Subgroups — #33083463

Order 18688.

1054 conjugacy classes of subgroups.

The subgroup lattice has 218 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×#17988433	C₂	1	1	yes	maximal, characteristic
3	#22609684	C₂	1	4	yes	maximal, complemented
4	#22609682	C₂	1	4	yes	maximal, complemented
5	#22609624	C₂	1	4	yes	maximal
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	C₂×#14556502	C₂²	1	2	yes	—
11	C₂×C₄×#10596831	C₂²	1	2	yes	—
12	#17988433	C₄	1	4	yes	—
13	#17988433	C₂²	1	2	yes	—
14	#17988433	C₂²	1	2	yes	—
15	C₂×#14556498	C₄	1	4	yes	—
16	#17988341	C₂²	1	16	yes	—
17	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₇₃	C₄	1	8	yes	complemented
20	C₂×#97×C₇₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
22	C₂³×Dic₇₃	C₂×C₄	1	4	yes	—
23	C₂×#11924012	—	2	8	no	—
24	C₂×#11924012	C₂×C₄	1	8	yes	—
25	C₂²×#10596831	C₂³	1	4	yes	—
26	C₂×C₄×Dic₇₃	—	2	8	no	—
27	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
28	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
29	#14556502	C₂×C₄	1	8	yes	complemented
30	C₄×#10596831	C₂×C₄	1	8	yes	complemented
31	#14556498	C₂×C₄	1	8	yes	—
32	#14556498	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₇₃	—	2	8	no	—
36	C₂×M₁₆×C₇₃	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
38	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	#97×C₇₃	C₂×C₄	1	8	yes	complemented
44	C₄×C₈×C₇₃	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₇₃	—	2	8	no	—
46	C₂²×Dic₇₃	—	4	8	no	—
47	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
48	#11924012	—	2	16	no	—
49	C₂×#10596831	—	2	8	no	—
50	C₂×#10596831	C₂²×C₄	1	8	yes	—
51	C₄×Dic₇₃	—	2	8	no	—
52	C₄×Dic₇₃	—	2	8	no	—
53	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	—	4	4	no	—
55	C₂²×C₄×C₇₃	C₄²	1	8	yes	—
56	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₇₃	#41	1	8	yes	—
62	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₇₃	—	2	16	no	—
64	C₂×C₈×C₇₃	—	2	8	no	—
65	C₂×C₈×C₇₃	C₂²×C₄	1	8	yes	—
66	C₄²×C₇₃	C₄²	1	8	yes	—
67	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
71	C₂³×C₇₃	—	2	4	no	—
72	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
73	C₂³×C₇₃	#117	1	4	yes	—
74	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	—	2	4	no	—
76	C₂×C₄×C₇₃	—	2	4	no	—
77	C₂×C₄×C₇₃	—	4	8	no	—
78	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
79	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
80	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₇₃	#117	1	8	yes	—
82	C₂×C₄×C₇₃	#117	1	4	yes	—
83	C₂×C₄×C₇₃	#117	1	8	yes	—
84	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂×#41	1	4	yes	—
87	C₂×Dic₇₃	—	2	8	no	—
88	C₂×Dic₇₃	—	2	16	no	—
89	C₂×Dic₇₃	—	4	16	no	—
90	C₈×C₇₃	—	2	8	no	—
91	#10596831	—	2	8	no	—
92	C₂²×C₇₃	—	2	4	no	—
93	C₂²×C₇₃	—	2	8	no	—
94	C₂²×C₇₃	#376	1	4	yes	—
95	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
96	C₂²×C₇₃	C₂×#117	1	2	yes	—
97	C₂²×C₇₃	#568	1	4	yes	—
98	C₄×C₇₃	—	2	4	no	—
99	C₄×C₇₃	—	2	4	no	—
100	C₄×C₇₃	#376	1	4	yes	—
101	C₄×C₇₃	C₂×#117	1	2	yes	—
102	C₄×C₇₃	C₂×#117	1	2	yes	—
103	Dic₇₃	—	4	8	no	—
104	#20365	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	—	2	4	no	—
106	C₂×C₇₃	C₂×#376	1	1	yes	characteristic
107	C₂×C₇₃	#2873	1	2	yes	—
108	C₂×C₇₃	#2921	1	4	yes	—
109	C₂×C₄×M₁₆	—	73	1	no	—
110	C₂×C₄×M₁₆	D₇₃	1	1	yes	characteristic
111	C₂×#376	—	73	1	no	—
112	#1746	—	73	4	no	—
113	#1746	—	73	4	no	—
114	#1728	—	73	4	no	—
115	C₇₃	#20365	1	1	yes	characteristic, complemented, Sylow
116	C₂²×M₁₆	—	73	2	no	—
117	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
118	C₂²×C₄²	—	73	2	no	—
119	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
