Subgroups — C₂²×#97×D₇₃ #33949365

Order 18688.

4126 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₇₃	C₂	1	2	yes	maximal
2	C₂²×C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14556502	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14556501	C₂	1	2	yes	maximal, complemented
5	C₂×#97×D₇₃	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#97×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₇₃	C₄	1	2	yes	—
8	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
9	C₂³×#10596831	C₂²	1	2	yes	—
10	C₂²×C₈×D₇₃	C₄	1	32	yes	complemented
11	C₂²×C₈×D₇₃	C₂²	1	24	yes	—
12	C₂²×C₄×Dic₇₃	C₄	1	2	yes	—
13	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×D₇₃	C₄	1	12	yes	—
15	C₂×C₄²×D₇₃	C₂²	1	12	yes	—
16	C₂×#14556502	C₂²	1	12	yes	complemented
17	C₂×#14556501	C₂²	1	24	yes	complemented
18	#97×D₇₃	C₂²	1	64	yes	direct product factor
19	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₇₃	C₂²	1	12	yes	complemented
22	C₂⁴×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×D₇₃	—	2	16	no	—
26	C₂²×C₄×D₇₃	C₂×C₄	1	24	yes	—
27	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
28	C₂²×C₄×D₇₃	C₂×C₄	1	16	yes	—
29	C₂²×C₄×D₇₃	C₂³	1	6	yes	—
30	C₂²×#10596831	C₂×C₄	1	16	yes	complemented
31	C₂²×#10596831	C₂³	1	12	yes	—
32	C₂×C₈×D₇₃	C₂×C₄	1	192	yes	complemented
33	C₂×C₈×D₇₃	C₂³	1	32	yes	—
34	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₇₃	C₂×C₄	1	24	yes	—
36	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
37	C₄²×D₇₃	C₂×C₄	1	48	yes	—
38	C₄²×D₇₃	C₂³	1	16	yes	—
39	#14556502	C₂³	1	16	yes	complemented
40	#14556501	C₂³	1	32	yes	complemented
41	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₇₃	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₇₃	C₂³	1	12	yes	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	6	yes	—
46	C₂×C₄²×C₇₃	C₂³	1	6	yes	—
47	#97×C₇₃	C₂³	1	16	yes	complemented
48	C₂³×D₇₃	C₄²	1	8	yes	—
49	C₂³×D₇₃	M₁₆	1	16	yes	—
50	C₂³×D₇₃	C₂²×C₄	1	6	yes	—
51	C₂²×Dic₇₃	—	2	8	no	—
52	C₂²×Dic₇₃	C₄²	1	8	yes	—
53	C₂²×Dic₇₃	C₂²×C₄	1	12	yes	—
54	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
55	C₂×C₄×D₇₃	—	2	96	no	—
56	C₂×C₄×D₇₃	C₄²	1	48	yes	—
57	C₂×C₄×D₇₃	C₂²×C₄	1	32	yes	—
58	C₂×C₄×D₇₃	C₂²×C₄	1	12	yes	—
59	C₂×C₄×D₇₃	C₂²×C₄	1	48	yes	—
60	C₂×C₄×D₇₃	C₂⁴	1	4	yes	—
61	C₂×#10596831	C₂²×C₄	1	48	yes	complemented
62	C₂×#10596831	C₂⁴	1	8	yes	—
63	C₈×D₇₃	C₂²×C₄	1	128	yes	complemented
64	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
65	C₄×Dic₇₃	C₂²×C₄	1	32	yes	—
66	C₄×Dic₇₃	C₂⁴	1	4	yes	—
67	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	—	2	8	no	—
69	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₇₃	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
73	C₂×C₈×C₇₃	C₂²×C₄	1	48	yes	complemented
74	C₂×C₈×C₇₃	C₂⁴	1	8	yes	—
75	C₄²×C₇₃	C₂²×C₄	1	12	yes	—
76	C₄²×C₇₃	C₂⁴	1	4	yes	—
77	C₂³×C₇₃	C₂×C₄²	1	4	yes	—
78	C₂³×C₇₃	C₂×M₁₆	1	8	yes	—
79	C₂³×C₇₃	C₂³×C₄	1	3	yes	—
80	C₂²×D₇₃	#97	1	64	yes	direct product factor
81	C₂²×D₇₃	C₂×C₄²	1	24	yes	—
82	C₂²×D₇₃	C₂×M₁₆	1	48	yes	—
83	C₂²×D₇₃	C₂³×C₄	1	4	yes	—
84	C₂×C₄×C₇₃	—	2	24	no	—
