Subgroups — C₂×C₂₃×#2255 #19181353

Order 5888.

518 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	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2255	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#367	C₂	1	2	yes	maximal
5	C₂×C₂₃×#366	C₂	1	2	yes	maximal
6	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₂₃×#125	—	2	1	no	—
9	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂₃×#500	—	2	2	no	—
11	C₂₃×#500	C₂²	1	4	yes	—
12	C₂₃×#499	—	2	2	no	—
13	C₂₃×#499	C₂²	1	4	yes	—
14	C₂×C₂₃×#107	—	2	1	no	—
15	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#107	C₂²	1	2	yes	—
17	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#106	—	2	1	no	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#367	C₂²	1	8	yes	—
21	C₂₃×#366	C₂²	1	8	yes	—
22	C₂₃×#346	C₂²	1	4	yes	—
23	C₂²×Q₈×C₂₃	—	4	1	no	—
24	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×C₄⋊Q₈	—	2	4	no	—
27	C₂₃×C₄⋊Q₈	—	4	2	no	—
28	C₂₃×#125	—	2	4	no	—
29	C₂₃×#125	—	4	2	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
34	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
35	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₂₃	Q₈	1	8	yes	complemented
38	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
39	C₂₃×#107	—	2	4	no	—
40	C₂₃×#107	—	2	1	no	—
41	C₂₃×#107	—	2	1	no	—
42	C₂₃×#107	—	2	2	no	—
43	C₂₃×#107	C₂³	1	2	yes	—
44	C₂₃×#107	C₂³	1	4	yes	—
45	C₂₃×#107	C₂³	1	2	yes	—
46	C₂₃×#106	—	2	4	no	—
47	C₄×C₈×C₂₃	D₄	1	2	yes	—
48	C₄×C₈×C₂₃	C₂³	1	2	yes	—
49	C₂×Q₈×C₂₃	—	4	4	no	—
50	C₂×Q₈×C₂₃	—	8	4	no	—
51	C₂²×C₄×C₂₃	—	4	1	no	—
52	C₂²×C₄×C₂₃	—	4	1	no	—
53	C₂²×C₄×C₂₃	—	4	2	no	—
54	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	Q₁₆	1	4	yes	—
58	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
61	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃	—	2	4	no	—
65	C₄⋊C₄×C₂₃	—	4	1	no	—
66	C₄⋊C₄×C₂₃	—	4	1	no	—
67	C₄⋊C₄×C₂₃	—	4	2	no	—
68	C₄⋊C₄×C₂₃	—	4	4	no	—
69	C₄⋊C₄×C₂₃	—	4	4	no	—
70	C₄⋊C₄×C₂₃	—	4	4	no	—
71	C₄²×C₂₃	D₈	1	2	yes	—
72	C₄²×C₂₃	C₂×D₄	1	2	yes	—
73	C₁₆×C₂₃	C₂×Q₈	1	8	yes	complemented
74	C₂×#2255	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₂₃	—	8	8	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	4	2	no	—
80	C₂×C₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	4	4	no	—
82	C₂×C₄×C₂₃	—	8	2	no	—
83	C₂×C₄×C₂₃	—	8	4	no	—
84	C₂×C₄×C₂₃	—	8	2	no	—
85	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
91	C₈×C₂₃	C₄⋊Q₈	1	2	yes	—
92	C₈×C₂₃	C₂×Q₁₆	1	4	yes	—
93	C₈×C₂₃	C₂²×Q₈	1	2	yes	—
94	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
95	C₂×#499	C₂×C₂₃	1	1	yes	characteristic
96	#2255	C₂×C₂₃	1	8	yes	direct product factor
97	C₂×#367	C₂×C₂₃	1	2	yes	—
98	C₂×#366	C₂×C₂₃	1	2	yes	—
99	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
100	C₂²×C₂₃	#500	1	2	yes	—
101	C₂²×C₂₃	#509	1	2	yes	—
102	C₂²×C₂₃	#510	1	2	yes	—
103	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	8	2	no	—
107	C₄×C₂₃	—	8	4	no	—
108	C₄×C₂₃	—	8	2	no	—
109	C₄×C₂₃	#500	1	2	yes	—
110	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
111	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×#125	—	2	1	no	—
114	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
