Subgroups — #33086347

Order 18688.

616 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₇₃	C₂	1	1	yes	maximal, characteristic
2	#22610078	C₂	1	2	yes	maximal, complemented
3	#22610054	C₂	1	2	yes	maximal, complemented
4	#22610032	C₂	1	2	yes	maximal
5	#22609981	C₂	1	1	yes	maximal, characteristic
6	#22609812	C₂	1	2	yes	maximal, complemented
7	#22609733	C₂	1	2	yes	maximal
8	#22609685	C₂	1	2	yes	maximal, complemented
9	C₇₃×#1812	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556598	C₂²	1	1	yes	characteristic
11	#41×Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×#14556527	C₂²	1	2	yes	—
13	#17988571	C₂²	1	2	yes	—
14	#17988545	C₄	1	2	yes	complemented
15	#17988545	C₂²	1	1	yes	characteristic
16	#17988543	C₂²	1	4	yes	—
17	#17988542	C₄	1	2	yes	complemented
18	#17988542	C₂²	1	1	yes	characteristic
19	M₁₆×Dic₇₃	C₄	1	2	yes	complemented
20	M₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₇₃	C₄	1	2	yes	complemented
22	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
23	#17988357	C₂²	1	4	yes	—
24	#17988346	C₂²	1	4	yes	—
25	#17988345	C₂²	1	2	yes	—
26	#17988343	C₂²	1	4	yes	—
27	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
28	#427×C₇₃	C₂²	1	2	yes	—
29	#410×C₇₃	C₂²	1	2	yes	—
30	H(3,ℤ/4)×C₇₃	C₂²	1	2	yes	—
31	#14556598	—	2	2	no	—
32	#14556598	—	2	2	no	—
33	C₂×#11924012	C₂×C₄	1	2	yes	—
34	C₂×#11924012	C₂³	1	4	yes	—
35	C₂×#11924012	C₂³	1	1	yes	characteristic
36	C₂²×#10596831	C₂×C₄	1	2	yes	—
37	C₂²×#10596831	C₂³	1	1	yes	characteristic
38	Q₈×Dic₇₃	—	2	1	no	—
39	D₄×Dic₇₃	—	2	2	no	—
40	D₄×Dic₇₃	—	2	1	no	—
41	#14556550	C₂×C₄	1	2	yes	—
42	#14556550	C₂³	1	1	yes	characteristic
43	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
45	#14556527	—	2	2	no	—
46	#14556527	—	2	2	no	—
47	#14556510	—	2	2	no	—
48	#14556508	—	2	2	no	—
49	#14556502	—	2	4	no	—
50	#14556501	C₂×C₄	1	2	yes	—
51	#14556501	C₂×C₄	1	2	yes	—
52	#14556501	C₂×C₄	1	2	yes	—
53	C₄×#10596831	—	2	4	no	—
54	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
61	C₂×#41×C₇₃	C₂³	1	1	yes	characteristic
62	#131×C₇₃	—	2	2	no	—
63	#131×C₇₃	—	2	2	no	—
64	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₇₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	#117×C₇₃	C₂³	1	4	yes	—
69	#107×C₇₃	—	2	2	no	—
70	#106×C₇₃	—	2	2	no	—
71	#99×C₇₃	—	2	4	no	—
72	C₂²×Dic₇₃	—	2	2	no	—
73	C₂²×Dic₇₃	—	2	1	no	—
74	#11924012	—	2	2	no	—
75	#11924012	—	2	4	no	—
76	#11924012	—	2	4	no	—
77	#11924012	—	2	4	no	—
78	C₂×#10596831	—	2	2	no	—
79	C₂×#10596831	—	2	2	no	—
80	C₂×#10596831	—	2	4	no	—
81	C₂×#10596831	C₂²×C₄	1	2	yes	—
82	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
83	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
84	C₂×#10596831	#41	1	2	yes	—
85	C₂×#10596831	#41	1	2	yes	—
86	C₄⋊Dic₇₃	—	2	2	no	—
87	C₄⋊Dic₇₃	—	2	1	no	—
88	C₂²⋊Dic₇₃	—	2	2	no	—
89	C₂²⋊Dic₇₃	—	2	1	no	—
90	C₄×Dic₇₃	—	2	2	no	—
91	C₄×Dic₇₃	—	2	1	no	—
92	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
95	#41×C₇₃	—	2	2	no	—
96	#41×C₇₃	—	2	2	no	—
97	C₂×Q₈×C₇₃	#41	1	1	yes	characteristic
98	C₂×D₄×C₇₃	#41	1	2	yes	—
99	C₂×D₄×C₇₃	#41	1	1	yes	characteristic
