Subgroups — C₂²×#17989671 #34034486

Order 18688.

1182 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#10596842	C₂	1	2	yes	maximal
2	C₂×#17989671	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂³×#10596842	C₄	1	4	yes	—
5	C₂×C₄×#10596842	C₄	1	12	yes	—
6	C₂×C₄×#10596842	C₂²	1	12	yes	—
7	#17989671	C₂²	1	16	yes	direct product factor
8	C₂²×#11924036	C₄	1	1	yes	characteristic
9	C₂²×C₄×#64408	C₂²	1	1	yes	characteristic
10	C₂×C₈×#64408	C₄	1	6	yes	—
11	C₂×C₈×#64408	C₂²	1	6	yes	—
12	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
13	C₂²×#10596842	—	2	16	no	—
14	C₂²×#10596842	C₂×C₄	1	24	yes	—
15	C₄×#10596842	C₂×C₄	1	24	yes	—
16	C₄×#10596842	C₂³	1	8	yes	—
17	C₂³×#64408	C₂×C₄	1	2	yes	—
18	C₂×#11924036	C₈	1	6	yes	complemented
19	C₂×#11924036	C₂×C₄	1	6	yes	—
20	C₂²×#10596833	C₈	1	2	yes	complemented
21	C₂×C₄×#64408	C₂×C₄	1	6	yes	—
22	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
23	C₂×C₄×#64408	C₂³	1	3	yes	—
24	C₈×#64408	C₂×C₄	1	12	yes	—
25	C₈×#64408	C₂³	1	4	yes	—
26	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₈	1	1	yes	characteristic, complemented
28	C₂×C₈×D₇₃	C₈	1	6	yes	complemented
29	C₂×C₈×D₇₃	C₂×C₄	1	6	yes	—
30	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic, complemented
31	C₂×#10596842	—	2	48	no	—
32	C₂×#10596842	C₂²×C₄	1	16	yes	—
33	C₂²×#64408	C₄²	1	6	yes	—
34	C₂²×#64408	M₁₆	1	16	yes	—
35	C₂²×#64408	C₂²×C₄	1	6	yes	—
36	#11924036	C₂×C₈	1	12	yes	complemented
37	#11924036	C₂²×C₄	1	4	yes	—
38	C₂×#10596833	C₂×C₈	1	12	yes	complemented
39	C₄×#64408	C₄²	1	6	yes	—
40	C₄×#64408	C₂²×C₄	1	6	yes	—
41	C₄×#64408	C₂²×C₄	1	3	yes	—
42	C₄×#64408	C₂⁴	1	1	yes	characteristic
43	C₂³×D₇₃	C₄²	1	1	yes	characteristic
44	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×D₇₃	C₄²	1	3	yes	—
46	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
47	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
48	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
49	C₂×#10596831	C₂×C₈	1	6	yes	complemented
50	C₈×D₇₃	C₂×C₈	1	12	yes	complemented
51	C₈×D₇₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₂×C₈	1	6	yes	complemented
54	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
55	C₂²×D₇₃	C₄×C₈	1	3	yes	—
56	C₂²×D₇₃	#97	1	8	yes	—
57	C₂²×D₇₃	C₂×C₄²	1	3	yes	—
58	C₂×#64408	#97	1	24	yes	—
59	C₂×#64408	C₂×C₄²	1	6	yes	—
60	C₂×#64408	C₂×M₁₆	1	24	yes	—
61	C₂×#64408	C₂³×C₄	1	2	yes	—
62	#10596842	—	2	16	no	—
