Subgroups — C₂×C₂₃×#3407 #19210249

Order 5888.

2074 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	C₂²×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3407	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#504	C₂	1	8	yes	maximal, complemented
4	C₂²×C₂₃×M₃₂	C₂	1	6	yes	maximal, complemented
5	C₂²×#41×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₂₃×#131	C₄	1	4	yes	—
7	C₂×C₂₃×#131	C₄	1	8	yes	—
8	C₂×C₂₃×#131	C₂²	1	4	yes	—
9	C₂×C₂₃×#131	C₂²	1	8	yes	—
10	C₂²×M₁₆×C₂₃	C₄	1	3	yes	—
11	C₂³×C₈×C₂₃	C₂²	1	3	yes	—
12	C₂₃×#504	C₂²	1	64	yes	complemented
13	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
14	C₂×C₂₃×M₃₂	C₂²	1	12	yes	complemented
15	C₂×C₂₃×M₃₂	C₂²	1	48	yes	complemented
16	C₂²×C₁₆×C₂₃	C₂²	1	12	yes	complemented
17	C₂×#41×C₂₃	C₈	1	4	yes	—
18	C₂×#41×C₂₃	C₈	1	8	yes	—
19	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
20	C₂×#41×C₂₃	C₂×C₄	1	8	yes	—
21	C₂²×Q₈×C₂₃	C₈	1	1	yes	characteristic
22	C₂²×D₄×C₂₃	C₈	1	3	yes	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	3	yes	—
24	C₂₃×#131	C₂×C₄	1	16	yes	—
25	C₂₃×#131	C₂×C₄	1	32	yes	—
26	C₂₃×#131	C₂³	1	16	yes	—
27	C₂×M₁₆×C₂₃	C₂×C₄	1	12	yes	—
28	C₂×M₁₆×C₂₃	C₂×C₄	1	24	yes	—
29	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	6	yes	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	12	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
35	C₂₃×M₃₂	C₂³	1	16	yes	—
36	C₂₃×M₃₂	C₂³	1	48	yes	complemented
37	C₂×C₁₆×C₂₃	C₂³	1	8	yes	—
38	C₂×C₁₆×C₂₃	C₂³	1	48	yes	complemented
39	C₂⁴×C₂₃	C₂×C₈	1	3	yes	—
40	#41×C₂₃	C₂×C₈	1	16	yes	—
41	#41×C₂₃	C₂×C₈	1	32	yes	—
42	#41×C₂₃	C₂²×C₄	1	16	yes	—
43	C₂×Q₈×C₂₃	C₂×C₈	1	8	yes	—
44	C₂×Q₈×C₂₃	C₂×C₈	1	4	yes	—
45	C₂×D₄×C₂₃	C₂×C₈	1	12	yes	—
46	C₂×D₄×C₂₃	C₂×C₈	1	24	yes	—
47	C₂²×C₄×C₂₃	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₂₃	C₂×C₈	1	6	yes	—
49	C₂²×C₄×C₂₃	C₂×C₈	1	12	yes	—
50	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
53	M₁₆×C₂₃	C₂²×C₄	1	48	yes	—
54	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₂₃	C₂²×C₄	1	12	yes	—
57	C₂×C₈×C₂₃	C₂²×C₄	1	6	yes	—
58	C₂×C₈×C₂₃	C₂²×C₄	1	24	yes	—
59	C₂×C₈×C₂₃	C₂²×C₄	1	12	yes	—
60	C₂×C₈×C₂₃	C₂⁴	1	12	yes	—
61	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
62	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
63	C₁₆×C₂₃	C₂⁴	1	16	yes	—
64	C₂×#3407	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₂₃	—	4	6	no	—
66	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
67	C₂³×C₂₃	C₂²×C₈	1	6	yes	—
68	C₂³×C₂₃	C₂²×C₈	1	12	yes	—
69	Q₈×C₂₃	C₂²×C₈	1	16	yes	—
70	D₄×C₂₃	C₂²×C₈	1	48	yes	—
71	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂²×C₈	1	12	yes	—
74	C₂×C₄×C₂₃	C₂²×C₈	1	6	yes	—
75	C₂×C₄×C₂₃	C₂²×C₈	1	24	yes	—
76	C₂×C₄×C₂₃	C₂²×C₈	1	12	yes	—
