Subgroups — #32979647

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610048	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
3	#22607765	C₂	1	4	yes	maximal
4	C₇₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
6	C₂×C₄×#10596831	C₄	1	2	yes	—
7	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	2	yes	—
10	#17988342	—	2	2	no	—
11	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
12	C₄³×C₇₃	C₂²	1	1	yes	characteristic
13	#336×C₇₃	—	2	2	no	—
14	C₂²×#10596831	—	2	2	no	—
15	C₂²×#10596831	C₂×C₄	1	2	yes	—
16	C₂²×#10596831	C₂×C₄	1	2	yes	—
17	C₄⋊#10596831	—	2	4	no	—
18	C₄×#10596831	—	2	4	no	—
19	C₄×#10596831	—	2	2	no	—
20	C₄×#10596831	—	2	2	no	—
21	C₄×#10596831	—	2	2	no	—
22	C₄×#10596831	C₈	1	4	yes	complemented
23	C₂²×C₈×C₇₃	—	2	2	no	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₃	D₄	1	2	yes	—
27	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
28	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	4	no	—
30	C₂×#10596831	—	2	2	no	—
31	C₂×#10596831	—	2	2	no	—
32	C₂×#10596831	—	2	2	no	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	—	4	2	no	—
35	C₂×#10596831	—	4	4	no	—
36	C₂×#10596831	C₂×C₈	1	4	yes	—
37	C₂×#10596831	M₁₆	1	4	yes	—
38	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₇₃	—	4	2	no	—
44	C₂×C₈×C₇₃	—	4	4	no	—
45	C₄²×C₇₃	—	2	4	no	—
46	C₄²×C₇₃	—	2	2	no	—
47	C₄²×C₇₃	—	2	4	no	—
48	C₄²×C₇₃	C₂×C₈	1	2	yes	—
49	C₄²×C₇₃	M₁₆	1	2	yes	—
50	C₄²×C₇₃	#41	1	2	yes	—
51	C₄²×C₇₃	#41	1	2	yes	—
52	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₇₃	#108	1	4	yes	—
60	C₂×C₄×C₇₃	#117	1	2	yes	—
61	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#125	1	2	yes	—
64	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#135	1	4	yes	—
68	C₈×C₇₃	—	8	2	no	—
69	#10596831	—	2	4	no	—
70	#10596831	—	2	4	no	—
71	#10596831	—	8	2	no	—
72	C₂²×C₇₃	#382	1	1	yes	characteristic
73	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₂²×C₇₃	#423	1	1	yes	characteristic
75	C₂²×C₇₃	#427	1	2	yes	—
76	C₂²×C₇₃	C₂×#108	1	2	yes	—
77	C₄×C₇₃	—	2	4	no	—
78	C₄×C₇₃	—	2	4	no	—
79	C₄×C₇₃	—	2	2	no	—
80	C₄×C₇₃	#431	1	2	yes	—
81	C₄×C₇₃	#432	1	2	yes	—
82	C₄×C₇₃	#488	1	2	yes	—
83	C₄×C₇₃	#488	1	2	yes	—
84	#8785	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	#1564	1	2	yes	—
86	C₂×C₇₃	#1568	1	2	yes	—
87	C₂×C₇₃	#1831	1	1	yes	characteristic
88	C₂×C₇₃	#1835	1	1	yes	characteristic
89	C₂×C₇₃	#1837	1	1	yes	characteristic
90	#1831	—	73	1	no	—
91	#1831	D₇₃	1	1	yes	characteristic
92	C₄²×C₈	—	73	1	no	—
93	#1277	—	73	4	no	—
94	C₇₃	#8785	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄⋊C₈	—	73	2	no	—
96	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
97	C₂×C₄×C₈	—	73	1	no	—
98	C₂×C₄×C₈	—	73	2	no	—
99	C₂×C₄×C₈	—	73	1	no	—
100	C₂×C₄×C₈	—	73	2	no	—
101	C₄³	C₂×D₇₃	1	1	yes	characteristic
102	#336	—	2	2	no	—
103	#336	—	146	2	no	—
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	73	2	no	—
106	C₂²×C₈	—	73	2	no	—
107	C₂²×C₈	—	146	2	no	—
108	C₂×C₄²	Dic₁₄₆	1	2	yes	—
109	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
110	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
111	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
112	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	146	4	no	—
115	C₄×C₈	—	73	4	no	—
116	C₄×C₈	—	146	4	no	—
117	C₄×C₈	—	146	2	no	—
118	C₄×C₈	—	146	2	no	—
119	C₄×C₈	—	146	2	no	—
120	C₂²×C₄	#11924007	1	2	yes	—
121	C₂²×C₄	#11924007	1	2	yes	—
122	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	4	4	no	—
127	C₂×C₈	—	292	2	no	—
128	C₂×C₈	—	292	4	no	—
129	C₂×C₈	—	73	4	no	—
130	C₂×C₈	—	73	4	no	—
131	C₂×C₈	—	146	2	no	—
132	C₂×C₈	—	146	2	no	—
133	C₂×C₈	—	146	2	no	—
134	C₂×C₈	—	146	2	no	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	4	no	—
138	C₄²	C₈×D₇₃	1	2	yes	—
139	C₄²	#11924011	1	2	yes	—
140	C₄²	#11924027	1	2	yes	—
141	C₄²	#11924029	1	2	yes	—
142	C₂³	C₂×#11924007	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#14556520	1	4	yes	—
149	C₂×C₄	#14556526	1	4	yes	—
150	C₂×C₄	#14556549	1	2	yes	—
151	C₂×C₄	#14556578	1	1	yes	characteristic
152	C₂×C₄	#14556580	1	1	yes	characteristic
153	C₂×C₄	#14556582	1	2	yes	—
154	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
155	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
156	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
157	C₂×C₄	#14556611	1	4	yes	—
158	C₈	—	8	2	no	—
159	C₈	—	584	2	no	—
160	C₈	—	146	4	no	—
161	C₈	—	146	4	no	—
162	C₂²	#17988452	1	1	yes	characteristic
163	C₂²	C₂×#14556520	1	1	yes	characteristic
164	C₂²	#17988610	1	1	yes	characteristic
165	C₂²	#17988623	1	2	yes	—
166	C₂²	C₂×#14556526	1	2	yes	—
167	C₄	—	2	4	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	#17988639	1	2	yes	—
171	C₄	#17988643	1	2	yes	—
172	C₄	#17988941	1	2	yes	—
173	C₄	#17988944	1	2	yes	—
174	C₂	#22608677	1	2	yes	—
175	C₂	#22608692	1	2	yes	—
176	C₂	#22610047	1	1	yes	characteristic
177	C₂	#22610061	1	1	yes	characteristic
178	C₂	#22610068	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1831	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1831	D₇₃
hypocenter	C₇₃	#8785
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
22. C₄×#10596831	C₈	semidirect
94. C₇₃	#8785	semidirect

Maximal subgroups

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

subgroup	index
#22610048	2
C₄²×#10596831	2
#22607765	2
C₇₃×#1831	2
#8785	73