Subgroups — C₂×C₂₃×#2161 #19181039

Order 5888.

642 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	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#108	C₂	1	2	yes	maximal
3	C₂₃×#2161	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#368	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	—	2	2	no	—
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×M₃₂	C₄	1	4	yes	—
8	C₂×C₂₃×M₃₂	C₄	1	4	yes	—
9	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
10	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#108	—	2	4	no	—
12	C₂×C₂₃×#108	—	2	2	no	—
13	C₂×C₂₃×#108	C₂²	1	4	yes	—
14	C₂×C₂₃×#108	C₂²	1	8	yes	—
15	C₂₃×#368	C₂²	1	16	yes	complemented
16	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂×M₁₆×C₂₃	—	2	8	no	—
18	C₂×M₁₆×C₂₃	—	2	4	no	—
19	C₂×M₁₆×C₂₃	—	4	4	no	—
20	C₂×M₁₆×C₂₃	—	4	2	no	—
21	C₂²×C₈×C₂₃	—	4	2	no	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₂₃	D₄	1	2	yes	—
25	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
27	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
28	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
30	C₂₃×M₃₂	C₂×C₄	1	16	yes	—
31	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
32	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
34	C₂₃×#108	—	2	4	no	—
35	C₂₃×#108	—	2	8	no	—
36	C₂₃×#108	C₂³	1	8	yes	—
37	C₂⁴×C₂₃	Q₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₂₃	D₈	1	2	yes	—
41	C₂²×C₄×C₂₃	D₈	1	2	yes	—
42	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₂₃	—	2	8	no	—
47	M₁₆×C₂₃	—	4	8	no	—
48	M₁₆×C₂₃	—	4	4	no	—
49	C₂×C₈×C₂₃	—	4	8	no	—
50	C₂×C₈×C₂₃	—	4	4	no	—
51	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
64	C₁₆×C₂₃	C₂²×C₄	1	8	yes	—
65	C₂×#2161	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	—	4	2	no	—
67	C₂³×C₂₃	#107	1	2	yes	—
68	C₂³×C₂₃	#107	1	2	yes	—
69	C₂³×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂³×C₂₃	C₂×Q₁₆	1	2	yes	—
71	C₂×C₄×C₂₃	#107	1	2	yes	—
72	C₂×C₄×C₂₃	#107	1	4	yes	—
73	C₂×C₄×C₂₃	#107	1	2	yes	—
74	C₂×C₄×C₂₃	#107	1	4	yes	—
75	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
84	C₈×C₂₃	—	4	8	no	—
85	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
88	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₂₃	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic
91	C₂²×#108	C₂×C₂₃	1	2	yes	—
92	#2161	C₂×C₂₃	1	8	yes	direct product factor
93	C₂×#368	C₂×C₂₃	1	4	yes	complemented
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	—	4	2	no	—
98	C₂²×C₂₃	C₂×#107	1	4	yes	—
99	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
100	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
101	C₂²×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
102	C₄×C₂₃	C₂×#107	1	4	yes	—
103	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
104	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
105	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
107	C₂²×M₁₆	—	2	2	no	—
108	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×M₃₂	C₄×C₂₃	1	4	yes	—
110	C₂×M₃₂	C₄×C₂₃	1	4	yes	—
111	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
112	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
113	C₂×#108	—	2	4	no	—
114	C₂×#108	—	2	2	no	—
115	C₂×#108	C₂²×C₂₃	1	4	yes	—
116	C₂×#108	C₂²×C₂₃	1	8	yes	—
117	#368	C₂²×C₂₃	1	16	yes	complemented
118	C₂×C₂₃	—	2	1	no	—
119	C₂×C₂₃	—	2	4	no	—
120	C₂×C₂₃	—	2	1	no	—
121	C₂×C₂₃	#2161	1	2	yes	direct product factor
122	C₂×C₂₃	C₂²×#107	1	1	yes	characteristic
123	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	—	2	8	no	—
126	C₂×M₁₆	—	4	2	no	—
127	C₂×M₁₆	—	4	4	no	—
128	C₂²×C₈	—	4	2	no	—
129	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
131	C₂²×C₈	D₄×C₂₃	1	2	yes	—
132	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
133	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
134	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
135	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
137	M₃₂	C₂×C₄×C₂₃	1	16	yes	—
138	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
139	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
140	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
141	#108	—	2	4	no	—
142	#108	—	2	8	no	—
143	#108	C₂³×C₂₃	1	8	yes	—
144	C₂₃	C₂×#2161	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁴	Q₁₆×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
147	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
148	C₂²×C₄	D₈×C₂₃	1	2	yes	—
149	C₂²×C₄	D₈×C₂₃	1	2	yes	—
150	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
154	M₁₆	—	2	8	no	—
155	M₁₆	—	4	8	no	—
156	M₁₆	—	4	4	no	—
157	C₂×C₈	—	4	4	no	—
158	C₂×C₈	—	4	8	no	—
159	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
160	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
161	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
162	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
163	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
164	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
166	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
167	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
168	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
170	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
171	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
172	C₁₆	C₂²×C₄×C₂₃	1	8	yes	—
173	C₂³	—	4	2	no	—
174	C₂³	C₂₃×#107	1	2	yes	—
175	C₂³	C₂₃×#107	1	2	yes	—
176	C₂³	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
177	C₂³	C₂×Q₁₆×C₂₃	1	2	yes	—
178	C₂×C₄	C₂₃×#107	1	4	yes	—
179	C₂×C₄	C₂₃×#107	1	2	yes	—
180	C₂×C₄	C₂₃×#107	1	4	yes	—
181	C₂×C₄	C₂₃×#107	1	2	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
187	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
188	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
190	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
191	C₈	—	4	8	no	—
192	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
193	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
194	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
195	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
196	C₈	C₂²×Q₈×C₂₃	1	1	yes	characteristic
197	C₂²	—	2	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	C₂×C₂₃×#107	1	4	yes	—
202	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
203	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
204	C₂²	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
205	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
206	C₄	C₂×C₂₃×#107	1	4	yes	—
207	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
208	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
209	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	4	no	—
212	C₂	—	2	1	no	—
213	C₂	C₂₃×#2161	1	2	yes	direct product factor
214	C₂	C₂²×C₂₃×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₈	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#107
prime radical	C₂⁴×C₂₃	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2161	C₂	direct (N × H)
4. C₂×C₂₃×#368	C₂	semidirect
15. C₂₃×#368	C₂²	semidirect
65. C₂×#2161	C₂₃	direct (N × H)
92. #2161	C₂×C₂₃	direct (N × H)
93. C₂×#368	C₂×C₂₃	semidirect
117. #368	C₂²×C₂₃	semidirect
121. C₂×C₂₃	#2161	direct (N × H)
144. C₂₃	C₂×#2161	direct (N × H)
213. C₂	C₂₃×#2161	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×M₃₂	2
C₂²×C₂₃×#108	2
C₂₃×#2161	2
C₂×C₂₃×#368	2
C₂×#2161	23

Subgroups — C2×C23×#2161 #19181039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2161 #19181039