Subgroups — C₇₃×#33775 #32943179

Order 18688.

582 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	C₇₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×#494×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2247	C₂	1	2	yes	maximal, complemented
4	C₇₃×#2241	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2209	C₂	1	4	yes	maximal, complemented
6	C₇₃×#2188	C₂	1	4	yes	maximal
7	C₇₃×#2162	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₇₃	—	2	1	no	—
9	C₂²×Q₁₆×C₇₃	C₂²	1	2	yes	—
10	#533×C₇₃	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₇₃	—	2	1	no	—
12	C₂×Q₃₂×C₇₃	C₂²	1	8	yes	—
13	#500×C₇₃	C₂²	1	1	yes	characteristic
14	#494×C₇₃	—	2	1	no	—
15	#494×C₇₃	—	2	1	no	—
16	#494×C₇₃	C₂²	1	2	yes	—
17	#494×C₇₃	C₂²	1	2	yes	—
18	#494×C₇₃	C₂²	1	1	yes	characteristic
19	#487×C₇₃	—	2	1	no	—
20	#467×C₇₃	—	2	1	no	—
21	#467×C₇₃	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₇₃	—	2	1	no	—
23	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
26	#358×C₇₃	C₂²	1	8	yes	—
27	C₂²⋊C₁₆×C₇₃	C₂²	1	2	yes	—
28	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
29	C₂²×Q₈×C₇₃	—	2	2	no	—
30	C₂×Q₁₆×C₇₃	—	2	1	no	—
31	C₂×Q₁₆×C₇₃	—	2	2	no	—
32	C₂×Q₁₆×C₇₃	—	2	2	no	—
33	C₂×Q₁₆×C₇₃	—	2	2	no	—
34	C₂×Q₁₆×C₇₃	—	4	4	no	—
35	C₂×Q₁₆×C₇₃	—	4	4	no	—
36	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
37	C₂×Q₁₆×C₇₃	C₂³	1	4	yes	—
38	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₇₃	—	2	1	no	—
41	C₄⋊Q₈×C₇₃	—	2	1	no	—
42	C₄⋊Q₈×C₇₃	—	2	2	no	—
43	C₄⋊Q₈×C₇₃	—	2	2	no	—
44	C₄⋊Q₈×C₇₃	—	4	1	no	—
45	C₄⋊Q₈×C₇₃	—	4	1	no	—
46	#125×C₇₃	—	4	1	no	—
47	C₂²⋊Q₈×C₇₃	—	2	2	no	—
48	C₂²⋊Q₈×C₇₃	—	4	1	no	—
49	C₄×Q₈×C₇₃	—	2	2	no	—
50	C₄×Q₈×C₇₃	—	4	1	no	—
51	#117×C₇₃	—	4	1	no	—
52	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
53	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
54	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
55	Q₃₂×C₇₃	—	4	8	no	—
56	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
57	#107×C₇₃	—	2	1	no	—
58	#107×C₇₃	C₂³	1	2	yes	—
59	#103×C₇₃	—	2	2	no	—
60	#103×C₇₃	—	2	2	no	—
61	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₇₃	—	2	2	no	—
66	C₂×Q₈×C₇₃	—	2	4	no	—
67	C₂×Q₈×C₇₃	—	4	4	no	—
68	C₂×Q₈×C₇₃	—	4	4	no	—
69	C₂×Q₈×C₇₃	—	4	1	no	—
70	C₂×Q₈×C₇₃	—	4	2	no	—
71	C₂²×C₄×C₇₃	—	4	2	no	—
72	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	Q₁₆×C₇₃	—	2	4	no	—
76	Q₁₆×C₇₃	—	2	8	no	—
77	Q₁₆×C₇₃	—	4	4	no	—
78	Q₁₆×C₇₃	—	4	4	no	—
79	Q₁₆×C₇₃	—	4	2	no	—
80	Q₁₆×C₇₃	—	4	4	no	—
81	C₂×C₈×C₇₃	—	2	2	no	—
82	C₂×C₈×C₇₃	—	2	2	no	—
83	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₇₃	—	2	2	no	—
88	C₄⋊C₄×C₇₃	—	4	2	no	—
89	C₄⋊C₄×C₇₃	—	4	2	no	—
90	C₄⋊C₄×C₇₃	—	4	2	no	—
91	C₄⋊C₄×C₇₃	—	4	2	no	—
92	C₄⋊C₄×C₇₃	—	4	1	no	—
93	C₄⋊C₄×C₇₃	—	4	1	no	—
94	C₄⋊C₄×C₇₃	—	4	2	no	—
95	C₄⋊C₄×C₇₃	—	4	2	no	—
96	C₄⋊C₄×C₇₃	—	4	1	no	—
