Subgroups — #34012085

Order 18688.

478 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989474	C₂	1	2	yes	maximal
2	C₄×#14556664	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×#64408	C₂	1	2	yes	maximal
4	#22625587	C₂	1	1	yes	maximal, characteristic
5	#346×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556664	C₂²	1	1	yes	characteristic
7	C₂×#14556664	C₂²	1	1	yes	characteristic
8	#17989546	C₄	1	1	yes	characteristic
9	C₂×#14556645	C₄	1	2	yes	—
10	C₂×C₈×#64408	C₂²	1	2	yes	—
11	#17989523	C₄	1	1	yes	characteristic
12	#17989474	C₄	1	8	yes	complemented
13	#17989473	C₄	1	2	yes	complemented
14	C₄×#11924031	C₄	1	2	yes	complemented
15	C₁₆×#64408	C₄	1	8	yes	complemented
16	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
17	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
18	#17988363	C₄	1	1	yes	characteristic, complemented
19	#17988362	C₄	1	2	yes	complemented
20	#346×C₇₃	C₄	1	1	yes	characteristic, complemented
21	C₂×#11924036	C₂×C₄	1	2	yes	—
22	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
23	#14556664	C₂×C₄	1	4	yes	—
24	#14556664	C₂×C₄	1	4	yes	—
25	C₂×#11924031	C₂×C₄	1	2	yes	—
26	C₂×#11924031	C₂×C₄	1	2	yes	—
27	#14556646	C₈	1	4	yes	—
28	#14556645	C₈	1	8	yes	—
29	C₈×#64408	C₂×C₄	1	4	yes	—
30	C₈×#64408	C₂×C₄	1	4	yes	—
31	(C₂×#64408)⋊C₄	C₈	1	4	yes	—
32	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
34	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#11924003	C₂×C₄	1	2	yes	—
36	C₁₆×D₇₃	C₂×C₄	1	4	yes	—
37	C₁₆×D₇₃	C₂×C₄	1	4	yes	—
38	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#64408	C₂×C₈	1	4	yes	—
44	#11924036	C₄²	1	4	yes	—
45	#11924036	C₂×C₈	1	4	yes	—
46	C₂×#10596833	C₂×C₈	1	4	yes	—
47	C₄×#64408	C₄²	1	4	yes	—
48	C₄×#64408	C₂×C₈	1	4	yes	—
49	#11924031	C₄²	1	4	yes	—
50	#11924031	C₄²	1	4	yes	—
51	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
52	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×#10596831	C₄²	1	1	yes	characteristic
54	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
55	C₈×D₇₃	C₂×C₈	1	4	yes	—
56	C₈×D₇₃	C₂²×C₄	1	2	yes	—
57	C₈×D₇₃	C₂²×C₄	1	2	yes	—
58	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
59	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
60	#11924003	C₄²	1	4	yes	—
61	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₇₃	C₄²	1	1	yes	characteristic
64	C₁₆×C₇₃	C₄²	1	4	yes	—
65	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×#64408	—	2	8	no	—
67	C₂×#64408	C₄×C₈	1	8	yes	—
68	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
71	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
72	C₄×D₇₃	—	2	4	no	—
73	C₄×D₇₃	C₂×C₄²	1	2	yes	—
74	C₄×D₇₃	C₂²×C₈	1	2	yes	—
75	#10596833	C₄×C₈	1	8	yes	—
76	C₈×C₇₃	C₄×C₈	1	2	yes	—
77	C₈×C₇₃	C₂×C₄²	1	2	yes	—
78	#10596831	C₄×C₈	1	2	yes	—
79	#10596831	C₂×C₄²	1	1	yes	characteristic
80	#10596831	C₂×C₄²	1	1	yes	characteristic
81	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂×D₇₃	C₂×C₄×C₈	1	2	yes	—
83	C₂×D₇₃	#504	1	4	yes	—
84	#64408	—	2	8	no	—
85	C₄×C₇₃	—	2	2	no	—
86	C₄×C₇₃	C₄³	1	1	yes	characteristic
87	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
88	Dic₇₃	—	2	2	no	—