63	C₂×C₄×C₇₃	C₄×C₈	1	3	yes	—
64	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
65	C₂×Dic₇₃	C₄×C₈	1	3	yes	—
66	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
67	C₄×D₇₃	C₄×C₈	1	6	yes	—
68	C₄×D₇₃	C₂×C₄²	1	3	yes	—
69	C₄×D₇₃	C₂²×C₈	1	3	yes	—
70	C₄×D₇₃	C₂²×C₈	1	3	yes	—
71	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
72	#10596833	C₂²×C₈	1	8	yes	complemented
73	C₈×C₇₃	C₂²×C₈	1	4	yes	complemented
74	#10596831	C₂²×C₈	1	4	yes	complemented
75	C₂²×C₇₃	#322	1	4	yes	complemented
76	C₂²×C₇₃	C₂×C₄×C₈	1	3	yes	—
77	C₂×D₇₃	#322	1	12	yes	complemented
78	C₂×D₇₃	C₂×C₄×C₈	1	3	yes	—
79	C₂×D₇₃	C₂×#97	1	12	yes	—
80	C₂×D₇₃	C₂²×C₄²	1	1	yes	characteristic
81	#64408	C₂×#97	1	12	yes	—
82	#64408	C₂²×M₁₆	1	4	yes	—
83	C₄×C₇₃	C₂×C₄×C₈	1	3	yes	—
84	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
85	Dic₇₃	C₂×C₄×C₈	1	3	yes	—
86	Dic₇₃	C₂³×C₈	1	1	yes	characteristic
87	C₂²×#322	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	C₂×#322	1	6	yes	complemented
89	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
90	D₇₃	C₂×#322	1	6	yes	complemented
91	D₇₃	C₂²×#97	1	2	yes	—
92	C₂²×C₄×C₈	—	73	2	no	—
93	C₂²×C₄×C₈	—	73	1	no	—
94	C₂×#322	—	73	12	no	—
95	C₇₃	C₂²×#322	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₈	—	73	1	no	—
97	C₂³×C₈	—	73	1	no	—
98	C₂³×C₈	—	73	4	no	—
99	C₂²×C₄²	—	73	1	no	—
100	C₂×C₄×C₈	—	73	12	no	—
101	C₂×C₄×C₈	—	73	12	no	—
102	C₂×C₄×C₈	—	73	6	no	—
103	C₂×C₄×C₈	—	73	6	no	—
104	#322	—	73	16	no	—
105	C₂³×C₄	—	73	2	no	—
106	C₂³×C₄	—	73	1	no	—
107	C₂²×C₈	—	73	2	no	—
108	C₂²×C₈	—	73	6	no	—
109	C₂²×C₈	—	73	6	no	—
110	C₂²×C₈	—	73	1	no	—
111	C₂²×C₈	—	73	6	no	—
112	C₂²×C₈	—	73	6	no	—
113	C₂²×C₈	—	73	24	no	—
114	C₂²×C₈	—	146	16	no	—
115	C₂²×C₈	#10596842	1	1	yes	characteristic, complemented
116	C₂×C₄²	—	73	3	no	—
117	C₂×C₄²	—	73	3	no	—
118	C₂×C₄²	—	73	6	no	—
119	C₄×C₈	—	73	24	no	—
120	C₄×C₈	—	73	8	no	—
121	C₄×C₈	—	73	4	no	—
122	C₄×C₈	—	73	12	no	—
123	C₂⁴	—	73	1	no	—
124	C₂²×C₄	—	73	6	no	—
125	C₂²×C₄	—	73	16	no	—
126	C₂²×C₄	—	73	6	no	—
127	C₂²×C₄	—	73	1	no	—
128	C₂²×C₄	—	73	3	no	—
129	C₂²×C₄	—	73	3	no	—
130	C₂²×C₄	—	73	3	no	—
131	C₂²×C₄	—	73	3	no	—
132	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
133	C₂×C₈	—	73	12	no	—
134	C₂×C₈	—	73	12	no	—
135	C₂×C₈	—	73	4	no	—
136	C₂×C₈	—	73	6	no	—
137	C₂×C₈	—	73	12	no	—
138	C₂×C₈	—	73	4	no	—
139	C₂×C₈	—	73	16	no	—
140	C₂×C₈	—	146	48	no	—
