Subgroups — C₇₃×#32806 #32942210

Order 18688.

980 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#2858	C₂	1	2	yes	maximal, complemented
2	C₇₃×#2054	C₂	1	4	yes	maximal, complemented
3	C₇₃×#2023	C₂	1	4	yes	maximal, complemented
4	C₇₃×#2011	C₂	1	4	yes	maximal, complemented
5	C₇₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₇₃	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
8	C₂×#127×C₇₃	C₂²	1	2	yes	—
9	C₂×#124×C₇₃	—	2	2	no	—
10	C₂×#124×C₇₃	—	2	2	no	—
11	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
12	C₂×#102×C₇₃	C₂²	1	8	yes	—
13	#390×C₇₃	—	2	2	no	—
14	#390×C₇₃	—	2	2	no	—
15	#390×C₇₃	C₂²	1	4	yes	—
16	#390×C₇₃	C₂²	1	4	yes	—
17	C₄³×C₇₃	C₂²	1	1	yes	characteristic
18	#340×C₇₃	C₂²	1	4	yes	—
19	#336×C₇₃	C₂²	1	4	yes	—
20	C₂²×Q₈×C₇₃	—	2	2	no	—
21	C₂²×D₄×C₇₃	—	2	2	no	—
22	C₂²×D₄×C₇₃	—	2	2	no	—
23	C₂²×D₄×C₇₃	C₂³	1	4	yes	—
24	C₂×D₈×C₇₃	—	4	8	no	—
25	C₂×D₈×C₇₃	—	4	4	no	—
26	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
27	C₄⋊Q₈×C₇₃	—	4	4	no	—
28	#127×C₇₃	—	4	4	no	—
29	#124×C₇₃	—	4	8	no	—
30	#124×C₇₃	—	4	8	no	—
31	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
33	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
37	C₂×C₄²×C₇₃	D₄	1	2	yes	—
38	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
42	C₄⋊C₈×C₇₃	—	4	2	no	—
43	#102×C₇₃	—	4	8	no	—
44	C₂⁴×C₇₃	—	2	4	no	—
45	C₂×Q₈×C₇₃	—	4	4	no	—
46	C₂×Q₈×C₇₃	—	4	4	no	—
47	C₂×Q₈×C₇₃	—	4	4	no	—
48	C₂×D₄×C₇₃	—	2	8	no	—
49	C₂×D₄×C₇₃	—	2	8	no	—
50	C₂×D₄×C₇₃	—	2	4	no	—
51	C₂×D₄×C₇₃	—	2	4	no	—
52	C₂×D₄×C₇₃	—	4	4	no	—
53	C₂×D₄×C₇₃	—	4	4	no	—
54	C₂×D₄×C₇₃	—	4	4	no	—
55	C₂×D₄×C₇₃	—	4	4	no	—
56	C₂×D₄×C₇₃	—	4	4	no	—
57	C₂×D₄×C₇₃	—	4	4	no	—
58	C₂²×C₄×C₇₃	—	2	4	no	—
59	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
64	D₈×C₇₃	—	4	16	no	—
65	C₂×C₈×C₇₃	—	2	8	no	—
66	C₂×C₈×C₇₃	—	2	4	no	—
67	C₄⋊C₄×C₇₃	—	2	8	no	—
68	C₄⋊C₄×C₇₃	—	4	4	no	—
69	C₂²⋊C₄×C₇₃	—	4	8	no	—
70	C₂²⋊C₄×C₇₃	—	4	8	no	—
71	C₂²⋊C₄×C₇₃	—	4	8	no	—
72	C₂²⋊C₄×C₇₃	—	4	8	no	—
73	C₄²×C₇₃	—	2	4	no	—
74	C₄²×C₇₃	—	2	1	no	—
75	C₄²×C₇₃	—	2	4	no	—
76	C₄²×C₇₃	—	2	1	no	—
77	C₄²×C₇₃	—	4	1	no	—
78	C₄²×C₇₃	D₈	1	4	yes	complemented
79	C₂³×C₇₃	—	4	4	no	—
80	C₂³×C₇₃	—	4	4	no	—
81	C₂³×C₇₃	—	4	4	no	—
82	C₂³×C₇₃	—	4	4	no	—
83	C₂³×C₇₃	—	4	4	no	—
84	C₂³×C₇₃	—	4	4	no	—
85	C₂³×C₇₃	—	4	4	no	—
86	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
87	Q₈×C₇₃	—	8	8	no	—
88	D₄×C₇₃	—	4	8	no	—
89	D₄×C₇₃	—	4	8	no	—
90	D₄×C₇₃	—	8	8	no	—
91	D₄×C₇₃	—	8	8	no	—
92	C₂×C₄×C₇₃	—	2	2	no	—
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	1	no	—
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	—	2	1	no	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	1	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	4	4	no	—