97	C₂²⋊C₄×C₇₃	—	4	2	no	—
98	C₄²×C₇₃	—	4	2	no	—
99	C₄²×C₇₃	D₈	1	1	yes	characteristic
100	C₄²×C₇₃	D₈	1	1	yes	characteristic
101	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₁₆×C₇₃	—	2	4	no	—
104	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₇₃	—	4	4	no	—
106	Q₈×C₇₃	—	4	2	no	—
107	Q₈×C₇₃	—	4	4	no	—
108	Q₈×C₇₃	—	4	2	no	—
109	Q₈×C₇₃	—	8	4	no	—
110	Q₈×C₇₃	—	8	2	no	—
111	C₂×C₄×C₇₃	—	2	1	no	—
112	C₂×C₄×C₇₃	—	2	1	no	—
113	C₂×C₄×C₇₃	—	2	2	no	—
114	C₂×C₄×C₇₃	—	4	2	no	—
115	C₂×C₄×C₇₃	—	4	2	no	—
116	C₂×C₄×C₇₃	—	4	4	no	—
117	C₂×C₄×C₇₃	—	8	4	no	—
118	C₂×C₄×C₇₃	Q₃₂	1	4	yes	complemented
119	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₇₃	—	2	1	no	—
126	C₈×C₇₃	—	2	1	no	—
127	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₈×C₇₃	H(5,𝔽₂)	1	2	yes	—
129	C₂²×C₇₃	—	2	1	no	—
130	C₂²×C₇₃	—	2	1	no	—
131	C₂²×C₇₃	C₂×Q₃₂	1	2	yes	—
132	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₇₃	—	2	1	no	—
134	C₄×C₇₃	—	2	1	no	—
135	C₄×C₇₃	—	8	2	no	—
136	C₄×C₇₃	—	8	4	no	—
137	C₄×C₇₃	—	8	2	no	—
138	C₄×C₇₃	C₂×Q₃₂	1	2	yes	—
139	C₄×C₇₃	C₂×Q₃₂	1	2	yes	—
140	C₄×C₇₃	#510	1	2	yes	—
141	C₄×C₇₃	#535	1	1	yes	characteristic
142	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
143	#33775	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₇₃	—	2	2	no	—
145	C₂×C₇₃	#3205	1	1	yes	characteristic
146	C₂×C₇₃	C₂²×Q₃₂	1	1	yes	characteristic
147	C₂×C₇₃	C₂×#510	1	1	yes	characteristic
148	#3170	C₂×C₇₃	1	1	yes	characteristic
149	C₂×#494	C₂×C₇₃	1	1	yes	characteristic
150	#2247	C₂×C₇₃	1	2	yes	complemented
151	#2241	C₂×C₇₃	1	2	yes	complemented
152	#2209	C₂×C₇₃	1	4	yes	complemented
153	#2188	C₂×C₇₃	1	4	yes	—
154	#2162	C₂×C₇₃	1	1	yes	characteristic
155	C₇₃	#33775	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×Q₁₆	—	2	1	no	—
157	C₂²×Q₁₆	C₂²×C₇₃	1	2	yes	—
158	#533	—	2	1	no	—
159	C₂×C₄⋊Q₈	—	2	1	no	—
160	C₂×Q₃₂	C₂²×C₇₃	1	8	yes	—
161	#500	C₂²×C₇₃	1	1	yes	characteristic
162	#494	—	2	1	no	—
163	#494	—	2	1	no	—
164	#494	C₂²×C₇₃	1	2	yes	—
165	#494	C₂²×C₇₃	1	2	yes	—
166	#494	C₂²×C₇₃	1	1	yes	characteristic
167	#487	—	2	1	no	—
168	#467	—	2	1	no	—
169	#467	C₂²×C₇₃	1	2	yes	—
170	C₄×Q₁₆	—	2	1	no	—
171	C₄×Q₁₆	C₂²×C₇₃	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
173	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
174	#358	C₂²×C₇₃	1	8	yes	—
175	C₂²⋊C₁₆	C₂²×C₇₃	1	2	yes	—
176	C₄×C₁₆	C₂²×C₇₃	1	2	yes	—
177	C₂²×Q₈	—	2	2	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	2	1	no	—
182	C₂×Q₁₆	—	4	4	no	—
183	C₂×Q₁₆	—	4	4	no	—
184	C₂×Q₁₆	C₂³×C₇₃	1	4	yes	—
185	C₂×Q₁₆	C₂³×C₇₃	1	2	yes	—
186	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	1	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	1	no	—
192	C₄⋊Q₈	—	4	1	no	—
193	C₄⋊Q₈	—	4	1	no	—
194	#125	—	4	1	no	—
195	C₂²⋊Q₈	—	2	2	no	—
196	C₂²⋊Q₈	—	4	1	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₄×Q₈	—	4	1	no	—
