Subgroups — #33086450

Order 18688.

954 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988634	C₂	1	2	yes	maximal
2	C₂×#108×D₇₃	C₂	1	2	yes	maximal
3	#22618493	C₂	1	2	yes	maximal
4	C₂×#17988541	C₂	1	1	yes	maximal, characteristic
5	#22610105	C₂	1	2	yes	maximal, complemented
6	#22610104	C₂	1	2	yes	maximal, complemented
7	#22610103	C₂	1	1	yes	maximal, characteristic, complemented
8	#22610101	C₂	1	2	yes	maximal, complemented
9	C₇₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11924011	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₇₃	C₂²	1	4	yes	—
12	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×#14556548	C₂²	1	1	yes	characteristic
14	#17988634	—	2	8	no	—
15	C₂×#14556527	C₂²	1	2	yes	—
16	C₂×#14556527	C₂²	1	2	yes	—
17	C₂×#14556526	C₂²	1	4	yes	—
18	C₂×#14556526	C₂²	1	4	yes	—
19	#108×D₇₃	—	2	8	no	—
20	#17988612	—	2	8	no	—
21	#17988611	C₂²	1	2	yes	—
22	#17988610	C₂²	1	4	yes	—
23	#17988541	C₄	1	8	yes	—
24	C₂×#14556502	C₂²	1	1	yes	characteristic
25	C₂×#14556501	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₇₃	C₂²	1	2	yes	—
28	C₂×#108×C₇₃	C₂²	1	2	yes	—
29	#423×C₇₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
32	C₂×#11924012	C₂³	1	4	yes	—
33	C₂×#11924011	—	2	4	no	—
34	C₂×#11924011	C₂×C₄	1	4	yes	—
35	C₂×#11924011	D₄	1	4	yes	—
36	C₂×#11924011	Q₈	1	4	yes	—
37	M₁₆×D₇₃	—	2	8	no	—
38	M₁₆×D₇₃	—	2	8	no	—
39	C₂²×#10596831	C₂³	1	1	yes	characteristic
40	C₂²×#10596831	C₂³	1	1	yes	characteristic
41	C₂×C₈×D₇₃	—	2	4	no	—
42	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
43	C₂×C₈×D₇₃	D₄	1	4	yes	—
44	C₂×C₈×D₇₃	Q₈	1	4	yes	—
45	#14556548	C₂×C₄	1	4	yes	—
46	#14556548	D₄	1	4	yes	—
47	C₂×#11924007	Q₈	1	2	yes	—
48	C₂×#11924005	Q₈	1	2	yes	—
49	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
50	#14556527	—	2	4	no	—
51	#14556527	—	2	4	no	—
52	#14556526	—	2	8	no	—
53	#14556526	—	2	8	no	—
54	C₄⋊#10596831	—	2	4	no	—
55	#14556520	—	2	8	no	—
56	#14556502	C₂×C₄	1	4	yes	—
57	#14556501	C₂×C₄	1	4	yes	—
58	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
59	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
60	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
63	#108×C₇₃	—	2	4	no	—
64	#108×C₇₃	—	2	4	no	—
65	C₄⋊C₈×C₇₃	—	2	4	no	—
66	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
67	C₂³×D₇₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×Dic₇₃	C₂×Q₈	1	2	yes	—
69	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
75	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
76	#11924012	—	2	4	no	—
77	#11924012	—	2	4	no	—
78	#11924011	—	4	8	no	—
79	#11924011	C₄⋊C₄	1	16	yes	—
80	C₂×#10596831	—	2	4	no	—
81	C₂×#10596831	C₂²×C₄	1	2	yes	—
82	C₂×#10596831	C₂²×C₄	1	2	yes	—
83	C₂×#10596831	C₂×D₄	1	2	yes	—
84	C₂×#10596831	C₂×D₄	1	2	yes	—
85	C₂×#10596831	C₂×Q₈	1	2	yes	—
86	C₂×#10596831	C₂×Q₈	1	2	yes	—
87	C₈×D₇₃	—	4	8	no	—
88	C₈×D₇₃	C₄⋊C₄	1	16	yes	—
89	#11924007	C₄⋊C₄	1	8	yes	—
90	#11924005	C₄⋊C₄	1	8	yes	—
91	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
92	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₇₃	—	2	4	no	—
96	M₁₆×C₇₃	—	2	4	no	—
