Subgroups — C₂₃×#2897 #15670516

Order 2944.

778 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#418	C₂	1	3	yes	maximal, complemented
5	C₂₃×#417	C₂	1	3	yes	maximal, complemented
6	C₂₃×#416	C₂	1	3	yes	maximal, complemented
7	C₂×C₂₃×#102	C₂	1	3	yes	maximal, complemented
8	C₂₃×H(5,𝔽₂)	—	2	4	no	—
9	C₂₃×H(5,𝔽₂)	C₄	1	8	yes	complemented
10	C₂×#41×C₂₃	—	2	1	no	—
11	C₂×#41×C₂₃	C₂²	1	3	yes	—
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₂₃	—	2	3	no	—
14	C₂²×D₄×C₂₃	C₂²	1	3	yes	—
15	C₂₃×#131	—	2	4	no	—
16	C₂×M₁₆×C₂₃	C₂²	1	3	yes	complemented
17	C₂²×C₈×C₂₃	C₂²	1	3	yes	complemented
18	C₄×Q₈×C₂₃	—	2	1	no	—
19	C₄×D₄×C₂₃	—	2	3	no	—
20	C₂₃×#117	C₂²	1	3	yes	complemented
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	3	yes	complemented
22	C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
23	C₂₃×#103	C₂²	1	3	yes	complemented
24	C₂₃×#102	C₂²	1	3	yes	complemented
25	C₂₃×#102	C₂²	1	6	yes	complemented
26	C₂₃×#102	C₂²	1	3	yes	complemented
27	C₂⁴×C₂₃	—	2	3	no	—
28	#41×C₂₃	—	2	8	no	—
29	#41×C₂₃	—	2	6	no	—
30	#41×C₂₃	C₂×C₄	1	4	yes	—
31	#41×C₂₃	C₂×C₄	1	12	yes	complemented
32	#41×C₂₃	D₄	1	4	yes	—
33	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×D₄×C₂₃	—	2	3	no	—
36	C₂×D₄×C₂₃	—	2	24	no	—
37	C₂×D₄×C₂₃	—	2	3	no	—
38	C₂×D₄×C₂₃	—	2	6	no	—
39	C₂×D₄×C₂₃	C₂×C₄	1	12	yes	complemented
40	C₂×D₄×C₂₃	D₄	1	3	yes	—
41	C₂×D₄×C₂₃	C₂³	1	3	yes	—
42	C₂²×C₄×C₂₃	—	2	3	no	—
43	C₂²×C₄×C₂₃	—	2	3	no	—
44	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
45	M₁₆×C₂₃	—	2	6	no	—
46	C₂×C₈×C₂₃	—	2	6	no	—
47	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₂³	1	3	yes	—
49	C₄⋊C₄×C₂₃	—	2	3	no	—
50	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
51	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²⋊C₄×C₂₃	—	2	3	no	—
53	C₄²×C₂₃	—	2	3	no	—
54	C₂³×C₂₃	—	2	3	no	—
55	C₂³×C₂₃	—	2	12	no	—
56	C₂³×C₂₃	—	8	3	no	—
57	C₂³×C₂₃	—	8	3	no	—
58	C₂³×C₂₃	C₂×D₄	1	3	yes	—
59	Q₈×C₂₃	—	2	1	no	—
60	Q₈×C₂₃	C₂²⋊C₄	1	4	yes	—
61	Q₈×C₂₃	C₂²×C₄	1	2	yes	—
62	D₄×C₂₃	—	2	12	no	—
63	D₄×C₂₃	—	2	12	no	—
64	D₄×C₂₃	—	2	3	no	—
65	D₄×C₂₃	C₂²⋊C₄	1	12	yes	—
66	D₄×C₂₃	C₂²×C₄	1	6	yes	—
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	12	no	—
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	3	no	—
71	C₂×C₄×C₂₃	—	8	3	no	—
72	C₂×C₄×C₂₃	C₂²×C₄	1	6	yes	—
73	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
74	C₂×C₄×C₂₃	C₂×D₄	1	6	yes	—
75	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
76	C₈×C₂₃	—	2	1	no	—
77	C₈×C₂₃	—	2	3	no	—
78	#2897	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₂₃	—	2	3	no	—
80	C₂²×C₂₃	—	2	3	no	—
81	C₂²×C₂₃	—	2	6	no	—