141	C₂×C₈	C₂×#10596842	1	6	yes	complemented
142	C₄²	—	73	6	no	—
143	C₄²	—	73	6	no	—
144	C₄²	—	73	3	no	—
145	C₄²	—	73	1	no	—
146	C₂³	—	73	8	no	—
147	C₂³	—	73	3	no	—
148	C₂³	—	73	3	no	—
149	C₂³	C₄×#10596842	1	1	yes	characteristic
150	C₂×C₄	—	73	2	no	—
151	C₂×C₄	—	73	6	no	—
152	C₂×C₄	—	73	24	no	—
153	C₂×C₄	—	73	24	no	—
154	C₂×C₄	—	73	3	no	—
155	C₂×C₄	—	73	3	no	—
156	C₂×C₄	—	73	3	no	—
157	C₂×C₄	—	73	3	no	—
158	C₂×C₄	—	73	6	no	—
159	C₂×C₄	—	73	1	no	—
160	C₂×C₄	—	73	3	no	—
161	C₂×C₄	C₄×#10596842	1	3	yes	—
162	C₂×C₄	C₂²×#10596842	1	3	yes	—
163	C₈	—	73	8	no	—
164	C₈	—	73	4	no	—
165	C₈	—	146	16	no	—
166	C₈	C₂²×#10596842	1	4	yes	complemented
167	C₂²	—	73	1	no	—
168	C₂²	—	73	3	no	—
169	C₂²	—	73	12	no	—
170	C₂²	—	73	12	no	—
171	C₂²	#17989671	1	4	yes	direct product factor
172	C₂²	C₂×C₄×#10596842	1	3	yes	—
173	C₄	—	73	4	no	—
174	C₄	—	73	12	no	—
175	C₄	—	73	1	no	—
176	C₄	—	73	3	no	—
177	C₄	C₂×C₄×#10596842	1	3	yes	—
178	C₄	C₂³×#10596842	1	1	yes	characteristic
179	C₂	—	73	2	no	—
180	C₂	—	73	6	no	—
181	C₂	C₂×#17989671	1	6	yes	direct product factor
182	C₂	C₂²×C₄×#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#10596842
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₄	C₂³×#10596842
Fitting	C₂²×C₈×C₇₃	C₈
socle	C₂³×C₇₃	C₄×C₈
prime radical	C₂³×D₇₃	C₄²
cosocle	C₄×#64408	C₂⁴
hypercenter	C₂²×C₈	#10596842
hypocenter	C₇₃	C₂²×#322
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17989671	C₂	direct (N × H)
7. #17989671	C₂²	direct (N × H)
18. C₂×#11924036	C₈	semidirect
20. C₂²×#10596833	C₈	semidirect
27. C₂²×#10596831	C₈	semidirect
28. C₂×C₈×D₇₃	C₈	semidirect
30. C₂²×C₈×C₇₃	C₈	semidirect
36. #11924036	C₂×C₈	semidirect
38. C₂×#10596833	C₂×C₈	semidirect
49. C₂×#10596831	C₂×C₈	semidirect
50. C₈×D₇₃	C₂×C₈	semidirect
53. C₂×C₈×C₇₃	C₂×C₈	semidirect
72. #10596833	C₂²×C₈	semidirect
73. C₈×C₇₃	C₂²×C₈	semidirect
74. #10596831	C₂²×C₈	semidirect
75. C₂²×C₇₃	#322	semidirect
77. C₂×D₇₃	#322	semidirect
88. C₂×C₇₃	C₂×#322	semidirect
90. D₇₃	C₂×#322	semidirect
95. C₇₃	C₂²×#322	semidirect
115. C₂²×C₈	#10596842	semidirect
141. C₂×C₈	C₂×#10596842	semidirect
166. C₈	C₂²×#10596842	semidirect
171. C₂²	#17989671	direct (N × H)
181. C₂	C₂×#17989671	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×C₄×#10596842	2
C₂×#17989671	2
C₂²×C₈×#64408	2
C₂²×#322	73

Subgroups — C22×#17989671 #34034486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989671 #34034486