Subgroups — C₂³×#14556549 #33992543

Order 18688.

14334 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556549	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#11924005	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₇₃	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₃	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₇₃	C₂²	1	2	yes	—
9	C₂⁴×Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₇₃	C₄	1	32	yes	complemented
11	C₂³×C₄×D₇₃	C₂²	1	28	yes	—
12	C₂×#14556549	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊Dic₇₃	C₂²	1	28	yes	complemented
14	C₂²×#11924005	C₂²	1	56	yes	—
15	C₂²×C₄×Dic₇₃	C₂²	1	56	yes	complemented
16	C₂⁴×C₄×C₇₃	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₇₃	C₂²	1	28	yes	complemented
19	C₂⁴×D₇₃	—	2	8	no	—
20	C₂⁴×D₇₃	—	2	8	no	—
21	C₂⁴×D₇₃	C₂×C₄	1	16	yes	—
22	C₂⁴×D₇₃	C₂³	1	14	yes	—
23	C₂³×Dic₇₃	—	2	16	no	—
24	C₂³×Dic₇₃	C₂×C₄	1	16	yes	—
25	C₂³×Dic₇₃	C₂³	1	28	yes	—
26	C₂³×Dic₇₃	C₂³	1	14	yes	—
27	C₂²×C₄×D₇₃	C₂×C₄	1	448	yes	complemented
28	C₂²×C₄×D₇₃	C₂³	1	112	yes	—
29	#14556549	C₂³	1	512	yes	direct product factor
30	C₂×C₄⋊Dic₇₃	C₂³	1	112	yes	complemented
31	C₂×#11924005	C₂³	1	224	yes	—
32	C₂×C₄×Dic₇₃	C₂³	1	224	yes	complemented
33	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₇₃	—	2	16	no	—
35	C₂³×C₄×C₇₃	C₂×C₄	1	16	yes	—
36	C₂³×C₄×C₇₃	C₂³	1	28	yes	—
37	C₂³×C₄×C₇₃	C₂³	1	14	yes	—
38	C₂×C₄⋊C₄×C₇₃	C₂³	1	112	yes	complemented
39	C₂³×D₇₃	—	2	128	no	—
40	C₂³×D₇₃	—	2	56	no	—
41	C₂³×D₇₃	—	2	56	no	—
42	C₂³×D₇₃	C₂²×C₄	1	112	yes	—
43	C₂³×D₇₃	C₂⁴	1	28	yes	—
44	C₂²×Dic₇₃	—	2	112	no	—
45	C₂²×Dic₇₃	C₂²×C₄	1	112	yes	—
46	C₂²×Dic₇₃	C₂⁴	1	56	yes	—
47	C₂²×Dic₇₃	C₂⁴	1	28	yes	—
48	C₂×C₄×D₇₃	C₂²×C₄	1	896	yes	complemented
49	C₂×C₄×D₇₃	C₂⁴	1	64	yes	—
50	C₄⋊Dic₇₃	C₂⁴	1	64	yes	complemented
51	#11924005	C₂⁴	1	128	yes	—
52	C₄×Dic₇₃	C₂⁴	1	128	yes	complemented
53	C₂⁴×C₇₃	C₂²×C₄	1	8	yes	—
54	C₂⁴×C₇₃	#41	1	8	yes	—
55	C₂⁴×C₇₃	#41	1	8	yes	—
56	C₂⁴×C₇₃	C₂⁴	1	7	yes	—
57	C₂²×C₄×C₇₃	—	2	112	no	—
58	C₂²×C₄×C₇₃	C₂²×C₄	1	112	yes	—
59	C₂²×C₄×C₇₃	C₂⁴	1	56	yes	—
60	C₂²×C₄×C₇₃	C₂⁴	1	28	yes	—
61	C₄⋊C₄×C₇₃	C₂⁴	1	64	yes	complemented
62	C₂³×C₇₃	#117	1	64	yes	complemented
63	C₂³×C₇₃	C₂³×C₄	1	28	yes	—
64	C₂³×C₇₃	C₂×#41	1	28	yes	—
65	C₂³×C₇₃	C₂×#41	1	28	yes	—
66	C₂³×C₇₃	C₂⁵	1	7	yes	—
67	C₂²×D₇₃	—	2	448	no	—
68	C₂²×D₇₃	—	2	56	no	—
69	C₂²×D₇₃	—	2	56	no	—
70	C₂²×D₇₃	C₂³×C₄	1	112	yes	—
71	C₂²×D₇₃	C₂⁵	1	8	yes	—
72	C₂×C₄×C₇₃	—	2	112	no	—
73	C₂×C₄×C₇₃	C₂³×C₄	1	112	yes	—
74	C₂×C₄×C₇₃	C₂⁵	1	16	yes	—
75	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
76	C₂×Dic₇₃	—	2	112	no	—
