Subgroups — #34012310

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556664	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
3	#22624968	C₂	1	4	yes	maximal
4	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556664	C₂²	1	2	yes	—
6	C₄×#11924036	C₄	1	1	yes	characteristic
7	C₂×C₈×#64408	C₄	1	2	yes	—
8	C₂×C₈×#64408	C₂²	1	2	yes	—
9	C₄²×#64408	C₄	1	1	yes	characteristic
10	#17989473	C₄	1	2	yes	complemented
11	C₄×#11924031	C₄	1	2	yes	complemented
12	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
13	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
14	C₄×#11924003	C₄	1	1	yes	characteristic, complemented
15	C₁₆×Dic₇₃	C₄	1	2	yes	complemented
16	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂×#11924036	C₂×C₄	1	2	yes	—
18	C₂×C₄×#64408	C₈	1	2	yes	—
19	C₂×C₄×#64408	C₈	1	2	yes	—
20	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
21	#14556664	—	2	4	no	—
22	C₂×#11924031	C₂×C₄	1	4	yes	—
23	C₄×#10596833	C₈	1	2	yes	—
24	C₄×#10596833	C₈	1	2	yes	—
25	C₈×#64408	—	2	4	no	—
26	C₈×#64408	—	2	4	no	—
27	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
29	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#11924003	C₂×C₄	1	2	yes	—
31	C₁₆×D₇₃	—	2	4	no	—
32	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×#64408	C₂×C₈	1	4	yes	—
38	#11924036	—	2	2	no	—
39	#11924036	—	2	2	no	—
40	C₂×#10596833	C₂×C₈	1	4	yes	—
41	C₄×#64408	—	2	4	no	—
42	C₄×#64408	—	2	2	no	—
43	C₄×#64408	—	2	2	no	—
44	C₄×#64408	C₁₆	1	8	yes	complemented
45	#11924031	—	2	4	no	—
46	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×#10596831	C₄²	1	1	yes	characteristic
50	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
51	C₈×D₇₃	M₁₆	1	4	yes	—
52	C₈×D₇₃	#41	1	4	yes	—
53	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
54	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
55	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
56	#11924003	—	2	2	no	—
57	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₃	C₄²	1	1	yes	characteristic
60	C₁₆×C₇₃	—	2	2	no	—
61	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×#64408	—	2	4	no	—
63	C₂×#64408	—	2	4	no	—
64	C₂×#64408	C₂×C₁₆	1	8	yes	—
65	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
69	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
70	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
71	C₄×D₇₃	C₂×C₁₆	1	4	yes	—
72	C₄×D₇₃	M₃₂	1	4	yes	—
73	C₄×D₇₃	#117	1	2	yes	—
74	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
75	#10596833	—	2	4	no	—
76	C₈×C₇₃	#97	1	2	yes	—
77	C₈×C₇₃	#117	1	2	yes	—
78	#10596831	#97	1	2	yes	—
79	#10596831	#117	1	2	yes	—
80	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
81	C₂×D₇₃	#431	1	2	yes	—
82	C₂×D₇₃	C₂²×C₁₆	1	2	yes	—
83	C₂×D₇₃	C₂×M₃₂	1	2	yes	—
84	#64408	—	2	8	no	—
85	C₄×C₇₃	C₄×C₁₆	1	2	yes	—
86	C₄×C₇₃	#346	1	2	yes	—
87	C₄×C₇₃	#376	1	1	yes	characteristic
88	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
89	Dic₇₃	C₄×C₁₆	1	2	yes	—
90	Dic₇₃	#346	1	2	yes	—
91	Dic₇₃	#431	1	2	yes	—
92	#12590	—	73	1	no	maximal, Sylow
93	C₂×C₇₃	#1744	1	1	yes	characteristic
94	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
