Subgroups — C₂×C₂₃×#1332 #19159216

Order 5888.

406 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1332	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
7	C₂²×C₁₆×C₂₃	—	2	1	no	—
8	C₄×Q₁₆×C₂₃	—	2	1	no	—
9	C₄×Q₁₆×C₂₃	—	2	1	no	—
10	C₄×Q₁₆×C₂₃	C₄	1	2	yes	—
11	C₄×Q₁₆×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×#103	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₂₃	—	2	1	no	—
17	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#335	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂×Q₁₆×C₂₃	—	2	1	no	—
22	C₂×Q₁₆×C₂₃	—	2	2	no	—
23	C₂×Q₁₆×C₂₃	C₈	1	4	yes	complemented
24	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₂₃	—	4	1	no	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₂₃	—	2	2	no	—
29	C₄×Q₈×C₂₃	—	2	2	no	—
30	C₄×Q₈×C₂₃	—	4	1	no	—
31	C₄×Q₈×C₂₃	—	4	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
34	C₂×C₄²×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×#103	—	2	2	no	—
43	C₂₃×#103	—	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	C₂×Q₈×C₂₃	—	2	4	no	—
49	C₂×Q₈×C₂₃	—	2	2	no	—
50	C₂×Q₈×C₂₃	—	4	1	no	—
51	C₂×Q₈×C₂₃	—	4	2	no	—
52	C₂²×C₄×C₂₃	—	4	1	no	—
53	C₂²×C₄×C₂₃	—	4	1	no	—
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
57	Q₁₆×C₂₃	—	2	4	no	—
58	Q₁₆×C₂₃	—	2	2	no	—
59	Q₁₆×C₂₃	C₂×C₈	1	4	yes	complemented
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	—	8	2	no	—
63	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
67	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₄⋊C₄×C₂₃	—	4	1	no	—
71	C₄⋊C₄×C₂₃	—	4	2	no	—
72	C₄⋊C₄×C₂₃	—	4	2	no	—
73	C₄²×C₂₃	—	4	2	no	—
74	C₄²×C₂₃	—	4	2	no	—
75	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₂₃	D₈	1	1	yes	characteristic
77	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
78	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₁₆×C₂₃	—	2	4	no	—
80	C₂×#1332	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₃	#102	1	1	yes	characteristic
82	Q₈×C₂₃	—	2	4	no	—
83	Q₈×C₂₃	—	4	4	no	—
84	Q₈×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	8	2	no	—
90	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
94	C₂×C₄×C₂₃	#104	1	2	yes	—
95	C₂×C₄×C₂₃	SD₃₂	1	2	yes	—
96	C₂×C₄×C₂₃	Q₃₂	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₈×C₂₃	—	2	1	no	—
101	C₈×C₂₃	—	2	1	no	—
102	C₈×C₂₃	—	8	2	no	—
103	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
104	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
105	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
107	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
108	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
109	#1332	C₂×C₂₃	1	4	yes	direct product factor
110	C₂²×C₂₃	D₄⋊C₈	1	2	yes	—
111	C₂²×C₂₃	#358	1	2	yes	—
112	C₂²×C₂₃	#359	1	2	yes	—
113	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
114	C₄×C₂₃	—	2	1	no	—
115	C₄×C₂₃	—	2	1	no	—
116	C₄×C₂₃	—	4	4	no	—
117	C₄×C₂₃	—	8	2	no	—
118	C₄×C₂₃	D₄⋊C₈	1	2	yes	—
119	C₄×C₂₃	#358	1	2	yes	—
120	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
122	C₄×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
123	C₄×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
124	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
125	C₂×C₄×Q₈	—	2	1	no	—
126	C₂²×C₁₆	—	2	1	no	—
127	C₄×Q₁₆	—	2	1	no	—
128	C₄×Q₁₆	—	2	1	no	—
129	C₄×Q₁₆	C₄×C₂₃	1	2	yes	—
130	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
131	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
132	C₂×C₄⋊C₈	—	2	1	no	—
133	C₂×#103	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
135	C₄×C₁₆	—	2	1	no	—
136	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
137	#335	—	2	1	no	—
138	#335	C₂²×C₂₃	1	2	yes	—
139	C₂×C₂₃	#1332	1	4	yes	direct product factor
140	C₂×C₂₃	C₂×D₄⋊C₈	1	1	yes	characteristic
141	C₂×C₂₃	C₂×#358	1	1	yes	characteristic
142	C₂×C₂₃	C₂×#359	1	1	yes	characteristic
143	C₂²×Q₈	—	2	1	no	—
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	C₈×C₂₃	1	4	yes	complemented
147	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
148	C₂²×C₈	—	4	1	no	—
149	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
150	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
151	C₄×Q₈	—	2	2	no	—
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	4	1	no	—
154	C₄×Q₈	—	4	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	4	1	no	—
157	C₂×C₄²	—	4	1	no	—
158	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
159	C₂×C₁₆	—	2	4	no	—
160	C₂×C₁₆	—	2	2	no	—
161	#107	—	2	1	no	—
162	#107	C₂×C₄×C₂₃	1	2	yes	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	4	1	no	—
165	#103	—	2	2	no	—
166	#103	—	2	2	no	—
167	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
168	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
169	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
170	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₂₃	C₂×#1332	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
180	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
181	Q₁₆	—	2	2	no	—
182	Q₁₆	—	2	4	no	—
183	Q₁₆	C₂×C₈×C₂₃	1	4	yes	complemented
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	8	2	no	—
187	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
190	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
191	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
200	C₄²	D₈×C₂₃	1	1	yes	characteristic
201	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₁₆	—	2	4	no	—
204	C₂³	C₂₃×#102	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	4	4	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
214	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
215	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
216	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
217	C₂×C₄	C₂₃×#104	1	2	yes	—
218	C₂×C₄	C₂₃×SD₃₂	1	2	yes	—
219	C₂×C₄	C₂₃×Q₃₂	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
221	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	8	2	no	—
226	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
227	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
228	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
229	C₂²	C₂₃×D₄⋊C₈	1	2	yes	—
230	C₂²	C₂₃×#358	1	2	yes	—
231	C₂²	C₂₃×#359	1	2	yes	—
232	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	C₂₃×D₄⋊C₈	1	2	yes	—
238	C₄	C₂₃×#358	1	2	yes	—
239	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
240	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
241	C₄	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
242	C₄	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
243	C₂	C₂₃×#1332	1	4	yes	direct product factor
244	C₂	C₂×C₂₃×D₄⋊C₈	1	1	yes	characteristic
245	C₂	C₂×C₂₃×#358	1	1	yes	characteristic
246	C₂	C₂×C₂₃×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	D₈
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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
4. C₂₃×#1332	C₂	direct (N × H)
23. C₂×Q₁₆×C₂₃	C₈	semidirect
59. Q₁₆×C₂₃	C₂×C₈	semidirect
80. C₂×#1332	C₂₃	direct (N × H)
109. #1332	C₂×C₂₃	direct (N × H)
139. C₂×C₂₃	#1332	direct (N × H)
146. C₂×Q₁₆	C₈×C₂₃	semidirect
171. C₂₃	C₂×#1332	direct (N × H)
183. Q₁₆	C₂×C₈×C₂₃	semidirect
243. C₂	C₂₃×#1332	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×Q₁₆×C₂₃	2
C₂×C₄×C₁₆×C₂₃	2
C₂×C₂₃×#335	2
C₂₃×#1332	2
C₂×#1332	23

Subgroups — C2×C23×#1332 #19159216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1332 #19159216