Subgroups — #32977112

Order 18688.

502 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556520	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556520	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988309	C₂	1	1	yes	maximal, characteristic
5	C₂×#17988308	C₂	1	1	yes	maximal, characteristic
6	#22607758	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#1273	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556520	C₄	1	2	yes	—
9	C₂×#14556520	C₄	1	2	yes	—
10	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
14	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
15	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
16	#17988341	C₄	1	2	yes	—
17	#17988341	C₄	1	2	yes	—
18	#17988309	—	2	2	no	—
19	#17988308	—	2	2	no	—
20	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂²×#10596831	C₂×C₄	1	2	yes	—
24	C₂²×#10596831	C₂×C₄	1	2	yes	—
25	C₂²×#10596831	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₇₃	C₂×C₄	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	#14556520	—	2	4	no	—
33	#14556520	C₈	1	8	yes	complemented
34	C₄×#10596831	—	2	1	no	—
35	C₄×#10596831	—	2	1	no	—
36	C₄×#10596831	—	2	1	no	—
37	C₄×#10596831	—	2	1	no	—
38	C₄×#10596831	D₄	1	2	yes	—
39	C₄×#10596831	Q₈	1	2	yes	—
40	C₈×Dic₇₃	—	2	4	no	—
41	C₈×Dic₇₃	—	2	4	no	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₇₃	—	2	1	no	—
47	C₄×C₈×C₇₃	—	2	1	no	—
48	C₄×C₈×C₇₃	—	2	1	no	—
49	C₄×C₈×C₇₃	—	2	1	no	—
50	C₄×C₈×C₇₃	—	2	1	no	—
51	C₄×C₈×C₇₃	—	2	1	no	—
52	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
54	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
56	C₂×#10596831	—	2	2	no	—
57	C₂×#10596831	—	2	2	no	—
58	C₂×#10596831	—	2	2	no	—
59	C₂×#10596831	—	2	2	no	—
60	C₂×#10596831	—	2	2	no	—
61	C₂×#10596831	—	2	2	no	—
62	C₂×#10596831	—	2	2	no	—
63	C₂×#10596831	C₄⋊C₄	1	4	yes	—
64	C₂×#10596831	C₂×C₈	1	4	yes	—
65	C₄×Dic₇₃	—	2	2	no	—
66	C₄×Dic₇₃	—	2	2	no	—
67	C₄×Dic₇₃	—	2	2	no	—
68	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
69	C₄×Dic₇₃	M₁₆	1	4	yes	—
70	C₄×Dic₇₃	M₁₆	1	4	yes	—
71	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	—	2	2	no	—
75	C₂×C₈×C₇₃	—	2	2	no	—
76	C₂×C₈×C₇₃	—	2	2	no	—
77	C₂×C₈×C₇₃	—	2	2	no	—
78	C₂×C₈×C₇₃	—	2	2	no	—
79	C₂×C₈×C₇₃	—	2	2	no	—
80	C₂×C₈×C₇₃	—	2	2	no	—
81	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
82	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
83	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₇₃	#41	1	1	yes	characteristic
86	C₄²×C₇₃	#41	1	1	yes	characteristic
87	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₇₃	#131	1	2	yes	—
99	C₂×C₄×C₇₃	#131	1	2	yes	—
100	C₂×Dic₇₃	—	2	1	no	—
101	C₂×Dic₇₃	—	2	1	no	—
102	C₂×Dic₇₃	—	2	1	no	—
103	C₂×Dic₇₃	—	2	1	no	—
104	C₂×Dic₇₃	—	2	1	no	—
