Subgroups — C₂×C₂₃×#465 #15670666

Order 2944.

564 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#465	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₂₃	—	2	1	no	—
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₃	—	2	4	no	—
14	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
15	C₂²×C₈×C₂₃	—	2	4	no	—
16	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	4	yes	complemented
19	C₂₃×C₂²⋊Q₈	—	2	2	no	—
20	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
21	C₂₃×#121	—	2	2	no	—
22	C₂₃×#121	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
24	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
28	C₂₃×#106	C₂²	1	4	yes	complemented
29	C₂₃×#103	C₂²	1	4	yes	complemented
30	C₂₃×#102	C₂²	1	4	yes	complemented
31	C₂⁴×C₂₃	—	2	1	no	—
32	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
33	C₂×Q₈×C₂₃	—	2	1	no	—
34	C₂×Q₈×C₂₃	—	2	2	no	—
35	C₂×Q₈×C₂₃	—	2	2	no	—
36	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
37	C₂×D₄×C₂₃	—	2	4	no	—
38	C₂×D₄×C₂₃	—	2	2	no	—
39	C₂×D₄×C₂₃	—	2	1	no	—
40	C₂×D₄×C₂₃	—	2	2	no	—
41	C₂×D₄×C₂₃	—	4	4	no	—
42	C₂×D₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	—	2	1	no	—
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	—	2	1	no	—
48	C₂²×C₄×C₂₃	D₄	1	2	yes	—
49	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	SD₁₆×C₂₃	—	2	8	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	2	8	no	—
55	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
56	C₂×C₈×C₂₃	C₂³	1	2	yes	—
57	C₂×C₈×C₂₃	C₂³	1	2	yes	—
58	C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₂₃	—	2	4	no	—
64	C₂²⋊C₄×C₂₃	—	2	4	no	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	4	2	no	—
69	C₂³×C₂₃	—	4	2	no	—
70	C₂³×C₂₃	—	4	2	no	—
71	C₂³×C₂₃	SD₁₆	1	4	yes	complemented
72	C₂³×C₂₃	C₂×D₄	1	2	yes	—
73	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₂₃	—	2	2	no	—
75	Q₈×C₂₃	—	2	2	no	—
76	Q₈×C₂₃	—	4	2	no	—
77	D₄×C₂₃	—	2	2	no	—
78	D₄×C₂₃	—	2	2	no	—
79	D₄×C₂₃	—	4	8	no	—
80	D₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	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	2	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	#41	1	2	yes	—
95	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
96	C₈×C₂₃	—	2	2	no	—
97	C₈×C₂₃	C₂×D₄	1	4	yes	complemented
98	C₂×#465	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₂₃	—	2	2	no	—
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	8	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	—	4	2	no	—
105	C₂²×C₂₃	—	8	4	no	—
106	C₂²×C₂₃	#121	1	2	yes	—
107	C₂²×C₂₃	C₂×SD₁₆	1	2	yes	—
108	C₂²×C₂₃	C₂×SD₁₆	1	4	yes	complemented
109	C₂²×C₂₃	#135	1	2	yes	—
110	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	4	2	no	—
113	C₄×C₂₃	—	4	2	no	—
114	C₄×C₂₃	—	4	2	no	—
115	C₄×C₂₃	#121	1	2	yes	—
116	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₂²×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
119	C₂³×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
121	C₂×#121	C₂×C₂₃	1	1	yes	characteristic
122	#465	C₂×C₂₃	1	8	yes	direct product factor
123	C₂×#106	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
125	C₂×#102	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂×C₂₃	—	2	4	no	—
127	C₂×C₂₃	—	8	2	no	—
128	C₂×C₂₃	#465	1	4	yes	direct product factor
129	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
130	C₂×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
131	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
132	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
133	C₂²×D₄	—	2	1	no	—
134	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
135	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×SD₁₆	—	2	4	no	—
