Subgroups — C₂₃×#17814 #19165440

Order 5888.

730 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#423	C₂	1	2	yes	maximal
3	C₂₃×#1571	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1567	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	—	2	2	no	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#431	C₄	1	4	yes	—
8	C₂₃×#431	C₂²	1	4	yes	—
9	C₂₃×#423	—	2	2	no	—
10	C₂₃×#423	—	2	2	no	—
11	C₂₃×#423	C₂²	1	4	yes	—
12	C₂₃×#423	C₂²	1	4	yes	—
13	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
14	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
16	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#335	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
22	C₂×M₁₆×C₂₃	—	2	4	no	—
23	C₂×M₁₆×C₂₃	—	2	4	no	—
24	C₂×M₁₆×C₂₃	—	4	8	no	—
25	C₂²×C₈×C₂₃	—	2	4	no	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	D₄	1	2	yes	—
30	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
35	C₂₃×C₄⋊C₈	—	2	4	no	—
36	C₂₃×C₄⋊C₈	—	2	4	no	—
37	C₂₃×C₄⋊C₈	—	2	4	no	—
38	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
39	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
40	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₂₃	C₂³	1	4	yes	—
43	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	—	4	2	no	—
45	C₂²×C₄×C₂₃	—	4	2	no	—
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	D₈	1	4	yes	—
53	C₂²×C₄×C₂₃	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
60	M₁₆×C₂₃	—	4	8	no	—
61	M₁₆×C₂₃	—	4	8	no	—
62	C₂×C₈×C₂₃	—	2	8	no	—
63	C₂×C₈×C₂₃	—	4	4	no	—
64	C₂×C₈×C₂₃	—	4	8	no	—
65	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
66	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₂₃	C₂×D₄	1	2	yes	—
72	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
75	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
76	#17814	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	—	4	1	no	—
78	C₂³×C₂₃	—	4	1	no	—
79	C₂³×C₂₃	#107	1	4	yes	—
80	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	2	4	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	4	2	no	—
87	C₂×C₄×C₂₃	—	4	1	no	—
88	C₂×C₄×C₂₃	—	4	1	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	#107	1	4	yes	—
91	C₂×C₄×C₂₃	#107	1	8	yes	—
92	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
99	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
101	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
105	C₈×C₂₃	—	2	8	no	—
106	C₈×C₂₃	—	4	8	no	—
107	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
108	C₂×#423	C₂×C₂₃	1	2	yes	—
109	#1571	C₂×C₂₃	1	8	yes	complemented
110	#1567	C₂×C₂₃	1	4	yes	complemented
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	—	2	2	no	—
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	4	4	no	—
116	C₂²×C₂₃	C₂×#107	1	4	yes	—
117	C₂²×C₂₃	C₂×#107	1	2	yes	—
118	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₂₃	—	2	1	no	—
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	C₂×#107	1	4	yes	—
124	C₄×C₂₃	C₂×#107	1	2	yes	—
125	C₄×C₂₃	#430	1	2	yes	—
126	C₄×C₂₃	#568	1	2	yes	—
127	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
128	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
129	C₂²×M₁₆	—	2	2	no	—
130	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
131	#431	C₄×C₂₃	1	4	yes	—
132	#431	C₂²×C₂₃	1	4	yes	—
133	#423	—	2	2	no	—
134	#423	—	2	2	no	—
135	#423	C₂²×C₂₃	1	4	yes	—
136	#423	C₂²×C₂₃	1	4	yes	—
137	C₂×C₄⋊C₈	—	2	2	no	—
138	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
139	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
140	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
142	#335	C₂²×C₂₃	1	16	yes	complemented
143	C₂×C₂₃	—	2	2	no	—
144	C₂×C₂₃	—	2	4	no	—
145	C₂×C₂₃	#2905	1	1	yes	characteristic
146	C₂×C₂₃	C₂²×#107	1	1	yes	characteristic
147	C₂×C₂₃	C₂×#430	1	1	yes	characteristic
148	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
149	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
150	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	4	8	no	—
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
156	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
158	C₂×C₄²	D₄×C₂₃	1	2	yes	—
159	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
161	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
162	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
168	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
169	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
170	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
171	C₄×C₈	C₂³×C₂₃	1	4	yes	—
172	C₂₃	#17814	1	1	yes	characteristic, direct product factor, Sylow
173	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
177	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
179	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
182	C₂²×C₄	D₈×C₂₃	1	4	yes	—
183	C₂²×C₄	Q₁₆×C₂₃	1	4	yes	—
184	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
190	M₁₆	—	4	8	no	—
191	M₁₆	—	4	8	no	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	—	4	8	no	—
195	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
196	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
197	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
198	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
199	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
200	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
201	C₄²	C₂×D₄×C₂₃	1	2	yes	—
202	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
204	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
205	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
206	C₂³	—	4	1	no	—
207	C₂³	—	4	1	no	—
208	C₂³	C₂₃×#107	1	4	yes	—
209	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
210	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₂₃×#107	1	4	yes	—
220	C₂×C₄	C₂₃×#107	1	8	yes	—
221	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
223	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
227	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
228	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
229	C₂×C₄	C₂×Q₁₆×C₂₃	1	4	yes	—
230	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
231	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
233	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
234	C₈	—	2	8	no	—
235	C₈	—	4	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	4	no	—
241	C₂²	C₂×C₂₃×#107	1	4	yes	—
242	C₂²	C₂×C₂₃×#107	1	2	yes	—
243	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	C₂×C₂₃×#107	1	4	yes	—
249	C₄	C₂×C₂₃×#107	1	2	yes	—
250	C₄	C₂₃×#430	1	2	yes	—
251	C₄	C₂₃×#568	1	2	yes	—
252	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
253	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	—	2	2	no	—
256	C₂	C₂₃×#2905	1	1	yes	characteristic
257	C₂	C₂²×C₂₃×#107	1	1	yes	characteristic
258	C₂	C₂×C₂₃×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1571	C₂	semidirect
4. C₂₃×#1567	C₂	semidirect
18. C₂₃×#335	C₂²	semidirect
76. #17814	C₂₃	direct (N × H)
109. #1571	C₂×C₂₃	semidirect
110. #1567	C₂×C₂₃	semidirect
142. #335	C₂²×C₂₃	semidirect
172. C₂₃	#17814	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×#431	2
C₂×C₂₃×#423	2
C₂₃×#1571	2
C₂₃×#1567	2
#17814	23

Subgroups — C23×#17814 #19165440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17814 #19165440