105	C₂×Dic₇₃	—	2	1	no	—
106	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
107	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
108	C₂×Dic₇₃	#97	1	2	yes	—
109	C₂×Dic₇₃	#97	1	2	yes	—
110	C₂×Dic₇₃	#97	1	2	yes	—
111	C₂×Dic₇₃	#97	1	2	yes	—
112	C₈×C₇₃	—	2	4	no	—
113	C₈×C₇₃	—	2	4	no	—
114	C₈×C₇₃	—	4	2	no	—
115	#10596831	—	2	4	no	—
116	#10596831	—	2	4	no	—
117	#10596831	—	4	2	no	—
118	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
120	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
121	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
122	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
123	C₂²×C₇₃	#405	1	1	yes	characteristic
124	C₂²×C₇₃	#405	1	1	yes	characteristic
125	C₄×C₇₃	#423	1	2	yes	—
126	C₄×C₇₃	#432	1	2	yes	—
127	C₄×C₇₃	C₈×D₄	1	2	yes	—
128	C₄×C₇₃	#436	1	2	yes	—
129	C₄×C₇₃	C₈×Q₈	1	2	yes	—
130	C₄×C₇₃	#446	1	2	yes	—
131	Dic₇₃	—	2	2	no	—
132	Dic₇₃	—	2	2	no	—
133	Dic₇₃	—	2	2	no	—
134	Dic₇₃	#322	1	4	yes	complemented
135	#8261	—	73	1	no	maximal, Sylow
136	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
137	C₂×C₇₃	C₈×M₁₆	1	1	yes	characteristic
138	C₂×C₇₃	#1451	1	1	yes	characteristic
139	C₂×C₇₃	#1452	1	1	yes	characteristic
140	C₂×C₇₃	#1767	1	1	yes	characteristic
141	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₇₃	#1770	1	1	yes	characteristic
143	C₄×C₄⋊C₈	—	73	1	no	—
144	C₄×C₄⋊C₈	—	73	1	no	—
145	C₄²×C₈	—	73	1	no	—
146	C₂×#322	—	73	1	no	—
147	C₂×#322	—	73	1	no	—
148	#1273	—	73	1	no	—
149	#1273	D₇₃	1	1	yes	characteristic
150	C₇₃	#8261	1	1	yes	characteristic, complemented, Sylow
151	C₂×C₄⋊C₈	—	73	2	no	—
152	C₂×C₄⋊C₈	—	73	2	no	—
153	C₂×C₄×C₈	—	73	1	no	—
154	C₂×C₄×C₈	—	73	2	no	—
155	C₂×C₄×C₈	—	73	2	no	—
156	C₂×C₄×C₈	—	73	1	no	—
157	C₂×C₄×C₈	—	73	1	no	—
158	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
161	C₄³	—	73	1	no	—
162	#336	—	73	2	no	—
163	#336	—	73	2	no	—
164	#322	—	146	2	no	—
165	#322	—	146	2	no	—
166	C₂²×C₈	—	73	2	no	—
167	C₂²×C₈	—	73	2	no	—
168	C₂²×C₈	—	73	2	no	—
169	C₂²×C₈	C₄×D₇₃	1	2	yes	—
170	C₂²×C₈	C₄×D₇₃	1	2	yes	—
171	C₂²×C₈	C₄×D₇₃	1	2	yes	—
172	C₂×C₄²	—	73	1	no	—
173	C₂×C₄²	—	73	1	no	—
174	C₂×C₄²	—	73	1	no	—
175	C₂×C₄²	—	73	1	no	—
176	C₂×C₄²	—	73	1	no	—
177	C₂×C₄²	—	73	1	no	—
178	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
179	C₄⋊C₈	—	73	8	no	—
180	C₄⋊C₈	—	146	4	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	73	2	no	—
188	C₄×C₈	—	73	2	no	—
189	C₄×C₈	—	146	4	no	—
190	C₄×C₈	—	146	4	no	—
191	C₄×C₈	—	146	1	no	—
192	C₄×C₈	—	146	1	no	—
193	C₄×C₈	—	146	1	no	—
194	C₄×C₈	—	146	1	no	—
195	C₂²×C₄	—	73	1	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	73	1	no	—
198	C₂²×C₄	—	73	1	no	—
199	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	73	4	no	—