199	#117	—	4	1	no	—
200	C₂×C₄⋊C₄	—	4	1	no	—
201	C₂×C₄⋊C₄	—	4	1	no	—
202	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
203	Q₃₂	—	4	8	no	—
204	C₂×C₁₆	C₂³×C₇₃	1	4	yes	—
205	#107	—	2	1	no	—
206	#107	C₂³×C₇₃	1	2	yes	—
207	#103	—	2	2	no	—
208	#103	—	2	2	no	—
209	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
210	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	4	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	D₈×C₇₃	1	1	yes	characteristic
221	C₂²×C₄	D₈×C₇₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
223	Q₁₆	—	2	4	no	—
224	Q₁₆	—	2	8	no	—
225	Q₁₆	—	4	4	no	—
226	Q₁₆	—	4	4	no	—
227	Q₁₆	—	4	2	no	—
228	Q₁₆	—	4	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
232	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
233	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
234	C₂×C₈	C₂⁴×C₇₃	1	1	yes	characteristic
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₄²	—	4	2	no	—
247	C₄²	D₈×C₇₃	1	1	yes	characteristic
248	C₄²	D₈×C₇₃	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
251	C₁₆	—	2	4	no	—
252	C₂³	C₂×D₈×C₇₃	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	2	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	4	2	no	—
257	Q₈	—	8	4	no	—
258	Q₈	—	8	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	Q₃₂×C₇₃	1	4	yes	complemented
267	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
268	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
269	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
270	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
271	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	AGL(1,ℤ/8)×C₇₃	1	2	yes	—
276	C₈	H(5,𝔽₂)×C₇₃	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₂×Q₃₂×C₇₃	1	2	yes	—
280	C₂²	C₂²×D₈×C₇₃	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	—	8	2	no	—
286	C₄	C₂×Q₃₂×C₇₃	1	2	yes	—
287	C₄	C₂×Q₃₂×C₇₃	1	2	yes	—
288	C₄	#510×C₇₃	1	2	yes	—
289	C₄	#535×C₇₃	1	1	yes	characteristic
290	C₄	C₂×AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	C₇₃×#3205	1	1	yes	characteristic
293	C₂	C₂²×Q₃₂×C₇₃	1	1	yes	characteristic
294	C₂	C₂×#510×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₃
prime radical	C₂³×C₇₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₃×#2247	C₂	semidirect
4. C₇₃×#2241	C₂	semidirect
5. C₇₃×#2209	C₂	semidirect
118. C₂×C₄×C₇₃	Q₃₂	semidirect
143. #33775	C₇₃	direct (N × H)
150. #2247	C₂×C₇₃	semidirect
151. #2241	C₂×C₇₃	semidirect
152. #2209	C₂×C₇₃	semidirect
155. C₇₃	#33775	direct (N × H)
266. C₂×C₄	Q₃₂×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₇₃×#3170	2
C₂×#494×C₇₃	2
C₇₃×#2247	2
C₇₃×#2241	2
C₇₃×#2209	2
C₇₃×#2188	2
C₇₃×#2162	2
#33775	73

Subgroups — C73×#33775 #32943179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#33775 #32943179