100	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
102	M₁₆×C₇₃	—	2	2	no	—
103	M₁₆×C₇₃	—	2	4	no	—
104	C₂×C₈×C₇₃	—	2	2	no	—
105	C₂×C₈×C₇₃	—	2	2	no	—
106	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	#41	1	2	yes	—
110	C₂×C₈×C₇₃	#41	1	2	yes	—
111	C₄⋊C₄×C₇₃	—	2	4	no	—
112	C₂²⋊C₄×C₇₃	—	2	4	no	—
113	C₄²×C₇₃	—	2	4	no	—
114	C₂³×C₇₃	#117	1	2	yes	—
115	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
116	Q₈×C₇₃	—	2	2	no	—
117	D₄×C₇₃	—	2	4	no	—
118	D₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	4	4	no	—
122	C₂×C₄×C₇₃	#117	1	2	yes	—
123	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₇₃	—	2	1	no	—
128	C₂×Dic₇₃	—	2	2	no	—
129	C₂×Dic₇₃	—	2	1	no	—
130	C₂×Dic₇₃	—	4	2	no	—
131	C₂×Dic₇₃	—	4	4	no	—
132	C₈×C₇₃	—	2	2	no	—
133	C₈×C₇₃	#117	1	1	yes	characteristic
134	C₈×C₇₃	#117	1	1	yes	characteristic
135	C₈×C₇₃	#117	1	2	yes	—
136	#10596831	—	2	2	no	—
137	#10596831	—	4	4	no	—
138	#10596831	#117	1	1	yes	characteristic
139	#10596831	#117	1	1	yes	characteristic
140	#10596831	#117	1	2	yes	—
141	C₂²×C₇₃	—	2	2	no	—
142	C₂²×C₇₃	—	4	1	no	—
143	C₂²×C₇₃	—	4	2	no	—
144	C₂²×C₇₃	C₂×#117	1	1	yes	characteristic
145	C₂²×C₇₃	#529	1	2	yes	—
146	C₄×C₇₃	—	2	2	no	—
147	C₄×C₇₃	—	8	4	no	—
148	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
149	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
150	C₄×C₇₃	#529	1	2	yes	—
151	Dic₇₃	—	2	2	no	—
152	Dic₇₃	—	4	2	no	—
153	Dic₇₃	—	4	1	no	—
154	#20787	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	1	no	—
156	C₂×C₇₃	—	2	2	no	—
157	C₂×C₇₃	—	4	2	no	—
158	C₂×C₇₃	#2283	1	1	yes	characteristic
159	C₄×#131	—	73	1	no	—
160	#1841	—	73	2	no	—
161	#1833	—	73	2	no	—
162	#1827	—	73	2	no	—
163	#1813	—	73	1	no	—
164	#1812	D₇₃	1	1	yes	characteristic
165	#1772	—	73	2	no	—
166	#1757	—	73	2	no	—
167	#1747	—	73	2	no	—
168	C₇₃	#20787	1	1	yes	characteristic, complemented, Sylow
169	C₂×#131	—	73	1	no	—
170	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
171	C₄×#41	—	73	1	no	—
172	C₂×#108	—	73	2	no	—
173	#427	C₂×D₇₃	1	2	yes	—
174	#413	—	73	2	no	—
175	#410	C₂×D₇₃	1	2	yes	—
176	#405	—	73	2	no	—
177	#405	—	73	1	no	—
178	#405	—	73	2	no	—
179	#405	—	73	1	no	—
180	#405	—	73	4	no	—
181	C₄×M₁₆	—	73	2	no	—
182	C₄×M₁₆	—	73	1	no	—
183	C₂×C₄×C₈	—	73	2	no	—
184	C₂×C₄×C₈	—	73	1	no	—
185	#344	—	73	4	no	—
186	#339	—	73	4	no	—
187	#338	—	73	2	no	—
188	H(3,ℤ/4)	—	73	4	no	—
189	H(3,ℤ/4)	C₂×D₇₃	1	2	yes	—
190	C₂×#41	C₂²×D₇₃	1	1	yes	characteristic
191	#131	—	2	2	no	—
192	#131	—	2	2	no	—
193	#131	—	146	2	no	—
194	#131	—	146	2	no	—
195	C₂×M₁₆	—	73	2	no	—
196	C₂×M₁₆	—	73	1	no	—
197	C₂×M₁₆	—	73	4	no	—
198	C₂×M₁₆	C₄×D₇₃	1	2	yes	—
199	C₂×M₁₆	C₂²×D₇₃	1	1	yes	characteristic
200	C₂²×C₈	—	73	1	no	—
201	C₂²×C₈	—	73	2	no	—
202	C₂²×C₈	C₄×D₇₃	1	2	yes	—
203	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
204	C₄×Q₈	—	146	1	no	—
205	C₄×D₄	—	146	2	no	—
206	C₄×D₄	—	146	1	no	—
207	#117	—	73	2	no	—
208	#117	—	73	1	no	—
209	#117	C₂²×D₇₃	1	4	yes	—
210	C₂×C₄²	—	73	2	no	—
211	C₂×C₄²	—	73	1	no	—
212	#108	—	146	2	no	—
213	#108	—	146	2	no	—
214	#107	—	2	2	no	—
215	#106	—	2	2	no	—
216	#101	—	146	2	no	—
217	#100	—	146	2	no	—