97	C₂×C₈×C₇₃	—	2	4	no	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
103	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
104	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₇₃	C₂×D₄	1	2	yes	—
106	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×D₇₃	—	4	2	no	—
108	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂²×D₇₃	C₄⋊Q₈	1	2	yes	—
110	C₂²×D₇₃	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₇₃	—	4	2	no	—
112	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×Dic₇₃	—	4	2	no	—
120	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×Dic₇₃	C₄⋊Q₈	1	2	yes	—
123	C₂×Dic₇₃	C₄⋊Q₈	1	2	yes	—
124	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
126	C₄×D₇₃	#127	1	4	yes	—
127	C₄×D₇₃	C₄⋊Q₈	1	4	yes	—
128	C₈×C₇₃	—	4	4	no	—
129	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
130	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
131	#10596831	—	4	4	no	—
132	#10596831	C₂×C₄⋊C₄	1	4	yes	—
133	#10596831	C₂×C₄⋊C₄	1	4	yes	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₂×D₇₃	—	2	4	no	—
139	C₂×D₇₃	—	4	2	no	—
140	C₂×D₇₃	—	4	2	no	—
141	C₂×D₇₃	#384	1	4	yes	—
142	C₄×C₇₃	—	2	4	no	—
143	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₇₃	C₂×#127	1	1	yes	characteristic
146	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	Dic₇₃	—	2	4	no	—
148	Dic₇₃	#384	1	4	yes	—
149	#20807	—	73	1	no	maximal, Sylow
150	C₂×C₇₃	—	2	1	no	—
151	C₂×C₇₃	—	2	1	no	—
152	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
153	C₂×C₇₃	#2916	1	2	yes	—
154	D₇₃	—	2	4	no	—
155	C₂×#430	—	73	2	no	—
156	C₂²×#108	—	73	2	no	—
157	#2905	—	73	2	no	—
158	C₂×#405	—	73	1	no	—
159	#1849	—	73	2	no	—
160	#1849	—	73	2	no	—
161	#1848	—	73	2	no	—
162	#1848	—	73	1	no	—
163	#1848	D₇₃	1	1	yes	characteristic, complemented
164	C₇₃	#20807	1	1	yes	characteristic, complemented, Sylow
165	C₂²×M₁₆	—	73	1	no	—
166	C₂²×M₁₆	—	73	4	no	—
167	C₂³×C₈	—	73	1	no	—
168	C₂×#117	—	73	1	no	—
169	#430	—	146	8	no	—
170	C₂×#108	—	73	4	no	—
171	C₂×#108	—	73	4	no	—
172	C₂×#108	—	73	2	no	—
173	C₂×#108	—	73	2	no	—
174	C₂×#108	—	146	8	no	—
175	C₂×#108	C₂×D₇₃	1	2	yes	—
176	C₂×#108	C₂×D₇₃	1	2	yes	—
177	#424	—	146	8	no	—
178	#423	—	73	4	no	—
179	#423	—	73	2	no	—
180	#423	C₂×D₇₃	1	2	yes	—
181	#405	—	73	8	no	—
182	C₂×#97	—	73	1	no	—
183	C₂×#97	—	73	1	no	—
184	C₂×C₄×C₈	—	73	1	no	—
185	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
186	C₂³×C₄	—	73	1	no	—
187	C₂×M₁₆	—	73	4	no	—
188	C₂×M₁₆	—	73	4	no	—
189	C₂×M₁₆	—	73	4	no	—
190	C₂×M₁₆	—	73	4	no	—
191	C₂×M₁₆	—	146	4	no	—
192	C₂×M₁₆	—	146	8	no	—
193	C₂×M₁₆	—	146	8	no	—
194	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
195	C₂²×C₈	—	73	1	no	—
196	C₂²×C₈	—	73	1	no	—
197	C₂²×C₈	—	73	4	no	—
198	C₂²×C₈	—	73	4	no	—
199	C₂²×C₈	—	73	4	no	—
200	C₂²×C₈	—	146	4	no	—
201	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
203	#117	—	73	4	no	—
204	#117	—	73	4	no	—
205	C₂×C₄⋊C₄	—	73	2	no	—
206	C₂×C₂²⋊C₄	—	73	2	no	—
207	C₂×C₄²	—	73	1	no	—
208	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
209	#108	—	2	4	no	—
210	#108	—	2	4	no	—
211	#108	—	146	8	no	—
212	#108	—	146	8	no	—
