Subgroups — C₂₃×#3287 #15670906

Order 2944.

504 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#135	C₂	1	2	yes	maximal, complemented
2	C₂₃×#550	C₂	1	2	yes	maximal, complemented
3	C₂₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#466	C₂	1	2	yes	maximal, complemented
5	C₂₃×#460	C₂	1	2	yes	maximal, complemented
6	C₂₃×#449	C₂	1	4	yes	maximal, complemented
7	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₃	C₂²	1	4	yes	—
10	C₂₃×#135	—	2	8	no	—
11	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	—
13	C₂×D₈×C₂₃	C₂²	1	2	yes	—
14	C₂×D₈×C₂₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#127	—	2	2	no	—
17	C₂₃×#127	C₂²	1	2	yes	—
18	C₂₃×#121	—	2	4	no	—
19	C₂₃×#121	C₂²	1	4	yes	—
20	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
21	C₄×D₄×C₂₃	—	2	2	no	—
22	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#107	C₂²	1	1	yes	characteristic
24	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂₃×#103	C₂²	1	2	yes	—
26	C₂₃×#102	C₂²	1	4	yes	—
27	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
29	#41×C₂₃	—	2	8	no	—
30	#41×C₂₃	—	4	4	no	—
31	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
32	C₂×D₄×C₂₃	—	2	4	no	—
33	C₂×D₄×C₂₃	—	2	4	no	—
34	C₂×D₄×C₂₃	—	2	2	no	—
35	C₂×D₄×C₂₃	—	2	2	no	—
36	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
37	C₂×D₄×C₂₃	C₂³	1	4	yes	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	Q₁₆×C₂₃	D₄	1	4	yes	complemented
41	SD₁₆×C₂₃	—	2	8	no	—
42	D₈×C₂₃	—	2	4	no	—
43	D₈×C₂₃	—	4	2	no	—
44	C₂×C₈×C₂₃	—	2	4	no	—
45	C₂×C₈×C₂₃	C₂³	1	2	yes	—
46	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
53	C₄²×C₂₃	—	2	2	no	—
54	C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂³×C₂₃	—	2	4	no	—
56	C₂³×C₂₃	C₂×D₄	1	2	yes	—
57	Q₈×C₂₃	—	2	2	no	—
58	Q₈×C₂₃	C₂×D₄	1	4	yes	—
59	D₄×C₂₃	—	2	2	no	—
60	D₄×C₂₃	—	2	4	no	—
61	D₄×C₂₃	—	2	4	no	—
62	D₄×C₂₃	—	4	8	no	—
63	D₄×C₂₃	—	4	4	no	—
64	D₄×C₂₃	—	8	2	no	—
65	D₄×C₂₃	—	8	4	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	4	8	no	—
71	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
75	C₈×C₂₃	—	2	2	no	—
76	C₈×C₂₃	—	2	1	no	—
77	C₈×C₂₃	C₂×D₄	1	2	yes	—
78	#3287	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₂₃	—	2	2	no	—
80	C₂²×C₂₃	—	2	2	no	—
81	C₂²×C₂₃	—	2	2	no	—
82	C₂²×C₂₃	—	4	4	no	—
83	C₂²×C₂₃	—	4	4	no	—
84	C₂²×C₂₃	—	4	4	no	—
85	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₂₃	—	2	2	no	—
87	C₄×C₂₃	—	2	4	no	—
88	C₄×C₂₃	—	2	1	no	—
89	C₄×C₂₃	—	4	2	no	—
90	C₄×C₂₃	#135	1	2	yes	complemented
91	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
93	C₂×#135	C₂×C₂₃	1	2	yes	complemented
94	#550	C₂×C₂₃	1	2	yes	complemented
95	#493	C₂×C₂₃	1	1	yes	characteristic, complemented
96	#466	C₂×C₂₃	1	2	yes	complemented
97	#460	C₂×C₂₃	1	2	yes	complemented
98	#449	C₂×C₂₃	1	4	yes	complemented
99	C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
100	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
101	C₂×C₂₃	—	4	2	no	—
102	C₂×C₂₃	—	8	4	no	—
103	C₂×C₂₃	D₄²	1	1	yes	characteristic
104	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
105	C₂×C₂₃	#576	1	1	yes	characteristic
106	C₂×#41	C₂²×C₂₃	1	4	yes	—
107	#135	—	2	8	no	—
