Subgroups — C₂₃×#33326 #19180952

Order 5888.

818 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#504	C₂	1	2	yes	maximal
2	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2108	C₂	1	6	yes	maximal, complemented
4	C₄×C₂₃×M₃₂	C₂	1	3	yes	maximal, complemented
5	C₂×C₂₃×#346	C₂	1	3	yes	maximal, complemented
6	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
8	C₄×#41×C₂₃	C₄	1	1	yes	characteristic
9	C₂₃×#504	C₄	1	16	yes	complemented
10	C₂×C₂₃×M₃₂	C₂²	1	6	yes	—
11	C₂²×C₁₆×C₂₃	C₂²	1	6	yes	—
12	C₂₃×#405	C₄	1	3	yes	—
13	C₂₃×#405	C₂²	1	3	yes	—
14	C₄×M₁₆×C₂₃	C₄	1	3	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
16	C₂₃×#346	C₂²	1	1	yes	characteristic
17	C₂₃×#346	C₂²	1	3	yes	—
18	C₂₃×#346	C₂²	1	6	yes	—
19	C₄×C₁₆×C₂₃	C₂²	1	6	yes	—
20	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂₃×#131	C₈	1	8	yes	—
22	C₂₃×#131	C₂×C₄	1	8	yes	—
23	C₂×M₁₆×C₂₃	C₂×C₄	1	3	yes	—
24	C₂×M₁₆×C₂₃	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
27	C₄×Q₈×C₂₃	C₈	1	2	yes	—
28	C₄×D₄×C₂₃	C₈	1	6	yes	—
29	C₂₃×#117	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
31	C₂₃×M₃₂	C₂×C₄	1	24	yes	complemented
32	C₂×C₁₆×C₂₃	C₂×C₄	1	24	yes	complemented
33	C₂×C₁₆×C₂₃	C₂³	1	6	yes	—
34	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
35	C₂₃×#97	C₂×C₄	1	6	yes	—
36	C₄×C₈×C₂₃	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₂₃	C₂×C₄	1	3	yes	—
38	C₄×C₈×C₂₃	C₂³	1	3	yes	—
39	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	#41×C₂₃	C₄²	1	4	yes	—
41	#41×C₂₃	C₂×C₈	1	4	yes	—
42	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
43	C₂×D₄×C₂₃	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₂₃	C₂×C₈	1	6	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
46	M₁₆×C₂₃	C₄²	1	12	yes	—
47	M₁₆×C₂₃	C₂×C₈	1	12	yes	—
48	C₂×C₈×C₂₃	C₂×C₈	1	12	yes	—
49	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₂₃	C₂²×C₄	1	12	yes	—
51	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	C₂×C₈	1	6	yes	—
55	C₂²⋊C₄×C₂₃	C₂×C₈	1	6	yes	—
56	C₄²×C₂₃	C₂×C₈	1	6	yes	—
57	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
59	C₁₆×C₂₃	C₂²×C₄	1	4	yes	—
60	C₁₆×C₂₃	C₂²×C₄	1	12	yes	—
61	#33326	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₂₃	C₂²×C₈	1	3	yes	—
63	Q₈×C₂₃	C₄×C₈	1	4	yes	—
64	D₄×C₂₃	C₄×C₈	1	12	yes	—
65	C₂×C₄×C₂₃	—	4	6	no	—
66	C₂×C₄×C₂₃	C₂×C₄²	1	6	yes	—
67	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₂₃	C₂²×C₈	1	6	yes	—
69	C₂×C₄×C₂₃	C₂²×C₈	1	6	yes	—
70	C₂×C₄×C₂₃	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₂₃	C₂×C₄²	1	6	yes	—
73	C₈×C₂₃	C₂²×C₈	1	2	yes	—
74	C₈×C₂₃	C₂²×C₈	1	6	yes	—
75	C₈×C₂₃	C₂³×C₄	1	2	yes	—
76	C₂×#504	C₂×C₂₃	1	2	yes	—
77	C₄×#131	C₂×C₂₃	1	1	yes	characteristic
78	#2108	C₂×C₂₃	1	6	yes	complemented
79	C₄×M₃₂	C₂×C₂₃	1	3	yes	complemented
80	C₂×#346	C₂×C₂₃	1	3	yes	complemented
81	C₂²×C₂₃	—	2	6	no	—
82	C₂²×C₂₃	C₂×C₄×C₈	1	6	yes	—
83	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
84	C₄×C₂₃	—	2	2	no	—
85	C₄×C₂₃	—	2	6	no	—
86	C₄×C₂₃	C₂×C₄×C₈	1	6	yes	—
87	C₄×C₂₃	C₂²×C₄²	1	1	yes	characteristic
88	C₄×C₂₃	C₂³×C₈	1	1	yes	characteristic
89	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
90	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
91	C₄×#41	C₄×C₂₃	1	1	yes	characteristic
92	#504	C₄×C₂₃	1	16	yes	complemented
93	C₂×M₃₂	C₂²×C₂₃	1	6	yes	—
94	C₂²×C₁₆	C₂²×C₂₃	1	6	yes	—
95	#405	C₄×C₂₃	1	3	yes	—
96	#405	C₂²×C₂₃	1	3	yes	—
97	C₄×M₁₆	C₄×C₂₃	1	3	yes	—
98	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
99	#346	C₂²×C₂₃	1	1	yes	characteristic
