Subgroups — C₂₃×#3231 #15670850

Order 2944.

326 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#485	C₂	1	4	yes	maximal
8	C₂₃×#479	C₂	1	4	yes	maximal, complemented
9	C₂₃×#431	C₂	1	1	yes	maximal, characteristic
10	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂₃×#125	—	2	1	no	—
15	C₂₃×#125	C₂²	1	2	yes	—
16	C₂₃×C₂²⋊Q₈	—	2	1	no	—
17	C₂₃×C₂²⋊Q₈	—	2	1	no	—
18	C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₄×Q₈×C₂₃	—	2	1	no	—
21	C₄×Q₈×C₂₃	—	2	1	no	—
22	C₄×Q₈×C₂₃	—	2	1	no	—
23	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₂₃×#117	—	2	1	no	—
25	C₂₃×#117	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#107	C₂²	1	4	yes	—
28	C₂₃×#106	C₂²	1	4	yes	—
29	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂₃×#103	C₂²	1	4	yes	—
31	C₂₃×#103	C₂²	1	4	yes	—
32	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₂₃	—	2	1	no	—
36	C₂×Q₈×C₂₃	—	2	1	no	—
37	C₂×Q₈×C₂₃	—	2	1	no	—
38	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₂³	1	2	yes	—
43	C₂×C₈×C₂₃	C₂³	1	2	yes	—
44	C₄⋊C₄×C₂₃	—	2	1	no	—
45	C₄⋊C₄×C₂₃	—	2	1	no	—
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₃	—	2	1	no	—
49	C₄⋊C₄×C₂₃	—	2	1	no	—
50	C₄⋊C₄×C₂₃	—	2	1	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₄²×C₂₃	—	2	2	no	—
59	C₄²×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	D₄	1	1	yes	characteristic
61	C₄²×C₂₃	D₄	1	1	yes	characteristic
62	C₄²×C₂₃	C₂³	1	1	yes	characteristic
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₂₃	—	2	4	no	—
66	Q₈×C₂₃	—	4	2	no	—
67	Q₈×C₂₃	—	4	2	no	—
68	Q₈×C₂₃	—	4	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₈×C₂₃	—	2	2	no	—
83	C₈×C₂₃	—	2	2	no	—
84	#3231	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₂₃	—	2	1	no	—
86	C₂²×C₂₃	—	2	1	no	—
87	C₂²×C₂₃	—	2	1	no	—
88	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₂₃	—	2	1	no	—
90	C₄×C₂₃	—	2	1	no	—
91	C₄×C₂₃	—	2	1	no	—
92	C₄×C₂₃	—	4	2	no	—
93	C₄×C₂₃	—	4	2	no	—
94	C₄×C₂₃	—	4	2	no	—
95	C₄×C₂₃	—	4	2	no	—
96	C₄×C₂₃	#135	1	2	yes	—
97	C₄×C₂₃	#137	1	2	yes	—
98	C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
99	#533	C₂×C₂₃	1	1	yes	characteristic
100	#533	C₂×C₂₃	1	1	yes	characteristic
101	#500	C₂×C₂₃	1	1	yes	characteristic, complemented
102	#498	C₂×C₂₃	1	1	yes	characteristic, complemented
103	#487	C₂×C₂₃	1	1	yes	characteristic, complemented
104	#487	C₂×C₂₃	1	1	yes	characteristic, complemented
105	#485	C₂×C₂₃	1	4	yes	—
106	#479	C₂×C₂₃	1	4	yes	complemented
107	#431	C₂×C₂₃	1	1	yes	characteristic
108	C₂×C₂₃	—	4	1	no	—
109	C₂×C₂₃	#536	1	1	yes	characteristic
110	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
111	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
112	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
113	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
114	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
115	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
116	#125	—	2	1	no	—
117	#125	C₂²×C₂₃	1	2	yes	—
118	C₂²⋊Q₈	—	2	1	no	—
119	C₂²⋊Q₈	—	2	1	no	—
120	C₂²⋊Q₈	—	2	1	no	—
121	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	—	2	1	no	—
124	C₄×Q₈	—	2	1	no	—
125	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
126	#117	—	2	1	no	—
127	#117	C₂²×C₂₃	1	2	yes	—
128	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
129	#107	C₂²×C₂₃	1	4	yes	—
130	#106	C₂²×C₂₃	1	4	yes	—
131	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
132	#103	C₂²×C₂₃	1	4	yes	—
133	#103	C₂²×C₂₃	1	4	yes	—
134	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
135	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
136	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
137	C₂₃	#3231	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
142	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
143	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
145	C₂×C₈	C₂³×C₂₃	1	2	yes	—
146	C₂×C₈	C₂³×C₂₃	1	2	yes	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
157	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
158	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	D₄×C₂₃	1	1	yes	characteristic
164	C₄²	D₄×C₂₃	1	1	yes	characteristic
165	C₄²	C₂³×C₂₃	1	1	yes	characteristic
166	C₄²	C₂³×C₂₃	1	1	yes	characteristic
167	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
168	Q₈	—	2	4	no	—
169	Q₈	—	4	2	no	—
170	Q₈	—	4	2	no	—
171	Q₈	—	4	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	C₂₃×#135	1	2	yes	—
199	C₄	C₂₃×#137	1	2	yes	—
200	C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
201	C₂	—	4	1	no	—
202	C₂	C₂₃×#536	1	1	yes	characteristic
203	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
204	C₂	C₂×C₂₃×#137	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#500	C₂	semidirect
4. C₂₃×#498	C₂	semidirect
5. C₂₃×#487	C₂	semidirect
6. C₂₃×#487	C₂	semidirect
8. C₂₃×#479	C₂	semidirect
84. #3231	C₂₃	direct (N × H)
101. #500	C₂×C₂₃	semidirect
102. #498	C₂×C₂₃	semidirect
103. #487	C₂×C₂₃	semidirect
104. #487	C₂×C₂₃	semidirect
106. #479	C₂×C₂₃	semidirect
137. C₂₃	#3231	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×#533	2
C₂₃×#500	2
C₂₃×#498	2
C₂₃×#487	2
C₂₃×#485	2
C₂₃×#479	2
C₂₃×#431	2
#3231	23

Subgroups — C23×#3231 #15670850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3231 #15670850