Subgroups — C₂×C₄×#14556686 #34034660

Order 18688.

1022 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556686	C₂	1	2	yes	maximal
2	C₄×#14556686	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄×#11924040	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×#11924036	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556686	C₄	1	16	yes	direct product factor
6	C₂×#14556686	C₂²	1	8	yes	—
7	C₂²×#11924040	C₂²	1	4	yes	—
8	C₄×#11924040	C₂²	1	16	yes	complemented
9	C₂²×#11924036	C₄	1	1	yes	characteristic
10	C₂²×#11924036	C₂²	1	1	yes	characteristic
11	C₄×#11924036	C₄	1	4	yes	—
12	C₄×#11924036	C₂²	1	4	yes	—
13	C₂×C₄×#10596833	C₄	1	2	yes	—
14	C₂×C₄×#10596833	C₂²	1	2	yes	—
15	C₂×C₄²×D₇₃	C₄	1	1	yes	characteristic
16	#14556686	C₂×C₄	1	32	yes	direct product factor
17	C₂×#11924040	C₂×C₄	1	16	yes	complemented
18	C₂×#11924040	C₂³	1	8	yes	—
19	C₂×#11924036	C₈	1	8	yes	—
20	C₂×#11924036	C₂×C₄	1	8	yes	—
21	C₂×#11924036	C₂×C₄	1	2	yes	—
22	C₂×#11924036	C₂×C₄	1	4	yes	—
23	C₂×#11924036	C₂³	1	2	yes	—
24	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10596833	C₂×C₄	1	2	yes	—
26	C₂²×#10596833	C₂³	1	1	yes	characteristic
27	C₄×#10596833	C₂×C₄	1	8	yes	—
28	C₄×#10596833	C₂×C₄	1	4	yes	—
29	C₄×#10596833	C₂³	1	4	yes	—
30	C₂²×C₄×D₇₃	C₈	1	2	yes	—
31	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
34	C₄²×D₇₃	C₈	1	4	yes	—
35	C₄²×D₇₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
37	#11924040	C₂²×C₄	1	16	yes	complemented
38	#11924036	C₄²	1	8	yes	—
39	#11924036	C₂×C₈	1	16	yes	—
40	#11924036	C₂²×C₄	1	8	yes	—
41	C₂×#10596833	C₄²	1	4	yes	—
42	C₂×#10596833	C₂×C₈	1	8	yes	—
43	C₂×#10596833	C₂²×C₄	1	2	yes	—
44	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
45	C₂×#10596833	C₂²×C₄	1	4	yes	—
46	C₂×#10596833	C₂²×C₄	1	4	yes	—
47	C₂×#10596833	C₂⁴	1	1	yes	characteristic
48	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
50	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×D₇₃	C₁₆	1	16	yes	complemented
52	C₂×C₄×D₇₃	C₄²	1	4	yes	—
53	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
54	C₂×C₄×D₇₃	C₂×C₈	1	4	yes	—
55	C₂×C₄×D₇₃	C₂×C₈	1	8	yes	—
56	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
57	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
58	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
59	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
62	C₄²×C₇₃	C₂×C₈	1	4	yes	—
63	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₂²×D₇₃	C₄×C₈	1	4	yes	—
65	C₂²×D₇₃	C₂×C₁₆	1	8	yes	—
66	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
