Subgroups — #430×Dic₇₃ #33084093

Order 18688.

1034 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988636	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₇₃	C₂	1	2	yes	maximal
3	C₂×M₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	#22609839	C₂	1	4	yes	maximal, complemented
5	#108×Dic₇₃	C₂	1	4	yes	maximal, complemented
6	#22609657	C₂	1	2	yes	maximal
7	C₂×#430×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11924012	C₂²	1	2	yes	—
9	C₂²×#11924012	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
11	#17988636	C₄	1	8	yes	complemented
12	C₂×#14556527	C₂²	1	4	yes	—
13	M₁₆×Dic₇₃	—	2	2	no	—
14	M₁₆×Dic₇₃	—	2	2	no	—
15	M₁₆×Dic₇₃	C₄	1	8	yes	—
16	M₁₆×Dic₇₃	C₂²	1	4	yes	—
17	M₁₆×Dic₇₃	C₂²	1	4	yes	—
18	C₂×#14556501	—	2	2	no	—
19	C₂×#14556501	C₂²	1	2	yes	—
20	C₂×C₈×Dic₇₃	—	2	2	no	—
21	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
22	#17988351	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
24	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
25	#430×C₇₃	C₄	1	8	yes	complemented
26	C₂×#108×C₇₃	C₂²	1	4	yes	—
27	C₂³×Dic₇₃	Q₈	1	2	yes	—
28	C₂×#11924012	—	2	2	no	—
29	C₂×#11924012	—	2	2	no	—
30	C₂×#11924012	—	2	4	no	—
31	C₂×#11924012	—	2	2	no	—
32	C₂×#11924012	C₂×C₄	1	8	yes	—
33	C₂×#11924012	C₂×C₄	1	4	yes	—
34	C₂×#11924012	C₂³	1	4	yes	—
35	C₂²×#10596831	—	2	2	no	—
36	C₂²×#10596831	C₂³	1	2	yes	—
37	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
39	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
40	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
41	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
43	#14556527	C₂×C₄	1	16	yes	complemented
44	#14556501	—	2	4	no	—
45	#14556501	—	2	4	no	—
46	#14556501	C₂×C₄	1	8	yes	—
47	C₈×Dic₇₃	—	2	4	no	—
48	C₈×Dic₇₃	—	2	4	no	—
49	C₈×Dic₇₃	C₂×C₄	1	8	yes	—
50	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₇₃	—	2	2	no	—
52	C₂×M₁₆×C₇₃	—	2	4	no	—
53	C₂×M₁₆×C₇₃	C₂×C₄	1	8	yes	—
54	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
55	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
56	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
57	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
58	C₂²×C₈×C₇₃	—	2	2	no	—
59	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
60	#108×C₇₃	C₂×C₄	1	16	yes	complemented
61	C₂²×Dic₇₃	—	4	4	no	—
62	C₂²×Dic₇₃	C₄⋊C₄	1	4	yes	—
63	C₂²×Dic₇₃	C₄⋊C₄	1	4	yes	—
64	C₂²×Dic₇₃	C₂×Q₈	1	2	yes	—
65	C₂²×Dic₇₃	C₂×Q₈	1	2	yes	—
66	#11924012	—	2	8	no	—
67	#11924012	—	2	8	no	—
68	#11924012	—	2	4	no	—
69	#11924012	C₂²×C₄	1	8	yes	—
70	C₂×#10596831	—	2	8	no	—
71	C₂×#10596831	—	2	4	no	—
72	C₂×#10596831	—	2	2	no	—
73	C₂×#10596831	—	2	2	no	—
74	C₂×#10596831	C₂²×C₄	1	4	yes	—
75	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
76	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
77	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
78	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
79	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
80	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
81	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	#41	1	2	yes	—
89	C₂²×C₄×C₇₃	#41	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
91	M₁₆×C₇₃	—	2	8	no	—
92	M₁₆×C₇₃	—	2	4	no	—
93	M₁₆×C₇₃	C₄²	1	16	yes	—
94	M₁₆×C₇₃	C₂²×C₄	1	8	yes	—
