Subgroups — #33001301

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988407	C₂	1	2	yes	maximal
2	M₃₂×Dic₇₃	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	#22608013	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
13	#17988407	C₄	1	8	yes	complemented
14	#17988362	—	2	2	no	—
15	C₁₆×Dic₇₃	—	2	2	no	—
16	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₃	—	2	2	no	—
21	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	#14556531	—	2	2	no	—
24	#14556531	C₂×C₄	1	4	yes	—
25	C₂×#11924003	—	2	2	no	—
26	C₄×#10596831	C₈	1	4	yes	—
27	C₈×Dic₇₃	—	2	2	no	—
28	C₈×Dic₇₃	C₈	1	4	yes	—
29	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
30	C₈×Dic₇₃	D₄	1	1	yes	characteristic
31	C₈×Dic₇₃	D₄	1	1	yes	characteristic
32	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
33	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₇₃	—	2	2	no	—
38	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₇₃	—	2	2	no	—
40	C₄×C₈×C₇₃	—	2	2	no	—
41	C₂²×Dic₇₃	—	2	1	no	—
42	C₂²×Dic₇₃	M₁₆	1	2	yes	—
43	C₂×#10596831	C₄²	1	2	yes	—
44	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
46	C₂×#10596831	C₂×C₈	1	2	yes	—
47	C₂×#10596831	C₂×C₈	1	2	yes	—
48	C₂×#10596831	C₂×C₈	1	2	yes	—
49	C₂×#10596831	M₁₆	1	2	yes	—
50	C₄×Dic₇₃	—	2	8	no	—
51	C₄×Dic₇₃	—	2	1	no	—
52	C₄×Dic₇₃	C₄²	1	2	yes	—
53	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
56	#11924003	—	2	4	no	—
57	C₂²×C₄×C₇₃	—	2	1	no	—
58	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	—	2	1	no	—
60	C₂×C₈×C₇₃	—	2	1	no	—
61	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₇₃	#41	1	2	yes	—
66	C₄²×C₇₃	—	2	1	no	—
67	C₄²×C₇₃	C₂×C₈	1	2	yes	—
68	C₁₆×C₇₃	—	2	4	no	—
69	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	4	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₇₃	—	2	4	no	—
79	C₂×Dic₇₃	—	2	2	no	—
80	C₂×Dic₇₃	—	2	4	no	—
81	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
82	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	C₄×D₄	1	2	yes	—
85	C₈×C₇₃	C₄×Q₈	1	2	yes	—
86	#10596831	C₄×C₈	1	4	yes	—
87	#10596831	C₄⋊C₈	1	2	yes	—
88	#10596831	C₄⋊C₈	1	2	yes	—
89	C₂²×C₇₃	—	2	2	no	—
90	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
91	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₇₃	#431	1	1	yes	characteristic
93	C₄×C₇₃	—	2	4	no	—
94	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₇₃	#431	1	1	yes	characteristic
98	Dic₇₃	—	2	4	no	—
99	Dic₇₃	—	2	2	no	—
100	Dic₇₃	#364	1	4	yes	complemented
101	C₄×#364	—	73	1	no	maximal, Sylow
102	C₂×C₇₃	—	2	1	no	—
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₇₃	C₂×#364	1	2	yes	—
106	C₂×#364	—	73	2	no	—
107	C₄×M₃₂	—	73	2	no	—
108	C₄²×C₈	—	73	1	no	—
109	#1376	—	73	1	no	—
110	#1376	D₇₃	1	1	yes	characteristic
111	C₇₃	C₄×#364	1	1	yes	characteristic, complemented, Sylow
112	C₂×M₃₂	—	73	2	no	—
113	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
114	C₂×C₄×C₈	—	73	1	no	—
115	C₂×C₄×C₈	—	73	1	no	—
116	C₂×C₄×C₈	—	73	1	no	—
117	C₂×C₄×C₈	—	73	1	no	—
118	C₂×C₄×C₈	—	73	1	no	—
119	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
120	C₄³	—	73	1	no	—
121	#364	—	73	8	no	—
122	#346	—	146	2	no	—
123	C₄×C₁₆	—	146	2	no	—
124	C₂²×C₈	—	73	1	no	—
125	C₂²×C₈	—	73	1	no	—
126	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
128	C₂×C₄²	—	73	1	no	—
129	C₂×C₄²	—	73	1	no	—
130	C₂×C₄²	—	146	2	no	—
131	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	—	73	4	no	—
134	M₃₂	—	146	2	no	—
135	M₃₂	C₄×D₇₃	1	4	yes	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	146	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	73	4	no	—
140	C₄×C₈	—	73	4	no	—
141	C₄×C₈	—	73	1	no	—
142	C₄×C₈	—	73	1	no	—
143	C₄×C₈	—	73	1	no	—
144	C₄×C₈	—	73	1	no	—
145	C₄×C₈	—	73	4	no	—
146	C₄×C₈	—	146	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	73	2	no	—
149	C₂²×C₄	—	146	1	no	—
150	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	73	2	no	—
154	C₂×C₈	—	73	1	no	—
155	C₂×C₈	—	73	1	no	—
156	C₂×C₈	—	73	2	no	—
157	C₂×C₈	—	73	2	no	—
158	C₂×C₈	—	73	2	no	—
159	C₂×C₈	—	73	2	no	—
160	C₂×C₈	C₈×D₇₃	1	2	yes	—
161	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
162	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
164	C₂×C₈	#11924027	1	2	yes	—
165	C₄²	—	2	1	no	—
166	C₄²	—	73	1	no	—
167	C₄²	—	73	1	no	—
168	C₄²	—	73	2	no	—
169	C₄²	—	73	2	no	—
170	C₄²	—	146	8	no	—
171	C₄²	—	146	1	no	—
172	C₄²	C₈×D₇₃	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	146	4	no	—
175	C₂³	M₁₆×D₇₃	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	73	2	no	—
179	C₂×C₄	—	73	2	no	—
180	C₂×C₄	—	146	4	no	—
181	C₂×C₄	—	146	4	no	—
182	C₂×C₄	—	146	2	no	—
183	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
184	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
185	C₂×C₄	#14556547	1	1	yes	characteristic
186	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
188	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	73	4	no	—
191	C₈	—	73	2	no	—
192	C₈	—	73	2	no	—
193	C₈	#14556551	1	2	yes	—
194	C₈	#14556556	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	#17988539	1	1	yes	characteristic
197	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
198	C₂²	#17988637	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	73	4	no	—
201	C₄	—	146	4	no	—
202	C₄	—	146	2	no	—
203	C₄	#17988438	1	1	yes	characteristic
204	C₄	C₄×C₈×D₇₃	1	1	yes	characteristic
205	C₄	C₄⋊C₈×D₇₃	1	1	yes	characteristic
206	C₄	#17988637	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	#22609779	1	1	yes	characteristic
210	C₂	#364×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1376	C₂
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1376	D₇₃
hypocenter	C₇₃	C₄×#364
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
13. #17988407	C₄	semidirect
100. Dic₇₃	#364	semidirect
111. C₇₃	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988407	2
M₃₂×Dic₇₃	2
C₄×C₈×Dic₇₃	2
#22608013	2
C₇₃×#1376	2
C₄×#364	73