77	C₂×Dic₇₃	C₂³×C₄	1	112	yes	—
78	C₂×Dic₇₃	C₂⁵	1	16	yes	—
79	C₂×Dic₇₃	C₂⁵	1	8	yes	—
80	C₄×D₇₃	C₂³×C₄	1	256	yes	complemented
81	C₂²×C₇₃	C₂×#117	1	112	yes	complemented
82	C₂²×C₇₃	C₂⁴×C₄	1	14	yes	—
83	C₂²×C₇₃	C₂²×#41	1	14	yes	—
84	C₂²×C₇₃	C₂²×#41	1	14	yes	—
85	C₂²×C₇₃	C₂⁶	1	1	yes	characteristic
86	C₂×D₇₃	—	2	8	no	—
87	C₂×D₇₃	—	2	8	no	—
88	C₂×D₇₃	—	2	224	no	—
89	C₂×D₇₃	C₂⁴×C₄	1	16	yes	—
90	C₄×C₇₃	—	2	16	no	—
91	C₄×C₇₃	C₂⁴×C₄	1	16	yes	—
92	Dic₇₃	—	2	16	no	—
93	Dic₇₃	C₂⁴×C₄	1	16	yes	—
94	C₂³×#117	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	C₂²×#117	1	28	yes	complemented
96	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
97	C₂×C₇₃	C₂³×#41	1	1	yes	characteristic
98	C₂×C₇₃	C₂³×#41	1	1	yes	characteristic
99	D₇₃	—	2	16	no	—
100	C₂⁵×C₄	—	73	1	no	—
101	C₂²×#117	—	73	56	no	—
102	C₂³×C₄⋊C₄	—	73	1	no	—
103	C₂³×C₄⋊C₄	D₇₃	1	1	yes	characteristic, complemented
104	C₂³×C₂²⋊C₄	—	73	2	no	—
105	C₂³×C₄²	—	73	2	no	—
106	C₇₃	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
107	C₂⁶	—	73	1	no	—
108	C₂⁴×C₄	—	73	1	no	—
109	C₂⁴×C₄	—	73	32	no	—
110	C₂⁴×C₄	—	73	28	no	—
111	C₂⁴×C₄	—	73	2	no	—
112	C₂⁴×C₄	C₂×D₇₃	1	2	yes	—
113	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
114	C₂×#117	—	73	448	no	—
115	C₂²×C₄⋊C₄	—	73	28	no	—
116	C₂²×C₄⋊C₄	C₂×D₇₃	1	28	yes	complemented
117	C₂²×C₂²⋊C₄	—	73	56	no	—
118	C₂²×C₄²	—	73	56	no	—
119	C₂⁵	—	73	16	no	—
120	C₂⁵	—	73	14	no	—
121	C₂⁵	—	146	8	no	—
122	C₂⁵	—	146	8	no	—
123	C₂⁵	C₂²×D₇₃	1	1	yes	characteristic
124	C₂³×C₄	—	2	16	no	—
125	C₂³×C₄	—	73	16	no	—
126	C₂³×C₄	—	73	14	no	—
127	C₂³×C₄	—	73	448	no	—
128	C₂³×C₄	—	73	112	no	—
129	C₂³×C₄	—	73	28	no	—
130	C₂³×C₄	—	146	16	no	—
131	C₂³×C₄	C₄×D₇₃	1	16	yes	—
132	C₂³×C₄	C₂²×D₇₃	1	28	yes	—
133	C₂³×C₄	C₂²×D₇₃	1	14	yes	—
134	#117	—	73	512	no	—
135	C₂×C₄⋊C₄	—	73	112	no	—
136	C₂×C₄⋊C₄	C₂²×D₇₃	1	112	yes	complemented
137	C₂×C₂²⋊C₄	—	73	224	no	—
138	C₂×C₄²	—	73	224	no	—
139	C₂⁴	—	73	112	no	—
140	C₂⁴	—	73	28	no	—
141	C₂⁴	—	146	128	no	—
142	C₂⁴	—	146	56	no	—
143	C₂⁴	—	146	56	no	—
144	C₂⁴	C₂×C₄×D₇₃	1	8	yes	—
145	C₂⁴	#11924027	1	8	yes	—
146	C₂⁴	#11924029	1	8	yes	—
147	C₂⁴	C₂³×D₇₃	1	7	yes	—
148	C₂²×C₄	—	2	112	no	—
149	C₂²×C₄	—	73	112	no	—
150	C₂²×C₄	—	73	28	no	—
151	C₂²×C₄	—	73	896	no	—
152	C₂²×C₄	—	73	64	no	—
153	C₂²×C₄	—	73	56	no	—
154	C₂²×C₄	—	146	112	no	—
155	C₂²×C₄	C₂×C₄×D₇₃	1	112	yes	—
156	C₂²×C₄	C₂³×D₇₃	1	56	yes	—
157	C₂²×C₄	C₂³×D₇₃	1	28	yes	—
158	C₄⋊C₄	—	73	64	no	—
159	C₄⋊C₄	C₂³×D₇₃	1	64	yes	complemented
160	C₂²⋊C₄	—	73	128	no	—
161	C₄²	—	73	128	no	—
162	C₂³	—	73	112	no	—
163	C₂³	—	73	8	no	—
164	C₂³	—	146	56	no	—
