Subgroups — #32977150

Order 18688.

390 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608679	C₂	1	2	yes	maximal
2	#22608214	C₂	1	2	yes	maximal
3	#22607764	C₂	1	2	yes	maximal
4	C₄×#97×C₇₃	C₂	1	1	yes	maximal, characteristic
5	#17988640	C₂²	1	4	yes	—
6	#17988340	C₄	1	4	yes	complemented
7	#17988338	C₄	1	4	yes	complemented
8	#17988308	—	2	2	no	—
9	C₈×#10596831	—	2	2	no	—
10	C₂×#97×C₇₃	C₄	1	2	yes	—
11	C₂×#97×C₇₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₄³×C₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊#10596831	C₂×C₄	1	8	yes	—
16	C₂²⋊#10596831	—	2	4	no	—
17	C₄×#10596831	—	2	4	no	—
18	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₇₃	—	2	2	no	—
22	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
24	#97×C₇₃	—	2	4	no	—
25	#97×C₇₃	C₈	1	8	yes	complemented
26	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₇₃	D₄	1	2	yes	—
29	C₄×C₈×C₇₃	Q₈	1	2	yes	—
30	C₂×#10596831	—	2	8	no	—
31	C₂²×C₄×C₇₃	—	2	2	no	—
32	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	#41	1	2	yes	—
34	C₂×C₈×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₄²	1	2	yes	—
38	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
42	C₄²×C₇₃	—	2	2	no	—
43	C₄²×C₇₃	—	4	4	no	—
44	C₄²×C₇₃	C₂×C₈	1	4	yes	—
45	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
46	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂³×C₇₃	#117	1	1	yes	characteristic
49	C₂×C₄×C₇₃	—	2	4	no	—
50	C₂×C₄×C₇₃	—	2	4	no	—
51	C₂×C₄×C₇₃	—	4	4	no	—
52	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
55	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
56	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
58	C₈×C₇₃	—	2	1	no	—
59	C₈×C₇₃	—	2	1	no	—
60	C₈×C₇₃	C₄×C₈	1	4	yes	—
61	C₈×C₇₃	C₄⋊C₈	1	8	yes	—
62	#10596831	—	4	8	no	—
63	C₂²×C₇₃	—	2	2	no	—
64	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₇₃	#431	1	2	yes	—
66	C₄×C₇₃	—	2	2	no	—
67	C₄×C₇₃	—	4	4	no	—
68	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
70	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
71	C₄×C₇₃	#444	1	4	yes	—
72	#8268	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	2	2	no	—
74	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₇₃	#1773	1	2	yes	—
76	C₄×#97	D₇₃	1	1	yes	characteristic
77	#1564	—	73	2	no	—
78	#1450	—	73	2	no	—
79	#1276	—	73	2	no	—
80	C₇₃	#8268	1	1	yes	characteristic, complemented, Sylow
81	#431	—	73	4	no	—
82	C₂×#97	Dic₇₃	1	2	yes	—
83	C₂×#97	Dic₇₃	1	2	yes	—
84	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
86	C₄³	C₂×D₇₃	1	1	yes	characteristic
87	#335	—	73	4	no	—
88	#334	—	73	4	no	—
89	#322	—	146	2	no	—
90	C₈²	—	146	2	no	—
91	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
92	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
94	C₂×C₄²	—	2	2	no	—
95	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
96	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
97	C₄⋊C₈	—	73	8	no	—
98	C₂²⋊C₈	—	146	4	no	—
99	#97	—	2	4	no	—
100	#97	#10596831	1	8	yes	complemented
101	C₄×C₈	—	146	4	no	—
102	C₄×C₈	Dic₁₄₆	1	2	yes	—
103	C₄×C₈	C₄×D₇₃	1	2	yes	—
104	C₄×C₈	C₄×D₇₃	1	2	yes	—
105	C₄×C₈	D₂₉₂	1	2	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
108	C₂²×C₄	#11924028	1	2	yes	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	146	8	no	—
111	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
112	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
113	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
114	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
115	C₂×C₈	C₂×#10596831	1	4	yes	—
116	C₂×C₈	C₂×#10596831	1	4	yes	—
117	C₂×C₈	#11924012	1	4	yes	—
118	C₄²	—	2	2	no	—
119	C₄²	—	4	4	no	—
120	C₄²	C₂×#10596831	1	4	yes	—
121	C₄²	C₂×C₄×D₇₃	1	2	yes	—
122	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
123	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
124	C₂³	#14556550	1	1	yes	characteristic
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
130	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
131	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
132	C₂×C₄	C₂²×#10596831	1	2	yes	—
133	C₂×C₄	C₂×#11924012	1	2	yes	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	292	8	no	—
137	C₈	C₄×#10596831	1	4	yes	—
138	C₈	C₄⋊#10596831	1	8	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₄×C₄⋊Dic₇₃	1	1	yes	characteristic
141	C₂²	#17988640	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	4	4	no	—
144	C₄	C₂×C₄×#10596831	1	1	yes	characteristic
145	C₄	C₄×#11924012	1	1	yes	characteristic
146	C₄	C₂×C₄⋊#10596831	1	2	yes	—
147	C₄	#17988710	1	4	yes	—
148	C₂	—	2	2	no	—
149	C₂	C₄×C₄⋊#10596831	1	1	yes	characteristic
150	C₂	#22609815	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₂₉₂
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#97×C₇₃	C₂
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#97	D₇₃
hypocenter	C₇₃	#8268
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
6. #17988340	C₄	semidirect
7. #17988338	C₄	semidirect
25. #97×C₇₃	C₈	semidirect
80. C₇₃	#8268	semidirect
100. #97	#10596831	semidirect

Maximal subgroups

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

subgroup	index
#22608679	2
#22608214	2
#22607764	2
C₄×#97×C₇₃	2
#8268	73