101	C₂×C₄×C₇₃	—	4	4	no	—
102	C₂×C₄×C₇₃	—	4	4	no	—
103	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₇₃	C₂×D₈	1	4	yes	—
105	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
106	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×C₄×C₇₃	H(5,𝔽₂)	1	4	yes	—
108	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₈×C₇₃	—	4	4	no	—
110	C₂²×C₇₃	—	8	4	no	—
111	C₂²×C₇₃	—	8	4	no	—
112	C₂²×C₇₃	—	8	4	no	—
113	C₂²×C₇₃	—	8	4	no	—
114	C₂²×C₇₃	—	8	4	no	—
115	C₂²×C₇₃	—	8	4	no	—
116	C₂²×C₇₃	—	8	4	no	—
117	C₂²×C₇₃	#535	1	2	yes	—
118	C₂²×C₇₃	#537	1	1	yes	characteristic
119	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
120	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
121	C₂²×C₇₃	#575	1	1	yes	characteristic
122	C₄×C₇₃	—	2	4	no	—
123	C₄×C₇₃	—	2	4	no	—
124	C₄×C₇₃	—	2	2	no	—
125	C₄×C₇₃	—	4	1	no	—
126	C₄×C₇₃	—	8	4	no	—
127	C₄×C₇₃	H(3,𝔽₄)	1	2	yes	—
128	C₄×C₇₃	#562	1	2	yes	—
129	#32806	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₇₃	—	16	4	no	—
131	C₂×C₇₃	#2819	1	1	yes	characteristic
132	C₂×C₇₃	#3205	1	2	yes	—
133	C₂×C₇₃	#3213	1	2	yes	—
134	C₂×C₇₃	#3245	1	1	yes	characteristic
135	C₂×C₇₃	#3253	1	1	yes	characteristic
136	#2858	C₂×C₇₃	1	2	yes	complemented
137	#2054	C₂×C₇₃	1	4	yes	complemented
138	#2023	C₂×C₇₃	1	4	yes	complemented
139	#2011	C₂×C₇₃	1	4	yes	complemented
140	#1831	C₂×C₇₃	1	1	yes	characteristic, complemented
141	C₇₃	#32806	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×D₈	C₂²×C₇₃	1	4	yes	—
143	C₂×C₄⋊Q₈	C₂²×C₇₃	1	2	yes	—
144	C₂×#127	C₂²×C₇₃	1	2	yes	—
145	C₂×#124	—	2	2	no	—
146	C₂×#124	—	2	2	no	—
147	C₂×C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
148	C₂×#102	C₂²×C₇₃	1	8	yes	—
149	#390	—	2	2	no	—
150	#390	—	2	2	no	—
151	#390	C₂²×C₇₃	1	4	yes	—
152	#390	C₂²×C₇₃	1	4	yes	—
153	C₄³	C₂²×C₇₃	1	1	yes	characteristic
154	#340	C₂²×C₇₃	1	4	yes	—
155	#336	C₂²×C₇₃	1	4	yes	—
156	C₂²×Q₈	—	2	2	no	—
157	C₂²×D₄	—	2	2	no	—
158	C₂²×D₄	—	2	2	no	—
159	C₂²×D₄	C₂³×C₇₃	1	4	yes	—
160	C₂×D₈	—	4	8	no	—
161	C₂×D₈	—	4	4	no	—
162	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
163	C₄⋊Q₈	—	4	4	no	—
164	#127	—	4	4	no	—
165	#124	—	4	8	no	—
166	#124	—	4	8	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
169	C₂×C₂²⋊C₄	—	2	4	no	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₂²⋊C₄	—	2	4	no	—
172	C₂×C₂²⋊C₄	—	2	4	no	—
173	C₂×C₄²	D₄×C₇₃	1	2	yes	—
174	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
177	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
178	C₄⋊C₈	—	4	2	no	—
179	#102	—	4	8	no	—
180	C₂⁴	—	2	4	no	—