165	C₂³	—	146	56	no	—
166	C₂³	—	146	448	no	—
167	C₂³	#14556549	1	64	yes	direct product factor
168	C₂³	C₂²×C₄×D₇₃	1	28	yes	—
169	C₂³	C₂×#11924027	1	28	yes	—
170	C₂³	C₂×#11924029	1	28	yes	—
171	C₂³	C₂⁴×D₇₃	1	7	yes	—
172	C₂×C₄	—	2	112	no	—
173	C₂×C₄	—	73	112	no	—
174	C₂×C₄	—	73	8	no	—
175	C₂×C₄	—	73	256	no	—
176	C₂×C₄	—	73	16	no	—
177	C₂×C₄	—	146	112	no	—
178	C₂×C₄	C₂²×C₄×D₇₃	1	112	yes	—
179	C₂×C₄	C₂⁴×D₇₃	1	16	yes	—
180	C₂×C₄	C₂⁴×D₇₃	1	8	yes	—
181	C₂²	—	73	16	no	—
182	C₂²	—	146	224	no	—
183	C₂²	—	146	8	no	—
184	C₂²	—	146	8	no	—
185	C₂²	C₂×#14556549	1	112	yes	direct product factor
186	C₂²	C₂³×C₄×D₇₃	1	14	yes	—
187	C₂²	C₂²×#11924027	1	14	yes	—
188	C₂²	C₂²×#11924029	1	14	yes	—
189	C₂²	C₂⁵×D₇₃	1	1	yes	characteristic
190	C₄	—	2	16	no	—
191	C₄	—	73	16	no	—
192	C₄	—	146	16	no	—
193	C₄	C₂³×C₄×D₇₃	1	16	yes	—
194	C₂	—	146	16	no	—
195	C₂	C₂²×#14556549	1	28	yes	direct product factor
196	C₂	C₂⁴×C₄×D₇₃	1	1	yes	characteristic
197	C₂	C₂³×#11924027	1	1	yes	characteristic
198	C₂	C₂³×#11924029	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₇₃
commutator	C₂×C₇₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₇₃
Fitting	C₂³×C₄⋊C₄×C₇₃	C₂
socle	C₂⁵×C₇₃	C₂³
prime radical	C₂⁵×D₇₃	C₂²
cosocle	C₂²×C₇₃	C₂⁶
hypercenter	C₂³×C₄⋊C₄	D₇₃
hypocenter	C₇₃	C₂³×#117
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#14556549	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₇₃	C₂	semidirect
5. C₂³×C₄×Dic₇₃	C₂	semidirect
6. C₂³×C₄⋊C₄×C₇₃	C₂	semidirect
10. C₂³×C₄×D₇₃	C₄	semidirect
12. C₂×#14556549	C₂²	direct (N × H)
13. C₂²×C₄⋊Dic₇₃	C₂²	semidirect
15. C₂²×C₄×Dic₇₃	C₂²	semidirect
18. C₂²×C₄⋊C₄×C₇₃	C₂²	semidirect
27. C₂²×C₄×D₇₃	C₂×C₄	semidirect
29. #14556549	C₂³	direct (N × H)
30. C₂×C₄⋊Dic₇₃	C₂³	semidirect
32. C₂×C₄×Dic₇₃	C₂³	semidirect
38. C₂×C₄⋊C₄×C₇₃	C₂³	semidirect
48. C₂×C₄×D₇₃	C₂²×C₄	semidirect
50. C₄⋊Dic₇₃	C₂⁴	semidirect
52. C₄×Dic₇₃	C₂⁴	semidirect
61. C₄⋊C₄×C₇₃	C₂⁴	semidirect
62. C₂³×C₇₃	#117	semidirect
80. C₄×D₇₃	C₂³×C₄	semidirect
81. C₂²×C₇₃	C₂×#117	semidirect
95. C₂×C₇₃	C₂²×#117	semidirect
103. C₂³×C₄⋊C₄	D₇₃	semidirect
106. C₇₃	C₂³×#117	semidirect
116. C₂²×C₄⋊C₄	C₂×D₇₃	semidirect
136. C₂×C₄⋊C₄	C₂²×D₇₃	semidirect
159. C₄⋊C₄	C₂³×D₇₃	semidirect
167. C₂³	#14556549	direct (N × H)
185. C₂²	C₂×#14556549	direct (N × H)
195. C₂	C₂²×#14556549	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×D₇₃	2
C₂²×#14556549	2
C₂³×C₄⋊Dic₇₃	2
C₂³×#11924005	2
C₂³×C₄×Dic₇₃	2
C₂³×C₄⋊C₄×C₇₃	2
C₂³×#117	73

Subgroups — C23×#14556549 #33992543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556549 #33992543