Subgroups — C₂₃×#2990 #15670609

Order 2944.

398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#446	C₂	1	2	yes	maximal, complemented
3	C₂₃×#435	C₂	1	4	yes	maximal, complemented
4	C₈×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
5	C₂₃×#432	C₂	1	2	yes	maximal, complemented
6	C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₃	C₄	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₃	C₂²	1	4	yes	—
12	C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₂₃	C₄	1	2	yes	—
15	C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
16	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#117	C₄	1	1	yes	characteristic
19	C₂₃×#117	C₄	1	2	yes	—
20	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
25	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
29	C₂₃×#97	C₂²	1	4	yes	—
30	C₄×C₈×C₂₃	C₂²	1	2	yes	—
31	#41×C₂₃	—	2	4	no	—
32	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	M₁₆×C₂₃	—	4	2	no	—
42	C₂×C₈×C₂₃	—	4	4	no	—
43	C₂×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂×C₈×C₂₃	C₂³	1	4	yes	—
45	C₂×C₈×C₂₃	C₂³	1	2	yes	—
46	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
53	C₄²×C₂₃	C₂×C₄	1	2	yes	—
54	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₂₃	C₂×C₄	1	2	yes	—
56	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₂₃	C₂³	1	1	yes	characteristic
58	C₄²×C₂₃	C₂³	1	1	yes	characteristic
59	C₄²×C₂₃	C₂³	1	2	yes	—
60	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
62	Q₈×C₂₃	—	2	2	no	—
63	D₄×C₂₃	—	2	4	no	—
64	D₄×C₂₃	M₁₆	1	4	yes	complemented
65	C₂×C₄×C₂₃	—	2	4	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₈×C₂₃	—	2	2	no	—
83	C₈×C₂₃	—	2	4	no	—
84	C₈×C₂₃	—	2	2	no	—
85	#2990	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₂₃	—	2	2	no	—
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	—	2	1	no	—
89	C₂²×C₂₃	—	2	1	no	—
90	C₂²×C₂₃	—	2	1	no	—
91	C₂²×C₂₃	C₂×M₁₆	1	4	yes	—
92	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₂₃	—	2	2	no	—
94	C₄×C₂₃	—	2	1	no	—
95	C₄×C₂₃	—	2	1	no	—
96	C₄×C₂₃	—	2	2	no	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
100	C₄×C₂₃	#131	1	2	yes	—
101	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
102	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×#41	C₂×C₂₃	1	1	yes	characteristic
104	#446	C₂×C₂₃	1	2	yes	complemented
105	#435	C₂×C₂₃	1	4	yes	complemented
106	C₈×D₄	C₂×C₂₃	1	2	yes	complemented
107	#432	C₂×C₂₃	1	2	yes	complemented
108	#431	C₂×C₂₃	1	1	yes	characteristic, complemented
109	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
110	C₂×C₄⋊C₈	C₂×C₂₃	1	2	yes	complemented
111	C₂×C₂₃	—	2	4	no	—
112	C₂×C₂₃	—	4	1	no	—
113	C₂×C₂₃	#520	1	1	yes	characteristic
114	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
115	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
116	C₂×#41	C₄×C₂₃	1	1	yes	characteristic
117	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
118	C₂²×C₈	C₂²×C₂₃	1	4	yes	—
119	C₄×Q₈	C₄×C₂₃	1	1	yes	characteristic
120	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
121	C₄×D₄	C₄×C₂₃	1	2	yes	—
122	C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
123	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
124	C₄×D₄	C₂²×C₂₃	1	2	yes	—
125	#117	C₄×C₂₃	1	1	yes	characteristic
126	#117	C₄×C₂₃	1	2	yes	—
127	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
128	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
129	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
130	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
131	C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
132	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
133	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
135	C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
136	#97	C₂²×C₂₃	1	4	yes	—
137	C₄×C₈	C₂²×C₂₃	1	2	yes	—
138	C₂₃	#2990	1	1	yes	characteristic, direct product factor, Sylow
139	#41	—	2	4	no	—
140	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
142	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
144	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	4	4	no	—
151	C₂×C₈	C₂³×C₂₃	1	2	yes	—
152	C₂×C₈	C₂³×C₂₃	1	2	yes	—
153	C₂×C₈	C₂³×C₂₃	1	4	yes	—
154	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
155	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
156	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
157	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
158	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
159	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
160	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
161	C₄²	C₂×C₄×C₂₃	1	2	yes	—
162	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
163	C₄²	C₂×C₄×C₂₃	1	2	yes	—
164	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
165	C₄²	C₂³×C₂₃	1	1	yes	characteristic
166	C₄²	C₂³×C₂₃	1	2	yes	—
167	C₄²	C₂³×C₂₃	1	1	yes	characteristic
168	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
169	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	M₁₆×C₂₃	1	4	yes	complemented
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
183	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
186	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
187	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	2	4	no	—
192	C₈	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	C₂×M₁₆×C₂₃	1	4	yes	—
199	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×M₁₆×C₂₃	1	2	yes	—
207	C₄	C₂₃×#131	1	2	yes	—
208	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
209	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
210	C₂	—	2	4	no	—
211	C₂	—	4	1	no	—
212	C₂	C₂₃×#520	1	1	yes	characteristic
213	C₂	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
214	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	C₂×#41×C₂₃	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. C₂₃×#435	C₂	semidirect
4. C₈×D₄×C₂₃	C₂	semidirect
5. C₂₃×#432	C₂	semidirect
6. C₂₃×#431	C₂	semidirect
7. C₂₃×#423	C₂	semidirect
8. C₂×C₂₃×C₄⋊C₈	C₂	semidirect
64. D₄×C₂₃	M₁₆	semidirect
85. #2990	C₂₃	direct (N × H)
104. #446	C₂×C₂₃	semidirect
105. #435	C₂×C₂₃	semidirect
106. C₈×D₄	C₂×C₂₃	semidirect
107. #432	C₂×C₂₃	semidirect
108. #431	C₂×C₂₃	semidirect
109. #423	C₂×C₂₃	semidirect
110. C₂×C₄⋊C₈	C₂×C₂₃	semidirect
138. C₂₃	#2990	direct (N × H)
172. D₄	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#41×C₂₃	2
C₂₃×#446	2
C₂₃×#435	2
C₈×D₄×C₂₃	2
C₂₃×#432	2
C₂₃×#431	2
C₂₃×#423	2
C₂×C₂₃×C₄⋊C₈	2
#2990	23

Subgroups — C23×#2990 #15670609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2990 #15670609