Subgroups — #34001929

Order 18688.

606 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556646	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10596833	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989429	C₂	1	2	yes	maximal
4	C₂×C₈×#10596833	C₂	1	2	yes	maximal
5	#22609675	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556646	C₄	1	4	yes	—
7	C₂×C₄×#10596833	C₄	1	4	yes	—
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10596833	C₂²	1	2	yes	—
11	#17989429	C₄	1	8	yes	complemented
12	C₈×#10596833	C₄	1	8	yes	complemented
13	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
14	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
15	#17988342	C₄	1	1	yes	characteristic
16	#17988341	C₄	1	2	yes	—
17	#336×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	C₂×C₄	1	2	yes	—
19	C₂²×#10596833	C₂×C₄	1	4	yes	—
20	C₂²×#10596833	C₂×C₄	1	2	yes	—
21	#14556646	C₈	1	16	yes	complemented
22	C₄×#10596833	—	2	8	no	—
23	C₄×#10596833	C₂×C₄	1	4	yes	—
24	C₄×#10596833	C₂×C₄	1	4	yes	—
25	C₄×#10596833	C₂×C₄	1	4	yes	—
26	C₄×#10596833	C₂×C₄	1	4	yes	—
27	C₂²×#10596831	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
33	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#10596833	—	2	4	no	—
37	C₂×#10596833	C₄²	1	4	yes	—
38	C₂×#10596833	C₄²	1	8	yes	—
39	C₂×#10596833	C₄²	1	4	yes	—
40	C₂×#10596833	C₂×C₈	1	16	yes	—
41	C₂×#10596833	M₁₆	1	8	yes	—
42	C₂²×Dic₇₃	C₄²	1	2	yes	—
43	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×#10596831	—	2	2	no	—
46	C₂×#10596831	—	2	2	no	—
47	C₂×#10596831	C₄²	1	4	yes	—
48	C₄×Dic₇₃	—	2	1	no	—
49	C₄×Dic₇₃	—	2	1	no	—
50	C₄×Dic₇₃	—	2	4	no	—
51	C₄×Dic₇₃	C₂×C₈	1	8	yes	—
52	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
53	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	—	2	2	no	—
57	C₂×C₈×C₇₃	—	2	2	no	—
58	C₂×C₈×C₇₃	C₄²	1	4	yes	—
59	C₄²×C₇₃	—	2	2	no	—
60	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
64	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
65	C₂×C₄×C₇₃	#131	1	2	yes	—
66	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
68	C₂×Dic₇₃	—	2	2	no	—
69	C₂×Dic₇₃	—	2	2	no	—
70	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
71	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
72	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
75	C₂×Dic₇₃	C₂×M₁₆	1	4	yes	—
76	#10596833	—	2	4	no	—
77	#10596833	C₄×C₈	1	16	yes	complemented
78	#10596833	#97	1	8	yes	complemented
79	C₈×C₇₃	—	2	4	no	—
80	#10596831	—	2	4	no	—
81	C₂²×C₇₃	C₄³	1	1	yes	characteristic
82	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
83	C₂²×C₇₃	C₄×M₁₆	1	2	yes	—
84	C₂²×C₇₃	#405	1	1	yes	characteristic
85	C₂²×C₇₃	#405	1	1	yes	characteristic
86	C₄×C₇₃	—	2	4	no	—
87	C₄×C₇₃	#405	1	2	yes	—
88	C₄×C₇₃	#405	1	2	yes	—
89	Dic₇₃	—	2	4	no	—
90	Dic₇₃	C₂×C₄×C₈	1	4	yes	—
91	Dic₇₃	C₂×#97	1	2	yes	—
92	Dic₇₃	C₄×M₁₆	1	2	yes	—
93	C₈×#97	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	C₈×M₁₆	1	2	yes	—
95	C₂×C₇₃	C₈×M₁₆	1	2	yes	—
96	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
97	C₂×C₇₃	C₄×#97	1	1	yes	characteristic
