Subgroups — #34000202

Order 18688.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989450	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#10596833	—	2	2	no	—
4	#17989450	—	2	2	no	—
5	#17989450	C₄	1	4	yes	—
6	C₂²×#11924003	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
8	C₁₆×Dic₇₃	—	2	2	no	—
9	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
10	C₂²×#10596833	—	4	2	no	—
11	C₄⋊#10596833	—	2	4	no	—
12	C₄⋊#10596833	—	4	2	no	—
13	C₂²×#10596831	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
15	C₂×#11924003	—	2	1	no	—
16	C₂×#11924003	—	2	1	no	—
17	C₂×#11924003	C₈	1	2	yes	complemented
18	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
19	C₈×Dic₇₃	D₄	1	2	yes	—
20	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₃	—	2	1	no	—
22	C₂×C₁₆×C₇₃	—	2	1	no	—
23	C₂×C₁₆×C₇₃	C₈	1	2	yes	complemented
24	C₂×#10596833	—	4	4	no	—
25	C₂×#10596833	—	8	2	no	—
26	C₂×#10596833	—	8	2	no	—
27	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
28	C₂²×Dic₇₃	Q₁₆	1	1	yes	characteristic
29	C₂×#10596831	—	2	1	no	—
30	C₂×#10596831	—	2	1	no	—
31	C₂×#10596831	C₄⋊C₄	1	2	yes	—
32	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
33	C₄×Dic₇₃	SD₁₆	1	2	yes	—
34	#11924003	—	2	2	no	—
35	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
42	C₁₆×C₇₃	—	2	2	no	—
43	C₂³×C₇₃	#107	1	1	yes	characteristic
44	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
49	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	1	no	—
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	#102	1	2	yes	—
53	C₂×Dic₇₃	#103	1	2	yes	—
54	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
55	C₂×Dic₇₃	SD₃₂	1	2	yes	—
56	C₂×Dic₇₃	Q₃₂	1	1	yes	characteristic
57	#10596833	—	8	4	no	—
58	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
59	C₈×C₇₃	#97	1	1	yes	characteristic
60	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
61	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
62	#10596831	—	2	2	no	—
63	C₂²×C₇₃	#335	1	1	yes	characteristic
64	C₂²×C₇₃	#335	1	1	yes	characteristic
65	C₂²×C₇₃	#340	1	1	yes	characteristic
66	C₂²×C₇₃	#366	1	1	yes	characteristic
67	C₂²×C₇₃	#367	1	1	yes	characteristic
68	C₂²×C₇₃	#368	1	1	yes	characteristic
69	C₂²×C₇₃	#368	1	1	yes	characteristic
70	C₄×C₇₃	#334	1	1	yes	characteristic
71	C₄×C₇₃	#334	1	1	yes	characteristic
72	C₄×C₇₃	#336	1	1	yes	characteristic
73	C₄×C₇₃	#341	1	1	yes	characteristic
74	Dic₇₃	—	2	2	no	—
75	Dic₇₃	#357	1	2	yes	—
76	Dic₇₃	#358	1	2	yes	—
77	#7154	—	73	1	no	maximal, Sylow
78	C₂×C₇₃	#1277	1	1	yes	characteristic
79	C₂×C₇₃	#1371	1	1	yes	characteristic
80	C₂×C₇₃	#1371	1	1	yes	characteristic
81	C₂×C₇₃	#1372	1	1	yes	characteristic
82	C₂×C₇₃	#1372	1	1	yes	characteristic
83	C₂×C₇₃	#1380	1	1	yes	characteristic
84	C₂×C₇₃	#1381	1	1	yes	characteristic
85	C₂×C₄×C₁₆	—	73	1	no	—
86	C₂×#335	—	73	2	no	—
87	C₇₃	#7154	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	73	1	no	—