95	C₂×C₈×C₇₃	—	2	8	no	—
96	C₂×C₈×C₇₃	—	2	4	no	—
97	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
100	C₂³×C₇₃	—	4	2	no	—
101	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂³×C₇₃	C₄×Q₈	1	2	yes	—
103	C₂³×C₇₃	C₄×Q₈	1	2	yes	—
104	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₇₃	#117	1	4	yes	—
108	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
110	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₇₃	—	2	8	no	—
119	C₂×Dic₇₃	—	2	4	no	—
120	C₂×Dic₇₃	—	4	4	no	—
121	C₂×Dic₇₃	—	4	2	no	—
122	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
124	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₈×C₇₃	—	2	8	no	—
126	C₈×C₇₃	C₂×C₄²	1	8	yes	—
127	#10596831	—	2	4	no	—
128	#10596831	—	2	8	no	—
129	C₂²×C₇₃	—	2	4	no	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	4	2	no	—
132	C₂²×C₇₃	—	4	2	no	—
133	C₂²×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
134	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
136	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₄×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
138	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
139	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
140	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
142	Dic₇₃	—	2	4	no	—
143	Dic₇₃	—	2	2	no	—
144	Dic₇₃	#430	1	4	yes	direct product factor
145	C₄×#430	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	—	2	1	no	—
147	C₂×C₇₃	—	2	1	no	—
148	C₂×C₇₃	—	2	4	no	—
149	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₇₃	C₂×#430	1	2	yes	—
151	C₂×#430	—	73	1	no	—
152	C₂×#430	D₇₃	1	1	yes	characteristic
153	C₂×C₄×M₁₆	—	73	1	no	—
154	C₂×C₄×M₁₆	—	73	2	no	—
155	#1778	—	73	4	no	—
156	C₄×#108	—	73	4	no	—
157	#1738	—	73	2	no	—
158	C₇₃	C₄×#430	1	1	yes	characteristic, complemented, Sylow
159	C₂²×M₁₆	—	73	1	no	—
160	C₂²×M₁₆	—	73	2	no	—
161	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
162	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
163	C₂²×C₄²	—	73	1	no	—
164	#430	—	73	8	no	—
165	#430	Dic₇₃	1	8	yes	direct product factor
166	C₂×#108	—	73	4	no	—
167	C₂×#108	C₂×D₇₃	1	4	yes	—
168	C₄×M₁₆	—	73	8	no	—
169	C₄×M₁₆	—	73	4	no	—
170	C₄×M₁₆	—	73	4	no	—
171	C₄×M₁₆	—	146	2	no	—
172	C₄×M₁₆	—	146	2	no	—
173	C₂×#97	—	73	2	no	—
174	C₂×#97	—	146	2	no	—
175	C₂×C₄×C₈	—	73	2	no	—
176	C₂×C₄×C₈	—	146	2	no	—
177	#341	—	73	8	no	—
178	C₂³×C₄	—	73	2	no	—
179	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	73	4	no	—
183	C₂×M₁₆	—	73	8	no	—
184	C₂×M₁₆	—	73	4	no	—
185	C₂×M₁₆	—	146	2	no	—
186	C₂×M₁₆	—	146	2	no	—
187	C₂×M₁₆	—	146	4	no	—
188	C₂×M₁₆	—	146	2	no	—
189	C₂×M₁₆	C₄×D₇₃	1	4	yes	—
190	C₂×M₁₆	C₄×D₇₃	1	4	yes	—
191	C₂×M₁₆	C₂×Dic₇₃	1	8	yes	—
192	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
193	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	73	2	no	—
196	C₂²×C₈	—	146	2	no	—
197	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
198	C₂×C₄²	—	73	4	no	—
199	C₂×C₄²	—	73	2	no	—
200	C₂×C₄²	—	73	2	no	—
201	C₂×C₄²	—	73	2	no	—
202	C₂×C₄²	—	73	1	no	—
203	C₂×C₄²	—	73	1	no	—
204	#108	—	73	16	no	—
205	#108	C₂×Dic₇₃	1	16	yes	complemented
206	#97	—	73	8	no	—
207	#97	—	146	4	no	—
208	#97	—	146	4	no	—
209	C₄×C₈	—	73	8	no	—
210	C₄×C₈	—	146	4	no	—
211	C₄×C₈	—	146	4	no	—
212	C₂⁴	Q₈×D₇₃	1	1	yes	characteristic