100	#346	C₂²×C₂₃	1	3	yes	—
101	#346	C₂²×C₂₃	1	6	yes	—
102	C₄×C₁₆	C₂²×C₂₃	1	6	yes	—
103	C₂×C₂₃	—	2	6	no	—
104	C₂×C₂₃	C₂²×C₄×C₈	1	1	yes	characteristic
105	C₂×C₂₃	#3407	1	2	yes	—
106	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
107	#131	C₈×C₂₃	1	8	yes	—
108	#131	C₂×C₄×C₂₃	1	8	yes	—
109	C₂×M₁₆	C₂×C₄×C₂₃	1	3	yes	—
110	C₂×M₁₆	C₂×C₄×C₂₃	1	3	yes	—
111	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
112	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
113	C₄×Q₈	C₈×C₂₃	1	2	yes	—
114	C₄×D₄	C₈×C₂₃	1	6	yes	—
115	#117	C₂×C₄×C₂₃	1	3	yes	—
116	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
117	M₃₂	C₂×C₄×C₂₃	1	24	yes	complemented
118	C₂×C₁₆	C₂×C₄×C₂₃	1	24	yes	complemented
119	C₂×C₁₆	C₂³×C₂₃	1	6	yes	—
120	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
121	#97	C₂×C₄×C₂₃	1	6	yes	—
122	C₄×C₈	C₂×C₄×C₂₃	1	3	yes	—
123	C₄×C₈	C₂×C₄×C₂₃	1	3	yes	—
124	C₄×C₈	C₂³×C₂₃	1	3	yes	—
125	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
126	C₂₃	#33326	1	1	yes	characteristic, direct product factor, Sylow
127	#41	C₄²×C₂₃	1	4	yes	—
128	#41	C₂×C₈×C₂₃	1	4	yes	—
129	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
130	C₂×D₄	C₂×C₈×C₂₃	1	3	yes	—
131	C₂²×C₄	C₂×C₈×C₂₃	1	6	yes	—
132	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
133	M₁₆	C₄²×C₂₃	1	12	yes	—
134	M₁₆	C₂×C₈×C₂₃	1	12	yes	—
135	C₂×C₈	C₂×C₈×C₂₃	1	12	yes	—
136	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
137	C₂×C₈	C₂²×C₄×C₂₃	1	12	yes	—
138	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
140	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
141	C₄⋊C₄	C₂×C₈×C₂₃	1	6	yes	—
142	C₂²⋊C₄	C₂×C₈×C₂₃	1	6	yes	—
143	C₄²	C₂×C₈×C₂₃	1	6	yes	—
144	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
145	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
146	C₁₆	C₂²×C₄×C₂₃	1	4	yes	—
147	C₁₆	C₂²×C₄×C₂₃	1	12	yes	—
148	C₂³	C₂²×C₈×C₂₃	1	3	yes	—
149	Q₈	C₄×C₈×C₂₃	1	4	yes	—
150	D₄	C₄×C₈×C₂₃	1	12	yes	—
151	C₂×C₄	—	4	6	no	—
152	C₂×C₄	C₂×C₄²×C₂₃	1	6	yes	—
153	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
154	C₂×C₄	C₂²×C₈×C₂₃	1	6	yes	—
155	C₂×C₄	C₂²×C₈×C₂₃	1	6	yes	—
156	C₂×C₄	C₂²×C₈×C₂₃	1	3	yes	—
157	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
158	C₈	C₂×C₄²×C₂₃	1	6	yes	—
159	C₈	C₂²×C₈×C₂₃	1	2	yes	—
160	C₈	C₂²×C₈×C₂₃	1	6	yes	—
161	C₈	C₂³×C₄×C₂₃	1	2	yes	—
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×C₈×C₂₃	1	6	yes	—
164	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	6	no	—
167	C₄	C₂×C₄×C₈×C₂₃	1	6	yes	—
168	C₄	C₂²×C₄²×C₂₃	1	1	yes	characteristic
169	C₄	C₂³×C₈×C₂₃	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×C₈×C₂₃	1	1	yes	characteristic
172	C₂	C₂₃×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂	C₂²×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₈
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2108	C₂	semidirect
4. C₄×C₂₃×M₃₂	C₂	semidirect
5. C₂×C₂₃×#346	C₂	semidirect
9. C₂₃×#504	C₄	semidirect
31. C₂₃×M₃₂	C₂×C₄	semidirect
32. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
61. #33326	C₂₃	direct (N × H)
78. #2108	C₂×C₂₃	semidirect
79. C₄×M₃₂	C₂×C₂₃	semidirect
80. C₂×#346	C₂×C₂₃	semidirect
92. #504	C₄×C₂₃	semidirect
117. M₃₂	C₂×C₄×C₂₃	semidirect
118. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
126. C₂₃	#33326	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#504	2
C₄×C₂₃×#131	2
C₂₃×#2108	2
C₄×C₂₃×M₃₂	2
C₂×C₂₃×#346	2
#33326	23

Subgroups — C23×#33326 #19180952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33326 #19180952