Subgroups — #33083332

Order 18688.

1470 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988545	C₂	1	4	yes	maximal
2	C₂²×#14556502	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556550	C₂	1	1	yes	maximal, characteristic
4	#22609617	C₂	1	4	yes	maximal, complemented
5	#97×Dic₇₃	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10596831	C₂²	1	2	yes	—
8	C₂×#14556550	C₂²	1	2	yes	—
9	#17988545	C₄	1	32	yes	complemented
10	C₂×#14556502	C₄	1	8	yes	complemented
11	C₂×#14556502	C₂²	1	2	yes	—
12	C₂×#14556502	C₂²	1	2	yes	—
13	C₂×#14556501	C₂²	1	8	yes	—
14	C₂×C₈×Dic₇₃	C₂²	1	8	yes	—
15	C₄×C₄⋊Dic₇₃	C₄	1	4	yes	—
16	C₄×C₂²⋊Dic₇₃	C₄	1	4	yes	—
17	#17988433	C₂²	1	2	yes	—
18	C₄²×Dic₇₃	C₂²	1	2	yes	—
19	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₇₃	C₄	1	8	yes	complemented
22	C₂×#97×C₇₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
24	C₂²×#10596831	C₂×C₄	1	8	yes	—
25	C₂²×#10596831	C₂×C₄	1	16	yes	—
26	C₂²×#10596831	C₂³	1	4	yes	—
27	#14556550	C₂×C₄	1	16	yes	—
28	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	8	yes	—
31	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
32	#14556502	C₂×C₄	1	8	yes	—
33	#14556502	C₂×C₄	1	8	yes	—
34	#14556501	C₂×C₄	1	32	yes	complemented
35	C₈×Dic₇₃	C₂×C₄	1	32	yes	complemented
36	#14556498	C₂×C₄	1	8	yes	—
37	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₇₃	C₂×C₄	1	16	yes	—
40	C₂×M₁₆×C₇₃	C₂×C₄	1	8	yes	—
41	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
47	#97×C₇₃	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₇₃	C₂×C₄	1	8	yes	—
49	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
50	C₂×#10596831	C₄²	1	32	yes	complemented
51	C₂×#10596831	C₂²×C₄	1	8	yes	—
52	C₂×#10596831	C₂²×C₄	1	8	yes	—
53	C₂×#10596831	C₂²×C₄	1	8	yes	—
54	C₄⋊Dic₇₃	C₄²	1	16	yes	—
55	C₂²⋊Dic₇₃	C₄²	1	16	yes	—
56	C₄×Dic₇₃	—	4	8	no	—
57	C₄×Dic₇₃	C₂²×C₄	1	16	yes	—
58	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	—	4	4	no	—
60	C₂²×C₄×C₇₃	C₄²	1	8	yes	—
61	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₇₃	C₄²	1	32	yes	complemented
69	C₂×C₈×C₇₃	C₂²×C₄	1	16	yes	—
70	C₂×C₈×C₇₃	C₂²×C₄	1	8	yes	—
71	C₄²×C₇₃	C₄²	1	8	yes	—
72	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
76	C₂³×C₇₃	—	2	4	no	—
77	C₂³×C₇₃	C₂×C₄²	1	4	yes	—
78	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
79	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	—	2	4	no	—
81	C₂×C₄×C₇₃	—	2	4	no	—
82	C₂×C₄×C₇₃	—	4	8	no	—
83	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₇₃	C₂×C₄²	1	8	yes	—
85	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
86	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
87	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₇₃	C₂×C₄²	1	8	yes	—
89	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
91	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×Dic₇₃	—	2	16	no	—
93	C₂×Dic₇₃	C₂×C₄²	1	16	yes	—
94	C₈×C₇₃	C₂×C₄²	1	16	yes	—
95	#10596831	C₂×C₄²	1	16	yes	—
96	C₂²×C₇₃	—	2	4	no	—
97	C₂²×C₇₃	—	2	8	no	—
98	C₂²×C₇₃	C₄³	1	4	yes	—
99	C₂²×C₇₃	C₂²×C₄²	1	2	yes	—
100	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
101	C₂²×C₇₃	#568	1	4	yes	—
102	C₄×C₇₃	—	2	4	no	—
103	C₄×C₇₃	—	2	4	no	—
104	C₄×C₇₃	C₄³	1	4	yes	—
105	C₄×C₇₃	C₂²×C₄²	1	2	yes	—
106	C₄×C₇₃	C₂²×C₄²	1	2	yes	—
107	Dic₇₃	—	2	16	no	—
108	#20331	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	—	2	4	no	—
110	C₂×C₇₃	C₂×C₄³	1	1	yes	characteristic
111	C₂×C₇₃	#2873	1	2	yes	—
112	C₂×C₇₃	#2875	1	4	yes	—
113	C₂×#405	—	73	4	no	—
114	C₂×C₄×M₁₆	D₇₃	1	1	yes	characteristic
115	C₂²×#97	—	73	1	no	—
116	C₄×#117	—	73	1	no	—
117	#1726	—	73	4	no	—
118	C₄×#97	—	73	4	no	—
119	C₇₃	#20331	1	1	yes	characteristic, complemented, Sylow
120	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
121	C₂³×C₈	—	73	2	no	—
122	C₂×#117	—	73	2	no	—
123	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