218	#99	—	2	4	no	—
219	#97	—	73	2	no	—
220	#97	—	73	2	no	—
221	#97	—	73	2	no	—
222	#97	—	146	4	no	—
223	C₄×C₈	—	73	1	no	—
224	C₄×C₈	—	73	1	no	—
225	C₄×C₈	—	73	2	no	—
226	C₄×C₈	—	73	2	no	—
227	C₄×C₈	—	73	1	no	—
228	C₄×C₈	—	73	1	no	—
229	C₄×C₈	—	73	2	no	—
230	C₄×C₈	—	146	4	no	—
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	C₂×Q₈	#11924029	1	1	yes	characteristic
234	C₂×D₄	#11924027	1	2	yes	—
235	C₂×D₄	#11924029	1	1	yes	characteristic
236	C₂²×C₄	—	146	2	no	—
237	C₂²×C₄	—	146	1	no	—
238	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
239	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	146	4	no	—
243	M₁₆	—	146	4	no	—
244	M₁₆	—	146	4	no	—
245	M₁₆	—	146	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	73	1	no	—
249	C₂×C₈	—	73	1	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	73	2	no	—
253	C₂×C₈	—	146	4	no	—
254	C₂×C₈	—	146	2	no	—
255	C₂×C₈	—	146	2	no	—
256	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
259	C₂×C₈	#11924027	1	2	yes	—
260	C₂×C₈	#11924027	1	2	yes	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	146	2	no	—
263	C₄⋊C₄	—	146	1	no	—
264	C₂²⋊C₄	—	2	4	no	—
265	C₂²⋊C₄	—	146	2	no	—
266	C₂²⋊C₄	—	146	1	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	73	1	no	—
269	C₄²	—	73	2	no	—
270	C₄²	—	73	1	no	—
271	C₄²	—	146	2	no	—
272	C₄²	—	146	1	no	—
273	C₂³	#14556549	1	2	yes	—
274	C₂³	C₂×#11924027	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	292	2	no	—
282	C₂×C₄	—	292	4	no	—
283	C₂×C₄	—	146	2	no	—
284	C₂×C₄	—	146	1	no	—
285	C₂×C₄	—	146	1	no	—
286	C₂×C₄	#14556549	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
288	C₂×C₄	#41×D₇₃	1	2	yes	—
289	C₂×C₄	C₂×#11924027	1	2	yes	—
290	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	292	4	no	—
293	C₈	—	73	1	no	—
294	C₈	—	73	1	no	—
295	C₈	—	73	2	no	—
296	C₈	—	146	2	no	—
297	C₈	#14556547	1	1	yes	characteristic
298	C₈	#14556547	1	1	yes	characteristic
299	C₈	#14556547	1	2	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×#14556549	1	1	yes	characteristic
304	C₂²	#17989118	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	8	4	no	—
307	C₄	—	292	1	no	—
308	C₄	—	292	2	no	—
309	C₄	—	146	2	no	—
310	C₄	#117×D₇₃	1	1	yes	characteristic
311	C₄	C₂×#14556547	1	1	yes	characteristic
312	C₄	#17989123	1	2	yes	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	—	4	2	no	—
316	C₂	#22612326	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1812	C₂
socle	C₂×C₇₃	#2283
prime radical	C₂×#41×C₇₃	C₂³
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1812	D₇₃
hypocenter	C₇₃	#20787
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1812

Complements

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

N	H ≅ G/N	type
2. #22610078	C₂	semidirect
3. #22610054	C₂	semidirect
6. #22609812	C₂	semidirect
8. #22609685	C₂	semidirect
14. #17988545	C₄	semidirect
17. #17988542	C₄	semidirect
19. M₁₆×Dic₇₃	C₄	semidirect
21. C₂×C₈×Dic₇₃	C₄	semidirect
168. C₇₃	#20787	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₇₃	2
#22610078	2
#22610054	2
#22610032	2
#22609981	2
#22609812	2
#22609733	2
#22609685	2
C₇₃×#1812	2
#20787	73