95	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
96	D₇₃	#2162	1	4	yes	—
97	C₂×C₄×C₁₆	—	73	1	no	—
98	C₂×C₄×C₁₆	—	73	1	no	—
99	C₄²×C₈	—	73	1	no	—
100	#1374	—	73	4	no	—
101	C₇₃	#12590	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	73	2	no	—
103	C₂²×C₁₆	—	73	2	no	—
104	C₂×C₄×C₈	—	73	1	no	—
105	C₂×C₄×C₈	—	73	1	no	—
106	C₂×C₄×C₈	—	73	2	no	—
107	C₂×C₄×C₈	—	73	2	no	—
108	C₄³	—	73	1	no	—
109	C₄×C₁₆	—	73	2	no	—
110	C₄×C₁₆	—	73	2	no	—
111	C₄×C₁₆	—	73	2	no	—
112	C₄×C₁₆	—	73	1	no	—
113	C₄×C₁₆	#64408	1	1	yes	characteristic, complemented
114	C₂²×C₈	—	73	1	no	—
115	C₂²×C₈	—	73	1	no	—
116	C₂²×C₈	—	73	2	no	—
117	C₂×C₄²	—	73	2	no	—
118	C₂×C₄²	—	73	1	no	—
119	C₂×C₄²	—	73	2	no	—
120	C₂×C₄²	—	73	2	no	—
121	C₂×C₁₆	—	73	4	no	—
122	C₂×C₁₆	—	73	2	no	—
123	C₂×C₁₆	—	146	4	no	—
124	C₂×C₁₆	—	146	4	no	—
125	C₂×C₁₆	C₂×#64408	1	2	yes	—
126	C₄×C₈	—	73	2	no	—
127	C₄×C₈	—	73	2	no	—
128	C₄×C₈	—	73	1	no	—
129	C₄×C₈	—	73	1	no	—
130	C₄×C₈	—	73	1	no	—
131	C₄×C₈	—	146	4	no	—
132	C₄×C₈	—	146	4	no	—
133	C₄×C₈	C₂×#64408	1	1	yes	characteristic
134	C₂²×C₄	—	73	1	no	—
135	C₂²×C₄	—	73	1	no	—
136	C₂²×C₄	—	73	1	no	—
137	C₂²×C₄	—	73	4	no	—
138	C₂×C₈	—	73	1	no	—
139	C₂×C₈	—	73	1	no	—
140	C₂×C₈	—	73	4	no	—
141	C₂×C₈	—	73	4	no	—
142	C₂×C₈	—	73	4	no	—
143	C₂×C₈	—	146	2	no	—
144	C₂×C₈	—	146	2	no	—
145	C₂×C₈	C₄×#64408	1	1	yes	characteristic
146	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
147	C₄²	—	73	8	no	—
148	C₄²	—	73	1	no	—
149	C₄²	—	73	1	no	—
150	C₄²	—	73	1	no	—
151	C₄²	—	146	2	no	—
152	C₄²	—	146	2	no	—
153	C₄²	—	146	4	no	—
154	C₄²	C₄×#64408	1	1	yes	characteristic
155	C₁₆	—	2	2	no	—
156	C₁₆	—	146	4	no	—
157	C₁₆	—	146	2	no	—
158	C₂³	—	73	1	no	—
159	C₂×C₄	—	73	1	no	—
160	C₂×C₄	—	73	1	no	—
161	C₂×C₄	—	73	1	no	—
162	C₂×C₄	—	73	4	no	—
163	C₂×C₄	—	73	4	no	—
164	C₂×C₄	—	73	2	no	—
165	C₂×C₄	—	73	2	no	—
166	C₂×C₄	—	73	8	no	—
167	C₂×C₄	—	146	4	no	—
168	C₂×C₄	—	146	4	no	—
169	C₂×C₄	C₈×#64408	1	1	yes	characteristic
170	C₂×C₄	C₈×#64408	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
172	C₈	—	73	2	no	—
173	C₈	—	73	2	no	—
174	C₈	—	146	4	no	—
175	C₈	#14556645	1	2	yes	—
176	C₈	#14556670	1	2	yes	—
177	C₂²	—	73	2	no	—
178	C₂²	—	73	2	no	—
179	C₂²	—	73	2	no	—
180	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
181	C₄	—	73	2	no	—
182	C₄	—	73	2	no	—
183	C₄	—	73	2	no	—
184	C₄	—	146	8	no	—
185	C₄	C₁₆×#64408	1	2	yes	—
186	C₄	#17989474	1	2	yes	—
187	C₄	#17989523	1	1	yes	characteristic
188	C₄	C₂×#14556645	1	1	yes	characteristic
189	C₂	—	73	4	no	—
190	C₂	#22625644	1	1	yes	characteristic
191	C₂	C₂×C₁₆×#64408	1	1	yes	characteristic
192	C₂	C₂×#17989474	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#64408
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#12590
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #17989473	C₄	semidirect
11. C₄×#11924031	C₄	semidirect
14. C₄×#11924003	C₄	semidirect
15. C₁₆×Dic₇₃	C₄	semidirect
16. C₄×C₁₆×C₇₃	C₄	semidirect
44. C₄×#64408	C₁₆	semidirect
101. C₇₃	#12590	semidirect
113. C₄×C₁₆	#64408	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556664	2
C₄×C₈×#64408	2
#22624968	2
C₄×C₁₆×D₇₃	2
#12590	73