Subgroups — #34012080

Order 18688.

510 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	C₂×#17989476	C₂	1	2	yes	maximal
2	C₂×#17989475	C₂	1	2	yes	maximal
3	C₂×#17989473	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	#17989475	C₄	1	8	yes	complemented
8	#17989473	C₄	1	8	yes	complemented
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924031	C₂×C₄	1	8	yes	—
17	C₂×#11924031	C₂×C₄	1	8	yes	—
18	C₂×#11924031	C₂×C₄	1	8	yes	—
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596831	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	C₈	1	4	yes	—
22	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596831	C₂×C₄	1	4	yes	—
25	C₈×Dic₇₃	C₈	1	4	yes	—
26	C₈×Dic₇₃	C₈	1	4	yes	—
27	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
28	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₈	1	4	yes	—
33	#11924031	C₄²	1	16	yes	complemented
34	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
35	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
36	C₂×#10596831	C₄²	1	2	yes	—
37	C₂×#10596831	C₂×C₈	1	2	yes	—
38	C₂×#10596831	C₂×C₈	1	2	yes	—
39	C₂×#10596831	C₂²×C₄	1	2	yes	—
40	C₂×#10596831	C₂²×C₄	1	2	yes	—
41	C₂×#10596831	C₂²×C₄	1	2	yes	—
42	C₄×Dic₇₃	—	2	8	no	—
43	C₄×Dic₇₃	C₄²	1	2	yes	—
44	C₄×Dic₇₃	C₄²	1	2	yes	—
45	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
46	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₄²	1	2	yes	—
50	C₂×C₈×C₇₃	C₄²	1	2	yes	—
51	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
55	C₄²×C₇₃	C₄²	1	2	yes	—
56	C₄²×C₇₃	C₂×C₈	1	2	yes	—
57	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	—	2	4	no	—
59	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
60	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×Dic₇₃	—	2	4	no	—
67	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
68	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
69	C₂×Dic₇₃	M₃₂	1	8	yes	—
70	C₈×C₇₃	C₄×C₈	1	4	yes	—
71	C₈×C₇₃	C₄×C₈	1	4	yes	—
72	#10596831	C₄×C₈	1	4	yes	—
73	#10596831	C₂×C₄²	1	4	yes	—
74	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
75	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
76	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
77	C₂²×C₇₃	C₂×M₃₂	1	4	yes	—
78	C₄×C₇₃	—	2	4	no	—
79	C₄×C₇₃	C₄³	1	1	yes	characteristic
80	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
81	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
82	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
83	Dic₇₃	—	2	4	no	—
84	Dic₇₃	#346	1	8	yes	complemented
85	C₄×#346	—	73	1	no	maximal, Sylow
86	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
87	C₂×C₇₃	C₂×#346	1	2	yes	—
88	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
89	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
90	C₂×#346	—	73	2	no	—
91	C₂×#346	—	73	2	no	—
92	C₂×C₄×C₁₆	—	73	2	no	—
93	C₄²×C₈	—	73	1	no	—
94	C₇₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	73	4	no	—
96	C₂×C₄×C₈	—	73	1	no	—
97	C₂×C₄×C₈	—	73	1	no	—
98	C₂×C₄×C₈	—	73	1	no	—
99	C₂×C₄×C₈	—	73	1	no	—
100	C₂×C₄×C₈	—	73	1	no	—
101	C₂×C₄×C₈	#64408	1	1	yes	characteristic
102	C₄³	—	73	1	no	—
103	#346	—	73	8	no	—
104	#346	—	73	8	no	—
105	C₄×C₁₆	—	73	8	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
109	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
110	C₂×C₄²	—	73	1	no	—
111	C₂×C₄²	—	73	1	no	—
112	C₂×C₄²	—	73	4	no	—
113	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
114	C₂×C₁₆	—	73	8	no	—
115	C₂×C₁₆	—	73	8	no	—
116	C₂×C₁₆	—	73	8	no	—
117	C₄×C₈	—	73	4	no	—
118	C₄×C₈	—	73	4	no	—
119	C₄×C₈	—	73	4	no	—
120	C₄×C₈	—	73	4	no	—
121	C₄×C₈	—	73	4	no	—
122	C₄×C₈	#10596833	1	4	yes	—
123	C₂²×C₄	—	73	2	no	—
124	C₂²×C₄	—	73	2	no	—
125	C₂²×C₄	#11924036	1	2	yes	—
126	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	73	2	no	—
129	C₂×C₈	—	73	2	no	—
130	C₂×C₈	—	73	2	no	—
131	C₂×C₈	—	73	2	no	—
132	C₂×C₈	—	73	2	no	—
133	C₂×C₈	C₄×#64408	1	2	yes	—
134	C₂×C₈	C₄×#64408	1	2	yes	—
135	C₂×C₈	C₂×#10596833	1	2	yes	—
136	C₂×C₈	C₂×#10596833	1	2	yes	—
137	C₂×C₈	#11924036	1	2	yes	—
138	C₂×C₈	#11924036	1	2	yes	—
139	C₄²	—	73	2	no	—
140	C₄²	—	73	2	no	—
141	C₄²	—	73	2	no	—
142	C₄²	—	73	2	no	—
143	C₄²	—	146	8	no	—
144	C₄²	C₄×#64408	1	2	yes	—
145	C₄²	C₂×#10596833	1	2	yes	—
146	C₁₆	—	73	16	no	—
147	C₂³	C₂×#11924036	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	73	8	no	—
150	C₂×C₄	—	73	4	no	—
151	C₂×C₄	—	73	4	no	—
152	C₂×C₄	—	146	4	no	—
153	C₂×C₄	C₄×#10596833	1	4	yes	—
154	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
157	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
158	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
159	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
160	C₈	—	73	4	no	—
161	C₈	—	73	4	no	—
162	C₈	C₄×#10596833	1	4	yes	—
163	C₈	C₄×#10596833	1	4	yes	—
164	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
165	C₂²	C₄×#11924036	1	1	yes	characteristic
166	C₂²	C₄×#11924036	1	1	yes	characteristic
167	C₂²	#17989643	1	4	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	73	8	no	—
170	C₄	—	146	4	no	—
171	C₄	C₄²×#64408	1	1	yes	characteristic
172	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
173	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
174	C₄	C₄×#11924036	1	1	yes	characteristic
175	C₂	C₄²×#10596833	1	1	yes	characteristic
176	C₂	#22626132	1	2	yes	—
177	C₂	#22626137	1	2	yes	—
178	C₂	#22626138	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#64408
commutator	C₂×C₇₃	C₄²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₄×#346
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17989476	C₄	semidirect
7. #17989475	C₄	semidirect
8. #17989473	C₄	semidirect
33. #11924031	C₄²	semidirect
84. Dic₇₃	#346	semidirect
94. C₇₃	C₄×#346	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
C₂×#17989475	2
C₂×#17989473	2
C₄×C₈×Dic₇₃	2
C₄×#346	73