Subgroups — C₂³×C₈×#64408 #34031058

Order 18688.

4638 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924036	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×#64408	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×#64408	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×#64408	C₄	1	2	yes	—
6	C₂²×#11924036	C₄	1	14	yes	complemented
7	C₂²×#11924036	C₂²	1	14	yes	—
8	C₂³×#10596833	C₄	1	2	yes	complemented
9	C₂²×C₄×#64408	C₄	1	14	yes	—
10	C₂²×C₄×#64408	C₂²	1	14	yes	—
11	C₂×C₈×#64408	C₂²	1	112	yes	direct product factor
12	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
13	C₂³×#10596831	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₈×D₇₃	C₄	1	14	yes	complemented
15	C₂²×C₈×D₇₃	C₂²	1	14	yes	—
16	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂³×#64408	C₈	1	32	yes	direct product factor
18	C₂³×#64408	C₂×C₄	1	28	yes	—
19	C₂×#11924036	C₂×C₄	1	84	yes	complemented
20	C₂×#11924036	C₂³	1	28	yes	—
21	C₂²×#10596833	C₂×C₄	1	28	yes	complemented
22	C₂×C₄×#64408	C₂×C₄	1	84	yes	—
23	C₂×C₄×#64408	C₂³	1	28	yes	—
24	C₈×#64408	C₂³	1	64	yes	direct product factor
25	C₂⁴×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₇₃	C₂×C₄	1	7	yes	—
28	C₂²×C₄×D₇₃	C₂×C₄	1	7	yes	—
29	C₂²×C₄×D₇₃	C₂×C₄	1	7	yes	—
30	C₂²×C₄×D₇₃	C₂³	1	7	yes	—
31	C₂²×#10596831	C₂×C₄	1	14	yes	complemented
32	C₂×C₈×D₇₃	C₂×C₄	1	84	yes	complemented
33	C₂×C₈×D₇₃	C₂³	1	28	yes	—
34	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₃	C₂×C₄	1	14	yes	complemented
36	C₂²×#64408	C₂×C₈	1	224	yes	direct product factor
37	C₂²×#64408	C₂²×C₄	1	56	yes	—
38	#11924036	C₂²×C₄	1	56	yes	complemented
39	#11924036	C₂⁴	1	8	yes	—
40	C₂×#10596833	C₂²×C₄	1	56	yes	complemented
41	C₄×#64408	C₂²×C₄	1	56	yes	—
42	C₄×#64408	C₂⁴	1	8	yes	—
43	C₂³×D₇₃	C₄²	1	7	yes	—
44	C₂³×D₇₃	C₂×C₈	1	16	yes	—
45	C₂³×D₇₃	C₂²×C₄	1	7	yes	—
46	C₂²×Dic₇₃	C₄²	1	7	yes	—
47	C₂²×Dic₇₃	C₂²×C₄	1	7	yes	—
48	C₂×C₄×D₇₃	C₄²	1	42	yes	—
49	C₂×C₄×D₇₃	C₂²×C₄	1	21	yes	—
50	C₂×C₄×D₇₃	C₂²×C₄	1	21	yes	—
51	C₂×C₄×D₇₃	C₂²×C₄	1	21	yes	—
52	C₂×C₄×D₇₃	C₂⁴	1	7	yes	—
53	C₂×#10596831	C₂²×C₄	1	28	yes	complemented
54	C₈×D₇₃	C₂²×C₄	1	56	yes	complemented
55	C₈×D₇₃	C₂⁴	1	8	yes	—
56	C₂⁴×C₇₃	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	C₄²	1	7	yes	—
58	C₂²×C₄×C₇₃	C₂²×C₄	1	7	yes	—
59	C₂×C₈×C₇₃	C₂²×C₄	1	28	yes	complemented
60	C₂³×C₇₃	C₄×C₈	1	8	yes	complemented
61	C₂³×C₇₃	C₂×C₄²	1	7	yes	—
62	C₂²×D₇₃	C₄×C₈	1	56	yes	complemented
63	C₂²×D₇₃	C₂×C₄²	1	21	yes	—
64	C₂²×D₇₃	C₂²×C₈	1	56	yes	—
65	C₂²×D₇₃	C₂³×C₄	1	7	yes	—
66	C₂×#64408	C₂²×C₈	1	224	yes	direct product factor
67	C₂×#64408	C₂³×C₄	1	16	yes	—
68	C₂×C₄×C₇₃	C₂×C₄²	1	21	yes	—
69	C₂×C₄×C₇₃	C₂³×C₄	1	7	yes	—