120	C₄×M₁₆	—	73	8	no	—
121	C₄×M₁₆	Dic₇₃	1	8	yes	complemented
122	C₂×#97	—	73	2	no	—
123	C₂×#97	C₂×D₇₃	1	2	yes	—
124	C₂×C₄×C₈	—	73	2	no	—
125	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
126	#376	—	73	4	no	—
127	#376	—	73	2	no	—
128	#376	—	73	2	no	—
129	C₂×#95	—	73	4	no	—
130	#336	—	73	16	no	—
131	C₂³×C₄	—	73	4	no	—
132	C₂³×C₄	C₄×D₇₃	1	2	yes	—
133	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
134	C₂×M₁₆	—	2	8	no	—
135	C₂×M₁₆	—	73	8	no	—
136	C₂×M₁₆	—	146	8	no	—
137	C₂×M₁₆	C₂×Dic₇₃	1	8	yes	—
138	C₂²×C₈	—	73	4	no	—
139	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
140	C₂×C₄²	—	73	4	no	—
141	C₂×C₄²	—	73	4	no	—
142	C₂×C₄²	—	146	8	no	—
143	C₂×C₄²	C₄×D₇₃	1	2	yes	—
144	C₂×C₄²	C₄×D₇₃	1	4	yes	—
145	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
146	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
148	#97	—	73	8	no	—
149	#97	C₂×Dic₇₃	1	8	yes	complemented
150	C₄×C₈	—	73	8	no	—
151	C₄×C₈	C₂×Dic₇₃	1	8	yes	complemented
152	#95	—	73	8	no	—
153	#95	—	73	8	no	—
154	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	292	8	no	—
157	C₂²×C₄	—	73	8	no	—
158	C₂²×C₄	—	146	8	no	—
159	C₂²×C₄	C₄×Dic₇₃	1	8	yes	—
160	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
161	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
162	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
164	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
165	C₂²×C₄	#11924028	1	8	yes	—
166	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
167	M₁₆	—	2	16	no	—
168	M₁₆	—	146	16	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	73	8	no	—
171	C₂×C₈	—	146	8	no	—
172	C₂×C₈	C₂²×Dic₇₃	1	8	yes	—
173	C₄²	—	146	8	no	—
174	C₄²	—	146	8	no	—
175	C₄²	C₄×Dic₇₃	1	8	yes	—
176	C₄²	C₂×C₄×D₇₃	1	2	yes	—
177	C₄²	C₂×C₄×D₇₃	1	2	yes	—
178	C₄²	C₂²×Dic₇₃	1	2	yes	—
179	C₄²	C₂²×Dic₇₃	1	2	yes	—
180	C₂³	—	2	4	no	—
181	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
182	C₂³	#14556548	1	4	yes	—
183	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	—	292	16	no	—
188	C₂×C₄	—	146	8	no	—
189	C₂×C₄	—	146	16	no	—
190	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
191	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
192	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
193	C₂×C₄	#14556548	1	4	yes	—
194	C₂×C₄	#14556548	1	8	yes	—
195	C₂×C₄	#14556550	1	8	yes	—
196	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
197	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
198	C₂×C₄	C₂×#11924028	1	4	yes	—
199	C₈	—	2	8	no	—
200	C₈	—	146	8	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	#17988433	1	4	yes	—
204	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
205	C₂²	C₂×#14556548	1	2	yes	—
206	C₂²	#17989353	1	4	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	292	8	no	—
210	C₄	#17988433	1	4	yes	—
211	C₄	C₂×#14556548	1	2	yes	—
212	C₄	C₂×#14556550	1	2	yes	—
213	C₂	—	2	4	no	—
214	C₂	C₂×#17988433	1	1	yes	characteristic
215	C₂	#22618349	1	2	yes	—
216	C₂	#22618567	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂²×C₄²
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	#20365
LCS	G ▷ C₂²×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
3. #22609684	C₂	semidirect
4. #22609682	C₂	semidirect
9. C₄×#11924012	C₄	semidirect
19. C₄×M₁₆×C₇₃	C₄	semidirect
29. #14556502	C₂×C₄	semidirect
30. C₄×#10596831	C₂×C₄	semidirect
43. #97×C₇₃	C₂×C₄	semidirect
44. C₄×C₈×C₇₃	C₂×C₄	semidirect
115. C₇₃	#20365	semidirect
121. C₄×M₁₆	Dic₇₃	semidirect
149. #97	C₂×Dic₇₃	semidirect
151. 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×#17988433	2
#22609684	2
#22609682	2
#22609624	2
C₂×C₄×M₁₆×C₇₃	2
#20365	73