Subgroups — C₂₃×#3037 #15670656

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#532	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#464	C₂	1	4	yes	maximal, complemented
5	C₂₃×#462	C₂	1	2	yes	maximal, complemented
6	C₂₃×#460	C₂	1	2	yes	maximal, complemented
7	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#417	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₃	C₂²	1	2	yes	—
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#137	—	2	8	no	—
12	C₂×Q₁₆×C₂₃	C₂²	1	4	yes	—
13	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	—
14	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×#127	C₂²	1	1	yes	characteristic
17	C₂₃×#124	C₂²	1	2	yes	—
18	C₂₃×#121	—	2	1	no	—
19	C₂₃×#121	—	2	1	no	—
20	C₄×Q₈×C₂₃	—	2	1	no	—
21	C₄×Q₈×C₂₃	—	2	1	no	—
22	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
23	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₄×D₄×C₂₃	—	2	1	no	—
25	C₄×D₄×C₂₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	—	2	1	no	—
29	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
32	C₂₃×#103	C₂²	1	4	yes	—
33	C₂₃×#102	C₂²	1	4	yes	—
34	#41×C₂₃	—	2	4	no	—
35	#41×C₂₃	—	2	4	no	—
36	C₂×Q₈×C₂₃	—	2	1	no	—
37	C₂×Q₈×C₂₃	—	2	2	no	—
38	C₂×Q₈×C₂₃	D₄	1	4	yes	complemented
39	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
40	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
41	C₂×D₄×C₂₃	—	2	2	no	—
42	C₂×D₄×C₂₃	—	2	1	no	—
43	C₂×D₄×C₂₃	—	2	1	no	—
44	C₂×D₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	—	2	1	no	—
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	Q₁₆×C₂₃	—	2	8	no	—
52	SD₁₆×C₂₃	—	2	8	no	—
53	M₁₆×C₂₃	—	2	4	no	—
54	C₂×C₈×C₂₃	C₂³	1	4	yes	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₂₃	—	2	2	no	—
62	C₂²⋊C₄×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	2	2	no	—
65	C₄²×C₂₃	D₄	1	1	yes	characteristic
66	C₄²×C₂₃	D₄	1	1	yes	characteristic
67	C₄²×C₂₃	C₂³	1	1	yes	characteristic
68	C₄²×C₂₃	C₂³	1	1	yes	characteristic
69	C₂³×C₂₃	—	2	2	no	—
70	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₂₃	—	2	4	no	—
72	Q₈×C₂₃	—	2	2	no	—
73	Q₈×C₂₃	—	2	2	no	—
74	Q₈×C₂₃	—	2	2	no	—
75	Q₈×C₂₃	C₂×D₄	1	4	yes	—
76	D₄×C₂₃	—	2	2	no	—
77	D₄×C₂₃	—	2	2	no	—
78	D₄×C₂₃	—	4	2	no	—
79	D₄×C₂₃	—	4	4	no	—
80	D₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	2	2	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	—	4	1	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	4	4	no	—
89	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#41	1	2	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₈×C₂₃	—	2	4	no	—
98	#3037	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	#121	1	2	yes	—
105	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₂₃	—	2	4	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	#121	1	2	yes	—
113	C₄×C₂₃	#137	1	2	yes	complemented
114	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₂×#137	C₂×C₂₃	1	2	yes	complemented
117	#532	C₂×C₂₃	1	1	yes	characteristic
118	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#464	C₂×C₂₃	1	4	yes	complemented
120	#462	C₂×C₂₃	1	2	yes	complemented
121	#460	C₂×C₂₃	1	2	yes	complemented
122	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#417	C₂×C₂₃	1	2	yes	complemented
124	C₂×C₂₃	—	2	2	no	—
125	C₂×C₂₃	—	8	2	no	—
126	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
127	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
128	C₂×C₂₃	#575	1	1	yes	characteristic
129	C₂×#41	C₂²×C₂₃	1	2	yes	—
130	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
131	#137	—	2	8	no	—
132	C₂×Q₁₆	C₂²×C₂₃	1	4	yes	—
133	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	—
134	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
135	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
136	#127	C₂²×C₂₃	1	1	yes	characteristic
137	#124	C₂²×C₂₃	1	2	yes	—
138	#121	—	2	1	no	—
139	#121	—	2	1	no	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
143	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
144	C₄×D₄	—	2	1	no	—
145	C₄×D₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
151	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	#103	C₂²×C₂₃	1	4	yes	—
153	#102	C₂²×C₂₃	1	4	yes	—
154	C₂₃	#3037	1	1	yes	characteristic, direct product factor, Sylow
155	#41	—	2	4	no	—
156	#41	—	2	4	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	D₄×C₂₃	1	4	yes	complemented
160	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
161	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	1	no	—
165	C₂×D₄	C₂³×C₂₃	1	2	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	Q₁₆	—	2	8	no	—
173	SD₁₆	—	2	8	no	—
174	M₁₆	—	2	4	no	—
175	C₂×C₈	C₂³×C₂₃	1	4	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	D₄×C₂₃	1	1	yes	characteristic
187	C₄²	D₄×C₂₃	1	1	yes	characteristic
188	C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	C₄²	C₂³×C₂₃	1	1	yes	characteristic
190	C₂³	—	2	2	no	—
191	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
192	Q₈	—	2	4	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	2	2	no	—
196	Q₈	C₂×D₄×C₂₃	1	4	yes	—
197	D₄	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	4	no	—
201	D₄	—	4	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	#41×C₂₃	1	2	yes	—
217	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₂₃×#121	1	2	yes	—
225	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	4	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	C₂₃×#121	1	2	yes	—
233	C₄	C₂₃×#137	1	2	yes	complemented
234	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
235	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	8	2	no	—
238	C₂	C₂×C₂₃×#121	1	1	yes	characteristic
239	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
240	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. C₂×C₂₃×#137	C₂	semidirect
3. C₂×C₄×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#464	C₂	semidirect
5. C₂₃×#462	C₂	semidirect
6. C₂₃×#460	C₂	semidirect
7. C₂₃×#423	C₂	semidirect
8. C₂₃×#417	C₂	semidirect
38. C₂×Q₈×C₂₃	D₄	semidirect
98. #3037	C₂₃	direct (N × H)
113. C₄×C₂₃	#137	semidirect
116. C₂×#137	C₂×C₂₃	semidirect
118. C₂×C₄×Q₈	C₂×C₂₃	semidirect
119. #464	C₂×C₂₃	semidirect
120. #462	C₂×C₂₃	semidirect
121. #460	C₂×C₂₃	semidirect
122. #423	C₂×C₂₃	semidirect
123. #417	C₂×C₂₃	semidirect
154. C₂₃	#3037	direct (N × H)
159. C₂×Q₈	D₄×C₂₃	semidirect
233. C₄	C₂₃×#137	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×#532	2
C₂×C₄×Q₈×C₂₃	2
C₂₃×#464	2
C₂₃×#462	2
C₂₃×#460	2
C₂₃×#423	2
C₂₃×#417	2
#3037	23

Subgroups — C23×#3037 #15670656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3037 #15670656