77	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
78	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
80	C₈×C₂₃	C₂³×C₄	1	2	yes	—
81	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
82	C₈×C₂₃	C₂³×C₄	1	12	yes	—
83	C₈×C₂₃	C₂⁵	1	1	yes	characteristic
84	C₂²×#131	C₂×C₂₃	1	1	yes	characteristic
85	#3407	C₂×C₂₃	1	16	yes	direct product factor
86	C₂×#504	C₂×C₂₃	1	8	yes	complemented
87	C₂²×M₃₂	C₂×C₂₃	1	6	yes	complemented
88	C₂²×C₂₃	—	2	2	no	—
89	C₂²×C₂₃	—	2	12	no	—
90	C₂²×C₂₃	—	4	12	no	—
91	C₂²×C₂₃	C₂³×C₈	1	2	yes	—
92	C₂²×C₂₃	C₂³×C₈	1	12	yes	—
93	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₂₃	C₂³×C₈	1	2	yes	—
95	C₄×C₂₃	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₂₃	C₂³×C₈	1	12	yes	—
97	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
98	C₂²×#41	C₄×C₂₃	1	1	yes	characteristic
99	C₂×#131	C₄×C₂₃	1	4	yes	—
100	C₂×#131	C₄×C₂₃	1	8	yes	—
101	C₂×#131	C₂²×C₂₃	1	4	yes	—
102	C₂×#131	C₂²×C₂₃	1	8	yes	—
103	C₂²×M₁₆	C₄×C₂₃	1	3	yes	—
104	C₂³×C₈	C₂²×C₂₃	1	3	yes	—
105	#504	C₂²×C₂₃	1	64	yes	complemented
106	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
107	C₂×M₃₂	C₂²×C₂₃	1	12	yes	complemented
108	C₂×M₃₂	C₂²×C₂₃	1	48	yes	complemented
109	C₂²×C₁₆	C₂²×C₂₃	1	12	yes	complemented
110	C₂×C₂₃	—	2	2	no	—
111	C₂×C₂₃	—	2	12	no	—
112	C₂×C₂₃	#3407	1	2	yes	direct product factor
113	C₂×C₂₃	C₂⁴×C₈	1	1	yes	characteristic
114	C₂×#41	C₈×C₂₃	1	4	yes	—
115	C₂×#41	C₈×C₂₃	1	8	yes	—
116	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
117	C₂×#41	C₂×C₄×C₂₃	1	8	yes	—
118	C₂²×Q₈	C₈×C₂₃	1	1	yes	characteristic
119	C₂²×D₄	C₈×C₂₃	1	3	yes	—
120	C₂³×C₄	C₂×C₄×C₂₃	1	3	yes	—
121	#131	C₂×C₄×C₂₃	1	16	yes	—
122	#131	C₂×C₄×C₂₃	1	32	yes	—
123	#131	C₂³×C₂₃	1	16	yes	—
124	C₂×M₁₆	C₂×C₄×C₂₃	1	12	yes	—
125	C₂×M₁₆	C₂×C₄×C₂₃	1	24	yes	—
126	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
127	C₂²×C₈	C₂×C₄×C₂₃	1	6	yes	—
128	C₂²×C₈	C₂×C₄×C₂₃	1	12	yes	—
129	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
131	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
132	M₃₂	C₂³×C₂₃	1	16	yes	—
133	M₃₂	C₂³×C₂₃	1	48	yes	complemented
134	C₂×C₁₆	C₂³×C₂₃	1	8	yes	—
135	C₂×C₁₆	C₂³×C₂₃	1	48	yes	complemented
136	C₂₃	C₂×#3407	1	1	yes	characteristic, direct product factor, Sylow
137	C₂⁴	C₂×C₈×C₂₃	1	3	yes	—
138	#41	C₂×C₈×C₂₃	1	16	yes	—
139	#41	C₂×C₈×C₂₃	1	32	yes	—
140	#41	C₂²×C₄×C₂₃	1	16	yes	—
141	C₂×Q₈	C₂×C₈×C₂₃	1	8	yes	—
142	C₂×Q₈	C₂×C₈×C₂₃	1	4	yes	—
143	C₂×D₄	C₂×C₈×C₂₃	1	12	yes	—
144	C₂×D₄	C₂×C₈×C₂₃	1	24	yes	—
145	C₂²×C₄	C₂×C₈×C₂₃	1	3	yes	—
146	C₂²×C₄	C₂×C₈×C₂₃	1	6	yes	—
147	C₂²×C₄	C₂×C₈×C₂₃	1	12	yes	—
148	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
150	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
151	M₁₆	C₂²×C₄×C₂₃	1	48	yes	—