89	C₂²×C₁₆	#64408	1	1	yes	characteristic
90	C₂×C₄⋊C₈	—	146	2	no	—
91	C₂×C₄×C₈	—	73	1	no	—
92	C₄×C₁₆	—	146	2	no	—
93	#335	—	73	4	no	—
94	#335	—	146	2	no	—
95	C₂²×C₈	—	292	2	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
98	C₂×C₄²	—	73	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	73	2	no	—
102	C₂×C₁₆	—	146	1	no	—
103	C₂×C₁₆	—	146	1	no	—
104	C₂×C₁₆	#10596833	1	2	yes	complemented
105	C₄⋊C₈	—	292	2	no	—
106	C₄⋊C₈	—	146	4	no	—
107	C₄×C₈	—	73	2	no	—
108	C₄×C₈	—	73	2	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
112	C₂×C₈	—	292	4	no	—
113	C₂×C₈	—	584	2	no	—
114	C₂×C₈	—	584	2	no	—
115	C₂×C₈	—	73	2	no	—
116	C₂×C₈	—	146	1	no	—
117	C₂×C₈	—	146	1	no	—
118	C₂×C₈	C₄×#64408	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
120	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
121	C₂×C₈	#11924036	1	1	yes	characteristic
122	C₂×C₈	#11924037	1	1	yes	characteristic
123	C₂×C₈	#11924038	1	1	yes	characteristic
124	C₄²	—	73	2	no	—
125	C₄²	—	73	2	no	—
126	C₁₆	—	2	2	no	—
127	C₁₆	—	146	2	no	—
128	C₂³	#14556660	1	1	yes	characteristic
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	73	1	no	—
132	C₂×C₄	—	73	2	no	—
133	C₂×C₄	—	73	2	no	—
134	C₂×C₄	—	146	1	no	—
135	C₂×C₄	—	146	1	no	—
136	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
138	C₂×C₄	#14556658	1	1	yes	characteristic
139	C₂×C₄	#14556659	1	1	yes	characteristic
140	C₂×C₄	#14556661	1	1	yes	characteristic
141	C₂×C₄	#14556662	1	1	yes	characteristic
142	C₈	—	584	4	no	—
143	C₈	—	146	2	no	—
144	C₈	C₄×#10596833	1	1	yes	characteristic
145	C₈	#14556646	1	1	yes	characteristic
146	C₈	C₂²⋊#10596833	1	1	yes	characteristic
147	C₈	#14556648	1	1	yes	characteristic
148	C₂²	#17989450	1	1	yes	characteristic
149	C₂²	#17989451	1	1	yes	characteristic
150	C₂²	#17989461	1	1	yes	characteristic
151	C₂²	#17989513	1	1	yes	characteristic
152	C₂²	#17989514	1	1	yes	characteristic
153	C₂²	#17989515	1	1	yes	characteristic
154	C₂²	#17989516	1	1	yes	characteristic
155	C₄	—	73	2	no	—
156	C₄	—	73	2	no	—
157	C₄	—	146	2	no	—
158	C₄	#17989448	1	1	yes	characteristic
159	C₄	#17989449	1	1	yes	characteristic
160	C₄	#17989453	1	1	yes	characteristic
161	C₄	#17989463	1	1	yes	characteristic
162	C₂	#22624761	1	1	yes	characteristic
163	C₂	#22624962	1	1	yes	characteristic
164	C₂	#22624963	1	1	yes	characteristic
165	C₂	#22624964	1	1	yes	characteristic
166	C₂	#22624965	1	1	yes	characteristic
167	C₂	#22624989	1	1	yes	characteristic
168	C₂	#22624991	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556660
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	#107
cosocle	C₂×C₈×Dic₇₃	C₂²
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#7154
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
17. C₂×#11924003	C₈	semidirect
23. C₂×C₁₆×C₇₃	C₈	semidirect
87. C₇₃	#7154	semidirect
104. C₂×C₁₆	#10596833	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989450	2
C₂×C₁₆×Dic₇₃	2
#7154	73