Subgroups — C₂×C₂₃×#1420 #19161250

Order 5888.

286 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×M₆₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#368	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1420	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₂₃	—	2	1	no	—
6	C₂×C₂₃×Q₃₂	—	2	1	no	—
7	C₂×C₂₃×Q₃₂	—	2	1	no	—
8	C₂×C₂₃×Q₃₂	—	2	1	no	—
9	C₂×C₂₃×Q₃₂	C₄	1	2	yes	—
10	C₂×C₂₃×Q₃₂	C₄	1	2	yes	—
11	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#108	—	2	1	no	—
14	C₂₃×M₆₄	—	2	1	no	—
15	C₂₃×M₆₄	C₂²	1	2	yes	—
16	C₂×C₂₃×C₃₂	—	2	1	no	—
17	C₂₃×#368	—	2	1	no	—
18	C₂₃×#368	C₂²	1	2	yes	—
19	C₂²×Q₈×C₂₃	—	4	1	no	—
20	C₂×Q₁₆×C₂₃	—	2	2	no	—
21	C₂×Q₁₆×C₂₃	—	2	2	no	—
22	C₂×Q₁₆×C₂₃	—	2	2	no	—
23	C₂×Q₁₆×C₂₃	—	4	1	no	—
24	C₂×Q₁₆×C₂₃	—	4	1	no	—
25	C₂×Q₁₆×C₂₃	—	4	1	no	—
26	C₂×M₁₆×C₂₃	—	4	1	no	—
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₂₃×Q₃₂	—	2	2	no	—
29	C₂₃×Q₃₂	—	2	2	no	—
30	C₂₃×Q₃₂	—	2	2	no	—
31	C₂₃×Q₃₂	C₂×C₄	1	4	yes	—
32	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×#108	—	2	2	no	—
39	C₂₃×#108	—	4	1	no	—
40	C₂₃×C₃₂	—	2	2	no	—
41	C₂×Q₈×C₂₃	—	4	2	no	—
42	C₂×Q₈×C₂₃	—	4	2	no	—
43	C₂×Q₈×C₂₃	—	4	2	no	—
44	C₂×Q₈×C₂₃	—	8	1	no	—
45	C₂×Q₈×C₂₃	—	8	1	no	—
46	C₂×Q₈×C₂₃	—	8	1	no	—
47	C₂²×C₄×C₂₃	—	8	1	no	—
48	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
49	Q₁₆×C₂₃	—	2	4	no	—
50	Q₁₆×C₂₃	—	4	2	no	—
51	Q₁₆×C₂₃	—	4	2	no	—
52	Q₁₆×C₂₃	—	4	2	no	—
53	M₁₆×C₂₃	—	4	2	no	—
54	M₁₆×C₂₃	—	8	1	no	—
55	C₂×C₈×C₂₃	—	8	1	no	—
56	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
59	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
60	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
61	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₁₆×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₁₆×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂×#1420	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₃	SD₃₂	1	1	yes	characteristic
68	Q₈×C₂₃	—	4	4	no	—
69	Q₈×C₂₃	—	8	2	no	—
70	Q₈×C₂₃	—	8	2	no	—
71	Q₈×C₂₃	—	8	2	no	—
72	C₂×C₄×C₂₃	—	8	2	no	—
73	C₂×C₄×C₂₃	—	8	2	no	—
74	C₂×C₄×C₂₃	—	8	2	no	—
75	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
78	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
79	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
81	C₈×C₂₃	—	8	2	no	—
82	C₈×C₂₃	#102	1	1	yes	characteristic
83	C₈×C₂₃	#102	1	1	yes	characteristic
84	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂²×Q₃₂	C₂×C₂₃	1	1	yes	characteristic
87	C₂×M₆₄	C₂×C₂₃	1	1	yes	characteristic
88	C₂×#368	C₂×C₂₃	1	1	yes	characteristic
89	#1420	C₂×C₂₃	1	4	yes	direct product factor
90	C₂²×C₂₃	—	2	2	no	—
91	C₂²×C₂₃	#357	1	1	yes	characteristic
92	C₂²×C₂₃	#357	1	1	yes	characteristic
93	C₂²×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
94	C₄×C₂₃	—	8	4	no	—
95	C₄×C₂₃	#357	1	1	yes	characteristic
96	C₄×C₂₃	#357	1	1	yes	characteristic
97	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
98	C₄×C₂₃	C₂×D₁₆	1	1	yes	characteristic
99	C₂²×Q₁₆	—	2	1	no	—
100	C₂×Q₃₂	—	2	1	no	—
101	C₂×Q₃₂	—	2	1	no	—
102	C₂×Q₃₂	—	2	1	no	—
103	C₂×Q₃₂	C₄×C₂₃	1	2	yes	—
104	C₂×Q₃₂	C₄×C₂₃	1	2	yes	—
105	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	—
106	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
107	C₂×#108	—	2	1	no	—
108	M₆₄	—	2	1	no	—
109	M₆₄	C₂²×C₂₃	1	2	yes	—
110	C₂×C₃₂	—	2	1	no	—
