Subgroups — C₂₃×#22283 #19169909

Order 5888.

826 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3150	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2932	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1962	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1876	C₂	1	8	yes	maximal, complemented
5	C₂₃×#1838	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#135	C₂²	1	4	yes	—
8	C₂×C₂₃×#131	C₂²	1	4	yes	—
9	C₂₃×#532	C₂²	1	2	yes	—
10	C₂₃×#497	C₄	1	4	yes	complemented
11	C₂₃×#496	C₄	1	4	yes	complemented
12	C₂₃×#435	—	2	8	no	—
13	C₂₃×#423	C₂²	1	2	yes	—
14	C₂×C₂₃×#104	C₂²	1	8	yes	—
15	C₂₃×#408	C₂²	1	4	yes	—
16	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
18	C₄³×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#341	C₂²	1	4	yes	—
20	C₂₃×#336	C₂²	1	4	yes	—
21	C₂×#41×C₂₃	C₂³	1	4	yes	—
22	C₂₃×#135	—	4	8	no	—
23	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×SD₁₆×C₂₃	C₂×C₄	1	8	yes	—
25	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
26	C₂₃×#131	—	4	8	no	—
27	C₂×M₁₆×C₂₃	—	2	4	no	—
28	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
29	C₂²×C₈×C₂₃	—	2	4	no	—
30	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
32	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₂₃×#127	C₂×C₄	1	2	yes	—
34	C₂₃×#124	C₂×C₄	1	4	yes	—
35	C₂₃×#121	—	2	4	no	—
36	C₄×D₄×C₂₃	—	2	4	no	—
37	C₂×C₄²×C₂₃	—	2	2	no	—
38	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	—	2	4	no	—
41	C₂₃×#104	—	4	8	no	—
42	C₂₃×C₂²⋊C₈	—	2	8	no	—
43	C₂₃×#97	C₂×C₄	1	4	yes	—
44	C₂₃×#97	D₄	1	4	yes	—
45	#41×C₂₃	—	2	8	no	—
46	#41×C₂₃	—	2	8	no	—
47	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
48	C₂×D₄×C₂₃	—	2	4	no	—
49	C₂×D₄×C₂₃	—	2	4	no	—
50	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	#41	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
55	Q₁₆×C₂₃	—	4	4	no	—
56	SD₁₆×C₂₃	—	4	8	no	—
57	D₈×C₂₃	—	4	4	no	—
58	M₁₆×C₂₃	—	2	8	no	—
59	M₁₆×C₂₃	—	4	8	no	—
60	C₂×C₈×C₂₃	—	2	4	no	—
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	—	2	4	no	—
63	C₂×C₈×C₂₃	—	4	8	no	—
64	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
66	C₄⋊C₄×C₂₃	—	2	4	no	—
67	C₂²⋊C₄×C₂₃	—	2	8	no	—
68	C₄²×C₂₃	—	2	4	no	—
69	C₄²×C₂₃	—	4	4	no	—
70	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₂₃	C₂×D₄	1	2	yes	—
72	#22283	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	—	2	4	no	—
74	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
75	Q₈×C₂₃	—	2	8	no	—
76	D₄×C₂₃	—	2	8	no	—
77	D₄×C₂₃	—	4	8	no	—
78	D₄×C₂₃	—	8	4	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	—	4	4	no	—
85	C₂×C₄×C₂₃	—	4	4	no	—
86	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
87	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
90	C₈×C₂₃	—	2	4	no	—
91	C₈×C₂₃	—	2	4	no	—
92	C₈×C₂₃	—	4	4	no	—
93	#3150	C₂×C₂₃	1	1	yes	characteristic
94	#2932	C₂×C₂₃	1	2	yes	complemented
95	#1962	C₂×C₂₃	1	2	yes	complemented
96	#1876	C₂×C₂₃	1	8	yes	complemented
97	#1838	C₂×C₂₃	1	1	yes	characteristic, complemented
98	#1745	C₂×C₂₃	1	1	yes	characteristic, complemented
99	C₂²×C₂₃	—	2	2	no	—
100	C₂²×C₂₃	—	4	4	no	—
101	C₂²×C₂₃	—	4	8	no	—
102	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
103	C₂²×C₂₃	#540	1	2	yes	—
104	C₄×C₂₃	—	2	4	no	—
105	C₄×C₂₃	—	4	4	no	—
106	C₄×C₂₃	—	4	4	no	—
107	C₄×C₂₃	#518	1	2	yes	—
108	C₄×C₂₃	#535	1	2	yes	—
109	C₂×#135	C₂²×C₂₃	1	4	yes	—
110	C₂×#131	C₂²×C₂₃	1	4	yes	—
111	#532	C₂²×C₂₃	1	2	yes	—
112	#497	C₄×C₂₃	1	4	yes	complemented
113	#496	C₄×C₂₃	1	4	yes	complemented