108	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	—
110	C₂×D₈	C₂²×C₂₃	1	2	yes	—
111	C₂×D₈	C₂²×C₂₃	1	2	yes	—
112	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
113	#127	—	2	2	no	—
114	#127	C₂²×C₂₃	1	2	yes	—
115	#121	—	2	4	no	—
116	#121	C₂²×C₂₃	1	4	yes	—
117	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
118	C₄×D₄	—	2	2	no	—
119	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
120	#107	C₂²×C₂₃	1	1	yes	characteristic
121	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
122	#103	C₂²×C₂₃	1	2	yes	—
123	#102	C₂²×C₂₃	1	4	yes	—
124	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
125	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
126	C₂₃	#3287	1	1	yes	characteristic, direct product factor, Sylow
127	#41	—	2	8	no	—
128	#41	—	4	4	no	—
129	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
130	C₂×D₄	—	2	4	no	—
131	C₂×D₄	—	2	4	no	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
135	C₂×D₄	C₂³×C₂₃	1	4	yes	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
138	Q₁₆	D₄×C₂₃	1	4	yes	complemented
139	SD₁₆	—	2	8	no	—
140	D₈	—	2	4	no	—
141	D₈	—	4	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
144	C₂×C₈	C₂³×C₂₃	1	2	yes	—
145	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
148	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
151	C₄²	—	2	2	no	—
152	C₄²	C₂³×C₂₃	1	1	yes	characteristic
153	C₂³	—	2	4	no	—
154	C₂³	C₂×D₄×C₂₃	1	2	yes	—
155	Q₈	—	2	2	no	—
156	Q₈	C₂×D₄×C₂₃	1	4	yes	—
157	D₄	—	2	2	no	—
158	D₄	—	2	4	no	—
159	D₄	—	2	4	no	—
160	D₄	—	4	8	no	—
161	D₄	—	4	4	no	—
162	D₄	—	8	2	no	—
163	D₄	—	8	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
170	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
171	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
172	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	1	no	—
175	C₈	C₂×D₄×C₂₃	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	4	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂₃×#135	1	2	yes	complemented
188	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
189	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
190	C₂	—	4	2	no	—
191	C₂	—	8	4	no	—
192	C₂	D₄²×C₂₃	1	1	yes	characteristic
193	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
194	C₂	C₂₃×#576	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#135	C₂	semidirect
2. C₂₃×#550	C₂	semidirect
3. C₂₃×#493	C₂	semidirect
4. C₂₃×#466	C₂	semidirect
5. C₂₃×#460	C₂	semidirect
6. C₂₃×#449	C₂	semidirect
7. C₄×Q₁₆×C₂₃	C₂	semidirect
8. C₈×D₄×C₂₃	C₂	semidirect
40. Q₁₆×C₂₃	D₄	semidirect
78. #3287	C₂₃	direct (N × H)
90. C₄×C₂₃	#135	semidirect
93. C₂×#135	C₂×C₂₃	semidirect
94. #550	C₂×C₂₃	semidirect
95. #493	C₂×C₂₃	semidirect
96. #466	C₂×C₂₃	semidirect
97. #460	C₂×C₂₃	semidirect
98. #449	C₂×C₂₃	semidirect
99. C₄×Q₁₆	C₂×C₂₃	semidirect
100. C₈×D₄	C₂×C₂₃	semidirect
126. C₂₃	#3287	direct (N × H)
138. Q₁₆	D₄×C₂₃	semidirect
187. C₄	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#135	2
C₂₃×#550	2
C₂₃×#493	2
C₂₃×#466	2
C₂₃×#460	2
C₂₃×#449	2
C₄×Q₁₆×C₂₃	2
C₈×D₄×C₂₃	2
#3287	23

Subgroups — C23×#3287 #15670906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3287 #15670906