Subgroups — #33083318

Order 18688.

1502 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11924012	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₇₃	C₂	1	4	yes	maximal
3	C₂×#17988433	C₂	1	1	yes	maximal, characteristic
4	#22609619	C₂	1	4	yes	maximal, complemented
5	#22609617	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11924012	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₇₃	C₂²	1	2	yes	—
9	C₄×#11924012	C₄	1	8	yes	complemented
10	M₁₆×Dic₇₃	C₄	1	32	yes	complemented
11	C₂×#14556502	C₂²	1	2	yes	—
12	C₂×#14556501	C₂²	1	8	yes	—
13	C₂×C₄×#10596831	C₂²	1	2	yes	—
14	C₂×C₈×Dic₇₃	C₂²	1	8	yes	—
15	#17988433	C₄	1	4	yes	—
16	#17988433	C₂²	1	2	yes	—
17	#17988433	C₂²	1	2	yes	—
18	C₂×#14556498	C₄	1	4	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	C₂³×Dic₇₃	C₂×C₄	1	4	yes	—
25	C₂×#11924012	C₂×C₄	1	16	yes	—
26	C₂×#11924012	C₂×C₄	1	8	yes	—
27	C₂²×#10596831	C₂³	1	4	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	16	yes	—
30	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
31	#14556502	C₂×C₄	1	8	yes	complemented
32	#14556501	C₂×C₄	1	32	yes	complemented
33	C₄×#10596831	C₂×C₄	1	8	yes	complemented
34	C₈×Dic₇₃	C₂×C₄	1	32	yes	complemented
35	#14556498	C₂×C₄	1	8	yes	—
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	complemented
48	C₄×C₈×C₇₃	C₂×C₄	1	8	yes	complemented
49	C₂²×Dic₇₃	—	4	8	no	—
50	C₂²×Dic₇₃	C₄²	1	16	yes	—
51	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
52	#11924012	C₄²	1	32	yes	—
53	C₂×#10596831	C₂²×C₄	1	16	yes	—
54	C₂×#10596831	C₂²×C₄	1	8	yes	—
55	C₄×Dic₇₃	C₄²	1	16	yes	—
56	C₄×Dic₇₃	C₂²×C₄	1	8	yes	—
57	C₄×Dic₇₃	C₂²×C₄	1	8	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	—
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	—	4	16	no	—
94	C₂×Dic₇₃	C₂×C₄²	1	16	yes	—
95	C₈×C₇₃	C₂×C₄²	1	16	yes	—
96	#10596831	C₂×C₄²	1	16	yes	—
97	C₂²×C₇₃	—	2	8	no	—
98	C₂²×C₇₃	—	2	4	no	—
99	C₂²×C₇₃	C₄³	1	4	yes	—
100	C₂²×C₇₃	C₂²×C₄²	1	2	yes	—
101	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
102	C₂²×C₇₃	#568	1	4	yes	—
103	C₄×C₇₃	—	2	4	no	—
104	C₄×C₇₃	—	2	4	no	—
105	C₄×C₇₃	C₄³	1	4	yes	—
106	C₄×C₇₃	C₂²×C₄²	1	2	yes	—
107	C₄×C₇₃	C₂²×C₄²	1	2	yes	—
108	Dic₇₃	—	2	16	no	—
109	#20329	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	—	2	4	no	—
111	C₂×C₇₃	C₂×C₄³	1	1	yes	characteristic
112	C₂×C₇₃	#2873	1	4	yes	—
113	C₂×C₇₃	#2873	1	2	yes	—
114	C₂×C₄×M₁₆	—	73	1	no	—
115	C₂×C₄×M₁₆	—	73	4	no	—
116	C₂×C₄×M₁₆	D₇₃	1	1	yes	characteristic
117	C₂×#376	—	73	1	no	—
118	#1726	—	73	4	no	—
119	#1726	—	73	4	no	—
120	C₇₃	#20329	1	1	yes	characteristic, complemented, Sylow
121	C₂²×M₁₆	—	73	2	no	—
122	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
123	C₂²×C₄²	—	73	2	no	—
124	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
125	C₄×M₁₆	—	73	32	no	—
126	C₄×M₁₆	—	73	8	no	—
127	C₄×M₁₆	Dic₇₃	1	8	yes	complemented
128	C₂×#97	—	73	8	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	—	73	2	no	—
133	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
134	#376	—	73	4	no	—
135	#376	—	73	2	no	—
136	#376	—	73	2	no	—
137	C₂×#95	—	73	4	no	—
138	C₂³×C₄	—	73	4	no	—
139	C₂³×C₄	C₄×D₇₃	1	2	yes	—
