Subgroups — #33055131

Order 18688.

730 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988638	C₂	1	1	yes	maximal, characteristic
2	#423×D₇₃	C₂	1	2	yes	maximal
3	#22608704	C₂	1	8	yes	maximal, complemented
4	#22608690	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608689	C₂	1	2	yes	maximal, complemented
6	C₇₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₁₆×D₇₃	—	2	2	no	—
8	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
9	#17988638	C₄	1	4	yes	—
10	#17988638	C₂²	1	4	yes	—
11	#17988611	C₂²	1	2	yes	—
12	#17988610	—	2	2	no	—
13	#17988609	—	2	2	no	—
14	#17988609	C₂²	1	4	yes	—
15	C₂×#14556520	C₄	1	2	yes	—
16	C₄⋊C₈×D₇₃	—	2	2	no	—
17	C₄⋊C₈×D₇₃	C₂²	1	4	yes	—
18	C₂×#14556503	C₄	1	2	yes	—
19	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
20	#17988340	C₂²	1	4	yes	complemented
21	#17988339	C₂²	1	8	yes	complemented
22	#423×C₇₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#335×C₇₃	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₇₃	Q₈	1	1	yes	characteristic
28	C₂×#11924012	—	2	2	no	—
29	C₂×#11924011	—	2	4	no	—
30	M₁₆×D₇₃	—	4	8	no	—
31	C₂²×#10596831	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₇₃	—	2	4	no	—
33	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
36	C₄²×D₇₃	C₂×C₄	1	2	yes	—
37	C₄²×D₇₃	D₄	1	2	yes	—
38	C₄²×D₇₃	Q₈	1	2	yes	—
39	C₄²×D₇₃	C₂³	1	2	yes	—
40	C₄⋊#10596831	—	2	2	no	—
41	C₄⋊#10596831	C₂³	1	4	yes	—
42	#14556520	—	2	4	no	—
43	#14556520	—	2	4	no	—
44	#14556520	C₂×C₄	1	8	yes	—
45	#14556503	C₂×C₄	1	8	yes	—
46	C₄×#10596831	C₂×C₄	1	2	yes	—
47	C₄×#10596831	C₂³	1	2	yes	—
48	C₂×M₁₆×C₇₃	—	2	2	no	—
49	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₇₃	—	2	2	no	—
52	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
53	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₇₃	C₂³	1	2	yes	—
55	C₂³×D₇₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×D₇₃	—	4	2	no	—
60	C₂×C₄×D₇₃	—	4	2	no	—
61	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×D₇₃	D₈	1	4	yes	—
65	C₂×C₄×D₇₃	Q₁₆	1	4	yes	—
66	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
69	#11924012	—	4	4	no	—
70	#11924011	—	4	8	no	—
71	C₂×#10596831	—	2	4	no	—
72	C₂×#10596831	—	4	2	no	—
73	C₂×#10596831	C₂²×C₄	1	4	yes	—
74	C₈×D₇₃	—	4	8	no	—
75	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
76	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
77	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
78	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
79	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
80	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
84	M₁₆×C₇₃	—	4	4	no	—
85	C₂×C₈×C₇₃	—	2	4	no	—
86	C₂×C₈×C₇₃	—	4	2	no	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
88	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
92	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×D₇₃	—	4	1	no	—
94	C₂²×D₇₃	—	4	1	no	—
