Subgroups — C₂₃×#10822 #19158448

Order 5888.

718 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#341	C₂	1	2	yes	maximal
3	C₂×C₂₃×#341	C₂	1	2	yes	maximal
4	C₂₃×#1728	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	—	2	4	no	—
10	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	—
11	C₂₃×#341	—	2	4	no	—
12	C₂₃×#341	C₄	1	8	yes	complemented
13	C₂₃×#336	—	2	4	no	—
14	C₂³×C₄×C₂₃	Q₈	1	2	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	—	2	4	no	—
17	C₂×M₁₆×C₂₃	—	2	4	no	—
18	C₂×M₁₆×C₂₃	—	2	4	no	—
19	C₂×M₁₆×C₂₃	—	2	4	no	—
20	C₂×M₁₆×C₂₃	—	2	4	no	—
21	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	4	no	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₂₃	D₄	1	4	yes	—
31	C₂×C₄²×C₂₃	—	2	2	no	—
32	C₂×C₄²×C₂₃	—	2	2	no	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₄×C₈×C₂₃	—	2	8	no	—
36	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	2	8	no	—
39	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	#41	1	2	yes	—
45	C₂²×C₄×C₂₃	#41	1	2	yes	—
46	C₂²×C₄×C₂₃	#41	1	2	yes	—
47	C₂²×C₄×C₂₃	#41	1	2	yes	—
48	M₁₆×C₂₃	—	2	8	no	—
49	M₁₆×C₂₃	—	2	8	no	—
50	M₁₆×C₂₃	—	4	4	no	—
51	M₁₆×C₂₃	—	4	4	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	2	8	no	—
57	C₂×C₈×C₂₃	—	2	8	no	—
58	C₂×C₈×C₂₃	—	2	8	no	—
59	C₂×C₈×C₂₃	—	4	4	no	—
60	C₂×C₈×C₂₃	—	4	4	no	—
61	C₂×C₈×C₂₃	C₄²	1	4	yes	—
62	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	2	2	no	—
65	C₄²×C₂₃	—	2	4	no	—
66	#10822	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₃	—	2	4	no	—
68	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
70	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
71	C₂³×C₂₃	#125	1	2	yes	—
72	C₂×C₄×C₂₃	—	2	16	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	#108	1	8	yes	—
76	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
79	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#117	1	4	yes	—
81	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
84	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₂₃	#121	1	2	yes	—
86	C₂×C₄×C₂₃	#123	1	4	yes	—
87	C₂×C₄×C₂₃	#124	1	2	yes	—
88	C₂×C₄×C₂₃	#126	1	4	yes	—
89	C₈×C₂₃	—	2	4	no	—
90	C₈×C₂₃	—	2	4	no	—
91	C₈×C₂₃	—	4	8	no	—
92	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
93	C₂×#341	C₂×C₂₃	1	2	yes	—
94	C₂×#341	C₂×C₂₃	1	2	yes	—
95	#1728	C₂×C₂₃	1	2	yes	—
96	C₂²×C₂₃	—	2	4	no	—
97	C₂²×C₂₃	—	2	4	no	—
98	C₂²×C₂₃	—	2	4	no	—
99	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₂₃	#380	1	1	yes	characteristic
101	C₂²×C₂₃	#382	1	1	yes	characteristic
102	C₂²×C₂₃	#387	1	4	yes	—
103	C₂²×C₂₃	C₂×#108	1	4	yes	—
104	C₄×C₂₃	—	2	8	no	—
105	C₄×C₂₃	#376	1	1	yes	characteristic
106	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₂₃	#381	1	1	yes	characteristic
108	C₄×C₂₃	#383	1	1	yes	characteristic
109	C₄×C₂₃	#388	1	4	yes	—
110	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
111	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
112	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
113	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
114	C₂×C₄×C₈	—	2	4	no	—
115	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	—
116	#341	—	2	4	no	—
117	#341	C₄×C₂₃	1	8	yes	complemented
118	#336	—	2	4	no	—
119	C₂×C₂₃	—	2	4	no	—
120	C₂×C₂₃	#1434	1	1	yes	characteristic
121	C₂×C₂₃	C₄×#108	1	2	yes	—
122	C₂×C₂₃	#1810	1	2	yes	—
123	C₂×C₂₃	#1837	1	2	yes	—
124	C₂³×C₄	Q₈×C₂₃	1	2	yes	—
125	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
126	C₂×M₁₆	—	2	4	no	—
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	—	2	4	no	—
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	—	2	4	no	—
131	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
138	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
139	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
140	C₂²×C₈	D₄×C₂₃	1	4	yes	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
144	C₂×C₄²	D₄×C₂₃	1	2	yes	—
145	C₄×C₈	—	2	8	no	—
146	C₂₃	#10822	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
151	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
153	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
154	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
155	C₂²×C₄	#41×C₂₃	1	2	yes	—
156	C₂²×C₄	#41×C₂₃	1	2	yes	—
157	C₂²×C₄	#41×C₂₃	1	2	yes	—
158	C₂²×C₄	#41×C₂₃	1	2	yes	—
159	M₁₆	—	2	8	no	—
160	M₁₆	—	2	8	no	—
161	M₁₆	—	4	4	no	—
162	M₁₆	—	4	4	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	4	4	no	—
171	C₂×C₈	—	4	4	no	—
172	C₂×C₈	C₄²×C₂₃	1	4	yes	—
173	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	4	no	—
177	C₂³	—	2	4	no	—
178	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
180	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
181	C₂³	C₂₃×#125	1	2	yes	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	C₂₃×#108	1	8	yes	—
186	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
189	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
190	C₂×C₄	C₂₃×#117	1	4	yes	—
191	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
192	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
193	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
194	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
195	C₂×C₄	C₂₃×#121	1	2	yes	—
196	C₂×C₄	C₂₃×#123	1	4	yes	—
197	C₂×C₄	C₂₃×#124	1	2	yes	—
198	C₂×C₄	C₂₃×#126	1	4	yes	—
199	C₈	—	2	4	no	—
200	C₈	—	2	4	no	—
201	C₈	—	4	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂²	C₂₃×#380	1	1	yes	characteristic
207	C₂²	C₂₃×#382	1	1	yes	characteristic
208	C₂²	C₂₃×#387	1	4	yes	—
209	C₂²	C₂×C₂₃×#108	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	C₂₃×#376	1	1	yes	characteristic
212	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
213	C₄	C₂₃×#381	1	1	yes	characteristic
214	C₄	C₂₃×#383	1	1	yes	characteristic
215	C₄	C₂₃×#388	1	4	yes	—
216	C₂	—	2	4	no	—
217	C₂	C₂₃×#1434	1	1	yes	characteristic
218	C₂	C₄×C₂₃×#108	1	2	yes	—
219	C₂	C₂₃×#1810	1	2	yes	—
220	C₂	C₂₃×#1837	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂₃×#341	C₄	semidirect
66. #10822	C₂₃	direct (N × H)
117. #341	C₄×C₂₃	semidirect
146. C₂₃	#10822	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₈×C₂₃	2
C₂×C₂₃×#341	2
C₂₃×#1728	2
#10822	23

Subgroups — C23×#10822 #19158448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10822 #19158448