Subgroups — C₂₃×#10766 #19158392

Order 5888.

1180 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#568	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#410	C₂	1	2	yes	maximal
3	C₂₃×#1733	C₂	1	4	yes	maximal, complemented
4	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×#568	—	2	8	no	—
6	C₂×C₂₃×#131	—	2	4	no	—
7	C₂²×M₁₆×C₂₃	C₄	1	4	yes	complemented
8	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×#117	C₂²	1	4	yes	—
10	C₂₃×#410	—	2	4	no	—
11	C₂₃×#410	C₄	1	8	yes	complemented
12	C₂×C₂₃×#99	—	2	4	no	—
13	C₂₃×#339	—	2	8	no	—
14	C₂×#41×C₂₃	—	2	4	no	—
15	C₂×#41×C₂₃	—	2	4	no	—
16	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
17	C₂×#41×C₂₃	D₄	1	4	yes	—
18	C₂²×Q₈×C₂₃	Q₈	1	1	yes	characteristic
19	C₂²×D₄×C₂₃	D₄	1	2	yes	—
20	C₂²×D₄×C₂₃	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃×#131	—	2	32	no	—
24	C₂×M₁₆×C₂₃	—	2	4	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	4	no	—
27	C₂×M₁₆×C₂₃	—	2	16	no	—
28	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₂₃	D₄	1	8	yes	—
30	C₂²×C₈×C₂₃	—	2	4	no	—
31	C₂²×C₈×C₂₃	—	2	2	no	—
32	C₂₃×#117	—	2	4	no	—
33	C₂₃×#117	—	2	4	no	—
34	C₂₃×#117	—	2	4	no	—
35	C₂₃×#117	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
37	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
38	C₂×C₄²×C₂₃	—	2	4	no	—
39	C₂₃×#99	—	2	16	no	—
40	C₂⁴×C₂₃	C₄⋊C₄	1	2	yes	—
41	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
42	#41×C₂₃	—	2	8	no	—
43	#41×C₂₃	—	2	8	no	—
44	#41×C₂₃	—	2	8	no	—
45	#41×C₂₃	—	2	8	no	—
46	C₂×Q₈×C₂₃	—	2	4	no	—
47	C₂×Q₈×C₂₃	C₄⋊C₄	1	4	yes	—
48	C₂×D₄×C₂₃	—	2	8	no	—
49	C₂×D₄×C₂₃	—	2	4	no	—
50	C₂×D₄×C₂₃	C₂²⋊C₄	1	8	yes	—
51	C₂×D₄×C₂₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	—	2	4	no	—
54	C₂²×C₄×C₂₃	—	4	4	no	—
55	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
56	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	#41	1	4	yes	—
64	M₁₆×C₂₃	—	2	16	no	—
65	M₁₆×C₂₃	—	2	4	no	—
66	M₁₆×C₂₃	—	2	8	no	—
67	M₁₆×C₂₃	C₂²⋊C₄	1	8	yes	—
68	C₂×C₈×C₂₃	—	2	16	no	—
69	C₂×C₈×C₂₃	—	2	4	no	—
70	C₂×C₈×C₂₃	—	2	8	no	—
71	C₄⋊C₄×C₂₃	—	2	8	no	—
72	C₄⋊C₄×C₂₃	—	4	4	no	—
73	C₂²⋊C₄×C₂₃	—	2	8	no	—
74	C₂²⋊C₄×C₂₃	—	4	4	no	—
75	C₄²×C₂₃	—	2	8	no	—
76	C₄²×C₂₃	—	4	4	no	—
77	#10766	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	—	2	4	no	—
79	C₂³×C₂₃	—	8	1	no	—
80	C₂³×C₂₃	—	8	2	no	—
81	C₂³×C₂₃	#95	1	4	yes	—
82	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₂₃	#121	1	4	yes	—
85	Q₈×C₂₃	—	2	8	no	—
86	D₄×C₂₃	—	2	16	no	—
87	D₄×C₂₃	—	2	8	no	—
88	C₂×C₄×C₂₃	—	2	4	no	—
89	C₂×C₄×C₂₃	—	2	8	no	—
90	C₂×C₄×C₂₃	—	2	4	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	—	4	8	no	—
93	C₂×C₄×C₂₃	—	8	8	no	—
94	C₂×C₄×C₂₃	#95	1	4	yes	—
95	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₂₃	#117	1	2	yes	—
100	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
101	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₂₃	#120	1	4	yes	—
103	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
104	C₂×C₄×C₂₃	#123	1	4	yes	—
105	C₈×C₂₃	—	2	8	no	—
106	C₂×#568	C₂×C₂₃	1	1	yes	characteristic
107	C₂×#410	C₂×C₂₃	1	2	yes	—
108	#1733	C₂×C₂₃	1	4	yes	complemented
109	C₂²×C₂₃	—	2	4	no	—
110	C₂²×C₂₃	—	4	4	no	—
111	C₂²×C₂₃	—	4	1	no	—
112	C₂²×C₂₃	—	4	2	no	—
113	C₂²×C₂₃	—	4	4	no	—
114	C₂²×C₂₃	—	8	4	no	—
115	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
116	C₂²×C₂₃	#380	1	2	yes	—
117	C₂²×C₂₃	#386	1	4	yes	—
118	C₄×C₂₃	—	2	4	no	—
119	C₄×C₂₃	—	8	8	no	—
120	C₄×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
121	C₄×C₂₃	#379	1	1	yes	characteristic
122	C₄×C₂₃	#381	1	1	yes	characteristic
123	C₄×C₂₃	#385	1	4	yes	—
124	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
125	#568	—	2	8	no	—
126	C₂×#131	—	2	4	no	—
127	C₂²×M₁₆	C₄×C₂₃	1	4	yes	complemented
128	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
129	C₂×#117	C₂²×C₂₃	1	4	yes	—
130	#410	—	2	4	no	—
131	#410	C₄×C₂₃	1	8	yes	complemented
