Subgroups — #34009647

Order 18688.

330 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626137	C₂	1	1	yes	maximal, characteristic
2	#22625134	C₂	1	1	yes	maximal, characteristic
3	#22624842	C₂	1	2	yes	maximal, complemented
4	C₁₆×#10596833	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₄	1	2	yes	—
7	#17989603	C₄	1	2	yes	—
8	#17989476	C₄	1	2	yes	—
9	#17989473	C₂²	1	2	yes	—
10	#17989428	C₄	1	2	yes	—
11	C₈×#10596833	C₂²	1	2	yes	—
12	C₂²×#11924003	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
15	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
16	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
19	#14556665	C₈	1	8	yes	complemented
20	C₂×#11924031	C₂×C₄	1	4	yes	—
21	C₄⋊#10596833	C₈	1	4	yes	—
22	C₂²⋊#10596833	C₈	1	4	yes	—
23	C₄×#10596833	C₂×C₄	1	4	yes	—
24	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#11924003	C₈	1	4	yes	complemented
27	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₃	C₈	1	4	yes	complemented
35	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#10596833	C₂×C₈	1	8	yes	—
38	#11924031	C₂×C₈	1	8	yes	—
39	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
40	C₂×#10596831	C₄²	1	1	yes	characteristic
41	C₂×#10596831	C₂×C₈	1	4	yes	—
42	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
43	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
44	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
45	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
46	#11924003	C₂×C₈	1	2	yes	—
47	#11924003	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
54	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
55	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×Dic₇₃	—	2	4	no	—
61	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
62	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
63	#10596833	—	2	8	no	—
64	C₈×C₇₃	C₄×C₈	1	2	yes	—
65	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
67	#10596831	C₄×C₈	1	2	yes	—
68	#10596831	C₂²×C₈	1	2	yes	—
69	C₂²×C₇₃	—	2	2	no	—
70	C₂²×C₇₃	C₈²	1	2	yes	—
71	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
72	C₄×C₇₃	C₈²	1	2	yes	—
73	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
75	Dic₇₃	—	2	2	no	—
76	Dic₇₃	#504	1	4	yes	—
77	#11399	—	73	1	no	maximal, Sylow
78	C₂×C₇₃	—	2	2	no	—
79	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
80	C₂×C₇₃	#2108	1	2	yes	—
81	C₄×M₃₂	—	73	1	no	—
82	C₂×C₄×C₁₆	—	73	1	no	—
83	#1450	—	73	1	no	—
84	#1311	—	73	2	no	—
85	C₈×C₁₆	—	73	2	no	—
86	C₇₃	#11399	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₃₂	—	73	2	no	—
88	C₂²×C₁₆	—	73	1	no	—
89	C₂²×C₁₆	#64408	1	1	yes	characteristic
90	#431	—	73	2	no	—
91	C₂×C₄×C₈	—	73	1	no	—
92	#346	—	73	2	no	—
93	C₄×C₁₆	—	73	2	no	—
94	C₄×C₁₆	—	73	1	no	—
95	C₄×C₁₆	—	73	1	no	—
96	C₄×C₁₆	—	73	1	no	—
97	C₄×C₁₆	—	73	1	no	—
98	#322	—	73	2	no	—
99	C₈²	—	73	2	no	—
100	C₂²×C₈	—	73	1	no	—
101	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
102	C₂×C₄²	—	73	1	no	—
103	M₃₂	—	73	8	no	—
104	C₂×C₁₆	—	73	4	no	—
105	C₂×C₁₆	—	73	4	no	—
106	C₂×C₁₆	—	73	1	no	—
107	C₂×C₁₆	—	73	1	no	—
108	C₂×C₁₆	#10596833	1	4	yes	complemented
109	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
110	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
111	C₄⋊C₈	—	73	4	no	—
112	C₂²⋊C₈	—	73	4	no	—
113	C₄×C₈	—	73	4	no	—
114	C₄×C₈	—	73	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₂²×C₄	—	73	2	no	—
119	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
120	C₂×C₈	—	73	4	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	8	no	—
124	C₂×C₈	C₄×#64408	1	1	yes	characteristic
125	C₂×C₈	C₂×#10596833	1	2	yes	—
126	C₂×C₈	#11924036	1	2	yes	—
127	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
128	C₄²	—	73	2	no	—
129	C₄²	—	73	1	no	—
130	C₄²	—	73	1	no	—
131	C₁₆	—	73	8	no	—
132	C₁₆	—	73	2	no	—
133	C₁₆	—	73	2	no	—
134	C₁₆	C₂×#10596833	1	2	yes	—
135	C₁₆	C₂×#10596833	1	2	yes	—
136	C₂³	C₈×#64408	1	1	yes	characteristic
137	C₂×C₄	—	73	2	no	—
138	C₂×C₄	—	73	2	no	—
139	C₂×C₄	—	146	4	no	—
140	C₂×C₄	C₈×#64408	1	1	yes	characteristic
141	C₂×C₄	C₈×#64408	1	2	yes	—
142	C₂×C₄	C₄×#10596833	1	2	yes	—
143	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
144	C₈	—	73	2	no	—
145	C₈	—	73	2	no	—
146	C₈	—	146	8	no	—
147	C₈	C₄×#10596833	1	2	yes	—
148	C₈	C₂²×#10596833	1	1	yes	characteristic
149	C₈	C₂×#11924036	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	C₈×#10596833	1	2	yes	—
152	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
153	C₄	—	73	4	no	—
154	C₄	—	146	2	no	—
155	C₄	C₈×#10596833	1	2	yes	—
156	C₄	C₂×C₈×#64408	1	1	yes	characteristic
157	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
158	C₂	—	2	2	no	—
159	C₂	C₂×C₈×#10596833	1	1	yes	characteristic
160	C₂	#22626139	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂×#64408
commutator	C₂×C₇₃	C₂×C₈²
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#11399
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22624842	C₂	semidirect
4. C₁₆×#10596833	C₂	semidirect
19. #14556665	C₈	semidirect
26. C₂×#11924003	C₈	semidirect
34. C₂×C₁₆×C₇₃	C₈	semidirect
86. C₇₃	#11399	semidirect
108. 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
#22626137	2
#22625134	2
#22624842	2
C₁₆×#10596833	2
C₂×C₁₆×Dic₇₃	2
#11399	73