98	C₂×C₇₃	#1726	1	1	yes	characteristic
99	#1744	—	73	1	no	—
100	C₄×#97	—	73	1	no	—
101	C₄²×C₈	—	73	1	no	—
102	C₂×#322	—	73	2	no	—
103	C₂×C₈²	—	73	2	no	—
104	C₇₃	C₈×#97	1	1	yes	characteristic, complemented, Sylow
105	C₂×#97	—	73	4	no	—
106	C₂×C₄×C₈	—	73	1	no	—
107	C₂×C₄×C₈	—	73	2	no	—
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	—	73	2	no	—
110	C₂×C₄×C₈	—	73	4	no	—
111	C₄³	—	73	1	no	—
112	#336	—	73	2	no	—
113	#336	—	73	1	no	—
114	#336	#64408	1	1	yes	characteristic
115	#322	—	73	8	no	—
116	C₈²	—	73	8	no	—
117	C₂²×C₈	—	73	2	no	—
118	C₂²×C₈	—	73	4	no	—
119	C₂²×C₈	—	73	2	no	—
120	C₂²×C₈	—	73	2	no	—
121	C₂²×C₈	C₂×#64408	1	2	yes	—
122	C₂×C₄²	—	73	2	no	—
123	C₂×C₄²	—	73	1	no	—
124	C₂×C₄²	—	73	2	no	—
125	C₂×C₄²	—	73	1	no	—
126	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
127	#97	—	73	16	no	—
128	C₄×C₈	—	73	4	no	—
129	C₄×C₈	—	73	4	no	—
130	C₄×C₈	—	73	4	no	—
131	C₄×C₈	—	73	4	no	—
132	C₄×C₈	—	73	4	no	—
133	C₄×C₈	—	73	4	no	—
134	C₄×C₈	—	146	8	no	—
135	C₂²×C₄	—	73	1	no	—
136	C₂²×C₄	—	73	1	no	—
137	C₂²×C₄	—	73	2	no	—
138	C₂²×C₄	C₄×#64408	1	2	yes	—
139	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	73	8	no	—
143	C₂×C₈	—	73	4	no	—
144	C₂×C₈	—	73	16	no	—
145	C₂×C₈	—	73	8	no	—
146	C₂×C₈	—	73	4	no	—
147	C₂×C₈	—	73	4	no	—
148	C₂×C₈	—	146	4	no	—
149	C₂×C₈	—	146	2	no	—
150	C₂×C₈	—	146	2	no	—
151	C₂×C₈	C₄×#64408	1	4	yes	—
152	C₄²	—	2	2	no	—
153	C₄²	—	73	8	no	—
154	C₄²	—	73	2	no	—
155	C₄²	—	73	2	no	—
156	C₄²	—	146	4	no	—
157	C₄²	—	146	1	no	—
158	C₄²	—	146	1	no	—
159	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	73	2	no	—
163	C₂×C₄	—	73	1	no	—
164	C₂×C₄	—	73	1	no	—
165	C₂×C₄	—	73	4	no	—
166	C₂×C₄	—	73	4	no	—
167	C₂×C₄	—	73	4	no	—
168	C₂×C₄	—	146	2	no	—
169	C₂×C₄	—	146	2	no	—
170	C₂×C₄	C₈×#64408	1	4	yes	—
171	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
172	C₂×C₄	#14556678	1	1	yes	characteristic
173	C₂×C₄	#14556678	1	2	yes	—
174	C₂×C₄	#14556679	1	1	yes	characteristic
175	C₈	—	2	4	no	—
176	C₈	—	73	16	no	—
177	C₈	—	73	8	no	—
178	C₈	—	146	4	no	—
179	C₈	—	146	4	no	—
180	C₂²	C₄²×#64408	1	1	yes	characteristic
181	C₂²	C₂×C₈×#64408	1	2	yes	—
182	C₂²	M₁₆×#64408	1	2	yes	—
183	C₂²	#17989551	1	1	yes	characteristic
184	C₂²	#17989552	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	73	4	no	—
187	C₄	—	73	2	no	—
188	C₄	—	73	2	no	—
189	C₄	—	146	4	no	—
190	C₄	#17989551	1	2	yes	—
191	C₄	#17989552	1	2	yes	—
192	C₂	#22625132	1	2	yes	—
193	C₂	#22625133	1	2	yes	—
194	C₂	C₄×C₈×#64408	1	1	yes	characteristic
195	C₂	#97×#64408	1	1	yes	characteristic
196	C₂	#22625588	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂×C₇₃	C₄²×C₈
Fitting	#336×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#336	#64408
hypocenter	C₇₃	C₈×#97
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
11. #17989429	C₄	semidirect
12. C₈×#10596833	C₄	semidirect
21. #14556646	C₈	semidirect
77. #10596833	C₄×C₈	semidirect
78. #10596833	#97	semidirect
104. C₇₃	C₈×#97	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556646	2
C₄²×#10596833	2
C₂×#17989429	2
C₂×C₈×#10596833	2
#22609675	2
C₈×#97	73