Subgroups — #33083464

Order 18688.

1054 conjugacy classes of subgroups.

The subgroup lattice has 160 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	2	yes	maximal
2	#22609683	C₂	1	8	yes	maximal, complemented
3	#22609625	C₂	1	4	yes	maximal
4	C₂×#376×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₂²	1	4	yes	—
6	C₄×#11924012	C₄	1	16	yes	complemented
7	C₂×#14556502	C₂²	1	4	yes	—
8	C₂×C₄×#10596831	C₂²	1	4	yes	—
9	#17988342	C₂²	1	16	yes	—
10	C₂²×C₄²×C₇₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
12	#376×C₇₃	C₄	1	4	yes	—
13	#376×C₇₃	C₂²	1	4	yes	—
14	C₂×#95×C₇₃	C₄	1	4	yes	—
15	C₂×#11924012	—	2	16	no	—
16	C₂×#11924012	C₂×C₄	1	16	yes	—
17	C₂²×#10596831	C₂³	1	8	yes	—
18	#14556502	C₂×C₄	1	16	yes	complemented
19	C₄×#10596831	C₂×C₄	1	16	yes	complemented
20	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₇₃	—	2	8	no	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₇₃	C₂³	1	4	yes	—
28	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
29	#95×C₇₃	C₂×C₄	1	16	yes	—
30	#11924012	—	2	32	no	—
31	C₂×#10596831	—	2	16	no	—
32	C₂×#10596831	C₂²×C₄	1	16	yes	—
33	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	—	2	8	no	—
35	C₂²×C₄×C₇₃	—	4	4	no	—
36	C₂²×C₄×C₇₃	—	4	8	no	—
37	C₂²×C₄×C₇₃	C₄²	1	8	yes	—
38	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₇₃	C₂²×C₄	1	8	yes	—
40	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	#41	1	8	yes	—
43	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
44	C₄²×C₇₃	—	2	8	no	—
45	C₄²×C₇₃	—	2	8	no	—
46	C₄²×C₇₃	C₄²	1	8	yes	—
47	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
48	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
49	C₂³×C₇₃	—	2	4	no	—
50	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
51	C₂³×C₇₃	#117	1	4	yes	—
52	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	2	8	no	—
54	C₂×C₄×C₇₃	—	2	8	no	—
55	C₂×C₄×C₇₃	—	2	16	no	—
56	C₂×C₄×C₇₃	—	4	8	no	—
57	C₂×C₄×C₇₃	—	4	16	no	—
58	C₂×C₄×C₇₃	C₂×C₄²	1	8	yes	—
59	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₇₃	#117	1	16	yes	—
61	C₂×C₄×C₇₃	#117	1	4	yes	—
62	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
63	C₂×C₄×C₇₃	C₂×#41	1	4	yes	—
64	#10596831	—	2	16	no	—
65	C₂²×C₇₃	—	2	4	no	—
66	C₂²×C₇₃	—	2	8	no	—
67	C₂²×C₇₃	#376	1	4	yes	—
68	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
69	C₂²×C₇₃	C₂×#117	1	2	yes	—
70	C₂²×C₇₃	#568	1	4	yes	—
71	C₄×C₇₃	—	2	8	no	—
72	C₄×C₇₃	—	4	8	no	—
73	C₄×C₇₃	#376	1	4	yes	—
74	C₄×C₇₃	C₂×#117	1	4	yes	—
75	#20365	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	2	4	no	—
77	C₂×C₇₃	C₂×#376	1	1	yes	characteristic
78	C₂×C₇₃	#2873	1	2	yes	—
79	C₂×C₇₃	#2921	1	4	yes	—
80	C₂×C₄×M₁₆	—	73	2	no	—
81	C₂×#376	D₇₃	1	1	yes	characteristic
82	#1746	—	73	8	no	—
83	#1728	—	73	4	no	—
84	C₇₃	#20365	1	1	yes	characteristic, complemented, Sylow
85	C₂²×M₁₆	—	73	4	no	—
86	C₂²×C₄²	C₂×D₇₃	1	2	yes	—
87	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
88	C₄×M₁₆	—	73	16	no	—
89	C₂×#97	—	73	4	no	—
90	C₂×C₄×C₈	—	73	4	no	—
91	#376	Dic₇₃	1	4	yes	—
92	#376	C₂×D₇₃	1	4	yes	—
93	C₂×#95	Dic₇₃	1	4	yes	—
94	#336	—	73	16	no	—
95	C₂³×C₄	C₂×Dic₇₃	1	4	yes	—
96	C₂³×C₄	C₂×Dic₇₃	1	2	yes	—
97	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
98	C₂×M₁₆	—	73	16	no	—
99	C₂×M₁₆	—	146	16	no	—
100	C₂²×C₈	—	73	8	no	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	C₄×D₇₃	1	8	yes	—
103	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
104	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
105	C₂×C₄²	C₂²×D₇₃	1	4	yes	—
106	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
107	#97	—	73	16	no	—
108	C₄×C₈	—	73	16	no	—
109	#95	C₂×Dic₇₃	1	16	yes	—
110	C₂⁴	C₂²×Dic₇₃	1	1	yes	characteristic
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	4	4	no	—
113	C₂²×C₄	—	4	8	no	—
114	C₂²×C₄	C₄×Dic₇₃	1	8	yes	—
115	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
116	C₂²×C₄	C₂²×Dic₇₃	1	8	yes	—
117	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
118	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
119	C₂²×C₄	#11924028	1	8	yes	—
120	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
121	M₁₆	—	146	32	no	—
122	C₂×C₈	—	73	16	no	—
123	C₂×C₈	—	146	16	no	—
124	C₄²	—	2	8	no	—
125	C₄²	—	2	8	no	—
126	C₄²	C₄×Dic₇₃	1	8	yes	—
127	C₄²	C₂×C₄×D₇₃	1	4	yes	—
128	C₄²	C₂×C₄×D₇₃	1	4	yes	—
129	C₂³	—	2	4	no	—
130	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
131	C₂³	#14556550	1	4	yes	—
132	C₂³	C₂³×Dic₇₃	1	1	yes	characteristic
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	—	4	16	no	—
138	C₂×C₄	C₂×C₄×Dic₇₃	1	8	yes	—
139	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
140	C₂×C₄	#14556548	1	16	yes	—
141	C₂×C₄	#14556550	1	4	yes	—
142	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
143	C₂×C₄	C₂×#11924028	1	4	yes	—
144	C₈	—	146	16	no	—
145	C₂²	—	2	8	no	—
146	C₂²	—	2	4	no	—
147	C₂²	#17988433	1	4	yes	—
148	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
149	C₂²	C₂×#14556550	1	2	yes	—
150	C₂²	#17989354	1	4	yes	—
151	C₄	—	2	8	no	—
152	C₄	—	4	8	no	—
153	C₄	#17988433	1	4	yes	—
154	C₄	C₂×#14556548	1	4	yes	—
155	C₂	—	2	4	no	—
156	C₂	C₂×#17988433	1	1	yes	characteristic
157	C₂	#22618350	1	2	yes	—
158	C₂	#22618569	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂²×C₄²
Fitting	C₂×#376×C₇₃	C₂
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#376	D₇₃
hypocenter	C₇₃	#20365
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #22609683	C₂	semidirect
6. C₄×#11924012	C₄	semidirect
18. #14556502	C₂×C₄	semidirect
19. C₄×#10596831	C₂×C₄	semidirect
84. C₇₃	#20365	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
#22609683	2
#22609625	2
C₂×#376×C₇₃	2
#20365	73