181	C₂×Q₈	—	4	4	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×D₄	—	2	8	no	—
185	C₂×D₄	—	2	8	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	4	4	no	—
189	C₂×D₄	—	4	4	no	—
190	C₂×D₄	—	4	4	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂×D₄	—	4	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
199	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
200	D₈	—	4	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	4	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₂²⋊C₄	—	4	8	no	—
206	C₂²⋊C₄	—	4	8	no	—
207	C₂²⋊C₄	—	4	8	no	—
208	C₂²⋊C₄	—	4	8	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	D₈×C₇₃	1	4	yes	complemented
215	C₂³	—	4	4	no	—
216	C₂³	—	4	4	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	4	no	—
220	C₂³	—	4	4	no	—
221	C₂³	—	4	4	no	—
222	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
223	Q₈	—	8	8	no	—
224	D₄	—	4	8	no	—
225	D₄	—	4	8	no	—
226	D₄	—	8	8	no	—
227	D₄	—	8	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₂×D₈×C₇₃	1	2	yes	—
240	C₂×C₄	C₂×D₈×C₇₃	1	4	yes	—
241	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	4	yes	—
242	C₂×C₄	H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
243	C₂×C₄	H(5,𝔽₂)×C₇₃	1	4	yes	—
244	C₂×C₄	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
245	C₈	—	4	4	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	8	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	8	4	no	—
252	C₂²	—	8	4	no	—
253	C₂²	#535×C₇₃	1	2	yes	—
254	C₂²	#537×C₇₃	1	1	yes	characteristic
255	C₂²	C₂²×D₈×C₇₃	1	1	yes	characteristic
256	C₂²	C₂×AGL(1,ℤ/8)×C₇₃	1	2	yes	—
257	C₂²	#575×C₇₃	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	1	no	—
262	C₄	—	8	4	no	—
263	C₄	H(3,𝔽₄)×C₇₃	1	2	yes	—
264	C₄	#562×C₇₃	1	2	yes	—
265	C₂	—	16	4	no	—
266	C₂	C₇₃×#2819	1	1	yes	characteristic
267	C₂	C₇₃×#3205	1	2	yes	—
268	C₂	C₇₃×#3213	1	2	yes	—
269	C₂	C₇₃×#3245	1	1	yes	characteristic
270	C₂	C₇₃×#3253	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₄³×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₃×#2858	C₂	semidirect
2. C₇₃×#2054	C₂	semidirect
3. C₇₃×#2023	C₂	semidirect
4. C₇₃×#2011	C₂	semidirect
5. C₇₃×#1831	C₂	semidirect
78. C₄²×C₇₃	D₈	semidirect
129. #32806	C₇₃	direct (N × H)
136. #2858	C₂×C₇₃	semidirect
137. #2054	C₂×C₇₃	semidirect
138. #2023	C₂×C₇₃	semidirect
139. #2011	C₂×C₇₃	semidirect
140. #1831	C₂×C₇₃	semidirect
141. C₇₃	#32806	direct (N × H)
214. C₄²	D₈×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₇₃×#2858	2
C₇₃×#2054	2
C₇₃×#2023	2
C₇₃×#2011	2
C₇₃×#1831	2
#32806	73

Subgroups — C73×#32806 #32942210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#32806 #32942210