213	#108	—	146	4	no	—
214	#108	—	146	4	no	—
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	—	146	8	no	—
217	C₄⋊C₈	—	146	4	no	—
218	#97	—	73	4	no	—
219	#97	—	73	4	no	—
220	C₄×C₈	—	73	4	no	—
221	C₄×C₈	C₄×D₇₃	1	4	yes	—
222	C₂⁴	—	73	1	no	—
223	C₂²×C₄	—	73	2	no	—
224	C₂²×C₄	—	73	1	no	—
225	C₂²×C₄	—	73	2	no	—
226	C₂²×C₄	—	73	2	no	—
227	C₂²×C₄	—	73	2	no	—
228	C₂²×C₄	—	73	2	no	—
229	C₂²×C₄	—	73	2	no	—
230	C₂²×C₄	—	73	2	no	—
231	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
232	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	292	8	no	—
236	M₁₆	—	73	16	no	—
237	M₁₆	—	146	4	no	—
238	M₁₆	—	146	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	292	8	no	—
241	C₂×C₈	—	73	2	no	—
242	C₂×C₈	—	73	2	no	—
243	C₂×C₈	—	73	2	no	—
244	C₂×C₈	—	73	2	no	—
245	C₂×C₈	—	73	2	no	—
246	C₂×C₈	—	73	2	no	—
247	C₂×C₈	—	73	16	no	—
248	C₂×C₈	—	146	4	no	—
249	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
251	C₂×C₈	D₄×D₇₃	1	2	yes	—
252	C₂×C₈	D₄×D₇₃	1	2	yes	—
253	C₂×C₈	Q₈×D₇₃	1	2	yes	—
254	C₂×C₈	Q₈×D₇₃	1	2	yes	—
255	C₄⋊C₄	—	73	8	no	—
256	C₂²⋊C₄	—	73	8	no	—
257	C₄²	—	73	2	no	—
258	C₄²	—	73	2	no	—
259	C₄²	C₂×C₄×D₇₃	1	2	yes	—
260	C₄²	D₄×D₇₃	1	2	yes	—
261	C₂³	—	292	2	no	—
262	C₂³	—	73	2	no	—
263	C₂³	—	73	2	no	—
264	C₂³	—	73	2	no	—
265	C₂³	C₂×Q₈×D₇₃	1	1	yes	characteristic
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	292	2	no	—
268	C₂×C₄	—	73	2	no	—
269	C₂×C₄	—	73	2	no	—
270	C₂×C₄	—	73	2	no	—
271	C₂×C₄	—	73	4	no	—
272	C₂×C₄	—	73	4	no	—
273	C₂×C₄	—	73	4	no	—
274	C₂×C₄	—	73	4	no	—
275	C₂×C₄	—	73	4	no	—
276	C₂×C₄	C₄⋊C₄×D₇₃	1	4	yes	—
277	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
281	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
282	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
283	C₈	—	4	4	no	—
284	C₈	—	292	4	no	—
285	C₈	—	73	4	no	—
286	C₈	—	73	4	no	—
287	C₈	C₄⋊C₄×D₇₃	1	4	yes	—
288	C₈	C₄⋊C₄×D₇₃	1	4	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	292	2	no	—
291	C₂²	—	292	2	no	—
292	C₂²	—	73	4	no	—
293	C₂²	—	146	4	no	—
294	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
295	C₂²	C₄⋊Q₈×D₇₃	1	1	yes	characteristic
296	C₂²	C₄⋊Q₈×D₇₃	1	1	yes	characteristic
297	C₄	—	2	4	no	—
298	C₄	—	73	4	no	—
299	C₄	—	146	4	no	—
300	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
301	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
302	C₄	#127×D₇₃	1	1	yes	characteristic
303	C₄	C₄⋊Q₈×D₇₃	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	146	4	no	—
307	C₂	#384×D₇₃	1	1	yes	characteristic
308	C₂	#22618539	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1848	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1848	D₇₃
hypocenter	C₇₃	#20807
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
5. #22610105	C₂	semidirect
6. #22610104	C₂	semidirect
7. #22610103	C₂	semidirect
8. #22610101	C₂	semidirect
9. C₇₃×#1848	C₂	semidirect
163. #1848	D₇₃	semidirect
164. C₇₃	#20807	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988634	2
C₂×#108×D₇₃	2
#22618493	2
C₂×#17988541	2
#22610105	2
#22610104	2
#22610103	2
#22610101	2
C₇₃×#1848	2
#20807	73