213	C₂²×C₄	—	292	4	no	—
214	C₂²×C₄	—	73	4	no	—
215	C₂²×C₄	—	73	2	no	—
216	C₂²×C₄	—	73	4	no	—
217	C₂²×C₄	—	73	2	no	—
218	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
221	C₂²×C₄	D₄×D₇₃	1	2	yes	—
222	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
223	C₂²×C₄	#11924027	1	2	yes	—
224	C₂²×C₄	#11924029	1	2	yes	—
225	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
226	M₁₆	—	2	8	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	73	8	no	—
229	M₁₆	—	146	8	no	—
230	M₁₆	—	146	8	no	—
231	M₁₆	—	146	4	no	—
232	M₁₆	C₄×Dic₇₃	1	16	yes	—
233	M₁₆	C₂²×Dic₇₃	1	8	yes	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	73	4	no	—
237	C₂×C₈	—	146	8	no	—
238	C₂×C₈	—	146	4	no	—
239	C₂×C₈	—	146	2	no	—
240	C₂×C₈	—	146	2	no	—
241	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
242	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
243	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
244	C₄²	—	73	4	no	—
245	C₄²	—	73	4	no	—
246	C₄²	—	73	2	no	—
247	C₄²	—	73	2	no	—
248	C₄²	—	73	1	no	—
249	C₄²	—	73	2	no	—
250	C₄²	—	73	1	no	—
251	C₂³	—	4	2	no	—
252	C₂³	C₄⋊C₄×D₇₃	1	2	yes	—
253	C₂³	#14556556	1	2	yes	—
254	C₂³	Q₈×Dic₇₃	1	2	yes	—
255	C₂³	C₂×Q₈×D₇₃	1	1	yes	characteristic
256	C₂×C₄	—	292	4	no	—
257	C₂×C₄	—	292	2	no	—
258	C₂×C₄	—	73	2	no	—
259	C₂×C₄	—	73	2	no	—
260	C₂×C₄	—	73	4	no	—
261	C₂×C₄	—	146	8	no	—
262	C₂×C₄	—	146	4	no	—
263	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
264	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
265	C₂×C₄	#14556549	1	4	yes	—
266	C₂×C₄	#14556553	1	4	yes	—
267	C₂×C₄	D₄×Dic₇₃	1	4	yes	—
268	C₂×C₄	#14556556	1	2	yes	—
269	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
272	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
275	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
276	C₈	—	2	8	no	—
277	C₈	—	146	4	no	—
278	C₈	—	146	8	no	—
279	C₈	C₂×C₄×Dic₇₃	1	8	yes	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₄⋊C₄×Dic₇₃	1	4	yes	—
285	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
286	C₂²	C₂×#14556556	1	1	yes	characteristic
287	C₂²	C₂×Q₈×Dic₇₃	1	1	yes	characteristic
288	C₄	—	73	4	no	—
289	C₄	—	146	2	no	—
290	C₄	—	146	4	no	—
291	C₄	C₄⋊C₄×Dic₇₃	1	4	yes	—
292	C₄	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
293	C₄	C₂×#14556549	1	1	yes	characteristic
294	C₄	C₂×#14556553	1	1	yes	characteristic
295	C₄	C₂×D₄×Dic₇₃	1	1	yes	characteristic
296	C₂	—	2	4	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	C₂×C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
300	C₂	#430×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#430×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#430	D₇₃
hypocenter	C₇₃	C₄×#430
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
4. #22609839	C₂	semidirect
5. #108×Dic₇₃	C₂	semidirect
11. #17988636	C₄	semidirect
25. #430×C₇₃	C₄	semidirect
43. #14556527	C₂×C₄	semidirect
60. #108×C₇₃	C₂×C₄	semidirect
144. Dic₇₃	#430	direct (N × H)
158. C₇₃	C₄×#430	semidirect
165. #430	Dic₇₃	direct (N × H)
205. #108	C₂×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988636	2
C₂×M₁₆×Dic₇₃	2
#22609839	2
#108×Dic₇₃	2
#22609657	2
C₂×#430×C₇₃	2
C₄×#430	73

Subgroups — #430×Dic73 #33084093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #430×Dic₇₃ #33084093