Subgroups — #34030503

Order 18688.

462 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626355	C₂	1	6	yes	maximal, complemented
2	#22626354	C₂	1	2	yes	maximal, complemented
3	C₂×#17989520	C₂	1	3	yes	maximal, complemented
4	#22626352	C₂	1	3	yes	maximal, complemented
5	#504×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989520	C₂²	1	3	yes	—
7	#17989520	C₂²	1	1	yes	characteristic
8	#17989520	C₂²	1	6	yes	complemented
9	#17989519	C₂²	1	6	yes	complemented
10	#17989518	C₂²	1	6	yes	complemented
11	C₂×#14556641	C₂²	1	6	yes	complemented
12	#131×D₇₃	C₄	1	1	yes	characteristic
13	#17989005	C₂²	1	3	yes	—
14	#17989004	C₂²	1	1	yes	characteristic
15	#17989003	C₄	1	3	yes	—
16	M₃₂×D₇₃	C₄	1	3	yes	—
17	C₂×C₁₆×D₇₃	C₂²	1	3	yes	—
18	#504×C₇₃	C₄	1	1	yes	characteristic
19	#14556641	C₂³	1	2	yes	—
20	#14556641	C₂³	1	6	yes	—
21	#41×D₇₃	C₈	1	1	yes	characteristic
22	#14556599	C₈	1	3	yes	—
23	#14556598	C₈	1	1	yes	characteristic
24	#14556597	C₂×C₄	1	3	yes	—
25	M₁₆×D₇₃	C₈	1	3	yes	—
26	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
27	#14556531	C₂×C₄	1	3	yes	—
28	#14556530	C₂×C₄	1	6	yes	—
29	C₂×#11924003	C₂³	1	3	yes	—
30	C₁₆×D₇₃	C₂×C₄	1	3	yes	—
31	C₁₆×D₇₃	C₂×C₄	1	3	yes	—
32	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
33	C₁₆×D₇₃	C₂³	1	3	yes	—
34	#131×C₇₃	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₇₃	C₂×C₄	1	3	yes	—
36	C₂×C₁₆×C₇₃	C₂×C₄	1	3	yes	—
37	#11924029	C₁₆	1	1	yes	characteristic
38	#11924028	C₂×C₈	1	3	yes	—
39	#11924027	C₁₆	1	3	yes	—
40	Q₈×D₇₃	C₁₆	1	1	yes	characteristic
41	D₄×D₇₃	C₁₆	1	3	yes	—
42	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
43	#11924012	C₂×C₈	1	3	yes	—
44	#11924011	C₂×C₈	1	6	yes	—
45	C₂×#10596831	C₂×C₈	1	3	yes	—
46	C₈×D₇₃	C₂×C₈	1	3	yes	—
47	C₈×D₇₃	C₂×C₈	1	3	yes	—
48	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₈×D₇₃	C₂²×C₄	1	3	yes	—
50	#11924003	C₂²×C₄	1	3	yes	—
51	#11924003	C₂⁴	1	1	yes	characteristic
52	#41×C₇₃	C₂×C₈	1	1	yes	characteristic
53	M₁₆×C₇₃	C₂×C₈	1	3	yes	—
54	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
55	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₁₆×C₇₃	C₂²×C₄	1	3	yes	—
57	C₂²×D₇₃	C₂×C₁₆	1	3	yes	—
58	Q₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
59	D₄×C₇₃	C₂×C₁₆	1	3	yes	—
60	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
61	C₂²⋊D₇₃	C₂×C₁₆	1	6	yes	—
62	C₂×Dic₇₃	C₂×C₁₆	1	3	yes	—
63	D₂₉₂	C₂×C₁₆	1	3	yes	—
64	C₄×D₇₃	C₂×C₁₆	1	3	yes	—
65	C₄×D₇₃	C₂×C₁₆	1	3	yes	—
66	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
67	C₄×D₇₃	C₂²×C₈	1	3	yes	—
68	Dic₁₄₆	C₂×C₁₆	1	3	yes	—
69	C₈×C₇₃	C₂²×C₈	1	3	yes	—
70	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
71	#10596831	C₂²×C₈	1	1	yes	characteristic
72	#10596831	C₂²×C₈	1	3	yes	—
73	C₂²×C₇₃	C₂²×C₁₆	1	3	yes	—
74	C₂×D₇₃	—	4	3	no	—
75	C₂×D₇₃	C₂²×C₁₆	1	1	yes	characteristic
76	C₂×D₇₃	C₂²×C₁₆	1	3	yes	—
77	C₄×C₇₃	C₂²×C₁₆	1	3	yes	—
78	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
79	Dic₇₃	C₂²×C₁₆	1	3	yes	—
80	Dic₇₃	C₂²×C₁₆	1	1	yes	characteristic