115	#500	—	2	2	no	—
116	#500	C₂²×C₂₃	1	4	yes	—
117	#499	—	2	2	no	—
118	#499	C₂²×C₂₃	1	4	yes	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
121	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
122	C₂×#107	C₂²×C₂₃	1	2	yes	—
123	C₂×#106	—	2	1	no	—
124	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	#367	C₂²×C₂₃	1	8	yes	—
126	#366	C₂²×C₂₃	1	8	yes	—
127	#346	C₂²×C₂₃	1	4	yes	—
128	C₂×C₂₃	#2255	1	4	yes	direct product factor
129	C₂×C₂₃	C₂×#500	1	1	yes	characteristic
130	C₂×C₂₃	C₂×#509	1	1	yes	characteristic
131	C₂×C₂₃	C₂×#510	1	1	yes	characteristic
132	C₂²×Q₈	—	4	1	no	—
133	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	4	no	—
136	C₄⋊Q₈	—	4	2	no	—
137	#125	—	2	4	no	—
138	#125	—	4	2	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
146	C₂×C₁₆	Q₈×C₂₃	1	8	yes	complemented
147	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
148	#107	—	2	4	no	—
149	#107	—	2	1	no	—
150	#107	—	2	1	no	—
151	#107	—	2	2	no	—
152	#107	C₂³×C₂₃	1	2	yes	—
153	#107	C₂³×C₂₃	1	4	yes	—
154	#107	C₂³×C₂₃	1	2	yes	—
155	#106	—	2	4	no	—
156	C₄×C₈	D₄×C₂₃	1	2	yes	—
157	C₄×C₈	C₂³×C₂₃	1	2	yes	—
158	C₂₃	C₂×#2255	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	8	4	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
167	C₂×C₈	Q₁₆×C₂₃	1	4	yes	—
168	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
170	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
171	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄²	D₈×C₂₃	1	2	yes	—
182	C₄²	C₂×D₄×C₂₃	1	2	yes	—
183	C₁₆	C₂×Q₈×C₂₃	1	8	yes	complemented
184	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
185	Q₈	—	8	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	4	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
195	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
196	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
197	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
198	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
199	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
200	C₈	C₂₃×C₄⋊Q₈	1	2	yes	—
201	C₈	C₂×Q₁₆×C₂₃	1	4	yes	—
202	C₈	C₂²×Q₈×C₂₃	1	2	yes	—
203	C₂²	C₂₃×#500	1	2	yes	—
204	C₂²	C₂₃×#509	1	2	yes	—
205	C₂²	C₂₃×#510	1	2	yes	—
206	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	2	no	—
212	C₄	C₂₃×#500	1	2	yes	—
213	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
214	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
215	C₂	C₂₃×#2255	1	4	yes	direct product factor
216	C₂	C₂×C₂₃×#500	1	1	yes	characteristic
217	C₂	C₂×C₂₃×#509	1	1	yes	characteristic
218	C₂	C₂×C₂₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2255	C₂	direct (N × H)
37. C₂×C₁₆×C₂₃	Q₈	semidirect
73. C₁₆×C₂₃	C₂×Q₈	semidirect
74. C₂×#2255	C₂₃	direct (N × H)
96. #2255	C₂×C₂₃	direct (N × H)
128. C₂×C₂₃	#2255	direct (N × H)
146. C₂×C₁₆	Q₈×C₂₃	semidirect
158. C₂₃	C₂×#2255	direct (N × H)
183. C₁₆	C₂×Q₈×C₂₃	semidirect
215. C₂	C₂₃×#2255	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×#500	2
C₂×C₂₃×#499	2
C₂₃×#2255	2
C₂×C₂₃×#367	2
C₂×C₂₃×#366	2
C₂×C₂₃×#346	2
C₂×#2255	23

Subgroups — C2×C23×#2255 #19181353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2255 #19181353