67	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
68	C₂×C₄×C₇₃	C₂×C₁₆	1	8	yes	complemented
69	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
71	C₂×Dic₇₃	C₂×C₁₆	1	8	yes	complemented
72	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
73	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
74	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
76	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
77	C₄×D₇₃	C₄×C₈	1	8	yes	—
78	C₄×D₇₃	C₂×C₁₆	1	32	yes	complemented
79	C₄×D₇₃	C₂×C₄²	1	4	yes	—
80	C₄×D₇₃	C₂²×C₈	1	4	yes	—
81	#10596833	C₂×C₄²	1	4	yes	—
82	#10596833	C₂×C₄²	1	2	yes	—
83	#10596833	C₂²×C₈	1	8	yes	—
84	#10596833	C₂³×C₄	1	2	yes	—
85	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
86	C₂²×C₇₃	C₂²×C₁₆	1	4	yes	—
87	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
88	C₂×D₇₃	C₄×C₁₆	1	16	yes	complemented
89	C₂×D₇₃	C₂×C₄×C₈	1	4	yes	—
90	C₂×D₇₃	C₂²×C₁₆	1	8	yes	—
91	C₄×C₇₃	C₂×C₄×C₈	1	4	yes	—
92	C₄×C₇₃	C₂²×C₁₆	1	8	yes	complemented
93	Dic₇₃	C₂×C₄×C₈	1	2	yes	—
94	Dic₇₃	C₂²×C₁₆	1	8	yes	complemented
95	Dic₇₃	C₂²×C₄²	1	1	yes	characteristic
96	Dic₇₃	C₂³×C₈	1	1	yes	characteristic
97	C₂²×C₄×C₁₆	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	C₂×C₄×C₁₆	1	4	yes	complemented
99	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
100	C₂×C₇₃	C₂³×C₁₆	1	2	yes	—
101	D₇₃	C₂×C₄×C₁₆	1	8	yes	complemented
102	C₂³×C₁₆	—	73	2	no	—
103	C₂²×C₄×C₈	—	73	1	no	—
104	C₂×C₄×C₁₆	—	73	4	no	—
105	C₂×C₄×C₁₆	—	73	8	no	—
106	C₇₃	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₈	—	73	1	no	—
108	C₂³×C₈	—	73	1	no	—
109	C₂²×C₄²	—	73	1	no	—
110	C₂²×C₁₆	—	73	4	no	—
111	C₂²×C₁₆	—	73	8	no	—
112	C₂²×C₁₆	—	73	16	no	—
113	C₂×C₄×C₈	—	73	2	no	—
114	C₂×C₄×C₈	—	73	4	no	—
115	C₂×C₄×C₈	—	73	2	no	—
116	C₂×C₄×C₈	—	73	4	no	—
117	C₄×C₁₆	—	73	16	no	—
118	C₂³×C₄	—	73	1	no	—
119	C₂³×C₄	—	73	2	no	—
120	C₂²×C₈	—	73	1	no	—
121	C₂²×C₈	—	73	2	no	—
122	C₂²×C₈	—	73	1	no	—
123	C₂²×C₈	—	73	8	no	—
124	C₂²×C₈	—	73	2	no	—
125	C₂²×C₈	—	73	4	no	—
126	C₂²×C₈	—	73	8	no	—
127	C₂²×C₈	—	73	2	no	—
128	C₂×C₄²	—	73	1	no	—
129	C₂×C₄²	—	73	2	no	—
130	C₂×C₄²	—	73	4	no	—
131	C₂×C₄²	—	73	4	no	—
132	C₂×C₄²	#10596842	1	1	yes	characteristic
133	C₂×C₁₆	—	73	16	no	—
134	C₂×C₁₆	—	73	8	no	—
135	C₂×C₁₆	—	73	32	no	—
136	C₄×C₈	—	73	4	no	—
137	C₄×C₈	—	73	4	no	—
138	C₄×C₈	—	73	8	no	—
139	C₂⁴	—	73	1	no	—
140	C₂²×C₄	—	73	2	no	—
141	C₂²×C₄	—	73	1	no	—
142	C₂²×C₄	—	73	4	no	—
143	C₂²×C₄	—	73	4	no	—
144	C₂²×C₄	—	73	16	no	—
145	C₂²×C₄	—	73	2	no	—
146	C₂²×C₄	—	73	8	no	—
147	C₂²×C₄	—	73	2	no	—
148	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