140	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
141	C₂×M₁₆	—	73	8	no	—
142	C₂×M₁₆	—	73	16	no	—
143	C₂×M₁₆	C₄×D₇₃	1	16	yes	—
144	C₂×M₁₆	C₂×Dic₇₃	1	8	yes	—
145	C₂²×C₈	—	73	4	no	—
146	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
147	C₂×C₄²	—	73	4	no	—
148	C₂×C₄²	—	73	16	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	C₂×Dic₇₃	1	8	yes	complemented
158	C₄×C₈	—	73	32	no	—
159	C₄×C₈	—	73	8	no	—
160	C₄×C₈	C₂×Dic₇₃	1	8	yes	complemented
161	#95	—	73	8	no	—
162	#95	—	73	8	no	—
163	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	—	292	8	no	—
166	C₂²×C₄	—	73	16	no	—
167	C₂²×C₄	—	73	8	no	—
168	C₂²×C₄	C₄×Dic₇₃	1	8	yes	—
169	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
170	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
173	C₂²×C₄	C₂×C₄×D₇₃	1	8	yes	—
174	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
175	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
176	M₁₆	—	73	32	no	—
177	M₁₆	C₄×Dic₇₃	1	32	yes	—
178	C₂×C₈	—	73	8	no	—
179	C₂×C₈	—	73	16	no	—
180	C₂×C₈	C₂×C₄×D₇₃	1	16	yes	—
181	C₂×C₈	C₂²×Dic₇₃	1	8	yes	—
182	C₄²	—	73	16	no	—
183	C₄²	—	73	8	no	—
184	C₄²	—	73	8	no	—
185	C₄²	C₄×Dic₇₃	1	8	yes	—
186	C₄²	C₂×C₄×D₇₃	1	2	yes	—
187	C₄²	C₂×C₄×D₇₃	1	2	yes	—
188	C₄²	C₂²×Dic₇₃	1	2	yes	—
189	C₄²	C₂²×Dic₇₃	1	2	yes	—
190	C₂³	—	2	4	no	—
191	C₂³	C₄²×D₇₃	1	4	yes	—
192	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
193	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	4	8	no	—
197	C₂×C₄	—	292	16	no	—
198	C₂×C₄	—	73	16	no	—
199	C₂×C₄	—	146	16	no	—
200	C₂×C₄	C₄²×D₇₃	1	4	yes	—
201	C₂×C₄	C₄²×D₇₃	1	8	yes	—
202	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
203	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
204	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
205	C₂×C₄	C₂×C₄×Dic₇₃	1	8	yes	—
206	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₄×D₇₃	1	4	yes	—
208	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
209	C₈	—	73	16	no	—
210	C₈	C₂×C₄×Dic₇₃	1	16	yes	—
211	C₂²	—	2	8	no	—
212	C₂²	—	2	4	no	—
213	C₂²	C₄²×Dic₇₃	1	4	yes	—
214	C₂²	C₂×C₄²×D₇₃	1	2	yes	—
215	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
216	C₂²	#17989353	1	4	yes	—
217	C₄	—	2	4	no	—
218	C₄	—	2	4	no	—
219	C₄	—	146	16	no	—
220	C₄	C₄²×Dic₇₃	1	4	yes	—
221	C₄	C₂×C₄²×D₇₃	1	2	yes	—
222	C₄	C₂²×C₄×Dic₇₃	1	2	yes	—
223	C₂	—	2	4	no	—
224	C₂	C₂×C₄²×Dic₇₃	1	1	yes	characteristic
225	C₂	#22618348	1	4	yes	—
226	C₂	#22618349	1	2	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	#20329
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. #22609619	C₂	semidirect
5. #22609617	C₂	semidirect
9. C₄×#11924012	C₄	semidirect
10. M₁₆×Dic₇₃	C₄	semidirect
21. C₄×M₁₆×C₇₃	C₄	semidirect
31. #14556502	C₂×C₄	semidirect
32. #14556501	C₂×C₄	semidirect
33. C₄×#10596831	C₂×C₄	semidirect
34. C₈×Dic₇₃	C₂×C₄	semidirect
47. #97×C₇₃	C₂×C₄	semidirect
48. C₄×C₈×C₇₃	C₂×C₄	semidirect
120. C₇₃	#20329	semidirect
127. C₄×M₁₆	Dic₇₃	semidirect
157. #97	C₂×Dic₇₃	semidirect
160. C₄×C₈	C₂×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11924012	2
C₂×M₁₆×Dic₇₃	2
C₂×#17988433	2
#22609619	2
#22609617	2
C₂×C₄×M₁₆×C₇₃	2
#20329	73