152	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
153	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₈	C₂²×C₄×C₂₃	1	6	yes	—
155	C₂×C₈	C₂²×C₄×C₂₃	1	12	yes	—
156	C₂×C₈	C₂²×C₄×C₂₃	1	12	yes	—
157	C₂×C₈	C₂²×C₄×C₂₃	1	24	yes	—
158	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
159	C₂×C₈	C₂⁴×C₂₃	1	12	yes	—
160	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
161	C₁₆	C₂⁴×C₂₃	1	16	yes	—
162	C₂³	—	4	6	no	—
163	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
164	C₂³	C₂²×C₈×C₂₃	1	6	yes	—
165	C₂³	C₂²×C₈×C₂₃	1	12	yes	—
166	Q₈	C₂²×C₈×C₂₃	1	16	yes	—
167	D₄	C₂²×C₈×C₂₃	1	48	yes	—
168	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
169	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₈×C₂₃	1	12	yes	—
171	C₂×C₄	C₂²×C₈×C₂₃	1	6	yes	—
172	C₂×C₄	C₂²×C₈×C₂₃	1	24	yes	—
173	C₂×C₄	C₂²×C₈×C₂₃	1	12	yes	—
174	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
175	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
176	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
177	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
178	C₈	C₂³×C₄×C₂₃	1	2	yes	—
179	C₈	C₂³×C₄×C₂₃	1	12	yes	—
180	C₈	C₂⁵×C₂₃	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	2	12	no	—
183	C₂²	—	4	12	no	—
184	C₂²	C₂³×C₈×C₂₃	1	2	yes	—
185	C₂²	C₂³×C₈×C₂₃	1	12	yes	—
186	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
187	C₄	C₂³×C₈×C₂₃	1	1	yes	characteristic
188	C₄	C₂³×C₈×C₂₃	1	2	yes	—
189	C₄	C₂³×C₈×C₂₃	1	12	yes	—
190	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	12	no	—
193	C₂	C₂₃×#3407	1	2	yes	direct product factor
194	C₂	C₂⁴×C₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂	C₂⁴×C₈×C₂₃
Frattini, cosocle	C₈	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂³×C₈
prime radical	C₂²×#41×C₂₃	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
2. C₂₃×#3407	C₂	direct (N × H)
3. C₂×C₂₃×#504	C₂	semidirect
4. C₂²×C₂₃×M₃₂	C₂	semidirect
12. C₂₃×#504	C₂²	semidirect
14. C₂×C₂₃×M₃₂	C₂²	semidirect
15. C₂×C₂₃×M₃₂	C₂²	semidirect
16. C₂²×C₁₆×C₂₃	C₂²	semidirect
36. C₂₃×M₃₂	C₂³	semidirect
38. C₂×C₁₆×C₂₃	C₂³	semidirect
64. C₂×#3407	C₂₃	direct (N × H)
85. #3407	C₂×C₂₃	direct (N × H)
86. C₂×#504	C₂×C₂₃	semidirect
87. C₂²×M₃₂	C₂×C₂₃	semidirect
105. #504	C₂²×C₂₃	semidirect
107. C₂×M₃₂	C₂²×C₂₃	semidirect
108. C₂×M₃₂	C₂²×C₂₃	semidirect
109. C₂²×C₁₆	C₂²×C₂₃	semidirect
112. C₂×C₂₃	#3407	direct (N × H)
133. M₃₂	C₂³×C₂₃	semidirect
135. C₂×C₁₆	C₂³×C₂₃	semidirect
136. C₂₃	C₂×#3407	direct (N × H)
193. C₂	C₂₃×#3407	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×C₂₃×#131	2
C₂₃×#3407	2
C₂×C₂₃×#504	2
C₂²×C₂₃×M₃₂	2
C₂×#3407	23

Subgroups — C2×C23×#3407 #19210249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3407 #19210249