Subgroups — #34003350

Order 18688.

478 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#22624753	C₂	1	2	yes	maximal
4	#22624753	C₂	1	2	yes	maximal
5	#22609675	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556646	C₄	1	2	yes	—
7	C₂×#14556646	C₄	1	2	yes	—
8	C₂×C₄×#10596833	C₄	1	2	yes	—
9	C₂×C₄×#10596833	C₄	1	2	yes	—
10	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10596833	C₂²	1	2	yes	—
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	4	yes	—
19	C₂²×#10596833	C₂×C₄	1	2	yes	—
20	C₂²×#10596833	C₂×C₄	1	2	yes	—
21	#14556646	—	2	4	no	—
22	#14556646	C₈	1	8	yes	complemented
23	C₄×#10596833	—	2	4	no	—
24	C₄×#10596833	—	2	4	no	—
25	C₄×#10596833	—	2	2	no	—
26	C₄×#10596833	—	2	2	no	—
27	C₄×#10596833	—	2	2	no	—
28	C₄×#10596833	—	2	2	no	—
29	C₂²×#10596831	C₂×C₄	1	2	yes	—
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×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
36	C₈×Dic₇₃	—	2	2	no	—
37	C₈×Dic₇₃	—	2	2	no	—
38	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#10596833	—	2	4	no	—
41	C₂×#10596833	—	2	2	no	—
42	C₂×#10596833	—	2	2	no	—
43	C₂×#10596833	—	2	2	no	—
44	C₂×#10596833	—	2	2	no	—
45	C₂×#10596833	—	2	4	no	—
46	C₂×#10596833	C₂×C₈	1	8	yes	—
47	C₂×#10596833	M₁₆	1	4	yes	—
48	C₂×#10596833	M₁₆	1	4	yes	—
49	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
50	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
51	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×#10596831	—	2	2	no	—
54	C₂×#10596831	—	2	2	no	—
55	C₂×#10596831	—	2	2	no	—
56	C₄×Dic₇₃	—	2	1	no	—
57	C₄×Dic₇₃	—	2	1	no	—
58	C₄×Dic₇₃	—	2	2	no	—
59	C₄×Dic₇₃	—	2	2	no	—
60	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
61	C₄×Dic₇₃	M₁₆	1	4	yes	—
62	C₄×Dic₇₃	#41	1	2	yes	—
63	C₄×Dic₇₃	#41	1	2	yes	—
64	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₃	—	2	2	no	—
68	C₂×C₈×C₇₃	—	2	2	no	—
69	C₂×C₈×C₇₃	—	2	2	no	—
70	C₄²×C₇₃	—	2	2	no	—
71	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	1	no	—
74	C₂×C₄×C₇₃	—	2	1	no	—
75	C₂×C₄×C₇₃	—	2	1	no	—
76	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
77	C₂×C₄×C₇₃	#97	1	2	yes	—
78	C₂×C₄×C₇₃	#117	1	2	yes	—
79	C₂×C₄×C₇₃	#131	1	2	yes	—
80	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
82	C₂×Dic₇₃	—	2	1	no	—
83	C₂×Dic₇₃	—	2	1	no	—
84	C₂×Dic₇₃	—	2	1	no	—
85	C₂×Dic₇₃	—	2	1	no	—
86	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
87	C₂×Dic₇₃	#97	1	2	yes	—
88	C₂×Dic₇₃	#117	1	2	yes	—
89	C₂×Dic₇₃	#117	1	1	yes	characteristic
90	C₂×Dic₇₃	#117	1	1	yes	characteristic
91	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
92	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
93	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
94	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
95	#10596833	—	2	8	no	—
96	#10596833	—	2	4	no	—
97	#10596833	—	2	4	no	—
98	C₈×C₇₃	—	4	2	no	—
99	#10596831	—	4	2	no	—
100	C₂²×C₇₃	#376	1	1	yes	characteristic
101	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
102	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
103	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
104	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
105	C₂²×C₇₃	#405	1	1	yes	characteristic
106	C₂²×C₇₃	#405	1	1	yes	characteristic
107	C₄×C₇₃	—	2	2	no	—
108	C₄×C₇₃	—	2	2	no	—
109	C₄×C₇₃	#433	1	2	yes	—
110	C₄×C₇₃	#433	1	2	yes	—
111	Dic₇₃	—	2	2	no	—
112	Dic₇₃	—	2	2	no	—
113	Dic₇₃	#431	1	4	yes	—
114	Dic₇₃	#432	1	2	yes	—
115	Dic₇₃	#432	1	2	yes	—
116	#8132	—	73	1	no	maximal, Sylow
117	C₂×C₇₃	C₈×M₁₆	1	1	yes	characteristic
118	C₂×C₇₃	C₈×M₁₆	1	1	yes	characteristic
119	C₂×C₇₃	#1451	1	1	yes	characteristic
120	C₂×C₇₃	#1451	1	1	yes	characteristic
