Subgroups — C₂×C₂₃×#1333 #19159221

Order 5888.

462 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1333	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₂₃	—	2	1	no	—
7	C₂²×C₁₆×C₂₃	—	2	1	no	—
8	C₄×D₈×C₂₃	—	2	1	no	—
9	C₄×D₈×C₂₃	—	2	1	no	—
10	C₄×D₈×C₂₃	C₄	1	2	yes	—
11	C₄×D₈×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
14	C₂×C₂₃×#102	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#346	—	2	1	no	—
17	C₂₃×#346	C₂²	1	2	yes	complemented
18	C₂₃×#334	—	2	1	no	—
19	C₂₃×#334	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₂₃	—	2	1	no	—
21	C₂³×C₄×C₂₃	—	4	1	no	—
22	C₂×D₈×C₂₃	—	2	1	no	—
23	C₂×D₈×C₂₃	—	2	2	no	—
24	C₂×D₈×C₂₃	C₈	1	4	yes	complemented
25	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₂₃	—	4	1	no	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₄×D₄×C₂₃	—	2	2	no	—
30	C₄×D₄×C₂₃	—	2	2	no	—
31	C₄×D₄×C₂₃	—	4	1	no	—
32	C₄×D₄×C₂₃	—	4	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
35	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₂₃	—	2	4	no	—
37	C₂×C₁₆×C₂₃	—	2	2	no	—
38	C₂₃×#107	—	2	1	no	—
39	C₂₃×#107	C₂×C₄	1	2	yes	—
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	—	4	1	no	—
42	C₂₃×#102	—	2	2	no	—
43	C₂₃×#102	—	2	2	no	—
44	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂⁴×C₂₃	—	4	1	no	—
49	C₂×D₄×C₂₃	—	2	4	no	—
50	C₂×D₄×C₂₃	—	2	2	no	—
51	C₂×D₄×C₂₃	—	4	1	no	—
52	C₂×D₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	—	4	1	no	—
54	C₂²×C₄×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	—	4	2	no	—
56	C₂²×C₄×C₂₃	—	8	4	no	—
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
60	D₈×C₂₃	—	2	4	no	—
61	D₈×C₂₃	—	2	2	no	—
62	D₈×C₂₃	C₂×C₈	1	4	yes	complemented
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	—	8	2	no	—
66	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
70	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₄⋊C₄×C₂₃	—	4	1	no	—
74	C₂²⋊C₄×C₂₃	—	4	2	no	—
75	C₂²⋊C₄×C₂₃	—	4	2	no	—
76	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₂₃	D₈	1	1	yes	characteristic
78	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
79	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₁₆×C₂₃	—	2	4	no	—
81	C₂×#1333	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	—	4	4	no	—
83	C₂³×C₂₃	—	4	2	no	—
84	C₂³×C₂₃	—	8	2	no	—
85	C₂³×C₂₃	—	8	2	no	—
86	C₂³×C₂₃	#102	1	1	yes	characteristic
87	D₄×C₂₃	—	2	4	no	—
88	D₄×C₂₃	—	4	4	no	—
89	D₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	8	4	no	—
94	C₂×C₄×C₂₃	—	8	4	no	—
95	C₂×C₄×C₂₃	—	8	2	no	—
96	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#104	1	2	yes	—
101	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₈×C₂₃	—	2	1	no	—
105	C₈×C₂₃	—	2	1	no	—
106	C₈×C₂₃	—	8	2	no	—
107	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
108	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
109	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
111	C₂×#346	C₂×C₂₃	1	1	yes	characteristic, complemented
112	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
113	#1333	C₂×C₂₃	1	4	yes	direct product factor
114	C₂²×C₂₃	—	4	4	no	—
115	C₂²×C₂₃	—	8	8	no	—
116	C₂²×C₂₃	—	8	2	no	—
117	C₂²×C₂₃	—	8	2	no	—
118	C₂²×C₂₃	D₄⋊C₈	1	2	yes	—
119	C₂²×C₂₃	#360	1	2	yes	—
120	C₂²×C₂₃	#361	1	2	yes	—
121	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	2	1	no	—
125	C₄×C₂₃	—	2	1	no	—
126	C₄×C₂₃	—	8	2	no	—
127	C₄×C₂₃	D₄⋊C₈	1	2	yes	—
128	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
130	C₂²×D₈	C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₄×D₄	—	2	1	no	—
132	C₂²×C₁₆	—	2	1	no	—
133	C₄×D₈	—	2	1	no	—
134	C₄×D₈	—	2	1	no	—
135	C₄×D₈	C₄×C₂₃	1	2	yes	—
136	C₄×D₈	C₂²×C₂₃	1	2	yes	—
137	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
138	C₂×C₄⋊C₈	—	2	1	no	—
139	C₂×#102	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
141	#346	—	2	1	no	—
142	#346	C₂²×C₂₃	1	2	yes	complemented
143	#334	—	2	1	no	—
144	#334	C₂²×C₂₃	1	2	yes	complemented
145	C₂×C₂₃	—	8	4	no	—
146	C₂×C₂₃	#1333	1	4	yes	direct product factor
147	C₂×C₂₃	C₂×D₄⋊C₈	1	1	yes	characteristic
