Subgroups — #34034074

Order 18688.

390 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989681	C₂	1	2	yes	maximal
2	#22627001	C₂	1	2	yes	maximal
3	#22627001	C₂	1	2	yes	maximal
4	C₄²×#10596833	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924040	C₂²	1	4	yes	—
6	#17989681	C₄	1	8	yes	complemented
7	C₂×C₄×#10596833	C₄	1	1	yes	characteristic
8	C₂×C₄×#10596833	C₄	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₄	1	1	yes	characteristic
10	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
13	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×#11924040	—	2	4	no	—
15	C₂×#11924040	—	2	4	no	—
16	C₂×#11924040	C₂×C₄	1	8	yes	—
17	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
18	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596833	C₂³	1	1	yes	characteristic
21	C₄×#10596833	—	2	2	no	—
22	C₄×#10596833	—	2	2	no	—
23	C₄×#10596833	—	2	2	no	—
24	C₄×#10596833	—	2	2	no	—
25	C₄×#10596833	C₈	1	4	yes	—
26	C₄×#10596833	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
29	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
34	#11924040	—	2	8	no	—
35	C₂×#10596833	—	2	1	no	—
36	C₂×#10596833	—	2	1	no	—
37	C₂×#10596833	—	2	1	no	—
38	C₂×#10596833	—	2	1	no	—
39	C₂×#10596833	C₄²	1	2	yes	—
40	C₂×#10596833	C₂×C₈	1	2	yes	—
41	C₂×#10596833	C₂×C₈	1	2	yes	—
42	C₂×#10596833	M₁₆	1	2	yes	—
43	C₂×#10596833	M₁₆	1	2	yes	—
44	C₂×#10596833	C₂²×C₄	1	2	yes	—
45	C₂×#10596833	#41	1	2	yes	—
46	C₂×#10596833	#41	1	2	yes	—
47	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₄×Dic₇₃	—	2	2	no	—
52	C₄×Dic₇₃	—	2	2	no	—
53	C₄×Dic₇₃	—	2	2	no	—
54	C₄×Dic₇₃	—	2	1	no	—
55	C₄×Dic₇₃	—	2	1	no	—
56	C₄×Dic₇₃	—	2	1	no	—
57	C₄×Dic₇₃	—	2	1	no	—
58	C₄×Dic₇₃	C₄²	1	2	yes	—
59	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₇₃	—	2	2	no	—
64	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	2	1	no	—
66	C₂×C₄×C₇₃	—	2	1	no	—
67	C₂×C₄×C₇₃	—	2	1	no	—
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
72	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
73	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
74	C₂×Dic₇₃	—	2	1	no	—
75	C₂×Dic₇₃	—	2	1	no	—
76	C₂×Dic₇₃	—	2	1	no	—
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	—	2	1	no	—
79	C₂×Dic₇₃	—	2	1	no	—
80	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
81	C₂×Dic₇₃	M₃₂	1	2	yes	—
82	C₂×Dic₇₃	M₃₂	1	2	yes	—
83	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×Dic₇₃	#117	1	1	yes	characteristic
85	C₂×Dic₇₃	#117	1	1	yes	characteristic
86	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
89	#10596833	—	2	2	no	—
90	#10596833	—	2	2	no	—
91	#10596833	#97	1	4	yes	—
92	#10596833	#117	1	4	yes	—
93	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²×C₇₃	#431	1	1	yes	characteristic
95	C₂²×C₇₃	#431	1	1	yes	characteristic
96	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
97	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
98	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
99	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	—	2	2	no	—
102	C₄×C₇₃	—	2	2	no	—
103	Dic₇₃	—	2	2	no	—
104	Dic₇₃	—	2	2	no	—
105	Dic₇₃	—	2	2	no	—
106	Dic₇₃	#376	1	1	yes	characteristic
107	Dic₇₃	C₂×#97	1	1	yes	characteristic
108	Dic₇₃	#431	1	1	yes	characteristic
109	Dic₇₃	#431	1	1	yes	characteristic
110	#12591	—	73	1	no	maximal, Sylow
111	C₂×C₇₃	#1744	1	1	yes	characteristic
112	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
113	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