95	C₂²×D₇₃	#107	1	4	yes	—
96	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂×Dic₇₃	—	4	1	no	—
108	C₂×Dic₇₃	—	4	1	no	—
109	C₂×Dic₇₃	#107	1	4	yes	—
110	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₄×D₇₃	—	2	4	no	—
114	C₄×D₇₃	—	2	4	no	—
115	C₄×D₇₃	—	4	2	no	—
116	C₄×D₇₃	—	4	2	no	—
117	C₄×D₇₃	#107	1	8	yes	—
118	C₄×D₇₃	C₂×D₈	1	4	yes	—
119	C₄×D₇₃	C₂×Q₁₆	1	4	yes	—
120	C₈×C₇₃	—	2	4	no	—
121	C₈×C₇₃	—	4	4	no	—
122	#10596831	—	2	4	no	—
123	#10596831	—	4	4	no	—
124	C₂²×C₇₃	—	2	1	no	—
125	C₂²×C₇₃	—	2	1	no	—
126	C₂²×C₇₃	C₂×#107	1	2	yes	—
127	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂×D₇₃	—	2	2	no	—
129	C₂×D₇₃	—	2	2	no	—
130	C₂×D₇₃	—	4	4	no	—
131	C₂×D₇₃	C₂×#107	1	4	yes	—
132	C₄×C₇₃	—	2	1	no	—
133	C₄×C₇₃	—	2	1	no	—
134	C₄×C₇₃	C₂×#107	1	2	yes	—
135	C₄×C₇₃	#430	1	2	yes	—
136	C₄×C₇₃	#568	1	2	yes	—
137	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
138	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
139	Dic₇₃	—	2	2	no	—
140	Dic₇₃	—	2	2	no	—
141	Dic₇₃	C₂×#107	1	4	yes	—
142	#17814	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	2	2	no	—
144	C₂×C₇₃	#2905	1	1	yes	characteristic
145	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
146	C₂×C₇₃	C₂×#430	1	1	yes	characteristic
147	D₇₃	—	2	4	no	—
148	C₂×#431	—	73	1	no	—
149	C₂×#423	—	73	2	no	—
150	#1571	—	73	8	no	—
151	#1567	—	73	2	no	—
152	#1567	—	73	1	no	—
153	#1567	D₇₃	1	1	yes	characteristic, complemented
154	C₇₃	#17814	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	146	2	no	—
156	C₂²×C₄²	—	73	1	no	—
157	#431	—	73	4	no	—
158	#431	—	73	4	no	—
159	#423	—	73	2	no	—
160	#423	—	73	4	no	—
161	#423	—	146	2	no	—
162	#423	—	146	2	no	—
163	#423	C₂×D₇₃	1	2	yes	—
164	C₂×C₄⋊C₈	—	73	2	no	—
165	C₂×C₄⋊C₈	—	73	4	no	—
166	C₂×C₄⋊C₈	—	146	2	no	—
167	C₂×C₂²⋊C₈	—	73	2	no	—
168	C₂×C₄×C₈	—	73	1	no	—
169	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
170	#335	—	73	8	no	—
171	#335	—	73	4	no	—
172	#335	C₂×D₇₃	1	4	yes	complemented
173	C₂³×C₄	—	73	1	no	—
174	C₂³×C₄	—	73	1	no	—
175	C₂³×C₄	—	73	1	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	292	8	no	—
178	C₂×M₁₆	—	146	4	no	—
179	C₂×M₁₆	—	146	2	no	—
180	C₂²×C₈	—	73	2	no	—
181	C₂²×C₈	—	146	4	no	—
182	C₂²×C₈	C₄×D₇₃	1	2	yes	—
183	C₂×C₄²	—	73	1	no	—
184	C₂×C₄²	—	73	1	no	—
185	C₂×C₄²	—	73	1	no	—
186	C₂×C₄²	—	73	2	no	—
187	C₂×C₄²	—	73	2	no	—
188	C₂×C₄²	—	73	2	no	—
189	C₂×C₄²	—	73	2	no	—
190	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	73	8	no	—
193	C₄⋊C₈	—	73	4	no	—
194	C₄⋊C₈	—	146	4	no	—
195	C₄⋊C₈	—	146	4	no	—
196	C₄⋊C₈	—	146	2	no	—
197	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
198	C₂²⋊C₈	—	73	8	no	—
199	C₄×C₈	—	73	2	no	—
200	C₄×C₈	—	73	2	no	—
201	C₄×C₈	C₄×D₇₃	1	2	yes	—
202	C₄×C₈	C₂²×D₇₃	1	2	yes	—
203	C₂⁴	—	73	1	no	—
204	C₂²×C₄	—	292	2	no	—
205	C₂²×C₄	—	292	2	no	—
206	C₂²×C₄	—	73	1	no	—
207	C₂²×C₄	—	73	1	no	—