149	C₂²×C₄	C₂×#10596842	1	2	yes	—
150	C₂×C₈	—	73	4	no	—
151	C₂×C₈	—	73	8	no	—
152	C₂×C₈	—	73	2	no	—
153	C₂×C₈	—	73	1	no	—
154	C₂×C₈	—	73	4	no	—
155	C₂×C₈	—	73	4	no	—
156	C₂×C₈	—	73	1	no	—
157	C₂×C₈	—	73	8	no	—
158	C₂×C₈	—	73	16	no	—
159	C₂×C₈	—	73	8	no	—
160	C₄²	—	73	4	no	—
161	C₄²	—	73	4	no	—
162	C₄²	—	73	4	no	—
163	C₄²	C₂×#10596842	1	4	yes	—
164	C₁₆	—	73	16	no	—
165	C₂³	—	73	4	no	—
166	C₂³	—	73	8	no	—
167	C₂³	—	73	2	no	—
168	C₂³	C₂²×#10596842	1	1	yes	characteristic
169	C₂×C₄	—	73	2	no	—
170	C₂×C₄	—	73	2	no	—
171	C₂×C₄	—	73	8	no	—
172	C₂×C₄	—	73	1	no	—
173	C₂×C₄	—	73	1	no	—
174	C₂×C₄	—	73	4	no	—
175	C₂×C₄	—	73	4	no	—
176	C₂×C₄	—	73	4	no	—
177	C₂×C₄	—	73	8	no	—
178	C₂×C₄	—	73	32	no	—
179	C₂×C₄	C₄×#10596842	1	4	yes	—
180	C₂×C₄	#14556686	1	8	yes	direct product factor
181	C₂×C₄	C₂²×#10596842	1	2	yes	—
182	C₂×C₄	C₂²×#10596842	1	4	yes	—
183	C₈	—	73	2	no	—
184	C₈	—	73	4	no	—
185	C₈	—	73	2	no	—
186	C₈	—	73	8	no	—
187	C₂²	—	73	16	no	—
188	C₂²	—	73	4	no	—
189	C₂²	—	73	8	no	—
190	C₂²	C₂×C₄×#10596842	1	2	yes	—
191	C₂²	C₂×#14556686	1	4	yes	—
192	C₂²	C₂³×#10596842	1	1	yes	characteristic
193	C₄	—	73	1	no	—
194	C₄	—	73	1	no	—
195	C₄	—	73	2	no	—
196	C₄	—	73	8	no	—
197	C₄	C₂×C₄×#10596842	1	4	yes	—
198	C₄	C₂×#14556686	1	8	yes	direct product factor
199	C₂	—	73	8	no	—
200	C₂	C₄×#14556686	1	4	yes	direct product factor
201	C₂	C₂²×C₄×#10596842	1	1	yes	characteristic
202	C₂	C₂²×#14556686	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	#10596842
commutator, hypocenter	C₇₃	C₂²×C₄×C₁₆
Frattini	C₂²	C₂³×#10596842
Fitting	C₂×C₄²×C₇₃	C₈
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂×#10596833	C₂⁴
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#14556686	C₂	direct (N × H)
3. C₂×C₄×#11924040	C₂	semidirect
5. C₂×#14556686	C₄	direct (N × H)
8. C₄×#11924040	C₂²	semidirect
16. #14556686	C₂×C₄	direct (N × H)
17. C₂×#11924040	C₂×C₄	semidirect
37. #11924040	C₂²×C₄	semidirect
51. C₂×C₄×D₇₃	C₁₆	semidirect
68. C₂×C₄×C₇₃	C₂×C₁₆	semidirect
71. C₂×Dic₇₃	C₂×C₁₆	semidirect
78. C₄×D₇₃	C₂×C₁₆	semidirect
88. C₂×D₇₃	C₄×C₁₆	semidirect
92. C₄×C₇₃	C₂²×C₁₆	semidirect
94. Dic₇₃	C₂²×C₁₆	semidirect
98. C₂×C₇₃	C₂×C₄×C₁₆	semidirect
101. D₇₃	C₂×C₄×C₁₆	semidirect
106. C₇₃	C₂²×C₄×C₁₆	semidirect
180. C₂×C₄	#14556686	direct (N × H)
198. C₄	C₂×#14556686	direct (N × H)
200. C₂	C₄×#14556686	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×#14556686	2
C₄×#14556686	2
C₂×C₄×#11924040	2
C₂×C₄×#11924036	2
C₂²×C₄×C₁₆	73

Subgroups — C2×C4×#14556686 #34034660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14556686 #34034660