85	C₂×C₄×C₇₃	C₂×C₄²	1	12	yes	—
86	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
87	C₂×C₄×C₇₃	C₂³×C₄	1	12	yes	—
88	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
89	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
90	C₂×Dic₇₃	—	2	24	no	—
91	C₂×Dic₇₃	C₂×C₄²	1	24	yes	—
92	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
93	C₂×Dic₇₃	C₂³×C₄	1	4	yes	—
94	C₄×D₇₃	—	2	64	no	—
95	C₄×D₇₃	C₂×C₄²	1	48	yes	—
96	C₄×D₇₃	C₂³×C₄	1	16	yes	—
97	C₈×C₇₃	C₂³×C₄	1	16	yes	complemented
98	#10596831	C₂³×C₄	1	16	yes	complemented
99	C₂²×C₇₃	C₂×#97	1	16	yes	complemented
100	C₂²×C₇₃	C₂²×C₄²	1	6	yes	—
101	C₂²×C₇₃	C₂²×M₁₆	1	12	yes	—
102	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
103	C₂×D₇₃	C₂×#97	1	96	yes	direct product factor
104	C₂×D₇₃	C₂²×C₄²	1	8	yes	—
105	C₂×D₇₃	C₂²×M₁₆	1	16	yes	—
106	C₄×C₇₃	—	2	8	no	—
107	C₄×C₇₃	C₂²×C₄²	1	6	yes	—
108	C₄×C₇₃	C₂⁴×C₄	1	2	yes	—
109	Dic₇₃	—	2	8	no	—
110	Dic₇₃	C₂²×C₄²	1	8	yes	—
111	C₂³×#97	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	C₂²×#97	1	12	yes	complemented
113	C₂×C₇₃	C₂³×C₄²	1	1	yes	characteristic
114	C₂×C₇₃	C₂³×M₁₆	1	2	yes	—
115	D₇₃	C₂²×#97	1	16	yes	direct product factor
116	C₂⁴×C₈	—	73	2	no	—
117	C₂³×C₄²	—	73	1	no	—
118	C₂²×#97	—	73	1	no	—
119	C₂²×#97	—	73	24	no	—
120	C₂²×#97	—	73	2	no	—
121	C₂²×#97	D₇₃	1	1	yes	characteristic, direct product factor
122	C₇₃	C₂³×#97	1	1	yes	characteristic, complemented, Sylow
123	C₂⁴×C₄	—	73	1	no	—
124	C₂⁴×C₄	—	73	2	no	—
125	C₂³×C₈	—	73	2	no	—
126	C₂³×C₈	—	73	32	no	—
127	C₂³×C₈	—	73	24	no	—
128	C₂³×C₈	C₂×D₇₃	1	2	yes	—
129	C₂²×C₄²	—	73	2	no	—
130	C₂²×C₄²	—	73	1	no	—
131	C₂²×C₄²	—	73	12	no	—
132	C₂²×C₄²	—	73	12	no	—
133	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
134	C₂×#97	—	73	24	no	—
135	C₂×#97	—	73	12	no	—
136	C₂×#97	—	73	64	no	—
137	C₂×#97	C₂×D₇₃	1	12	yes	direct product factor
138	C₂⁵	—	73	1	no	—
139	C₂³×C₄	—	73	1	no	—
140	C₂³×C₄	—	73	16	no	—
141	C₂³×C₄	—	73	6	no	—
142	C₂³×C₄	—	73	6	no	—
143	C₂³×C₄	—	73	2	no	—
144	C₂³×C₄	—	73	24	no	—
145	C₂³×C₄	—	146	16	no	—
146	C₂³×C₄	C₄×D₇₃	1	2	yes	—
147	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
148	C₂²×C₈	—	73	16	no	—
149	C₂²×C₈	—	73	12	no	—
150	C₂²×C₈	—	73	192	no	—
151	C₂²×C₈	—	73	32	no	—
152	C₂²×C₈	C₄×D₇₃	1	16	yes	complemented
153	C₂²×C₈	C₂²×D₇₃	1	12	yes	—
154	C₂×C₄²	—	73	24	no	—
155	C₂×C₄²	—	73	6	no	—
156	C₂×C₄²	—	73	6	no	—
157	C₂×C₄²	—	73	48	no	—
158	C₂×C₄²	—	73	16	no	—
159	C₂×C₄²	C₄×D₇₃	1	6	yes	—
160	C₂×C₄²	C₂²×D₇₃	1	6	yes	—
161	#97	—	73	32	no	—
162	#97	—	73	16	no	—
163	#97	C₂²×D₇₃	1	16	yes	direct product factor
164	C₂⁴	—	73	8	no	—
165	C₂⁴	—	73	16	no	—
166	C₂⁴	—	73	6	no	—
167	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	73	8	no	—
170	C₂²×C₄	—	73	6	no	—
171	C₂²×C₄	—	73	48	no	—
172	C₂²×C₄	—	73	12	no	—
173	C₂²×C₄	—	73	48	no	—
174	C₂²×C₄	—	73	4	no	—
175	C₂²×C₄	—	73	12	no	—
176	C₂²×C₄	—	73	32	no	—
177	C₂²×C₄	—	146	96	no	—
178	C₂²×C₄	—	146	8	no	—
179	C₂²×C₄	C₂×C₄×D₇₃	1	12	yes	—
180	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
181	C₂²×C₄	C₂×C₄×D₇₃	1	8	yes	—
