Subgroups — #34009656

Order 18688.

350 conjugacy classes of subgroups.

The subgroup lattice has 176 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	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
3	#22624845	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556664	C₄	1	2	yes	—
6	C₄×#11924036	C₄	1	1	yes	characteristic
7	C₄×#11924036	C₄	1	1	yes	characteristic
8	#17989473	C₂²	1	2	yes	—
9	C₄×#11924031	C₄	1	2	yes	—
10	C₈×#10596833	C₄	1	2	yes	—
11	C₈×#10596833	C₂²	1	2	yes	—
12	C₂×C₁₆×D₇₃	C₄	1	2	yes	—
13	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
14	C₄×#11924003	C₄	1	1	yes	characteristic
15	C₁₆×Dic₇₃	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924036	C₈	1	2	yes	—
18	C₂×#11924036	C₈	1	2	yes	—
19	#14556664	—	2	4	no	—
20	C₂×#11924031	C₂×C₄	1	4	yes	—
21	C₄×#10596833	C₈	1	2	yes	—
22	C₄×#10596833	C₈	1	2	yes	—
23	C₄×#10596833	C₂×C₄	1	2	yes	—
24	C₄×#10596833	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924003	C₂×C₄	1	2	yes	—
29	C₁₆×D₇₃	—	2	4	no	—
30	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	#11924036	—	2	4	no	—
36	#11924036	C₁₆	1	8	yes	complemented
37	C₂×#10596833	C₂×C₈	1	4	yes	—
38	C₂×#10596833	C₂×C₈	1	4	yes	—
39	#11924031	—	2	4	no	—
40	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
41	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂×#10596831	C₄²	1	1	yes	characteristic
44	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
45	C₈×D₇₃	M₁₆	1	4	yes	—
46	C₈×D₇₃	M₁₆	1	4	yes	—
47	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
48	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
49	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
50	#11924003	—	2	2	no	—
51	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₇₃	C₄²	1	1	yes	characteristic
54	C₁₆×C₇₃	—	2	2	no	—
55	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
59	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
60	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
62	C₄×D₇₃	#97	1	2	yes	—
63	C₄×D₇₃	#97	1	2	yes	—
64	C₄×D₇₃	C₂×C₁₆	1	4	yes	—
65	C₄×D₇₃	M₃₂	1	4	yes	—
66	#10596833	—	2	4	no	—
67	#10596833	C₂×C₁₆	1	8	yes	complemented
68	C₈×C₇₃	#97	1	2	yes	—
69	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
70	#10596831	#97	1	2	yes	—
71	#10596831	C₂×M₁₆	1	2	yes	—
72	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
73	C₂×D₇₃	#322	1	2	yes	—
74	C₂×D₇₃	C₄×C₁₆	1	2	yes	—
75	C₂×D₇₃	#346	1	2	yes	—
76	C₄×C₇₃	C₄×C₁₆	1	2	yes	—
77	C₄×C₇₃	#346	1	2	yes	—
78	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
79	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
80	Dic₇₃	#322	1	2	yes	—
81	Dic₇₃	C₂²×C₁₆	1	2	yes	—
82	Dic₇₃	C₂×M₃₂	1	2	yes	—
83	C₂×#1312	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
85	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
86	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
87	D₇₃	#1312	1	4	yes	complemented
88	C₂×C₄×C₁₆	—	73	1	no	—
89	C₂×C₄×C₁₆	—	73	1	no	—
90	C₂×C₈²	—	73	1	no	—
91	#1312	—	73	4	no	—
92	C₇₃	C₂×#1312	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	—	73	2	no	—
94	C₂²×C₁₆	—	73	2	no	—
95	C₂×C₄×C₈	—	73	1	no	—
96	C₂×C₄×C₈	—	73	1	no	—
97	C₂×C₄×C₈	—	73	1	no	—
98	C₄×C₁₆	—	73	2	no	—
99	C₄×C₁₆	—	73	2	no	—
100	C₄×C₁₆	—	73	2	no	—
101	C₄×C₁₆	—	73	1	no	—
102	C₄×C₁₆	#64408	1	1	yes	characteristic
103	C₈²	—	73	2	no	—
104	C₈²	—	73	2	no	—
105	C₂²×C₈	—	73	2	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂²×C₈	—	73	2	no	—
109	C₂×C₄²	—	73	1	no	—
110	C₂×C₁₆	—	73	4	no	—
111	C₂×C₁₆	—	73	2	no	—
112	C₂×C₁₆	—	146	4	no	—
113	C₂×C₁₆	—	146	4	no	—
114	C₂×C₁₆	C₂×#64408	1	2	yes	—
115	C₄×C₈	—	73	2	no	—
116	C₄×C₈	—	73	2	no	—
117	C₄×C₈	—	73	2	no	—
118	C₄×C₈	—	73	2	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	C₂×#64408	1	1	yes	characteristic
123	C₂²×C₄	—	73	1	no	—
124	C₂²×C₄	—	73	1	no	—
125	C₂²×C₄	—	73	1	no	—
126	C₂×C₈	—	73	4	no	—
127	C₂×C₈	—	73	8	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	4	no	—
131	C₂×C₈	—	73	4	no	—
132	C₂×C₈	—	73	4	no	—
133	C₂×C₈	—	146	4	no	—
134	C₂×C₈	C₄×#64408	1	1	yes	characteristic
135	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
136	C₄²	—	73	1	no	—
137	C₄²	—	73	1	no	—
138	C₄²	—	73	1	no	—
139	C₄²	C₄×#64408	1	1	yes	characteristic
140	C₁₆	—	2	2	no	—
141	C₁₆	—	146	4	no	—
142	C₁₆	—	146	2	no	—
143	C₂³	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	4	no	—
148	C₂×C₄	—	73	4	no	—
149	C₂×C₄	—	73	2	no	—
150	C₂×C₄	—	73	2	no	—
151	C₂×C₄	C₈×#64408	1	1	yes	characteristic
152	C₂×C₄	C₈×#64408	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
154	C₈	—	73	8	no	—
155	C₈	—	73	2	no	—
156	C₈	—	73	2	no	—
157	C₈	—	146	4	no	—
158	C₈	#14556645	1	2	yes	—
159	C₈	D₇₃⋊M₁₆	1	2	yes	—
160	C₂²	—	73	2	no	—
161	C₂²	—	73	2	no	—
162	C₂²	—	73	2	no	—
163	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
164	C₄	—	73	2	no	—
165	C₄	—	73	2	no	—
166	C₄	—	73	2	no	—
167	C₄	C₁₆×#64408	1	2	yes	—
168	C₄	#17989474	1	2	yes	—
169	C₄	C₂×#14556645	1	1	yes	characteristic
170	C₄	#17989546	1	1	yes	characteristic
171	C₂	—	73	4	no	—
172	C₂	#22625127	1	1	yes	characteristic
173	C₂	C₂×C₁₆×#64408	1	1	yes	characteristic
174	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	C₄×C₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	C₂×#1312
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
3. #22624845	C₂	semidirect
36. #11924036	C₁₆	semidirect
67. #10596833	C₂×C₁₆	semidirect
87. D₇₃	#1312	semidirect
92. C₇₃	C₂×#1312	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
C₈×#11924036	2
#22624845	2
C₄×C₁₆×D₇₃	2
C₂×#1312	73