Subgroups — C₂₃×#3223 #15670842

Order 2944.

364 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#536	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#464	C₂	1	2	yes	maximal, complemented
8	C₂₃×#462	C₂	1	2	yes	maximal, complemented
9	C₂₃×#452	C₂	1	2	yes	maximal
10	C₂₃×#451	C₂	1	2	yes	maximal, complemented
11	C₂₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂₃×#126	—	2	2	no	—
22	C₂₃×#125	—	2	2	no	—
23	C₂₃×#125	C₂²	1	1	yes	characteristic
24	C₂₃×#124	C₂²	1	1	yes	characteristic
25	C₂₃×#124	C₂²	1	1	yes	characteristic
26	C₂₃×#123	—	2	2	no	—
27	C₂₃×C₂²⋊Q₈	—	2	2	no	—
28	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
30	C₂₃×#117	C₂²	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
32	C₂₃×#103	C₂²	1	2	yes	—
33	C₂₃×#103	C₂²	1	2	yes	—
34	C₂₃×#103	C₂²	1	2	yes	—
35	C₂₃×#102	C₂²	1	2	yes	—
36	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₂₃×#97	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	#41×C₂₃	—	4	2	no	—
40	C₂×Q₈×C₂₃	—	2	1	no	—
41	C₂×Q₈×C₂₃	—	4	2	no	—
42	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	Q₁₆×C₂₃	—	4	2	no	—
52	Q₁₆×C₂₃	—	4	2	no	—
53	Q₁₆×C₂₃	—	4	2	no	—
54	SD₁₆×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂×C₈×C₂₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
64	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₂₃	—	2	2	no	—
66	C₂²⋊C₄×C₂₃	—	2	2	no	—
67	C₂²⋊C₄×C₂₃	—	2	2	no	—
68	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
69	C₄²×C₂₃	—	2	2	no	—
70	C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₄²×C₂₃	C₂³	1	1	yes	characteristic
72	C₂³×C₂₃	—	2	1	no	—
73	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₂₃	—	2	2	no	—
75	Q₈×C₂₃	—	2	2	no	—
76	Q₈×C₂₃	—	2	2	no	—
77	Q₈×C₂₃	—	2	2	no	—
78	Q₈×C₂₃	—	2	2	no	—
79	Q₈×C₂₃	—	4	2	no	—
80	D₄×C₂₃	—	2	2	no	—
81	D₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃	—	2	2	no	—
95	C₈×C₂₃	—	2	2	no	—
96	#3223	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₂₃	—	2	1	no	—
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	4	1	no	—
100	C₂²×C₂₃	—	4	2	no	—
101	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₂₃	—	2	1	no	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	4	1	no	—
106	C₄×C₂₃	—	4	1	no	—
107	C₄×C₂₃	—	4	1	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
111	#541	C₂×C₂₃	1	1	yes	characteristic, complemented
112	#536	C₂×C₂₃	1	1	yes	characteristic
113	#497	C₂×C₂₃	1	1	yes	characteristic, complemented
114	#494	C₂×C₂₃	1	1	yes	characteristic, complemented
115	#491	C₂×C₂₃	1	1	yes	characteristic, complemented
116	#488	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#464	C₂×C₂₃	1	2	yes	complemented
118	#462	C₂×C₂₃	1	2	yes	complemented
119	#452	C₂×C₂₃	1	2	yes	—
120	#451	C₂×C₂₃	1	2	yes	complemented
121	#433	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂×C₂₃	—	4	1	no	—
123	C₂×C₂₃	—	8	1	no	—
124	C₂×C₂₃	#535	1	1	yes	characteristic
125	C₂×C₂₃	#578	1	2	yes	—
126	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
127	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
129	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
130	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
131	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
132	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
133	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
135	#126	—	2	2	no	—
136	#125	—	2	2	no	—
137	#125	C₂²×C₂₃	1	1	yes	characteristic
138	#124	C₂²×C₂₃	1	1	yes	characteristic
139	#124	C₂²×C₂₃	1	1	yes	characteristic
140	#123	—	2	2	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
143	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
144	#117	C₂²×C₂₃	1	1	yes	characteristic
145	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
146	#103	C₂²×C₂₃	1	2	yes	—
147	#103	C₂²×C₂₃	1	2	yes	—
148	#103	C₂²×C₂₃	1	2	yes	—
149	#102	C₂²×C₂₃	1	2	yes	—
150	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
151	#97	C₂²×C₂₃	1	1	yes	characteristic
152	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂₃	#3223	1	1	yes	characteristic, direct product factor, Sylow
154	#41	—	4	2	no	—
155	C₂×Q₈	—	2	1	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
158	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
159	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
160	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	2	no	—
168	Q₁₆	—	4	2	no	—
169	SD₁₆	—	4	2	no	—
170	C₂×C₈	C₂³×C₂₃	1	2	yes	—
171	C₂×C₈	C₂³×C₂₃	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
179	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	C₄²	C₂³×C₂₃	1	1	yes	characteristic
187	C₂³	—	2	1	no	—
188	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	4	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	4	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
206	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	1	no	—
215	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	2	no	—
224	C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
225	C₂	—	4	1	no	—
226	C₂	—	8	1	no	—
227	C₂	C₂₃×#535	1	1	yes	characteristic
228	C₂	C₂₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#541	C₂	semidirect
3. C₂₃×#497	C₂	semidirect
4. C₂₃×#494	C₂	semidirect
5. C₂₃×#491	C₂	semidirect
6. C₂₃×#488	C₂	semidirect
7. C₂₃×#464	C₂	semidirect
8. C₂₃×#462	C₂	semidirect
10. C₂₃×#451	C₂	semidirect
11. C₂₃×#433	C₂	semidirect
96. #3223	C₂₃	direct (N × H)
111. #541	C₂×C₂₃	semidirect
113. #497	C₂×C₂₃	semidirect
114. #494	C₂×C₂₃	semidirect
115. #491	C₂×C₂₃	semidirect
116. #488	C₂×C₂₃	semidirect
117. #464	C₂×C₂₃	semidirect
118. #462	C₂×C₂₃	semidirect
120. #451	C₂×C₂₃	semidirect
121. #433	C₂×C₂₃	semidirect
153. C₂₃	#3223	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×#541	2
C₂₃×#536	2
C₂₃×#497	2
C₂₃×#494	2
C₂₃×#491	2
C₂₃×#488	2
C₂₃×#464	2
C₂₃×#462	2
C₂₃×#452	2
C₂₃×#451	2
C₂₃×#433	2
#3223	23

Subgroups — C23×#3223 #15670842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3223 #15670842