208	C₂²×C₄	—	73	2	no	—
209	C₂²×C₄	—	73	2	no	—
210	C₂²×C₄	—	73	1	no	—
211	C₂²×C₄	—	73	2	no	—
212	C₂²×C₄	—	73	4	no	—
213	C₂²×C₄	—	73	4	no	—
214	C₂²×C₄	—	73	2	no	—
215	C₂²×C₄	—	73	2	no	—
216	C₂²×C₄	—	73	2	no	—
217	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
219	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
220	M₁₆	—	4	4	no	—
221	M₁₆	—	292	8	no	—
222	M₁₆	—	292	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	292	8	no	—
226	C₂×C₈	—	292	2	no	—
227	C₂×C₈	—	73	4	no	—
228	C₂×C₈	—	146	4	no	—
229	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
230	C₄²	—	73	2	no	—
231	C₄²	—	73	2	no	—
232	C₄²	—	73	2	no	—
233	C₄²	—	73	2	no	—
234	C₄²	—	73	2	no	—
235	C₄²	—	73	2	no	—
236	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
237	C₄²	D₄×D₇₃	1	1	yes	characteristic
238	C₄²	Q₈×D₇₃	1	1	yes	characteristic
239	C₄²	C₂³×D₇₃	1	1	yes	characteristic
240	C₂³	—	292	1	no	—
241	C₂³	—	292	1	no	—
242	C₂³	—	73	4	no	—
243	C₂³	—	73	2	no	—
244	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	292	2	no	—
248	C₂×C₄	—	292	2	no	—
249	C₂×C₄	—	292	1	no	—
250	C₂×C₄	—	292	1	no	—
251	C₂×C₄	—	73	4	no	—
252	C₂×C₄	—	73	2	no	—
253	C₂×C₄	—	73	8	no	—
254	C₂×C₄	—	73	4	no	—
255	C₂×C₄	—	73	4	no	—
256	C₂×C₄	—	73	2	no	—
257	C₂×C₄	—	73	2	no	—
258	C₂×C₄	—	146	4	no	—
259	C₂×C₄	—	146	4	no	—
260	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
261	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
262	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
263	C₂×C₄	D₈×D₇₃	1	2	yes	—
264	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
265	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
267	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	4	no	—
270	C₈	—	292	4	no	—
271	C₈	—	146	4	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	292	4	no	—
275	C₂²	—	73	4	no	—
276	C₂²	—	146	2	no	—
277	C₂²	—	146	2	no	—
278	C₂²	#107×D₇₃	1	2	yes	—
279	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	73	4	no	—
283	C₄	—	146	2	no	—
284	C₄	—	146	2	no	—
285	C₄	#107×D₇₃	1	2	yes	—
286	C₄	#17988634	1	2	yes	—
287	C₄	#17989353	1	2	yes	—
288	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	146	4	no	—
292	C₂	#22618493	1	1	yes	characteristic
293	C₂	C₂×#107×D₇₃	1	1	yes	characteristic
294	C₂	C₂×#17988634	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1567	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1567	D₇₃
hypocenter	C₇₃	#17814
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #22608704	C₂	semidirect
4. #22608690	C₂	semidirect
5. #22608689	C₂	semidirect
6. C₇₃×#1567	C₂	semidirect
20. #17988340	C₂²	semidirect
21. #17988339	C₂²	semidirect
24. #335×C₇₃	C₂²	semidirect
153. #1567	D₇₃	semidirect
154. C₇₃	#17814	semidirect
172. #335	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988638	2
#423×D₇₃	2
#22608704	2
#22608690	2
#22608689	2
C₇₃×#1567	2
#17814	73