210	C₂×C₈	—	73	4	no	—
211	C₂×C₈	—	146	2	no	—
212	C₂×C₈	—	146	2	no	—
213	C₂×C₈	—	146	2	no	—
214	C₂×C₈	—	146	2	no	—
215	C₂×C₈	—	146	2	no	—
216	C₂×C₈	—	146	2	no	—
217	C₂×C₈	—	146	2	no	—
218	C₂×C₈	C₈×D₇₃	1	4	yes	—
219	C₂×C₈	#11924011	1	4	yes	—
220	C₄²	—	73	4	no	—
221	C₄²	—	73	4	no	—
222	C₄²	—	73	4	no	—
223	C₄²	—	146	2	no	—
224	C₄²	—	146	2	no	—
225	C₄²	—	146	2	no	—
226	C₄²	D₄×D₇₃	1	1	yes	characteristic
227	C₄²	Q₈×D₇₃	1	1	yes	characteristic
228	C₄²	#11924027	1	1	yes	characteristic
229	C₄²	#11924027	1	1	yes	characteristic
230	C₂³	C₄²×D₇₃	1	1	yes	characteristic
231	C₂×C₄	—	73	2	no	—
232	C₂×C₄	—	73	2	no	—
233	C₂×C₄	—	73	2	no	—
234	C₂×C₄	—	73	2	no	—
235	C₂×C₄	—	73	2	no	—
236	C₂×C₄	—	73	2	no	—
237	C₂×C₄	—	146	1	no	—
238	C₂×C₄	—	146	1	no	—
239	C₂×C₄	—	146	1	no	—
240	C₂×C₄	—	146	1	no	—
241	C₂×C₄	—	146	1	no	—
242	C₂×C₄	—	146	1	no	—
243	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
244	C₂×C₄	#14556547	1	1	yes	characteristic
245	C₂×C₄	#14556551	1	1	yes	characteristic
246	C₂×C₄	#14556551	1	1	yes	characteristic
247	C₂×C₄	#14556556	1	1	yes	characteristic
248	C₂×C₄	#14556556	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
250	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
251	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
252	C₂×C₄	C₂×#11924011	1	2	yes	—
253	C₂×C₄	#14556597	1	2	yes	—
254	C₂×C₄	#14556599	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	—	2	4	no	—
257	C₈	—	4	2	no	—
258	C₈	—	292	2	no	—
259	C₈	—	146	4	no	—
260	C₈	—	146	4	no	—
261	C₂²	#17988438	1	1	yes	characteristic
262	C₂²	C₄×C₈×D₇₃	1	1	yes	characteristic
263	C₂²	#97×D₇₃	1	1	yes	characteristic
264	C₂²	#97×D₇₃	1	1	yes	characteristic
265	C₂²	C₄×#11924011	1	1	yes	characteristic
266	C₂²	#17988541	1	1	yes	characteristic
267	C₂²	#17988544	1	1	yes	characteristic
268	C₄	—	73	4	no	—
269	C₄	—	146	2	no	—
270	C₄	—	146	2	no	—
271	C₄	—	146	2	no	—
272	C₄	#17988609	1	2	yes	—
273	C₄	#17988641	1	2	yes	—
274	C₄	#17988650	1	2	yes	—
275	C₄	#17988662	1	2	yes	—
276	C₄	#17988711	1	2	yes	—
277	C₄	#17988716	1	2	yes	—
278	C₂	#322×D₇₃	1	1	yes	characteristic
279	C₂	C₈×#11924011	1	1	yes	characteristic
280	C₂	#22608217	1	1	yes	characteristic
281	C₂	#22608222	1	1	yes	characteristic
282	C₂	#22609783	1	1	yes	characteristic
283	C₂	#22609792	1	1	yes	characteristic
284	C₂	#22609797	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₇₃×#1273	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1273	D₇₃
hypocenter	C₇₃	#8261
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
33. #14556520	C₈	semidirect
134. Dic₇₃	#322	semidirect
150. C₇₃	#8261	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556520	2
C₄×C₈×Dic₇₃	2
C₂×#17988309	2
C₂×#17988308	2
#22607758	2
C₇₃×#1273	2
#8261	73