81	#33974	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	—	2	3	no	—
83	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
84	D₇₃	—	2	1	no	—
85	D₇₃	—	2	3	no	—
86	C₂×#504	—	73	1	no	—
87	#2258	—	73	2	no	—
88	#2258	—	73	6	no	—
89	C₂×M₆₄	—	73	3	no	—
90	C₂×M₆₄	—	73	3	no	—
91	C₇₃	#33974	1	1	yes	characteristic, complemented, Sylow
92	C₂×#131	—	73	1	no	—
93	#504	—	73	1	no	—
94	#504	—	73	3	no	—
95	#504	—	73	3	no	—
96	#504	#64408	1	1	yes	characteristic
97	C₂×M₃₂	—	73	3	no	—
98	C₂²×C₁₆	—	73	3	no	—
99	M₆₄	—	73	6	no	—
100	M₆₄	—	73	3	no	—
101	M₆₄	—	73	1	no	—
102	M₆₄	—	73	6	no	—
103	C₂×C₃₂	—	73	6	no	—
104	C₂×C₃₂	—	73	6	no	—
105	C₂×#41	—	73	1	no	—
106	#131	—	73	1	no	—
107	#131	—	73	3	no	—
108	#131	—	73	3	no	—
109	#131	C₂×#64408	1	1	yes	characteristic
110	C₂×M₁₆	—	73	3	no	—
111	C₂²×C₈	—	73	3	no	—
112	M₃₂	—	73	3	no	—
113	M₃₂	—	73	6	no	—
114	M₃₂	C₂×#64408	1	3	yes	—
115	C₂×C₁₆	—	73	3	no	—
116	C₂×C₁₆	—	73	3	no	—
117	C₂×C₁₆	—	73	3	no	—
118	C₂×C₁₆	—	73	3	no	—
119	C₂×C₁₆	—	73	1	no	—
120	C₂×C₁₆	C₂×#64408	1	3	yes	—
121	C₃₂	—	73	2	no	—
122	C₃₂	—	73	6	no	—
123	#41	—	73	1	no	—
124	#41	—	73	3	no	—
125	#41	—	73	3	no	—
126	#41	#11924036	1	1	yes	characteristic
127	C₂×Q₈	—	73	1	no	—
128	C₂×D₄	—	73	3	no	—
129	C₂²×C₄	—	73	3	no	—
130	M₁₆	—	73	3	no	—
131	M₁₆	—	73	6	no	—
132	M₁₆	#11924036	1	3	yes	—
133	C₂×C₈	—	73	3	no	—
134	C₂×C₈	—	73	3	no	—
135	C₂×C₈	—	73	3	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	3	no	—
138	C₂×C₈	C₂²×#64408	1	3	yes	—
139	C₁₆	—	73	3	no	—
140	C₁₆	—	73	1	no	—
141	C₁₆	C₂²×#64408	1	1	yes	characteristic
142	C₁₆	C₂²×#64408	1	3	yes	—
143	C₂³	—	73	3	no	—
144	Q₈	—	73	3	no	—
145	Q₈	#14556664	1	1	yes	characteristic
146	D₄	—	73	6	no	—
147	D₄	—	73	3	no	—
148	D₄	#14556664	1	3	yes	—
149	C₂×C₄	—	73	3	no	—
150	C₂×C₄	—	73	3	no	—
151	C₂×C₄	—	73	3	no	—
152	C₂×C₄	—	73	1	no	—
153	C₂×C₄	—	73	3	no	—
154	C₂×C₄	C₂×#11924036	1	3	yes	—
155	C₈	—	73	1	no	—
156	C₈	—	73	3	no	—
157	C₈	C₂×#11924036	1	3	yes	—
158	C₈	C₂³×#64408	1	1	yes	characteristic
159	C₂²	—	292	3	no	—
160	C₂²	—	73	1	no	—
161	C₂²	—	73	3	no	—
162	C₂²	C₂×#14556664	1	3	yes	—
163	C₄	—	73	1	no	—
164	C₄	—	73	3	no	—
165	C₄	C₂×#14556664	1	3	yes	—
166	C₄	C₂²×#11924036	1	1	yes	characteristic
167	C₂	—	2	3	no	—
168	C₂	—	146	1	no	—
169	C₂	—	146	3	no	—
170	C₂	C₂²×#14556664	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×#64408
commutator, socle	C₂×C₇₃	C₂³×C₁₆
Fitting	#504×C₇₃	C₄
prime radical	#41×D₇₃	C₈
cosocle	#11924003	C₂⁴
hypercenter	#504	#64408
hypocenter	C₇₃	#33974
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #22626355	C₂	semidirect
2. #22626354	C₂	semidirect
3. C₂×#17989520	C₂	semidirect
4. #22626352	C₂	semidirect
8. #17989520	C₂²	semidirect
9. #17989519	C₂²	semidirect
10. #17989518	C₂²	semidirect
11. C₂×#14556641	C₂²	semidirect
91. C₇₃	#33974	semidirect

Maximal subgroups

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

subgroup	index
#22626355	2
#22626354	2
C₂×#17989520	2
#22626352	2
#504×D₇₃	2
#33974	73