Subgroups — C₂₃×#3402 #15671021

Order 2944.

356 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×#556	C₂	1	2	yes	maximal, complemented
2	C₂₃×#497	C₂	1	2	yes	maximal, complemented
3	C₂₃×#495	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#464	C₂	1	2	yes	maximal, complemented
5	C₂₃×#463	C₂	1	2	yes	maximal
6	C₂₃×#462	C₂	1	2	yes	maximal, complemented
7	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#440	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂₃×#126	—	2	2	no	—
18	C₂₃×#126	—	2	2	no	—
19	C₂₃×#124	C₂²	1	2	yes	—
20	C₂₃×#124	C₂²	1	2	yes	—
21	C₂₃×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
23	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#117	—	2	2	no	—
27	C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂₃×#103	C₂²	1	2	yes	—
31	C₂₃×#103	C₂²	1	2	yes	—
32	C₂₃×#102	C₂²	1	2	yes	—
33	C₂₃×#97	C₂²	1	2	yes	—
34	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
37	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
38	C₂×D₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	Q₁₆×C₂₃	—	2	2	no	—
41	Q₁₆×C₂₃	—	4	1	no	—
42	Q₁₆×C₂₃	—	4	2	no	—
43	SD₁₆×C₂₃	—	2	4	no	—
44	SD₁₆×C₂₃	—	4	2	no	—
45	D₈×C₂₃	—	4	1	no	—
46	C₂×C₈×C₂₃	C₂³	1	2	yes	—
47	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
54	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₄²×C₂₃	—	2	2	no	—
62	C₄²×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	C₂³	1	2	yes	—
64	C₄²×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₂₃	—	2	2	no	—
66	Q₈×C₂₃	—	2	2	no	—
67	Q₈×C₂₃	—	2	2	no	—
68	Q₈×C₂₃	—	2	2	no	—
69	Q₈×C₂₃	—	2	2	no	—
70	Q₈×C₂₃	—	4	1	no	—
71	D₄×C₂₃	—	2	2	no	—
72	D₄×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	—	4	4	no	—
77	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₈×C₂₃	—	2	1	no	—
83	C₈×C₂₃	—	2	2	no	—
84	C₈×C₂₃	#41	1	2	yes	—
85	#3402	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₂₃	—	4	2	no	—
87	C₂²×C₂₃	—	4	2	no	—
88	C₂²×C₂₃	—	4	2	no	—
89	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₂₃	—	2	2	no	—
91	C₄×C₂₃	—	2	1	no	—
92	C₄×C₂₃	—	2	1	no	—
93	C₄×C₂₃	—	2	2	no	—
94	C₄×C₂₃	—	4	2	no	—
95	C₄×C₂₃	—	4	2	no	—
96	C₄×C₂₃	—	4	2	no	—
97	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
98	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
99	#556	C₂×C₂₃	1	2	yes	complemented
100	#497	C₂×C₂₃	1	2	yes	complemented
101	#495	C₂×C₂₃	1	1	yes	characteristic
102	#464	C₂×C₂₃	1	2	yes	complemented
103	#463	C₂×C₂₃	1	2	yes	—
104	#462	C₂×C₂₃	1	2	yes	complemented
105	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
106	#440	C₂×C₂₃	1	2	yes	complemented
107	C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
108	C₂×C₂₃	—	8	2	no	—
109	C₂×C₂₃	#550	1	1	yes	characteristic
110	C₂×C₂₃	#575	1	1	yes	characteristic
111	C₂×C₂₃	#578	1	1	yes	characteristic
112	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
113	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
114	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
115	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
116	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
117	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
118	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
119	#126	—	2	2	no	—
120	#126	—	2	2	no	—
121	#124	C₂²×C₂₃	1	2	yes	—
122	#124	C₂²×C₂₃	1	2	yes	—
123	C₂²⋊Q₈	—	2	2	no	—
124	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
125	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
126	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
127	C₄×D₄	C₂²×C₂₃	1	2	yes	—
128	#117	—	2	2	no	—
129	#107	C₂²×C₂₃	1	1	yes	characteristic
130	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
131	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
132	#103	C₂²×C₂₃	1	2	yes	—
133	#103	C₂²×C₂₃	1	2	yes	—
134	#102	C₂²×C₂₃	1	2	yes	—
135	#97	C₂²×C₂₃	1	2	yes	—
136	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
137	C₂₃	#3402	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
139	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
140	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
141	C₂×D₄	C₂³×C₂₃	1	2	yes	—
142	C₂²×C₄	—	2	2	no	—
143	Q₁₆	—	2	2	no	—
144	Q₁₆	—	4	1	no	—
145	Q₁₆	—	4	2	no	—
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	4	2	no	—
148	D₈	—	4	1	no	—
149	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
150	C₂×C₈	C₂³×C₂₃	1	2	yes	—
151	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
157	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
158	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	C₂³×C₂₃	1	1	yes	characteristic
167	C₄²	C₂³×C₂₃	1	2	yes	—
168	C₂³	—	2	2	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	2	2	no	—
173	Q₈	—	4	1	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
182	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
184	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	2	no	—
187	C₈	#41×C₂₃	1	2	yes	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
200	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
201	C₂	—	8	2	no	—
202	C₂	C₂₃×#550	1	1	yes	characteristic
203	C₂	C₂₃×#575	1	1	yes	characteristic
204	C₂	C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#556	C₂	semidirect
2. C₂₃×#497	C₂	semidirect
4. C₂₃×#464	C₂	semidirect
6. C₂₃×#462	C₂	semidirect
7. C₂₃×#446	C₂	semidirect
8. C₂₃×#440	C₂	semidirect
9. C₄×Q₁₆×C₂₃	C₂	semidirect
85. #3402	C₂₃	direct (N × H)
99. #556	C₂×C₂₃	semidirect
100. #497	C₂×C₂₃	semidirect
102. #464	C₂×C₂₃	semidirect
104. #462	C₂×C₂₃	semidirect
105. #446	C₂×C₂₃	semidirect
106. #440	C₂×C₂₃	semidirect
107. C₄×Q₁₆	C₂×C₂₃	semidirect
137. C₂₃	#3402	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×#556	2
C₂₃×#497	2
C₂₃×#495	2
C₂₃×#464	2
C₂₃×#463	2
C₂₃×#462	2
C₂₃×#446	2
C₂₃×#440	2
C₄×Q₁₆×C₂₃	2
#3402	23

Subgroups — C23×#3402 #15671021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3402 #15671021