes	—
15	C₂×C₂₃×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
17	C₂₃×#347	C₂²	1	16	yes	complemented
18	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	—	2	2	no	—
20	C₂×M₁₆×C₂₃	—	2	2	no	—
21	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
30	C₂₃×#117	—	2	2	no	—
31	C₂₃×#117	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
34	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
35	C₂₃×M₃₂	C₂×C₄	1	8	yes	—
36	C₂₃×M₃₂	C₂×C₄	1	16	yes	complemented
37	C₂₃×M₃₂	C₂³	1	8	yes	—
38	C₂×C₁₆×C₂₃	C₂×C₄	1	8	yes	—
39	C₂×C₁₆×C₂₃	C₂×C₄	1	16	yes	complemented
40	C₂₃×#97	C₂×C₄	1	4	yes	—
41	C₂₃×#97	C₂³	1	4	yes	—
42	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
43	C₂⁴×C₂₃	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
49	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
50	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
53	M₁₆×C₂₃	—	2	8	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	C₄²	1	4	yes	—
57	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₄²	1	4	yes	—
59	C₂×C₈×C₂₃	C₄²	1	2	yes	—
60	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	—	2	4	no	—
70	C₂²⋊C₄×C₂₃	—	2	4	no	—
71	C₄²×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
73	C₁₆×C₂₃	C₂²×C₄	1	16	yes	—
74	C₂×#2110	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₂₃	—	4	2	no	—
76	C₂³×C₂₃	#97	1	2	yes	—
77	C₂³×C₂₃	#97	1	2	yes	—
78	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂³×C₂₃	C₂×M₁₆	1	2	yes	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	4	4	no	—
82	C₂×C₄×C₂₃	#97	1	2	yes	—
83	C₂×C₄×C₂₃	#97	1	4	yes	—
84	C₂×C₄×C₂₃	#97	1	4	yes	—
85	C₂×C₄×C₂₃	#97	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
87	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
92	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₂₃	—	2	4	no	—
96	C₈×C₂₃	C₂×C₄²	1	2	yes	—
97	C₈×C₂₃	C₂×C₄²	1	2	yes	—
98	C₈×C₂₃	C₂×C₄²	1	2	yes	—
99	C₈×C₂₃	C₂³×C₄	1	2	yes	—
100	C₂²×M₃₂	C₂×C₂₃	1	2	yes	—
101	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
102	#2110	C₂×C₂₃	1	8	yes	direct product factor
103	C₂×#347	C₂×C₂₃	1	4	yes	complemented
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	4	no	—
106	C₂²×C₂₃	—	4	2	no	—
107	C₂²×C₂₃	—	4	2	no	—
108	C₂²×C₂₃	C₂×#97	1	1	yes	characteristic
109	C₂²×C₂₃	C₂×#97	1	4	yes	—
110	C₂²×C₂₃	C₂×#97	1	1	yes	characteristic
111	C₂²×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
112	C₄×C₂₃	—	4	4	no	—
113	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
114	C₄×C₂₃	C₂×#97	1	4	yes	—
115	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
116	C₄×C₂₃	C₂²×C₄²	1	1	yes	characteristic
117	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
118	C₂²×M₁₆	C₄×C₂₃	1	1	yes	characteristic
119	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
120	C₂×#117	C₄×C₂₃	1	1	yes	characteristic
121	C₂×M₃₂	C₄×C₂₃	1	4	yes	complemented
122	C₂×M₃₂	C₄×C₂₃	1	8	yes	complemented
123	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
124	C₂×M₃₂	C₂²×C₂₃	1	8	yes	—
125	C₂²×C₁₆	C₄×C₂₃	1	4	yes	complemented
126	#405	C₄×C₂₃	1	4	yes	—
127	#405	C₂²×C₂₃	1	4	yes	—
128	C₂×#97	C₂²×C₂₃	1	2	yes	—
129	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
130	#347	C₂²×C₂₃	1	16	yes	complemented
131	C₂×C₂₃	—	2	1	no	—
132	C₂×C₂₃	—	2	1	no	—
133	C₂×C₂₃	—	2	4	no	—
134	C₂×C₂₃	#2110	1	2	yes	direct product factor
135	C₂×C₂₃	C₂²×#97	1	1	yes	characteristic
136	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
140	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
141	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
142	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
143	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
144	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
145	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