124	#405	—	73	32	no	—
125	C₄×M₁₆	Dic₇₃	1	8	yes	complemented
126	C₂×#97	—	73	8	no	—
127	C₂×#97	—	73	8	no	—
128	C₂×#97	—	73	2	no	—
129	C₂×#97	—	73	2	no	—
130	C₂×#97	C₂×D₇₃	1	2	yes	—
131	C₂×C₄×C₈	—	73	8	no	—
132	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
133	C₄×C₄⋊C₄	—	73	4	no	—
134	C₄×C₂²⋊C₄	—	73	4	no	—
135	#376	—	73	2	no	—
136	C₄³	—	73	2	no	—
137	C₂³×C₄	C₄×D₇₃	1	2	yes	—
138	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
139	C₂×M₁₆	C₄×D₇₃	1	16	yes	—
140	C₂×M₁₆	C₂×Dic₇₃	1	8	yes	—
141	C₂²×C₈	—	73	8	no	—
142	C₂²×C₈	—	73	16	no	—
143	C₂²×C₈	—	73	4	no	—
144	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
145	#117	—	73	16	no	—
146	C₂×C₄⋊C₄	—	73	4	no	—
147	C₂×C₂²⋊C₄	—	73	4	no	—
148	C₂×C₄²	—	73	8	no	—
149	C₂×C₄²	—	73	4	no	—
150	C₂×C₄²	C₄×D₇₃	1	2	yes	—
151	C₂×C₄²	C₄×D₇₃	1	4	yes	—
152	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
153	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
155	#97	—	73	32	no	—
156	#97	—	73	8	no	—
157	#97	—	73	8	no	—
158	#97	C₂×Dic₇₃	1	8	yes	—
159	C₄×C₈	—	73	32	no	—
160	C₄×C₈	C₂×Dic₇₃	1	8	yes	—
161	#95	—	73	8	no	—
162	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	—	73	8	no	—
165	C₂²×C₄	C₄×Dic₇₃	1	8	yes	—
166	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
167	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
168	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
170	C₂²×C₄	C₂×C₄×D₇₃	1	8	yes	—
171	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
172	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
173	M₁₆	C₄×Dic₇₃	1	32	yes	complemented
174	C₂×C₈	—	73	32	no	—
175	C₂×C₈	—	73	8	no	—
176	C₂×C₈	—	73	8	no	—
177	C₂×C₈	—	73	8	no	—
178	C₂×C₈	C₂×C₄×D₇₃	1	16	yes	—
179	C₂×C₈	C₂²×Dic₇₃	1	8	yes	—
180	C₄⋊C₄	—	73	16	no	—
181	C₂²⋊C₄	—	73	16	no	—
182	C₄²	—	292	8	no	—
183	C₄²	—	73	16	no	—
184	C₄²	C₄×Dic₇₃	1	8	yes	—
185	C₄²	C₂×C₄×D₇₃	1	2	yes	—
186	C₄²	C₂×C₄×D₇₃	1	2	yes	—
187	C₄²	C₂²×Dic₇₃	1	2	yes	—
188	C₄²	C₂²×Dic₇₃	1	2	yes	—
189	C₂³	—	2	4	no	—
190	C₂³	C₄²×D₇₃	1	4	yes	—
191	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
192	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	—	73	16	no	—
197	C₂×C₄	—	146	16	no	—
198	C₂×C₄	C₄²×D₇₃	1	4	yes	—
199	C₂×C₄	C₄²×D₇₃	1	8	yes	—
200	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
201	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
202	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×Dic₇₃	1	8	yes	—
204	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₄×D₇₃	1	4	yes	—
206	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
207	C₈	—	73	16	no	—
208	C₈	C₂×C₄×Dic₇₃	1	16	yes	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	8	no	—
211	C₂²	C₄²×Dic₇₃	1	4	yes	—
212	C₂²	C₂×C₄²×D₇₃	1	2	yes	—
213	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
214	C₂²	#17989353	1	4	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	2	4	no	—
217	C₄	—	146	16	no	—
218	C₄	C₄²×Dic₇₃	1	4	yes	—
219	C₄	C₂×C₄²×D₇₃	1	2	yes	—
220	C₄	C₂²×C₄×Dic₇₃	1	2	yes	—
221	C₂	—	2	4	no	—
222	C₂	C₂×C₄²×Dic₇₃	1	1	yes	characteristic
223	C₂	#22618349	1	2	yes	—
224	C₂	#22618360	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator	C₂×C₇₃	C₂×C₄³
Fitting	C₂×C₄×M₁₆×C₇₃	C₂
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₇₃
hypocenter	C₇₃	#20331
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #22609617	C₂	semidirect
5. #97×Dic₇₃	C₂	semidirect
9. #17988545	C₄	semidirect
10. C₂×#14556502	C₄	semidirect
21. C₄×M₁₆×C₇₃	C₄	semidirect
34. #14556501	C₂×C₄	semidirect
35. C₈×Dic₇₃	C₂×C₄	semidirect
50. C₂×#10596831	C₄²	semidirect
68. M₁₆×C₇₃	C₄²	semidirect
119. C₇₃	#20331	semidirect
125. C₄×M₁₆	Dic₇₃	semidirect
173. M₁₆	C₄×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988545	2
C₂²×#14556502	2
C₄×#14556550	2
#22609617	2
#97×Dic₇₃	2
C₂×C₄×M₁₆×C₇₃	2
#20331	73