82	C₂²×C₂₃	—	4	3	no	—
83	C₂²×C₂₃	—	4	3	no	—
84	C₂²×C₂₃	—	8	12	no	—
85	C₂²×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
86	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₂₃	—	2	2	no	—
88	C₄×C₂₃	—	4	3	no	—
89	C₄×C₂₃	—	4	1	no	—
90	C₄×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
91	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×H(5,𝔽₂)	C₂×C₂₃	1	1	yes	characteristic
94	C₂×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
95	#520	C₂×C₂₃	1	1	yes	characteristic, complemented
96	#418	C₂×C₂₃	1	3	yes	complemented
97	#417	C₂×C₂₃	1	3	yes	complemented
98	#416	C₂×C₂₃	1	3	yes	complemented
99	C₂×#102	C₂×C₂₃	1	3	yes	complemented
100	C₂×C₂₃	—	2	6	no	—
101	C₂×C₂₃	—	4	6	no	—
102	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₂₃	#576	1	1	yes	characteristic
104	C₂×C₂₃	#577	1	1	yes	characteristic
105	H(5,𝔽₂)	—	2	4	no	—
106	H(5,𝔽₂)	C₄×C₂₃	1	8	yes	complemented
107	C₂×#41	—	2	1	no	—
108	C₂×#41	C₂²×C₂₃	1	3	yes	—
109	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
110	C₂²×D₄	—	2	3	no	—
111	C₂²×D₄	C₂²×C₂₃	1	3	yes	—
112	#131	—	2	4	no	—
113	C₂×M₁₆	C₂²×C₂₃	1	3	yes	complemented
114	C₂²×C₈	C₂²×C₂₃	1	3	yes	complemented
115	C₄×Q₈	—	2	1	no	—
116	C₄×D₄	—	2	3	no	—
117	#117	C₂²×C₂₃	1	3	yes	complemented
118	C₂×C₄⋊C₄	C₂²×C₂₃	1	3	yes	complemented
119	#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
120	#103	C₂²×C₂₃	1	3	yes	complemented
121	#102	C₂²×C₂₃	1	3	yes	complemented
122	#102	C₂²×C₂₃	1	6	yes	complemented
123	#102	C₂²×C₂₃	1	3	yes	complemented
124	C₂₃	#2897	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁴	—	2	3	no	—
126	#41	—	2	8	no	—
127	#41	—	2	6	no	—
128	#41	C₂×C₄×C₂₃	1	4	yes	—
129	#41	C₂×C₄×C₂₃	1	12	yes	complemented
130	#41	D₄×C₂₃	1	4	yes	—
131	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
132	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
133	C₂×D₄	—	2	3	no	—
134	C₂×D₄	—	2	24	no	—
135	C₂×D₄	—	2	3	no	—
136	C₂×D₄	—	2	6	no	—
137	C₂×D₄	C₂×C₄×C₂₃	1	12	yes	complemented
138	C₂×D₄	D₄×C₂₃	1	3	yes	—
139	C₂×D₄	C₂³×C₂₃	1	3	yes	—
140	C₂²×C₄	—	2	3	no	—
141	C₂²×C₄	—	2	3	no	—
142	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
143	M₁₆	—	2	6	no	—
144	C₂×C₈	—	2	6	no	—
145	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
146	C₂×C₈	C₂³×C₂₃	1	3	yes	—
147	C₄⋊C₄	—	2	3	no	—
148	C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
149	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
150	C₂²⋊C₄	—	2	3	no	—
151	C₄²	—	2	3	no	—
152	C₂³	—	2	3	no	—
153	C₂³	—	2	12	no	—
154	C₂³	—	8	3	no	—
155	C₂³	—	8	3	no	—
156	C₂³	C₂×D₄×C₂₃	1	3	yes	—
157	Q₈	—	2	1	no	—