148	C₂×C₂₃	C₂×#360	1	1	yes	characteristic
149	C₂×C₂₃	C₂×#361	1	1	yes	characteristic
150	C₂²×D₄	—	2	1	no	—
151	C₂³×C₄	—	4	1	no	—
152	C₂×D₈	—	2	1	no	—
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	C₈×C₂₃	1	4	yes	complemented
155	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
156	C₂²×C₈	—	4	1	no	—
157	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
159	C₄×D₄	—	2	2	no	—
160	C₄×D₄	—	2	2	no	—
161	C₄×D₄	—	4	1	no	—
162	C₄×D₄	—	4	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₂²⋊C₄	—	4	1	no	—
165	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
166	C₂×C₁₆	—	2	4	no	—
167	C₂×C₁₆	—	2	2	no	—
168	#107	—	2	1	no	—
169	#107	C₂×C₄×C₂₃	1	2	yes	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	4	1	no	—
172	#102	—	2	2	no	—
173	#102	—	2	2	no	—
174	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
176	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
177	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
178	C₂₃	C₂×#1333	1	1	yes	characteristic, direct product factor, Sylow
179	C₂⁴	—	4	1	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	8	4	no	—
188	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
190	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
191	D₈	—	2	2	no	—
192	D₈	—	2	4	no	—
193	D₈	C₂×C₈×C₂₃	1	4	yes	complemented
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	8	2	no	—
197	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
199	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
200	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
201	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
202	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
208	C₄²	D₈×C₂₃	1	1	yes	characteristic
209	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
210	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₁₆	—	2	4	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	8	2	no	—
215	C₂³	—	8	2	no	—
216	C₂³	C₂₃×#102	1	1	yes	characteristic
217	D₄	—	2	4	no	—
218	D₄	—	4	4	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
227	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
228	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
229	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
230	C₂×C₄	C₂₃×#104	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
233	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	8	2	no	—
237	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
238	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
239	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
240	C₂²	—	4	4	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	8	8	no	—
244	C₂²	C₂₃×D₄⋊C₈	1	2	yes	—
245	C₂²	C₂₃×#360	1	2	yes	—
246	C₂²	C₂₃×#361	1	2	yes	—
247	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	8	2	no	—
253	C₄	C₂₃×D₄⋊C₈	1	2	yes	—
254	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
255	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
256	C₂	—	8	4	no	—
257	C₂	C₂₃×#1333	1	4	yes	direct product factor
258	C₂	C₂×C₂₃×D₄⋊C₈	1	1	yes	characteristic
259	C₂	C₂×C₂₃×#360	1	1	yes	characteristic
260	C₂	C₂×C₂₃×#361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#102
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₄
LCS	G ▷ 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
2. C₂×C₂₃×#346	C₂	semidirect
3. C₂×C₂₃×#334	C₂	semidirect
4. C₂₃×#1333	C₂	direct (N × H)
17. C₂₃×#346	C₂²	semidirect
19. C₂₃×#334	C₂²	semidirect
24. C₂×D₈×C₂₃	C₈	semidirect
62. D₈×C₂₃	C₂×C₈	semidirect
81. C₂×#1333	C₂₃	direct (N × H)
111. C₂×#346	C₂×C₂₃	semidirect
112. C₂×#334	C₂×C₂₃	semidirect
113. #1333	C₂×C₂₃	direct (N × H)
142. #346	C₂²×C₂₃	semidirect
144. #334	C₂²×C₂₃	semidirect
146. C₂×C₂₃	#1333	direct (N × H)
154. C₂×D₈	C₈×C₂₃	semidirect
178. C₂₃	C₂×#1333	direct (N × H)
193. D₈	C₂×C₈×C₂₃	semidirect
257. C₂	C₂₃×#1333	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×C₄×D₈×C₂₃	2
C₂×C₂₃×#346	2
C₂×C₂₃×#334	2
C₂₃×#1333	2
C₂×#1333	23

Subgroups — C2×C23×#1333 #19159221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1333 #19159221