114	C₂×C₇₃	#2163	1	1	yes	characteristic
115	C₂×C₇₃	#2163	1	1	yes	characteristic
116	C₂×C₇₃	#2163	1	1	yes	characteristic
117	C₂×C₇₃	#2163	1	1	yes	characteristic
118	C₂×#346	—	73	2	no	—
119	C₄²×C₈	—	73	1	no	—
120	#1374	—	73	2	no	—
121	#1374	—	73	2	no	—
122	C₇₃	#12591	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₁₆	—	73	4	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₄³	—	73	1	no	—
131	#346	—	73	8	no	—
132	C₂²×C₈	—	73	1	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂×C₄²	—	73	1	no	—
137	C₂×C₄²	—	73	1	no	—
138	C₂×C₄²	—	73	1	no	—
139	C₂×C₄²	—	73	1	no	—
140	C₂×C₄²	—	73	1	no	—
141	C₂×C₄²	—	73	1	no	—
142	C₂×C₄²	#10596842	1	1	yes	characteristic
143	C₂×C₁₆	—	73	8	no	—
144	C₂×C₁₆	—	146	4	no	—
145	C₂×C₁₆	—	146	4	no	—
146	C₄×C₈	—	73	4	no	—
147	C₄×C₈	—	73	4	no	—
148	C₄×C₈	—	146	2	no	—
149	C₄×C₈	—	146	2	no	—
150	C₄×C₈	—	146	2	no	—
151	C₄×C₈	—	146	2	no	—
152	C₂²×C₄	—	73	1	no	—
153	C₂²×C₄	—	73	1	no	—
154	C₂²×C₄	—	73	1	no	—
155	C₂²×C₄	—	73	1	no	—
156	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
157	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
158	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
159	C₂×C₈	—	73	2	no	—
160	C₂×C₈	—	73	2	no	—
161	C₂×C₈	—	73	2	no	—
162	C₂×C₈	—	73	2	no	—
163	C₂×C₈	—	73	2	no	—
164	C₂×C₈	—	73	2	no	—
165	C₂×C₈	—	73	2	no	—
166	C₂×C₈	—	73	2	no	—
167	C₂×C₈	—	146	1	no	—
168	C₂×C₈	—	146	1	no	—
169	C₂×C₈	—	146	1	no	—
170	C₂×C₈	—	146	1	no	—
171	C₄²	—	2	2	no	—
172	C₄²	—	73	2	no	—
173	C₄²	—	73	2	no	—
174	C₄²	—	146	1	no	—
175	C₄²	—	146	1	no	—
176	C₄²	—	146	1	no	—
177	C₄²	—	146	1	no	—
178	C₄²	—	146	2	no	—
179	C₄²	—	146	2	no	—
180	C₄²	—	146	2	no	—
181	C₁₆	—	146	8	no	—
182	C₂³	C₂²×#10596842	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	73	1	no	—
190	C₂×C₄	—	73	1	no	—
191	C₂×C₄	—	73	1	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	2	no	—
196	C₂×C₄	—	73	2	no	—
197	C₂×C₄	—	73	2	no	—
198	C₂×C₄	—	146	1	no	—
199	C₂×C₄	—	146	1	no	—
200	C₂×C₄	—	146	1	no	—
201	C₂×C₄	—	146	1	no	—
202	C₂×C₄	—	146	1	no	—
203	C₂×C₄	—	146	1	no	—
204	C₂×C₄	C₄×#10596842	1	2	yes	—
205	C₂×C₄	#14556687	1	2	yes	—
206	C₂×C₄	#14556688	1	2	yes	—
207	C₈	—	73	4	no	—
208	C₈	—	73	4	no	—
209	C₈	—	146	2	no	—
210	C₈	—	146	2	no	—
211	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
212	C₂²	#17989700	1	1	yes	characteristic
213	C₂²	#17989700	1	1	yes	characteristic
214	C₂²	C₂×#14556687	1	1	yes	characteristic
215	C₂²	#17989706	1	1	yes	characteristic
216	C₂²	#17989706	1	1	yes	characteristic
217	C₂²	C₂×#14556688	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	73	1	no	—
222	C₄	—	73	1	no	—
223	C₄	—	73	1	no	—
224	C₄	—	73	1	no	—
225	C₄	—	146	2	no	—
226	C₄	—	146	2	no	—
227	C₄	—	146	2	no	—
228	C₂	#22627065	1	1	yes	characteristic
229	C₂	C₄×#14556687	1	1	yes	characteristic
230	C₂	C₄×#14556688	1	1	yes	characteristic
231	C₂	#22627132	1	1	yes	characteristic
232	C₂	#22627133	1	1	yes	characteristic
233	C₂	#22627134	1	1	yes	characteristic
234	C₂	#22627135	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10596842
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×C₄²×C₇₃	C₈
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596833	C₂³
hypercenter	C₂×C₄²	#10596842
hypocenter	C₇₃	#12591
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. #17989681	C₄	semidirect
122. C₇₃	#12591	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989681	2
#22627001	2
C₄²×#10596833	2
#12591	73