Subgroups — C₂₃×#2994 #15670613

Order 2944.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#435	C₂	1	2	yes	maximal, complemented
6	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
8	C₂₃×#433	C₂	1	2	yes	maximal, complemented
9	C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#424	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#126	C₄	1	2	yes	—
16	C₂₃×#125	C₄	1	1	yes	characteristic
17	C₂₃×#124	C₄	1	1	yes	characteristic
18	C₂₃×#123	C₄	1	2	yes	—
19	C₂₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
20	C₂₃×#121	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₂₃	C₂²	1	2	yes	—
23	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#117	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
27	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
32	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₂₃×#97	C₂²	1	2	yes	—
38	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₂₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	M₁₆×C₂₃	—	4	2	no	—
49	C₂×C₈×C₂₃	—	4	2	no	—
50	C₂×C₈×C₂₃	—	4	2	no	—
51	C₂×C₈×C₂₃	C₂³	1	2	yes	—
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
68	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₄²×C₂₃	C₂³	1	1	yes	characteristic
72	C₄²×C₂₃	C₂³	1	2	yes	—
73	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₂₃	—	2	2	no	—
76	D₄×C₂₃	—	2	4	no	—
77	D₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
93	C₈×C₂₃	—	2	1	no	—
94	C₈×C₂₃	—	2	2	no	—
95	C₈×C₂₃	—	2	2	no	—
96	C₈×C₂₃	—	2	1	no	—
97	C₈×C₂₃	—	2	1	no	—
98	C₈×C₂₃	—	2	1	no	—
99	#2994	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	—	2	1	no	—
102	C₂²×C₂₃	—	2	2	no	—
103	C₂²×C₂₃	—	2	2	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	#131	1	2	yes	—
106	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	2	1	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	—	2	2	no	—
114	C₄×C₂₃	#131	1	2	yes	—
115	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
116	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
117	#529	C₂×C₂₃	1	1	yes	characteristic
118	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
119	C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#436	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#435	C₂×C₂₃	1	2	yes	complemented
122	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₈×D₄	C₂×C₂₃	1	2	yes	complemented
124	#433	C₂×C₂₃	1	2	yes	complemented
125	#431	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#424	C₂×C₂₃	1	2	yes	complemented
127	C₂×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
128	C₂×C₂₃	—	2	2	no	—
129	C₂×C₂₃	—	4	2	no	—
130	C₂×C₂₃	#520	1	1	yes	characteristic
131	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
132	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
133	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
134	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
135	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
136	#126	C₄×C₂₃	1	2	yes	—
137	#125	C₄×C₂₃	1	1	yes	characteristic
138	#124	C₄×C₂₃	1	1	yes	characteristic
139	#123	C₄×C₂₃	1	2	yes	—
140	C₂²⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
141	#121	C₄×C₂₃	1	1	yes	characteristic
142	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₄×D₄	C₂²×C₂₃	1	2	yes	—
144	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
145	#117	C₂²×C₂₃	1	2	yes	—
146	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
147	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
148	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
153	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
157	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
158	#97	C₂²×C₂₃	1	2	yes	—
159	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
161	C₄×C₈	C₂²×C₂₃	1	2	yes	—
162	C₂₃	#2994	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
164	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
165	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
170	M₁₆	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
174	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
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	C₂×C₈	C₂³×C₂₃	1	2	yes	—
179	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
180	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
181	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
182	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
183	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
184	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
185	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
186	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
187	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
188	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
189	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
190	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
191	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
192	C₄²	C₂³×C₂₃	1	1	yes	characteristic
193	C₄²	C₂³×C₂₃	1	2	yes	—
194	C₄²	C₂³×C₂₃	1	1	yes	characteristic
195	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
196	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
211	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
212	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
213	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	C₂₃×#131	1	2	yes	—
227	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	C₂₃×#131	1	2	yes	—
236	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
237	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	C₂₃×#520	1	1	yes	characteristic
241	C₂	C₂×C₂₃×#131	1	1	yes	characteristic
242	C₂	C₂×C₂₃×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂₃×#121	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#446	C₂	semidirect
3. C₈×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#436	C₂	semidirect
5. C₂₃×#435	C₂	semidirect
6. C₈×D₄×C₂₃	C₂	semidirect
7. C₈×D₄×C₂₃	C₂	semidirect
8. C₂₃×#433	C₂	semidirect
9. C₂₃×#431	C₂	semidirect
10. C₂₃×#424	C₂	semidirect
11. C₂×C₂₃×C₄⋊C₈	C₂	semidirect
99. #2994	C₂₃	direct (N × H)
118. #446	C₂×C₂₃	semidirect
119. C₈×Q₈	C₂×C₂₃	semidirect
120. #436	C₂×C₂₃	semidirect
121. #435	C₂×C₂₃	semidirect
122. C₈×D₄	C₂×C₂₃	semidirect
123. C₈×D₄	C₂×C₂₃	semidirect
124. #433	C₂×C₂₃	semidirect
125. #431	C₂×C₂₃	semidirect
126. #424	C₂×C₂₃	semidirect
127. C₂×C₄⋊C₈	C₂×C₂₃	semidirect
162. C₂₃	#2994	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×#529	2
C₂₃×#446	2
C₈×Q₈×C₂₃	2
C₂₃×#436	2
C₂₃×#435	2
C₈×D₄×C₂₃	2
C₂₃×#433	2
C₂₃×#431	2
C₂₃×#424	2
C₂×C₂₃×C₄⋊C₈	2
#2994	23

Subgroups — C23×#2994 #15670613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2994 #15670613