Subgroups — C₂₃×#3277 #15670896

Order 2944.

476 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#137	C₂	1	2	yes	maximal, complemented
2	C₂₃×#548	C₂	1	2	yes	maximal, complemented
3	C₂₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#468	C₂	1	2	yes	maximal, complemented
5	C₂₃×#464	C₂	1	2	yes	maximal, complemented
6	C₂₃×#451	C₂	1	2	yes	maximal, complemented
7	C₂₃×#449	C₂	1	2	yes	maximal, complemented
8	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	2	yes	—
11	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#137	—	2	8	no	—
13	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
20	C₂₃×#124	—	2	2	no	—
21	C₂₃×#124	C₂²	1	2	yes	—
22	C₂₃×C₂²⋊Q₈	—	2	2	no	—
23	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₂₃×#121	—	2	2	no	—
25	C₂₃×#121	C₂²	1	2	yes	—
26	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
27	C₄×D₄×C₂₃	—	2	2	no	—
28	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂₃×#103	C₂²	1	2	yes	—
32	C₂₃×#103	C₂²	1	2	yes	—
33	C₂₃×#102	C₂²	1	2	yes	—
34	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
36	#41×C₂₃	—	2	4	no	—
37	#41×C₂₃	—	4	2	no	—
38	C₂×Q₈×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	—	2	4	no	—
40	C₂×Q₈×C₂₃	—	4	2	no	—
41	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
42	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
43	C₂×D₄×C₂₃	—	2	2	no	—
44	C₂×D₄×C₂₃	—	2	2	no	—
45	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂×D₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	Q₁₆×C₂₃	—	2	4	no	—
51	Q₁₆×C₂₃	—	4	1	no	—
52	Q₁₆×C₂₃	D₄	1	4	yes	complemented
53	SD₁₆×C₂₃	—	2	4	no	—
54	SD₁₆×C₂₃	—	2	4	no	—
55	D₈×C₂₃	—	4	1	no	—
56	M₁₆×C₂₃	—	2	4	no	—
57	C₂×C₈×C₂₃	C₂³	1	2	yes	—
58	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
64	C₂²⋊C₄×C₂₃	—	2	2	no	—
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	C₂³×C₂₃	—	2	2	no	—
72	C₂³×C₂₃	C₂×D₄	1	2	yes	—
73	Q₈×C₂₃	—	2	2	no	—
74	Q₈×C₂₃	—	2	2	no	—
75	Q₈×C₂₃	—	2	2	no	—
76	Q₈×C₂₃	—	4	4	no	—
77	Q₈×C₂₃	C₂×D₄	1	4	yes	—
78	D₄×C₂₃	—	2	2	no	—
79	D₄×C₂₃	—	2	2	no	—
80	D₄×C₂₃	—	4	1	no	—
81	D₄×C₂₃	—	4	4	no	—
82	D₄×C₂₃	—	4	4	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	4	4	no	—
89	C₂×C₄×C₂₃	—	4	4	no	—
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃	—	2	2	no	—
95	C₈×C₂₃	—	2	1	no	—
96	C₈×C₂₃	C₂×D₄	1	2	yes	—
97	#3277	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	2	no	—
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	2	4	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×#137	C₂×C₂₃	1	2	yes	complemented
114	#548	C₂×C₂₃	1	2	yes	complemented
115	#495	C₂×C₂₃	1	1	yes	characteristic, complemented
116	#468	C₂×C₂₃	1	2	yes	complemented
117	#464	C₂×C₂₃	1	2	yes	complemented
118	#451	C₂×C₂₃	1	2	yes	complemented
119	#449	C₂×C₂₃	1	2	yes	complemented
120	C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#436	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂×C₂₃	—	4	2	no	—
123	C₂×C₂₃	—	8	2	no	—
124	C₂×C₂₃	D₄²	1	1	yes	characteristic
125	C₂×C₂₃	#575	1	1	yes	characteristic
126	C₂×C₂₃	#578	1	1	yes	characteristic
127	C₂×#41	C₂²×C₂₃	1	2	yes	—
128	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
129	#137	—	2	8	no	—
130	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
131	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
132	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
134	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
135	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
137	#124	—	2	2	no	—
138	#124	C₂²×C₂₃	1	2	yes	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
141	#121	—	2	2	no	—
142	#121	C₂²×C₂₃	1	2	yes	—
143	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
144	C₄×D₄	—	2	2	no	—
145	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
146	#107	C₂²×C₂₃	1	1	yes	characteristic
147	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
148	#103	C₂²×C₂₃	1	2	yes	—
149	#103	C₂²×C₂₃	1	2	yes	—
150	#102	C₂²×C₂₃	1	2	yes	—
151	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂₃	#3277	1	1	yes	characteristic, direct product factor, Sylow
154	#41	—	2	4	no	—
155	#41	—	4	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
160	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
164	C₂×D₄	C₂³×C₂₃	1	2	yes	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
168	Q₁₆	—	2	4	no	—
169	Q₁₆	—	4	1	no	—
170	Q₁₆	D₄×C₂₃	1	4	yes	complemented
171	SD₁₆	—	2	4	no	—
172	SD₁₆	—	2	4	no	—
173	D₈	—	4	1	no	—
174	M₁₆	—	2	4	no	—
175	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₂₃	1	2	yes	—
177	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
187	C₄²	—	2	2	no	—
188	C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	C₂×D₄×C₂₃	1	2	yes	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	4	4	no	—
195	Q₈	C₂×D₄×C₂₃	1	4	yes	—
196	D₄	—	2	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	4	1	no	—
199	D₄	—	4	4	no	—
200	D₄	—	4	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
211	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₈	C₂×D₄×C₂₃	1	2	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	4	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
229	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
230	C₂	—	4	2	no	—
231	C₂	—	8	2	no	—
232	C₂	D₄²×C₂₃	1	1	yes	characteristic
233	C₂	C₂₃×#575	1	1	yes	characteristic
234	C₂	C₂₃×#578	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×#137	C₂	semidirect
2. C₂₃×#548	C₂	semidirect
3. C₂₃×#495	C₂	semidirect
4. C₂₃×#468	C₂	semidirect
5. C₂₃×#464	C₂	semidirect
6. C₂₃×#451	C₂	semidirect
7. C₂₃×#449	C₂	semidirect
8. C₄×Q₁₆×C₂₃	C₂	semidirect
9. C₂₃×#436	C₂	semidirect
52. Q₁₆×C₂₃	D₄	semidirect
97. #3277	C₂₃	direct (N × H)
113. C₂×#137	C₂×C₂₃	semidirect
114. #548	C₂×C₂₃	semidirect
115. #495	C₂×C₂₃	semidirect
116. #468	C₂×C₂₃	semidirect
117. #464	C₂×C₂₃	semidirect
118. #451	C₂×C₂₃	semidirect
119. #449	C₂×C₂₃	semidirect
120. C₄×Q₁₆	C₂×C₂₃	semidirect
121. #436	C₂×C₂₃	semidirect
153. C₂₃	#3277	direct (N × H)
170. Q₁₆	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#137	2
C₂₃×#548	2
C₂₃×#495	2
C₂₃×#468	2
C₂₃×#464	2
C₂₃×#451	2
C₂₃×#449	2
C₄×Q₁₆×C₂₃	2
C₂₃×#436	2
#3277	23

Subgroups — C23×#3277 #15670896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3277 #15670896