146	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
148	#117	—	2	2	no	—
149	#117	C₂×C₄×C₂₃	1	4	yes	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₂²⋊C₄	—	2	1	no	—
152	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
153	M₃₂	C₂×C₄×C₂₃	1	8	yes	—
154	M₃₂	C₂×C₄×C₂₃	1	16	yes	complemented
155	M₃₂	C₂³×C₂₃	1	8	yes	—
156	C₂×C₁₆	C₂×C₄×C₂₃	1	8	yes	—
157	C₂×C₁₆	C₂×C₄×C₂₃	1	16	yes	complemented
158	#97	C₂×C₄×C₂₃	1	4	yes	—
159	#97	C₂³×C₂₃	1	4	yes	—
160	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
161	C₂₃	C₂×#2110	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	M₁₆×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
165	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
166	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
167	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
168	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
172	M₁₆	—	2	8	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
176	C₂×C₈	C₄²×C₂₃	1	2	yes	—
177	C₂×C₈	C₄²×C₂₃	1	4	yes	—
178	C₂×C₈	C₄²×C₂₃	1	4	yes	—
179	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
180	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
181	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
184	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
185	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
186	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
187	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
192	C₁₆	C₂²×C₄×C₂₃	1	16	yes	—
193	C₂³	—	4	2	no	—
194	C₂³	C₂₃×#97	1	2	yes	—
195	C₂³	C₂₃×#97	1	2	yes	—
196	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
197	C₂³	C₂×M₁₆×C₂₃	1	2	yes	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₂₃×#97	1	2	yes	—
201	C₂×C₄	C₂₃×#97	1	4	yes	—
202	C₂×C₄	C₂₃×#97	1	4	yes	—
203	C₂×C₄	C₂₃×#97	1	2	yes	—
204	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
205	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
210	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
212	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	C₂×C₄²×C₂₃	1	2	yes	—
215	C₈	C₂×C₄²×C₂₃	1	2	yes	—
216	C₈	C₂×C₄²×C₂₃	1	2	yes	—
217	C₈	C₂³×C₄×C₂₃	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×C₂₃×#97	1	1	yes	characteristic
223	C₂²	C₂×C₂₃×#97	1	4	yes	—
224	C₂²	C₂×C₂₃×#97	1	1	yes	characteristic
225	C₂²	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
226	C₄	—	4	4	no	—
227	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
228	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
229	C₄	C₂×C₂₃×#97	1	4	yes	—
230	C₄	C₂²×C₄²×C₂₃	1	1	yes	characteristic
231	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	2	4	no	—
235	C₂	C₂₃×#2110	1	2	yes	direct product factor
236	C₂	C₂²×C₂₃×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₄²
commutator	C₄	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#97
prime radical	C₂⁴×C₂₃	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2110	C₂	direct (N × H)
4. C₂×C₂₃×#347	C₂	semidirect
8. C₂×C₂₃×M₃₂	C₄	semidirect
9. C₂×C₂₃×M₃₂	C₄	semidirect
12. C₂²×C₁₆×C₂₃	C₄	semidirect
17. C₂₃×#347	C₂²	semidirect
36. C₂₃×M₃₂	C₂×C₄	semidirect
39. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
74. C₂×#2110	C₂₃	direct (N × H)
102. #2110	C₂×C₂₃	direct (N × H)
103. C₂×#347	C₂×C₂₃	semidirect
121. C₂×M₃₂	C₄×C₂₃	semidirect
122. C₂×M₃₂	C₄×C₂₃	semidirect
125. C₂²×C₁₆	C₄×C₂₃	semidirect
130. #347	C₂²×C₂₃	semidirect
134. C₂×C₂₃	#2110	direct (N × H)
154. M₃₂	C₂×C₄×C₂₃	semidirect
157. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
161. C₂₃	C₂×#2110	direct (N × H)
235. C₂	C₂₃×#2110	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×M₃₂	2
C₂×C₂₃×#405	2
C₂₃×#2110	2
C₂×C₂₃×#347	2
C₂×#2110	23

Subgroups — C2×C23×#2110 #19180954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2110 #19180954