111	#368	—	2	1	no	—
112	#368	C₂²×C₂₃	1	2	yes	—
113	C₂×C₂₃	—	2	1	no	—
114	C₂×C₂₃	—	2	1	no	—
115	C₂×C₂₃	#1420	1	2	yes	direct product factor
116	C₂×C₂₃	C₂×#357	1	1	yes	characteristic
117	C₂²×Q₈	—	4	1	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	4	1	no	—
122	C₂×Q₁₆	—	4	1	no	—
123	C₂×Q₁₆	—	4	1	no	—
124	C₂×M₁₆	—	4	1	no	—
125	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
126	Q₃₂	—	2	2	no	—
127	Q₃₂	—	2	2	no	—
128	Q₃₂	—	2	2	no	—
129	Q₃₂	C₂×C₄×C₂₃	1	4	yes	—
130	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
132	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
133	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
134	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
135	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
136	#108	—	2	2	no	—
137	#108	—	4	1	no	—
138	C₃₂	—	2	2	no	—
139	C₂₃	C₂×#1420	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	8	1	no	—
144	C₂×Q₈	—	8	1	no	—
145	C₂×Q₈	—	8	1	no	—
146	C₂²×C₄	—	8	1	no	—
147	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
148	Q₁₆	—	2	4	no	—
149	Q₁₆	—	4	2	no	—
150	Q₁₆	—	4	2	no	—
151	Q₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	8	1	no	—
154	C₂×C₈	—	8	1	no	—
155	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
158	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
159	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
161	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
162	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
163	C₁₆	C₂²×C₄×C₂₃	1	1	yes	characteristic
164	C₁₆	C₂×D₄×C₂₃	1	1	yes	characteristic
165	C₂³	C₂₃×SD₃₂	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	Q₈	—	8	2	no	—
168	Q₈	—	8	2	no	—
169	Q₈	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
174	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
175	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
176	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
177	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
178	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
179	C₈	—	8	2	no	—
180	C₈	C₂₃×#102	1	1	yes	characteristic
181	C₈	C₂₃×#102	1	1	yes	characteristic
182	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
183	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	C₂₃×#357	1	1	yes	characteristic
186	C₂²	C₂₃×#357	1	1	yes	characteristic
187	C₂²	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	C₂₃×#357	1	1	yes	characteristic
190	C₄	C₂₃×#357	1	1	yes	characteristic
191	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
192	C₄	C₂×C₂₃×D₁₆	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₂₃×#1420	1	2	yes	direct product factor
196	C₂	C₂×C₂₃×#357	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#357
commutator	C₁₆	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
prime radical	C₂³×C₂₃	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ C₂²×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1420	C₂	direct (N × H)
66. C₂×#1420	C₂₃	direct (N × H)
89. #1420	C₂×C₂₃	direct (N × H)
115. C₂×C₂₃	#1420	direct (N × H)
139. C₂₃	C₂×#1420	direct (N × H)
195. C₂	C₂₃×#1420	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×Q₃₂	2
C₂×C₂₃×M₆₄	2
C₂×C₂₃×#368	2
C₂₃×#1420	2
C₂×#1420	23

Subgroups — C2×C23×#1420 #19161250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1420 #19161250