89	Dic₇₃	C₂×C₄×C₈	1	2	yes	—
90	#12539	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
92	C₂×C₇₃	#2108	1	2	yes	—
93	D₇₃	#2108	1	4	yes	—
94	C₂×#346	—	73	2	no	—
95	C₂×#346	—	73	1	no	—
96	C₂×C₄×C₁₆	—	73	2	no	—
97	C₂×C₄×C₁₆	—	73	1	no	—
98	#1726	—	73	1	no	—
99	C₇₃	#12539	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₁₆	—	73	1	no	—
101	C₂²×C₁₆	—	73	1	no	—
102	C₂²×C₁₆	—	73	2	no	—
103	C₂×#97	—	73	2	no	—
104	C₂×#97	—	73	1	no	—
105	C₂×C₄×C₈	—	73	2	no	—
106	C₂×C₄×C₈	—	73	1	no	—
107	#376	—	73	1	no	—
108	#346	—	73	8	no	—
109	#346	—	73	2	no	—
110	#346	—	73	1	no	—
111	#346	#64408	1	1	yes	characteristic, complemented
112	C₄×C₁₆	—	73	8	no	—
113	C₄×C₁₆	—	73	2	no	—
114	C₄×C₁₆	—	73	2	no	—
115	C₂²×C₈	—	73	2	no	—
116	C₂²×C₈	—	73	1	no	—
117	C₂²×C₈	—	73	1	no	—
118	C₂×C₄²	—	73	2	no	—
119	C₂×C₄²	—	73	1	no	—
120	C₂×C₁₆	—	73	2	no	—
121	C₂×C₁₆	—	73	2	no	—
122	C₂×C₁₆	—	73	4	no	—
123	C₂×C₁₆	—	73	4	no	—
124	C₂×C₁₆	—	73	2	no	—
125	C₂×C₁₆	—	73	4	no	—
126	C₂×C₁₆	—	73	4	no	—
127	C₂×C₁₆	C₂×#64408	1	2	yes	—
128	#97	—	73	8	no	—
129	#97	—	73	4	no	—
130	C₄×C₈	—	73	4	no	—
131	C₄×C₈	—	73	4	no	—
132	C₄×C₈	—	73	1	no	—
133	C₄×C₈	—	73	1	no	—
134	C₄×C₈	—	73	1	no	—
135	C₄×C₈	C₂×#64408	1	1	yes	characteristic
136	#95	—	73	4	no	—
137	C₂²×C₄	—	73	1	no	—
138	C₂²×C₄	—	73	4	no	—
139	C₂²×C₄	—	73	2	no	—
140	C₂×C₈	—	73	4	no	—
141	C₂×C₈	—	73	4	no	—
142	C₂×C₈	—	73	4	no	—
143	C₂×C₈	—	73	1	no	—
144	C₂×C₈	—	73	1	no	—
145	C₂×C₈	—	73	4	no	—
146	C₂×C₈	—	73	2	no	—
147	C₂×C₈	—	73	2	no	—
148	C₂×C₈	C₄×#64408	1	1	yes	characteristic
149	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
150	C₄²	—	73	4	no	—
151	C₄²	—	73	4	no	—
152	C₄²	—	73	2	no	—
153	C₄²	—	73	1	no	—
154	C₄²	C₄×#64408	1	1	yes	characteristic
155	C₁₆	—	73	4	no	—
156	C₁₆	—	73	4	no	—
157	C₁₆	—	73	4	no	—
158	C₁₆	C₄×#64408	1	4	yes	—
159	C₂³	—	73	1	no	—
160	C₂×C₄	—	73	1	no	—
161	C₂×C₄	—	73	2	no	—
162	C₂×C₄	—	73	2	no	—
163	C₂×C₄	—	73	8	no	—
164	C₂×C₄	—	73	2	no	—
165	C₂×C₄	—	146	4	no	—
166	C₂×C₄	—	146	8	no	—
167	C₂×C₄	C₈×#64408	1	2	yes	—
168	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
169	C₈	—	73	8	no	—
170	C₈	—	73	1	no	—
171	C₈	—	73	1	no	—
172	C₈	—	73	2	no	—
173	C₈	C₈×#64408	1	2	yes	—
174	C₈	C₂×C₄×#64408	1	2	yes	—
175	C₂²	—	73	2	no	—
176	C₂²	—	73	4	no	—
177	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	73	2	no	—
180	C₄	—	146	8	no	—
181	C₄	—	146	2	no	—
182	C₄	C₄²×#64408	1	1	yes	characteristic
183	C₄	C₂×C₈×#64408	1	1	yes	characteristic
184	C₂	—	73	4	no	—
185	C₂	C₄×C₈×#64408	1	1	yes	characteristic
186	C₂	#22626143	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator	C₂×C₇₃	C₄²×C₈
Fitting	#346×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	#12539
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
12. #17989474	C₄	semidirect
13. #17989473	C₄	semidirect
14. C₄×#11924031	C₄	semidirect
15. C₁₆×#64408	C₄	semidirect
18. #17988363	C₄	semidirect
19. #17988362	C₄	semidirect
20. #346×C₇₃	C₄	semidirect
99. C₇₃	#12539	semidirect
111. #346	#64408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989474	2
C₄×#14556664	2
C₂×C₁₆×#64408	2
#22625587	2
#346×D₇₃	2
#12539	73