182	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
183	C₂×C₈	—	73	48	no	—
184	C₂×C₈	—	73	8	no	—
185	C₂×C₈	—	73	128	no	—
186	C₂×C₈	C₂×C₄×D₇₃	1	48	yes	complemented
187	C₂×C₈	C₂³×D₇₃	1	8	yes	—
188	C₄²	—	73	32	no	—
189	C₄²	—	73	12	no	—
190	C₄²	—	73	4	no	—
191	C₄²	C₂×C₄×D₇₃	1	12	yes	—
192	C₄²	C₂³×D₇₃	1	4	yes	—
193	C₂³	—	73	4	no	—
194	C₂³	—	73	24	no	—
195	C₂³	—	73	64	no	—
196	C₂³	—	73	48	no	—
197	C₂³	C₄²×D₇₃	1	4	yes	—
198	C₂³	M₁₆×D₇₃	1	8	yes	—
199	C₂³	C₂²×C₄×D₇₃	1	3	yes	—
200	C₂×C₄	—	2	24	no	—
201	C₂×C₄	—	73	4	no	—
202	C₂×C₄	—	73	24	no	—
203	C₂×C₄	—	73	16	no	—
204	C₂×C₄	—	73	48	no	—
205	C₂×C₄	—	73	8	no	—
206	C₂×C₄	—	146	64	no	—
207	C₂×C₄	—	146	24	no	—
208	C₂×C₄	C₄²×D₇₃	1	12	yes	—
209	C₂×C₄	C₂²×C₄×D₇₃	1	8	yes	—
210	C₂×C₄	C₂²×C₄×D₇₃	1	12	yes	—
211	C₂×C₄	C₂²×C₄×D₇₃	1	3	yes	—
212	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
213	C₈	—	73	16	no	—
214	C₈	C₂²×C₄×D₇₃	1	16	yes	complemented
215	C₂²	—	73	8	no	—
216	C₂²	—	73	16	no	—
217	C₂²	—	73	96	no	—
218	C₂²	#97×D₇₃	1	16	yes	direct product factor
219	C₂²	C₂×C₄²×D₇₃	1	6	yes	—
220	C₂²	C₂×M₁₆×D₇₃	1	12	yes	—
221	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
222	C₄	—	2	8	no	—
223	C₄	—	73	8	no	—
224	C₄	—	146	8	no	—
225	C₄	C₂×C₄²×D₇₃	1	6	yes	—
226	C₄	C₂³×C₄×D₇₃	1	2	yes	—
227	C₂	—	73	16	no	—
228	C₂	C₂×#97×D₇₃	1	12	yes	direct product factor
229	C₂	C₂²×C₄²×D₇₃	1	1	yes	characteristic
230	C₂	C₂²×M₁₆×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₃
commutator	C₂×C₇₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×#97×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂⁴×D₇₃	C₂×C₄
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂²×#97	D₇₃
hypocenter	C₇₃	C₂³×#97
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. C₂²×#14556502	C₂	semidirect
4. C₂²×#14556501	C₂	semidirect
5. C₂×#97×D₇₃	C₂	direct (N × H)
6. C₂²×#97×C₇₃	C₂	semidirect
10. C₂²×C₈×D₇₃	C₄	semidirect
16. C₂×#14556502	C₂²	semidirect
17. C₂×#14556501	C₂²	semidirect
18. #97×D₇₃	C₂²	direct (N × H)
21. C₂×#97×C₇₃	C₂²	semidirect
30. C₂²×#10596831	C₂×C₄	semidirect
32. C₂×C₈×D₇₃	C₂×C₄	semidirect
39. #14556502	C₂³	semidirect
40. #14556501	C₂³	semidirect
43. C₂²×C₈×C₇₃	C₂×C₄	semidirect
47. #97×C₇₃	C₂³	semidirect
61. C₂×#10596831	C₂²×C₄	semidirect
63. C₈×D₇₃	C₂²×C₄	semidirect
73. C₂×C₈×C₇₃	C₂²×C₄	semidirect
80. C₂²×D₇₃	#97	direct (N × H)
97. C₈×C₇₃	C₂³×C₄	semidirect
98. #10596831	C₂³×C₄	semidirect
99. C₂²×C₇₃	C₂×#97	semidirect
103. C₂×D₇₃	C₂×#97	direct (N × H)
112. C₂×C₇₃	C₂²×#97	semidirect
115. D₇₃	C₂²×#97	direct (N × H)
121. C₂²×#97	D₇₃	direct (N × H)
122. C₇₃	C₂³×#97	semidirect
137. C₂×#97	C₂×D₇₃	direct (N × H)
152. C₂²×C₈	C₄×D₇₃	semidirect
163. #97	C₂²×D₇₃	direct (N × H)
186. C₂×C₈	C₂×C₄×D₇₃	semidirect
214. C₈	C₂²×C₄×D₇₃	semidirect
218. C₂²	#97×D₇₃	direct (N × H)
228. C₂	C₂×#97×D₇₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×D₇₃	2
C₂²×C₄²×D₇₃	2
C₂²×#14556502	2
C₂²×#14556501	2
C₂×#97×D₇₃	2
C₂²×#97×C₇₃	2
C₂³×#97	73

Subgroups — C22×#97×D73 #33949365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#97×D₇₃ #33949365