Subgroups — #34012316

Order 18688.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989476	C₂	1	2	yes	maximal
2	#22624970	C₂	1	2	yes	maximal
3	#22624969	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924031	C₂²	1	4	yes	—
6	#17989476	C₄	1	8	yes	complemented
7	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
14	C₂×#11924031	—	2	4	no	—
15	C₂×#11924031	—	2	4	no	—
16	C₂×#11924031	C₂×C₄	1	8	yes	—
17	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
18	C₂²×#10596831	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
20	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
21	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596831	—	2	2	no	—
24	C₈×Dic₇₃	—	2	2	no	—
25	C₈×Dic₇₃	—	2	2	no	—
26	C₈×Dic₇₃	C₈	1	4	yes	—
27	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	—	2	2	no	—
32	#11924031	—	2	8	no	—
33	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
36	C₂×#10596831	—	2	1	no	—
37	C₂×#10596831	—	2	1	no	—
38	C₂×#10596831	C₂×C₈	1	2	yes	—
39	C₂×#10596831	C₂²×C₄	1	2	yes	—
40	C₂×#10596831	#41	1	2	yes	—
41	C₂×#10596831	#41	1	2	yes	—
42	C₄×Dic₇₃	—	2	4	no	—
43	C₄×Dic₇₃	—	2	1	no	—
44	C₄×Dic₇₃	—	2	1	no	—
45	C₄×Dic₇₃	—	2	4	no	—
46	C₄×Dic₇₃	C₄²	1	2	yes	—
47	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₂×C₈×C₇₃	—	2	1	no	—
52	C₂×C₈×C₇₃	C₄²	1	2	yes	—
53	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
55	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
56	C₄²×C₇₃	—	2	1	no	—
57	C₄²×C₇₃	—	2	1	no	—
58	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
62	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×Dic₇₃	—	2	2	no	—
69	C₂×Dic₇₃	—	2	2	no	—
70	C₂×Dic₇₃	—	2	2	no	—
71	C₂×Dic₇₃	—	2	2	no	—
72	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
73	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
74	C₂×Dic₇₃	M₃₂	1	4	yes	—
75	C₈×C₇₃	—	2	2	no	—
76	C₈×C₇₃	#97	1	4	yes	—
77	#10596831	—	2	2	no	—
78	#10596831	#117	1	4	yes	—
79	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
80	C₂²×C₇₃	#431	1	1	yes	characteristic
81	C₂²×C₇₃	#431	1	1	yes	characteristic
82	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
83	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
84	C₄×C₇₃	—	2	4	no	—
85	C₄×C₇₃	#376	1	1	yes	characteristic
86	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
87	C₄×C₇₃	#431	1	1	yes	characteristic
88	C₄×C₇₃	#431	1	1	yes	characteristic
89	Dic₇₃	—	2	2	no	—
90	Dic₇₃	—	2	2	no	—
91	Dic₇₃	—	2	4	no	—
92	#12591	—	73	1	no	maximal, Sylow
93	C₂×C₇₃	#1744	1	1	yes	characteristic
94	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
95	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
96	C₂×C₇₃	#2163	1	2	yes	—
97	C₂×C₇₃	#2163	1	2	yes	—
98	C₂×#346	—	73	2	no	—
99	C₄²×C₈	—	73	1	no	—
100	#1374	—	73	2	no	—
101	#1374	—	73	2	no	—
102	C₇₃	#12591	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₁₆	—	73	4	no	—
104	C₂×C₄×C₈	—	73	1	no	—
105	C₂×C₄×C₈	—	73	1	no	—
106	C₂×C₄×C₈	—	73	1	no	—
107	C₂×C₄×C₈	—	73	1	no	—
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	#64408	1	1	yes	characteristic
110	C₄³	—	73	1	no	—
111	#346	—	73	8	no	—
112	C₂²×C₈	—	73	1	no	—
113	C₂²×C₈	—	73	1	no	—
114	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
115	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
116	C₂×C₄²	—	73	1	no	—
117	C₂×C₄²	—	73	2	no	—
118	C₂×C₄²	—	73	2	no	—
119	C₂×C₄²	—	73	1	no	—
120	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
121	C₂×C₁₆	—	73	8	no	—
122	C₂×C₁₆	—	146	4	no	—
123	C₂×C₁₆	—	146	4	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	73	4	no	—
126	C₄×C₈	—	73	4	no	—
127	C₄×C₈	—	146	2	no	—
128	C₄×C₈	—	146	2	no	—
129	C₄×C₈	—	146	2	no	—
130	C₂²×C₄	—	73	2	no	—
131	C₂²×C₄	—	73	1	no	—
132	C₂²×C₄	—	73	1	no	—
133	C₂²×C₄	#11924036	1	2	yes	—
134	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	73	2	no	—
138	C₂×C₈	—	73	2	no	—
139	C₂×C₈	—	73	2	no	—
140	C₂×C₈	—	73	2	no	—
141	C₂×C₈	—	146	1	no	—
142	C₂×C₈	—	146	1	no	—
143	C₂×C₈	C₄×#64408	1	2	yes	—
144	C₂×C₈	#11924036	1	2	yes	—
145	C₂×C₈	#11924037	1	2	yes	—
146	C₂×C₈	#11924038	1	2	yes	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	73	2	no	—
150	C₄²	—	73	2	no	—
151	C₄²	—	146	4	no	—
152	C₄²	—	146	1	no	—
153	C₄²	—	146	4	no	—
154	C₄²	—	146	1	no	—
155	C₁₆	—	146	8	no	—
156	C₂³	C₂×#11924036	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	73	4	no	—
160	C₂×C₄	—	73	2	no	—
161	C₂×C₄	—	73	2	no	—
162	C₂×C₄	—	146	2	no	—
163	C₂×C₄	—	146	2	no	—
164	C₂×C₄	—	146	2	no	—
165	C₂×C₄	—	146	2	no	—
166	C₂×C₄	#14556666	1	4	yes	—
167	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
168	C₂×C₄	#14556670	1	1	yes	characteristic
169	C₂×C₄	#14556670	1	1	yes	characteristic
170	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
171	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
172	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	73	4	no	—
175	C₈	—	146	2	no	—
176	C₈	#14556646	1	4	yes	—
177	C₂²	C₄×#11924036	1	1	yes	characteristic
178	C₂²	#17989604	1	1	yes	characteristic
179	C₂²	#17989606	1	1	yes	characteristic
180	C₂²	#17989643	1	2	yes	—
181	C₂²	C₂×#14556666	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	146	2	no	—
184	C₄	—	146	2	no	—
185	C₄	—	146	4	no	—
186	C₄	#17989523	1	1	yes	characteristic
187	C₄	C₂×#14556646	1	1	yes	characteristic
188	C₄	#17989604	1	1	yes	characteristic
189	C₄	#17989606	1	1	yes	characteristic
190	C₂	#22625646	1	1	yes	characteristic
191	C₂	C₄×#14556666	1	1	yes	characteristic
192	C₂	#22626138	1	1	yes	characteristic
193	C₂	#22626293	1	2	yes	—
194	C₂	#22626295	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12591
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17989476	C₄	semidirect
102. C₇₃	#12591	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989476	2
#22624970	2
#22624969	2
C₄×C₈×Dic₇₃	2
#12591	73