70	C₂×Dic₇₃	C₂×C₄²	1	21	yes	—
71	C₂×Dic₇₃	C₂³×C₄	1	7	yes	—
72	C₄×D₇₃	C₂×C₄²	1	42	yes	—
73	C₄×D₇₃	C₂³×C₄	1	7	yes	—
74	C₄×D₇₃	C₂³×C₄	1	7	yes	—
75	C₄×D₇₃	C₂³×C₄	1	7	yes	—
76	C₄×D₇₃	C₂⁵	1	1	yes	characteristic
77	#10596833	C₂³×C₄	1	16	yes	complemented
78	C₈×C₇₃	C₂³×C₄	1	8	yes	complemented
79	#10596831	C₂³×C₄	1	8	yes	complemented
80	C₂²×C₇₃	C₂×C₄×C₈	1	28	yes	complemented
81	C₂²×C₇₃	C₂²×C₄²	1	7	yes	—
82	C₂×D₇₃	C₂×C₄×C₈	1	84	yes	complemented
83	C₂×D₇₃	C₂²×C₄²	1	7	yes	—
84	C₂×D₇₃	C₂³×C₈	1	28	yes	—
85	C₂×D₇₃	C₂⁴×C₄	1	1	yes	characteristic
86	#64408	C₂³×C₈	1	32	yes	direct product factor
87	C₄×C₇₃	C₂²×C₄²	1	7	yes	—
88	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
89	Dic₇₃	C₂²×C₄²	1	7	yes	—
90	Dic₇₃	C₂⁴×C₄	1	1	yes	characteristic
91	C₂³×C₄×C₈	—	73	1	no	maximal, Sylow
92	C₂×C₇₃	C₂²×C₄×C₈	1	14	yes	complemented
93	C₂×C₇₃	C₂³×C₄²	1	1	yes	characteristic
94	D₇₃	C₂²×C₄×C₈	1	14	yes	complemented
95	D₇₃	C₂⁴×C₈	1	2	yes	—
96	C₂⁴×C₈	—	73	1	no	—
97	C₂⁴×C₈	—	73	1	no	—
98	C₂³×C₄²	—	73	1	no	—
99	C₂²×C₄×C₈	—	73	28	no	—
100	C₇₃	C₂³×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
101	C₂⁴×C₄	—	73	2	no	—
102	C₂⁴×C₄	—	73	1	no	—
103	C₂³×C₈	—	73	2	no	—
104	C₂³×C₈	—	73	14	no	—
105	C₂³×C₈	—	73	14	no	—
106	C₂³×C₈	—	73	1	no	—
107	C₂³×C₈	—	73	14	no	—
108	C₂³×C₈	—	73	14	no	—
109	C₂³×C₈	#64408	1	1	yes	characteristic, direct product factor
110	C₂²×C₄²	—	73	14	no	—
111	C₂²×C₄²	—	73	14	no	—
112	C₂×C₄×C₈	—	73	112	no	—
113	C₂⁵	—	73	1	no	—
114	C₂³×C₄	—	73	32	no	—
115	C₂³×C₄	—	73	28	no	—
116	C₂³×C₄	—	73	1	no	—
117	C₂³×C₄	—	73	7	no	—
118	C₂³×C₄	—	73	7	no	—
119	C₂³×C₄	—	73	7	no	—
120	C₂³×C₄	—	73	7	no	—
121	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
122	C₂²×C₈	—	73	28	no	—
123	C₂²×C₈	—	73	84	no	—
124	C₂²×C₈	—	73	28	no	—
125	C₂²×C₈	—	73	14	no	—
126	C₂²×C₈	—	73	28	no	—
127	C₂²×C₈	—	73	84	no	—
128	C₂²×C₈	C₂×#64408	1	14	yes	direct product factor
129	C₂×C₄²	—	73	28	no	—
130	C₂×C₄²	—	73	84	no	—
131	C₄×C₈	—	73	64	no	—
132	C₂⁴	—	73	16	no	—
133	C₂⁴	—	73	7	no	—
134	C₂⁴	—	73	7	no	—
135	C₂⁴	C₄×#64408	1	1	yes	characteristic
136	C₂²×C₄	—	73	224	no	—
137	C₂²×C₄	—	73	56	no	—
138	C₂²×C₄	—	73	7	no	—
139	C₂²×C₄	—	73	7	no	—
140	C₂²×C₄	—	73	21	no	—
141	C₂²×C₄	—	73	7	no	—
142	C₂²×C₄	—	73	42	no	—
143	C₂²×C₄	—	73	21	no	—
144	C₂²×C₄	—	73	21	no	—
145	C₂²×C₄	C₄×#64408	1	7	yes	—
146	C₂²×C₄	C₂²×#64408	1	7	yes	—
147	C₂×C₈	—	73	56	no	—
148	C₂×C₈	—	73	56	no	—
149	C₂×C₈	—	73	8	no	—
150	C₂×C₈	—	73	28	no	—
151	C₂×C₈	—	73	56	no	—
152	C₂×C₈	—	73	8	no	—
153	C₂×C₈	C₂²×#64408	1	28	yes	direct product factor
154	C₄²	—	73	56	no	—
155	C₄²	—	73	8	no	—
156	C₂³	—	73	7	no	—
157	C₂³	—	73	21	no	—
158	C₂³	—	73	56	no	—
159	C₂³	—	73	56	no	—
