Subgroups — #32979652

Order 18688.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 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	C₄×#14556502	C₂	1	1	yes	maximal, characteristic
3	#22607765	C₂	1	4	yes	maximal
4	C₇₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
6	C₂×#14556502	C₄	1	2	yes	—
7	C₂×#14556502	C₄	1	1	yes	characteristic
8	C₂×#14556502	C₄	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	2	yes	—
10	#17988342	—	2	2	no	—
11	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
12	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
13	C₄³×C₇₃	C₂²	1	1	yes	characteristic
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	#14556502	—	2	4	no	—
19	#14556502	—	2	2	no	—
20	#14556502	C₈	1	4	yes	complemented
21	C₄×#10596831	—	2	2	no	—
22	C₄×#10596831	—	2	2	no	—
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	C₄⋊C₈×C₇₃	—	4	4	no	—
31	C₂×#10596831	—	2	2	no	—
32	C₂×#10596831	—	2	2	no	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	—	2	2	no	—
35	C₂×#10596831	—	4	2	no	—
36	C₂×#10596831	—	4	2	no	—
37	C₂×#10596831	—	4	2	no	—
38	C₂×#10596831	C₂×C₈	1	4	yes	—
39	C₂×#10596831	M₁₆	1	4	yes	—
40	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	4	2	no	—
46	C₂×C₈×C₇₃	—	4	4	no	—
47	C₄²×C₇₃	—	2	2	no	—
48	C₄²×C₇₃	—	2	2	no	—
49	C₄²×C₇₃	—	2	2	no	—
50	C₄²×C₇₃	—	4	2	no	—
51	C₄²×C₇₃	C₂×C₈	1	2	yes	—
52	C₄²×C₇₃	M₁₆	1	2	yes	—
53	C₄²×C₇₃	#41	1	2	yes	—
54	C₄²×C₇₃	#41	1	2	yes	—
55	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	2	2	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
63	C₂×C₄×C₇₃	#117	1	2	yes	—
64	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#125	1	2	yes	—
67	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₇₃	#137	1	2	yes	—
72	C₈×C₇₃	—	8	2	no	—
73	#10596831	—	2	4	no	—
74	#10596831	—	2	4	no	—
75	#10596831	—	8	2	no	—
76	C₂²×C₇₃	#382	1	1	yes	characteristic
77	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₇₃	#423	1	1	yes	characteristic
79	C₂²×C₇₃	#428	1	2	yes	—
80	C₂²×C₇₃	#430	1	2	yes	—
81	C₄×C₇₃	—	2	4	no	—
82	C₄×C₇₃	—	2	2	no	—
83	C₄×C₇₃	—	4	2	no	—
84	C₄×C₇₃	#431	1	2	yes	—
85	C₄×C₇₃	#432	1	2	yes	—
86	C₄×C₇₃	#489	1	2	yes	—
87	C₄×C₇₃	#490	1	1	yes	characteristic
88	C₄×C₇₃	#491	1	1	yes	characteristic
89	#8786	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	#1565	1	2	yes	—
91	C₂×C₇₃	#1569	1	2	yes	—
92	C₂×C₇₃	#1831	1	1	yes	characteristic
93	C₂×C₇₃	#1836	1	1	yes	characteristic
94	C₂×C₇₃	#1838	1	1	yes	characteristic
95	#1842	D₇₃	1	1	yes	characteristic
96	#1831	—	73	1	no	—
97	C₄×#97	—	73	1	no	—
98	#1277	—	73	4	no	—
99	C₇₃	#8786	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊C₈	—	2	2	no	—
101	C₂×C₄⋊C₈	—	73	2	no	—
102	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
103	C₂×#97	—	73	1	no	—
104	C₂×#97	—	73	2	no	—
105	C₂×#97	—	73	1	no	—
106	C₂×C₄×C₈	—	73	2	no	—
107	C₄³	C₂×D₇₃	1	1	yes	characteristic
108	#336	—	146	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	73	2	no	—
111	C₂²×C₈	—	73	2	no	—
112	C₂²×C₈	—	146	2	no	—
113	C₂×C₄²	Dic₁₄₆	1	2	yes	—
114	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
117	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	4	4	no	—
120	C₄⋊C₈	—	146	4	no	—
121	#97	—	73	4	no	—
122	#97	—	146	4	no	—
123	#97	—	146	2	no	—
124	C₄×C₈	—	146	2	no	—
125	C₄×C₈	—	146	2	no	—
126	C₂²×C₄	#11924007	1	2	yes	—
127	C₂²×C₄	#11924007	1	2	yes	—
128	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	292	2	no	—
134	C₂×C₈	—	292	2	no	—
135	C₂×C₈	—	292	2	no	—
136	C₂×C₈	—	73	4	no	—
137	C₂×C₈	—	73	4	no	—
138	C₂×C₈	—	146	2	no	—
139	C₂×C₈	—	146	2	no	—
140	C₂×C₈	—	146	2	no	—
141	C₂×C₈	—	146	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₈×D₇₃	1	2	yes	—
147	C₄²	#11924011	1	2	yes	—
148	C₄²	#11924027	1	2	yes	—
149	C₄²	#11924029	1	2	yes	—
150	C₂³	C₂×#11924007	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#14556520	1	4	yes	—
158	C₂×C₄	#14556549	1	2	yes	—
159	C₂×C₄	#14556578	1	1	yes	characteristic
160	C₂×C₄	#14556580	1	1	yes	characteristic
161	C₂×C₄	#14556582	1	2	yes	—
162	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
163	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
164	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
165	C₂×C₄	#14556617	1	2	yes	—
166	C₂×C₄	#14556622	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	584	2	no	—
169	C₈	—	146	4	no	—
170	C₈	—	146	4	no	—
171	C₂²	#17988452	1	1	yes	characteristic
172	C₂²	C₂×#14556520	1	1	yes	characteristic
173	C₂²	#17988610	1	1	yes	characteristic
174	C₂²	#17988627	1	2	yes	—
175	C₂²	#17988635	1	2	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	#17988639	1	2	yes	—
180	C₄	#17988643	1	2	yes	—
181	C₄	#17988949	1	2	yes	—
182	C₄	#17988951	1	1	yes	characteristic
183	C₄	#17988955	1	1	yes	characteristic
184	C₂	#22608681	1	2	yes	—
185	C₂	#22608696	1	2	yes	—
186	C₂	#22610047	1	1	yes	characteristic
187	C₂	#22610066	1	1	yes	characteristic
188	C₂	#22610071	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×#1842	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1842	D₇₃
hypocenter	C₇₃	#8786
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
20. #14556502	C₈	semidirect
99. C₇₃	#8786	semidirect

Maximal subgroups

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

subgroup	index
#22610048	2
C₄×#14556502	2
#22607765	2
C₇₃×#1842	2
#8786	73