132	C₂×#99	—	2	4	no	—
133	#339	—	2	8	no	—
134	C₂×C₂₃	—	2	4	no	—
135	C₂×C₂₃	—	2	1	no	—
136	C₂×C₂₃	—	2	2	no	—
137	C₂×C₂₃	—	4	4	no	—
138	C₂×C₂₃	C₂²⋊#95	1	1	yes	characteristic
139	C₂×#41	—	2	4	no	—
140	C₂×#41	—	2	4	no	—
141	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
142	C₂×#41	D₄×C₂₃	1	4	yes	—
143	C₂²×Q₈	Q₈×C₂₃	1	1	yes	characteristic
144	C₂²×D₄	D₄×C₂₃	1	2	yes	—
145	C₂²×D₄	Q₈×C₂₃	1	1	yes	characteristic
146	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
147	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
148	#131	—	2	32	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	16	no	—
153	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
154	C₂×M₁₆	D₄×C₂₃	1	8	yes	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	—	2	2	no	—
157	#117	—	2	4	no	—
158	#117	—	2	4	no	—
159	#117	—	2	4	no	—
160	#117	C₂×C₄×C₂₃	1	8	yes	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₂²⋊C₄	—	2	4	no	—
163	C₂×C₄²	—	2	4	no	—
164	#99	—	2	16	no	—
165	C₂₃	#10766	1	1	yes	characteristic, direct product factor, Sylow
166	C₂⁴	C₄⋊C₄×C₂₃	1	2	yes	—
167	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
168	#41	—	2	8	no	—
169	#41	—	2	8	no	—
170	#41	—	2	8	no	—
171	#41	—	2	8	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	C₄⋊C₄×C₂₃	1	4	yes	—
174	C₂×D₄	—	2	8	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	C₂²⋊C₄×C₂₃	1	8	yes	—
177	C₂×D₄	C₄⋊C₄×C₂₃	1	4	yes	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
182	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
183	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
188	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	#41×C₂₃	1	4	yes	—
190	M₁₆	—	2	16	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	2	8	no	—
193	M₁₆	C₂²⋊C₄×C₂₃	1	8	yes	—
194	C₂×C₈	—	2	16	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	8	no	—
197	C₄⋊C₄	—	2	8	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	—	2	8	no	—
200	C₂²⋊C₄	—	4	4	no	—
201	C₄²	—	2	8	no	—
202	C₄²	—	4	4	no	—
203	C₂³	—	2	4	no	—
204	C₂³	—	8	1	no	—
205	C₂³	—	8	2	no	—
206	C₂³	C₂₃×#95	1	4	yes	—
207	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
208	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
209	C₂³	C₂₃×#121	1	4	yes	—
210	Q₈	—	2	8	no	—
211	D₄	—	2	16	no	—
212	D₄	—	2	8	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	—	8	8	no	—
219	C₂×C₄	C₂₃×#95	1	4	yes	—
220	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
224	C₂×C₄	C₂₃×#117	1	2	yes	—
225	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
226	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
227	C₂×C₄	C₂₃×#120	1	4	yes	—
228	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
229	C₂×C₄	C₂₃×#123	1	4	yes	—
230	C₈	—	2	8	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	4	4	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	4	no	—
236	C₂²	—	8	4	no	—
237	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
238	C₂²	C₂₃×#380	1	2	yes	—
239	C₂²	C₂₃×#386	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	8	8	no	—
242	C₄	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
243	C₄	C₂₃×#379	1	1	yes	characteristic
244	C₄	C₂₃×#381	1	1	yes	characteristic
245	C₄	C₂₃×#385	1	4	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	4	no	—
250	C₂	C₂₃×C₂²⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#379
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂×C₂₃	C₂²⋊#95
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1733	C₂	semidirect
7. C₂²×M₁₆×C₂₃	C₄	semidirect
11. C₂₃×#410	C₄	semidirect
77. #10766	C₂₃	direct (N × H)
108. #1733	C₂×C₂₃	semidirect
127. C₂²×M₁₆	C₄×C₂₃	semidirect
131. #410	C₄×C₂₃	semidirect
165. C₂₃	#10766	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×#568	2
C₂×C₂₃×#410	2
C₂₃×#1733	2
#10766	23

Subgroups — C23×#10766 #19158392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10766 #19158392