114	#435	—	2	8	no	—
115	#423	C₂²×C₂₃	1	2	yes	—
116	C₂×#104	C₂²×C₂₃	1	8	yes	—
117	#408	C₂²×C₂₃	1	4	yes	—
118	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
119	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
120	C₄³	C₂²×C₂₃	1	1	yes	characteristic
121	#341	C₂²×C₂₃	1	4	yes	—
122	#336	C₂²×C₂₃	1	4	yes	—
123	C₂×C₂₃	—	2	2	no	—
124	C₂×C₂₃	—	8	4	no	—
125	C₂×C₂₃	#2328	1	1	yes	characteristic
126	C₂×C₂₃	#2956	1	2	yes	—
127	C₂×#41	C₂³×C₂₃	1	4	yes	—
128	#135	—	4	8	no	—
129	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
130	C₂×SD₁₆	C₂×C₄×C₂₃	1	8	yes	—
131	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
132	#131	—	4	8	no	—
133	C₂×M₁₆	—	2	4	no	—
134	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
135	C₂²×C₈	—	2	4	no	—
136	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
137	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
138	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
139	#127	C₂×C₄×C₂₃	1	2	yes	—
140	#124	C₂×C₄×C₂₃	1	4	yes	—
141	#121	—	2	4	no	—
142	C₄×D₄	—	2	4	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
145	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
146	C₄⋊C₈	—	2	4	no	—
147	#104	—	4	8	no	—
148	C₂²⋊C₈	—	2	8	no	—
149	#97	C₂×C₄×C₂₃	1	4	yes	—
150	#97	D₄×C₂₃	1	4	yes	—
151	C₂₃	#22283	1	1	yes	characteristic, direct product factor, Sylow
152	#41	—	2	8	no	—
153	#41	—	2	8	no	—
154	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	#41×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
162	Q₁₆	—	4	4	no	—
163	SD₁₆	—	4	8	no	—
164	D₈	—	4	4	no	—
165	M₁₆	—	2	8	no	—
166	M₁₆	—	4	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	4	8	no	—
171	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
172	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
173	C₄⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	8	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	4	4	no	—
177	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
178	C₄²	C₂×D₄×C₂₃	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
181	Q₈	—	2	8	no	—
182	D₄	—	2	8	no	—
183	D₄	—	4	8	no	—
184	D₄	—	8	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
193	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
194	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
196	C₈	—	2	4	no	—
197	C₈	—	2	4	no	—
198	C₈	—	4	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	8	no	—
201	C₂²	—	4	4	no	—
202	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
203	C₂²	C₂₃×#540	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	4	4	no	—
206	C₄	—	4	4	no	—
207	C₄	C₂₃×#518	1	2	yes	—
208	C₄	C₂₃×#535	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	8	4	no	—
211	C₂	C₂₃×#2328	1	1	yes	characteristic
212	C₂	C₂₃×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2932	C₂	semidirect
3. C₂₃×#1962	C₂	semidirect
4. C₂₃×#1876	C₂	semidirect
5. C₂₃×#1838	C₂	semidirect
6. C₂₃×#1745	C₂	semidirect
10. C₂₃×#497	C₄	semidirect
11. C₂₃×#496	C₄	semidirect
72. #22283	C₂₃	direct (N × H)
94. #2932	C₂×C₂₃	semidirect
95. #1962	C₂×C₂₃	semidirect
96. #1876	C₂×C₂₃	semidirect
97. #1838	C₂×C₂₃	semidirect
98. #1745	C₂×C₂₃	semidirect
112. #497	C₄×C₂₃	semidirect
113. #496	C₄×C₂₃	semidirect
151. C₂₃	#22283	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3150	2
C₂₃×#2932	2
C₂₃×#1962	2
C₂₃×#1876	2
C₂₃×#1838	2
C₂₃×#1745	2
#22283	23

Subgroups — C23×#22283 #19169909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22283 #19169909