137	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
141	C₂²×C₈	C₂²×C₂₃	1	4	yes	complemented
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
144	#121	—	2	2	no	—
145	#121	C₂²×C₂₃	1	4	yes	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	—	2	1	no	—
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	#106	C₂²×C₂₃	1	4	yes	complemented
152	#103	C₂²×C₂₃	1	4	yes	complemented
153	#102	C₂²×C₂₃	1	4	yes	complemented
154	C₂₃	C₂×#465	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	—	2	1	no	—
156	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	4	4	no	—
166	C₂×D₄	C₂³×C₂₃	1	2	yes	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	D₄×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
175	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
176	SD₁₆	—	2	8	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	D₄×C₂₃	1	4	yes	complemented
180	C₂×C₈	C₂³×C₂₃	1	2	yes	—
181	C₂×C₈	C₂³×C₂₃	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
186	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	—	2	4	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	4	2	no	—
193	C₂³	—	4	2	no	—
194	C₂³	—	4	2	no	—
195	C₂³	SD₁₆×C₂₃	1	4	yes	complemented
196	C₂³	C₂×D₄×C₂₃	1	2	yes	—
197	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
198	Q₈	—	2	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	4	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	4	8	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
218	C₂×C₄	#41×C₂₃	1	2	yes	—
219	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	C₂×D₄×C₂₃	1	4	yes	complemented
222	C₂²	—	2	2	no	—
223	C₂²	—	2	8	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	8	4	no	—
229	C₂²	C₂₃×#121	1	2	yes	—
230	C₂²	C₂×SD₁₆×C₂₃	1	2	yes	—
231	C₂²	C₂×SD₁₆×C₂₃	1	4	yes	complemented
232	C₂²	C₂₃×#135	1	2	yes	—
233	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	C₂₃×#121	1	2	yes	—
239	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
240	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
241	C₂	—	2	4	no	—
242	C₂	—	8	2	no	—
243	C₂	C₂₃×#465	1	4	yes	direct product factor
244	C₂	C₂×C₂₃×#121	1	1	yes	characteristic
245	C₂	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
246	C₂	C₂×C₂₃×#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×SD₁₆×C₂₃	C₂	semidirect
2. C₂³×C₈×C₂₃	C₂	semidirect
3. C₂×C₂₃×C₂²⋊Q₈	C₂	semidirect
5. C₂₃×#465	C₂	direct (N × H)
6. C₂×C₂₃×#106	C₂	semidirect
7. C₂×C₂₃×#103	C₂	semidirect
8. C₂×C₂₃×#102	C₂	semidirect
14. C₂×SD₁₆×C₂₃	C₂²	semidirect
18. C₂²×C₈×C₂₃	C₂²	semidirect
20. C₂₃×C₂²⋊Q₈	C₂²	semidirect
28. C₂₃×#106	C₂²	semidirect
29. C₂₃×#103	C₂²	semidirect
30. C₂₃×#102	C₂²	semidirect
55. C₂×C₈×C₂₃	D₄	semidirect
71. C₂³×C₂₃	SD₁₆	semidirect
97. C₈×C₂₃	C₂×D₄	semidirect
98. C₂×#465	C₂₃	direct (N × H)
108. C₂²×C₂₃	C₂×SD₁₆	semidirect
118. C₂²×SD₁₆	C₂×C₂₃	semidirect
119. C₂³×C₈	C₂×C₂₃	semidirect
120. C₂×C₂²⋊Q₈	C₂×C₂₃	semidirect
122. #465	C₂×C₂₃	direct (N × H)
123. C₂×#106	C₂×C₂₃	semidirect
124. C₂×#103	C₂×C₂₃	semidirect
125. C₂×#102	C₂×C₂₃	semidirect
128. C₂×C₂₃	#465	direct (N × H)
137. C₂×SD₁₆	C₂²×C₂₃	semidirect
141. C₂²×C₈	C₂²×C₂₃	semidirect
143. C₂²⋊Q₈	C₂²×C₂₃	semidirect
151. #106	C₂²×C₂₃	semidirect
152. #103	C₂²×C₂₃	semidirect
153. #102	C₂²×C₂₃	semidirect
154. C₂₃	C₂×#465	direct (N × H)
179. C₂×C₈	D₄×C₂₃	semidirect
195. C₂³	SD₁₆×C₂₃	semidirect
221. C₈	C₂×D₄×C₂₃	semidirect
231. C₂²	C₂×SD₁₆×C₂₃	semidirect
243. C₂	C₂₃×#465	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×SD₁₆×C₂₃	2
C₂³×C₈×C₂₃	2
C₂×C₂₃×C₂²⋊Q₈	2
C₂×C₂₃×#121	2
C₂₃×#465	2
C₂×C₂₃×#106	2
C₂×C₂₃×#103	2
C₂×C₂₃×#102	2
C₂×#465	23

Subgroups — C2×C23×#465 #15670666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#465 #15670666