160	C₂³	C₈×#64408	1	8	yes	direct product factor
161	C₂³	C₂×C₄×#64408	1	7	yes	—
162	C₂×C₄	—	73	16	no	—
163	C₂×C₄	—	73	224	no	—
164	C₂×C₄	—	73	7	no	—
165	C₂×C₄	—	73	21	no	—
166	C₂×C₄	—	73	7	no	—
167	C₂×C₄	—	73	7	no	—
168	C₂×C₄	—	73	42	no	—
169	C₂×C₄	—	73	1	no	—
170	C₂×C₄	—	73	7	no	—
171	C₂×C₄	C₂×C₄×#64408	1	21	yes	—
172	C₂×C₄	C₂³×#64408	1	7	yes	—
173	C₈	—	73	16	no	—
174	C₈	—	73	8	no	—
175	C₈	C₂³×#64408	1	8	yes	direct product factor
176	C₂²	—	73	1	no	—
177	C₂²	—	73	7	no	—
178	C₂²	—	73	28	no	—
179	C₂²	—	73	84	no	—
180	C₂²	C₂×C₈×#64408	1	28	yes	direct product factor
181	C₂²	C₂²×C₄×#64408	1	7	yes	—
182	C₄	—	73	32	no	—
183	C₄	—	73	1	no	—
184	C₄	—	73	7	no	—
185	C₄	C₂²×C₄×#64408	1	7	yes	—
186	C₄	C₂⁴×#64408	1	1	yes	characteristic
187	C₂	—	73	2	no	—
188	C₂	—	73	14	no	—
189	C₂	C₂²×C₈×#64408	1	14	yes	direct product factor
190	C₂	C₂³×C₄×#64408	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	#64408
commutator, hypocenter	C₇₃	C₂³×C₄×C₈
Frattini	C₄	C₂⁴×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂⁴×C₇₃	C₄²
prime radical	C₂⁴×D₇₃	C₂×C₄
cosocle	C₄×D₇₃	C₂⁵
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×C₈×#64408	C₂	direct (N × H)
6. C₂²×#11924036	C₄	semidirect
8. C₂³×#10596833	C₄	semidirect
11. C₂×C₈×#64408	C₂²	direct (N × H)
13. C₂³×#10596831	C₄	semidirect
14. C₂²×C₈×D₇₃	C₄	semidirect
16. C₂³×C₈×C₇₃	C₄	semidirect
17. C₂³×#64408	C₈	direct (N × H)
19. C₂×#11924036	C₂×C₄	semidirect
21. C₂²×#10596833	C₂×C₄	semidirect
24. C₈×#64408	C₂³	direct (N × H)
31. C₂²×#10596831	C₂×C₄	semidirect
32. C₂×C₈×D₇₃	C₂×C₄	semidirect
35. C₂²×C₈×C₇₃	C₂×C₄	semidirect
36. C₂²×#64408	C₂×C₈	direct (N × H)
38. #11924036	C₂²×C₄	semidirect
40. C₂×#10596833	C₂²×C₄	semidirect
53. C₂×#10596831	C₂²×C₄	semidirect
54. C₈×D₇₃	C₂²×C₄	semidirect
59. C₂×C₈×C₇₃	C₂²×C₄	semidirect
60. C₂³×C₇₃	C₄×C₈	semidirect
62. C₂²×D₇₃	C₄×C₈	semidirect
66. C₂×#64408	C₂²×C₈	direct (N × H)
77. #10596833	C₂³×C₄	semidirect
78. C₈×C₇₃	C₂³×C₄	semidirect
79. #10596831	C₂³×C₄	semidirect
80. C₂²×C₇₃	C₂×C₄×C₈	semidirect
82. C₂×D₇₃	C₂×C₄×C₈	semidirect
86. #64408	C₂³×C₈	direct (N × H)
92. C₂×C₇₃	C₂²×C₄×C₈	semidirect
94. D₇₃	C₂²×C₄×C₈	semidirect
100. C₇₃	C₂³×C₄×C₈	semidirect
109. C₂³×C₈	#64408	direct (N × H)
128. C₂²×C₈	C₂×#64408	direct (N × H)
153. C₂×C₈	C₂²×#64408	direct (N × H)
160. C₂³	C₈×#64408	direct (N × H)
175. C₈	C₂³×#64408	direct (N × H)
180. C₂²	C₂×C₈×#64408	direct (N × H)
189. C₂	C₂²×C₈×#64408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11924036	2
C₂³×C₄×#64408	2
C₂²×C₈×#64408	2
C₂³×C₈×D₇₃	2
C₂³×C₄×C₈	73

Subgroups — C23×C8×#64408 #34031058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₈×#64408 #34031058