121	C₂×C₇₃	#1744	1	1	yes	characteristic
122	C₂×C₇₃	#1745	1	1	yes	characteristic
123	C₂×C₇₃	#1746	1	1	yes	characteristic
124	#1744	—	73	1	no	—
125	C₄×#97	—	73	1	no	—
126	C₄²×C₈	—	73	1	no	—
127	#1273	—	73	2	no	—
128	#1273	—	73	2	no	—
129	C₇₃	#8132	1	1	yes	characteristic, complemented, Sylow
130	C₂×#97	—	73	2	no	—
131	C₂×#97	—	73	2	no	—
132	C₂×C₄×C₈	—	73	2	no	—
133	C₂×C₄×C₈	—	73	2	no	—
134	C₂×C₄×C₈	—	73	1	no	—
135	C₂×C₄×C₈	—	73	1	no	—
136	C₂×C₄×C₈	—	73	2	no	—
137	C₂×C₄×C₈	—	73	2	no	—
138	C₄³	—	73	1	no	—
139	#336	—	73	2	no	—
140	#336	—	73	1	no	—
141	#336	#64408	1	1	yes	characteristic
142	C₂²×C₈	—	73	2	no	—
143	C₂²×C₈	—	73	2	no	—
144	C₂²×C₈	—	73	4	no	—
145	C₂²×C₈	—	73	2	no	—
146	C₂²×C₈	C₂×#64408	1	2	yes	—
147	C₂×C₄²	—	73	1	no	—
148	C₂×C₄²	—	73	1	no	—
149	C₂×C₄²	—	73	1	no	—
150	C₂×C₄²	—	73	1	no	—
151	C₂×C₄²	—	73	1	no	—
152	C₂×C₄²	—	73	1	no	—
153	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
154	#97	—	73	8	no	—
155	#97	—	146	4	no	—
156	C₄×C₈	—	146	2	no	—
157	C₄×C₈	—	146	2	no	—
158	C₄×C₈	—	146	2	no	—
159	C₄×C₈	—	146	2	no	—
160	C₄×C₈	—	146	2	no	—
161	C₄×C₈	—	146	2	no	—
162	C₄×C₈	—	146	4	no	—
163	C₄×C₈	—	146	4	no	—
164	C₂²×C₄	—	73	1	no	—
165	C₂²×C₄	—	73	1	no	—
166	C₂²×C₄	—	73	1	no	—
167	C₂²×C₄	—	73	1	no	—
168	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
169	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
170	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	73	8	no	—
175	C₂×C₈	—	73	4	no	—
176	C₂×C₈	—	73	4	no	—
177	C₂×C₈	—	146	2	no	—
178	C₂×C₈	—	146	2	no	—
179	C₂×C₈	—	146	2	no	—
180	C₂×C₈	—	146	2	no	—
181	C₂×C₈	—	146	2	no	—
182	C₂×C₈	—	146	4	no	—
183	C₂×C₈	—	146	2	no	—
184	C₂×C₈	—	146	2	no	—
185	C₂×C₈	—	146	4	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	73	4	no	—
188	C₄²	—	73	4	no	—
189	C₄²	—	73	2	no	—
190	C₄²	—	73	2	no	—
191	C₄²	—	146	2	no	—
192	C₄²	—	146	2	no	—
193	C₄²	—	146	1	no	—
194	C₄²	—	146	1	no	—
195	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	73	2	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	1	no	—
203	C₂×C₄	—	73	2	no	—
204	C₂×C₄	—	73	2	no	—
205	C₂×C₄	—	73	2	no	—
206	C₂×C₄	—	73	2	no	—
207	C₂×C₄	—	73	2	no	—
208	C₂×C₄	—	73	2	no	—
209	C₂×C₄	—	146	1	no	—
210	C₂×C₄	—	146	1	no	—
211	C₂×C₄	—	146	1	no	—
212	C₂×C₄	—	146	1	no	—
213	C₂×C₄	C₈×#64408	1	2	yes	—
214	C₂×C₄	#14556645	1	2	yes	—
215	C₂×C₄	#14556670	1	2	yes	—
216	C₂×C₄	#14556678	1	1	yes	characteristic
217	C₂×C₄	#14556678	1	2	yes	—
218	C₂×C₄	#14556679	1	1	yes	characteristic
219	C₈	—	4	2	no	—
220	C₈	—	292	2	no	—
221	C₈	—	146	8	no	—
222	C₈	—	146	4	no	—
223	C₈	—	146	4	no	—
224	C₂²	#17989523	1	1	yes	characteristic
225	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
226	C₂²	C₂×#14556645	1	1	yes	characteristic
227	C₂²	M₁₆×#64408	1	1	yes	characteristic
228	C₂²	M₁₆×#64408	1	1	yes	characteristic
229	C₂²	#17989551	1	1	yes	characteristic
230	C₂²	#17989552	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	73	4	no	—
234	C₄	—	73	2	no	—
235	C₄	—	73	2	no	—
236	C₄	—	146	2	no	—
237	C₄	—	146	2	no	—
238	C₄	#17989611	1	2	yes	—
239	C₄	#17989612	1	2	yes	—
240	C₂	#22625132	1	1	yes	characteristic
241	C₂	#22625133	1	1	yes	characteristic
242	C₂	#22625142	1	1	yes	characteristic
243	C₂	#22625143	1	1	yes	characteristic
244	C₂	#22625644	1	1	yes	characteristic
245	C₂	#22625647	1	1	yes	characteristic
246	C₂	#22625650	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	#336×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#336	#64408
hypocenter	C₇₃	#8132
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
22. #14556646	C₈	semidirect
129. C₇₃	#8132	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
#22624753	2
#22609675	2
#8132	73