158	Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
159	Q₈	C₂²×C₄×C₂₃	1	2	yes	—
160	D₄	—	2	12	no	—
161	D₄	—	2	12	no	—
162	D₄	—	2	3	no	—
163	D₄	C₂²⋊C₄×C₂₃	1	12	yes	—
164	D₄	C₂²×C₄×C₂₃	1	6	yes	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	12	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	3	no	—
169	C₂×C₄	—	8	3	no	—
170	C₂×C₄	C₂²×C₄×C₂₃	1	6	yes	—
171	C₂×C₄	C₂×D₄×C₂₃	1	3	yes	—
172	C₂×C₄	C₂×D₄×C₂₃	1	6	yes	—
173	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	3	no	—
176	C₂²	—	2	3	no	—
177	C₂²	—	2	3	no	—
178	C₂²	—	2	6	no	—
179	C₂²	—	4	3	no	—
180	C₂²	—	4	3	no	—
181	C₂²	—	8	12	no	—
182	C₂²	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
183	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	3	no	—
186	C₄	—	4	1	no	—
187	C₄	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
188	C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
189	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
190	C₂	—	2	6	no	—
191	C₂	—	4	6	no	—
192	C₂	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
193	C₂	C₂₃×#576	1	1	yes	characteristic
194	C₂	C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator	C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#131	C₂	semidirect
3. C₂₃×#520	C₂	semidirect
4. C₂₃×#418	C₂	semidirect
5. C₂₃×#417	C₂	semidirect
6. C₂₃×#416	C₂	semidirect
7. C₂×C₂₃×#102	C₂	semidirect
9. C₂₃×H(5,𝔽₂)	C₄	semidirect
16. C₂×M₁₆×C₂₃	C₂²	semidirect
17. C₂²×C₈×C₂₃	C₂²	semidirect
20. C₂₃×#117	C₂²	semidirect
21. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
22. C₂₃×#103	C₂²	semidirect
23. C₂₃×#103	C₂²	semidirect
24. C₂₃×#102	C₂²	semidirect
25. C₂₃×#102	C₂²	semidirect
26. C₂₃×#102	C₂²	semidirect
31. #41×C₂₃	C₂×C₄	semidirect
39. C₂×D₄×C₂₃	C₂×C₄	semidirect
78. #2897	C₂₃	direct (N × H)
94. C₂×#131	C₂×C₂₃	semidirect
95. #520	C₂×C₂₃	semidirect
96. #418	C₂×C₂₃	semidirect
97. #417	C₂×C₂₃	semidirect
98. #416	C₂×C₂₃	semidirect
99. C₂×#102	C₂×C₂₃	semidirect
106. H(5,𝔽₂)	C₄×C₂₃	semidirect
113. C₂×M₁₆	C₂²×C₂₃	semidirect
114. C₂²×C₈	C₂²×C₂₃	semidirect
117. #117	C₂²×C₂₃	semidirect
118. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
119. #103	C₂²×C₂₃	semidirect
120. #103	C₂²×C₂₃	semidirect
121. #102	C₂²×C₂₃	semidirect
122. #102	C₂²×C₂₃	semidirect
123. #102	C₂²×C₂₃	semidirect
124. C₂₃	#2897	direct (N × H)
129. #41	C₂×C₄×C₂₃	semidirect
137. C₂×D₄	C₂×C₄×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×H(5,𝔽₂)	2
C₂×C₂₃×#131	2
C₂₃×#520	2
C₂₃×#418	2
C₂₃×#417	2
C₂₃×#416	2
C₂×C₂₃×#102	2
#2897	23

Subgroups — C23×#2897 #15670516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2897 #15670516