Subgroups — C₂₃×#3216 #15670835

Order 2944.

390 conjugacy classes of subgroups.

The subgroup lattice has 220 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, complemented
2	C₂₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#497	C₂	1	2	yes	maximal, complemented
4	C₂₃×#489	C₂	1	2	yes	maximal, complemented
5	C₂₃×#462	C₂	1	2	yes	maximal, complemented
6	C₂₃×#460	C₂	1	2	yes	maximal, complemented
7	C₂₃×#452	C₂	1	4	yes	maximal, complemented
8	C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₃	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₂₃	C₂²	1	4	yes	—
11	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	—
12	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂₃×#125	—	2	1	no	—
17	C₂₃×#124	C₂²	1	2	yes	—
18	C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂₃×#121	—	2	1	no	—
21	C₂₃×#121	—	2	1	no	—
22	C₄×Q₈×C₂₃	—	2	1	no	—
23	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₄×D₄×C₂₃	—	2	1	no	—
25	C₄×D₄×C₂₃	—	2	1	no	—
26	C₂₃×#117	—	2	1	no	—
27	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₂₃×#103	C₂²	1	4	yes	—
30	C₂₃×#102	C₂²	1	4	yes	—
31	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂₃×#97	C₂²	1	2	yes	—
33	#41×C₂₃	—	4	4	no	—
34	C₂×Q₈×C₂₃	—	2	1	no	—
35	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
36	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
37	C₂×D₄×C₂₃	—	2	2	no	—
38	C₂×D₄×C₂₃	—	2	1	no	—
39	C₂×D₄×C₂₃	—	2	1	no	—
40	C₂×D₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	Q₁₆×C₂₃	—	4	4	no	—
46	SD₁₆×C₂₃	—	4	4	no	—
47	C₂×C₈×C₂₃	C₂³	1	4	yes	—
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	1	no	—
52	C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	2	2	no	—
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	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	—	2	2	no	—
65	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₂₃	—	2	2	no	—
67	Q₈×C₂₃	—	2	2	no	—
68	Q₈×C₂₃	—	2	4	no	—
69	Q₈×C₂₃	—	4	2	no	—
70	D₄×C₂₃	—	2	4	no	—
71	D₄×C₂₃	—	4	2	no	—
72	D₄×C₂₃	—	4	4	no	—
73	D₄×C₂₃	—	4	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	4	1	no	—
80	C₂×C₄×C₂₃	—	4	4	no	—
81	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
87	C₈×C₂₃	—	2	4	no	—
88	#3216	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₂₃	—	2	1	no	—
90	C₂²×C₂₃	—	2	1	no	—
91	C₂²×C₂₃	—	2	1	no	—
92	C₂²×C₂₃	—	4	2	no	—
93	C₂²×C₂₃	—	4	2	no	—
94	C₂²×C₂₃	—	4	2	no	—
95	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₂₃	—	2	1	no	—
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	2	1	no	—
100	C₄×C₂₃	—	4	2	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	—	4	2	no	—
103	C₄×C₂₃	#137	1	2	yes	—
104	C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
105	#533	C₂×C₂₃	1	1	yes	characteristic, complemented
106	#532	C₂×C₂₃	1	1	yes	characteristic
107	#497	C₂×C₂₃	1	2	yes	complemented
108	#489	C₂×C₂₃	1	2	yes	complemented
109	#462	C₂×C₂₃	1	2	yes	complemented
110	#460	C₂×C₂₃	1	2	yes	complemented
111	#452	C₂×C₂₃	1	4	yes	complemented
112	#432	C₂×C₂₃	1	1	yes	characteristic, complemented
113	C₂×C₂₃	—	4	1	no	—
114	C₂×C₂₃	—	8	2	no	—
115	C₂×C₂₃	#535	1	1	yes	characteristic
116	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
117	C₂×C₂₃	#575	1	1	yes	characteristic
118	C₂×#41	C₂²×C₂₃	1	2	yes	—
119	C₂×Q₁₆	C₂²×C₂₃	1	4	yes	—
120	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	—
121	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
122	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
123	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
124	#127	C₂²×C₂₃	1	1	yes	characteristic
125	#125	—	2	1	no	—
126	#124	C₂²×C₂₃	1	2	yes	—
127	C₂²⋊Q₈	—	2	1	no	—
128	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
129	#121	—	2	1	no	—
130	#121	—	2	1	no	—
131	C₄×Q₈	—	2	1	no	—
132	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
133	C₄×D₄	—	2	1	no	—
134	C₄×D₄	—	2	1	no	—
135	#117	—	2	1	no	—
136	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
137	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
138	#103	C₂²×C₂₃	1	4	yes	—
139	#102	C₂²×C₂₃	1	4	yes	—
140	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
141	#97	C₂²×C₂₃	1	2	yes	—
142	C₂₃	#3216	1	1	yes	characteristic, direct product factor, Sylow
143	#41	—	4	4	no	—
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
146	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	1	no	—
149	C₂×D₄	—	2	1	no	—
150	C₂×D₄	C₂³×C₂₃	1	2	yes	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
155	Q₁₆	—	4	4	no	—
156	SD₁₆	—	4	4	no	—
157	C₂×C₈	C₂³×C₂₃	1	4	yes	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₄²	D₄×C₂₃	1	1	yes	characteristic
171	C₄²	D₄×C₂₃	1	1	yes	characteristic
172	C₄²	C₂³×C₂₃	1	1	yes	characteristic
173	C₄²	C₂³×C₂₃	1	1	yes	characteristic
174	C₂³	—	2	2	no	—
175	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
176	Q₈	—	2	2	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	2	4	no	—
179	Q₈	—	4	2	no	—
180	D₄	—	2	4	no	—
181	D₄	—	4	2	no	—
182	D₄	—	4	4	no	—
183	D₄	—	4	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
194	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
196	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
197	C₈	—	2	4	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	C₂₃×#137	1	2	yes	—
213	C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
214	C₂	—	4	1	no	—
215	C₂	—	8	2	no	—
216	C₂	C₂₃×#535	1	1	yes	characteristic
217	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
218	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂
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
1. C₂₃×#533	C₂	semidirect
3. C₂₃×#497	C₂	semidirect
4. C₂₃×#489	C₂	semidirect
5. C₂₃×#462	C₂	semidirect
6. C₂₃×#460	C₂	semidirect
7. C₂₃×#452	C₂	semidirect
8. C₂₃×#432	C₂	semidirect
88. #3216	C₂₃	direct (N × H)
105. #533	C₂×C₂₃	semidirect
107. #497	C₂×C₂₃	semidirect
108. #489	C₂×C₂₃	semidirect
109. #462	C₂×C₂₃	semidirect
110. #460	C₂×C₂₃	semidirect
111. #452	C₂×C₂₃	semidirect
112. #432	C₂×C₂₃	semidirect
142. C₂₃	#3216	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×#532	2
C₂₃×#497	2
C₂₃×#489	2
C₂₃×#462	2
C₂₃×#460	2
C₂₃×#452	2
C₂₃×#432	2
#3216	23

Subgroups — C23×#3216 #15670835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3216 #15670835