Subgroups — C₇₃×#20718 #32930122

Order 18688.

1378 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₇₃	C₂	1	2	yes	maximal, complemented
3	C₇₃×#1818	C₂	1	2	yes	maximal, complemented
4	C₇₃×#1798	C₂	1	4	yes	maximal, complemented
5	C₇₃×#1780	C₂	1	2	yes	maximal
6	C₄×C₂²⋊C₈×C₇₃	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₇₃	C₂²	1	1	yes	characteristic
8	C₄×#41×C₇₃	—	2	8	no	—
9	C₂×C₄×Q₈×C₇₃	C₄	1	2	yes	—
10	C₂×C₄×D₄×C₇₃	C₄	1	4	yes	—
11	C₂×C₄×D₄×C₇₃	C₄	1	2	yes	—
12	C₂×#117×C₇₃	C₂²	1	2	yes	—
13	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₇₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	#431×C₇₃	—	2	8	no	—
17	C₂×C₄⋊C₈×C₇₃	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₈×C₇₃	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₈×C₇₃	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₇₃	C₂²	1	4	yes	—
21	#336×C₇₃	C₂²	1	8	yes	—
22	C₂×#41×C₇₃	—	2	4	no	—
23	C₂×#41×C₇₃	—	2	8	no	—
24	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×D₄×C₇₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₇₃	C₂³	1	8	yes	—
32	C₄×Q₈×C₇₃	—	2	8	no	—
33	C₄×D₄×C₇₃	—	2	16	no	—
34	C₄×D₄×C₇₃	—	2	8	no	—
35	#117×C₇₃	—	2	4	no	—
36	#117×C₇₃	—	2	8	no	—
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₇₃	—	2	8	no	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₃	D₄	1	8	yes	—
45	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₇₃	—	2	8	no	—
49	C₂²⋊C₈×C₇₃	—	2	8	no	—
50	C₂²⋊C₈×C₇₃	—	2	16	no	—
51	C₄×C₈×C₇₃	—	2	8	no	—
52	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂⁴×C₇₃	C₂²×C₄	1	2	yes	—
54	#41×C₇₃	—	2	32	no	—
55	C₂×Q₈×C₇₃	—	2	2	no	—
56	C₂×Q₈×C₇₃	—	2	4	no	—
57	C₂×D₄×C₇₃	—	2	4	no	—
58	C₂×D₄×C₇₃	—	2	4	no	—
59	C₂×D₄×C₇₃	—	2	4	no	—
60	C₂×D₄×C₇₃	—	2	2	no	—
61	C₂×D₄×C₇₃	—	2	4	no	—
62	C₂²×C₄×C₇₃	—	2	8	no	—
63	C₂²×C₄×C₇₃	—	2	4	no	—
64	C₂²×C₄×C₇₃	—	2	4	no	—
65	C₂²×C₄×C₇₃	—	2	4	no	—
66	C₂²×C₄×C₇₃	—	2	8	no	—
67	C₂²×C₄×C₇₃	—	2	16	no	—
68	C₂²×C₄×C₇₃	C₂²⋊C₄	1	8	yes	—
69	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₇₃	#41	1	8	yes	—
76	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₇₃	—	2	8	no	—
78	C₂×C₈×C₇₃	—	2	16	no	—
79	C₄⋊C₄×C₇₃	—	2	8	no	—
80	C₄⋊C₄×C₇₃	—	2	4	no	—
81	C₂²⋊C₄×C₇₃	—	2	4	no	—
82	C₂²⋊C₄×C₇₃	—	2	8	no	—
83	C₄²×C₇₃	—	2	8	no	—
84	C₄²×C₇₃	—	2	4	no	—
85	C₄²×C₇₃	C₂²⋊C₄	1	8	yes	—
86	C₄²×C₇₃	C₂×D₄	1	4	yes	—
87	C₄²×C₇₃	C₂×D₄	1	4	yes	—
88	C₂³×C₇₃	—	2	2	no	—
89	C₂³×C₇₃	—	2	2	no	—
90	C₂³×C₇₃	—	2	2	no	—
91	C₂³×C₇₃	—	2	4	no	—
92	C₂³×C₇₃	—	2	8	no	—
93	C₂³×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂³×C₇₃	#117	1	4	yes	—
95	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
96	Q₈×C₇₃	—	2	8	no	—
97	D₄×C₇₃	—	2	8	no	—
98	D₄×C₇₃	—	2	16	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	2	2	no	—
102	C₂×C₄×C₇₃	—	2	4	no	—
103	C₂×C₄×C₇₃	—	2	8	no	—
104	C₂×C₄×C₇₃	—	2	16	no	—
105	C₂×C₄×C₇₃	—	2	16	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	—	2	8	no	—
108	C₂×C₄×C₇₃	—	4	16	no	—
109	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂×C₄×C₇₃	#117	1	4	yes	—
113	C₂×C₄×C₇₃	#131	1	8	yes	—
114	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
115	C₂×C₄×C₇₃	C₂×#41	1	4	yes	—
116	C₈×C₇₃	—	4	8	no	—
117	C₂²×C₇₃	—	2	4	no	—
118	C₂²×C₇₃	—	2	8	no	—
119	C₂²×C₇₃	—	2	2	no	—
120	C₂²×C₇₃	—	2	4	no	—
121	C₂²×C₇₃	—	4	4	no	—
122	C₂²×C₇₃	—	4	4	no	—
123	C₂²×C₇₃	—	4	4	no	—
124	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₇₃	C₂×#117	1	2	yes	—
126	C₂²×C₇₃	C₂×#131	1	4	yes	—
127	C₄×C₇₃	—	2	4	no	—
128	C₄×C₇₃	—	2	2	no	—
129	C₄×C₇₃	—	2	4	no	—
130	C₄×C₇₃	—	4	4	no	—
131	C₄×C₇₃	#408	1	8	yes	—
132	C₄×C₇₃	#525	1	4	yes	—
133	#20718	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₇₃	—	4	2	no	—
135	C₂×C₇₃	—	4	4	no	—
136	C₂×C₇₃	#2280	1	1	yes	characteristic
137	C₂×C₇₃	C₂×#408	1	2	yes	—
138	C₂×C₇₃	#2922	1	4	yes	—
139	C₂×C₄×#41	C₂×C₇₃	1	1	yes	characteristic
140	C₂×#431	C₂×C₇₃	1	2	yes	complemented
141	#1818	C₂×C₇₃	1	2	yes	complemented
142	#1798	C₂×C₇₃	1	4	yes	complemented
143	#1780	C₂×C₇₃	1	2	yes	—
144	C₄×C₂²⋊C₈	C₂×C₇₃	1	4	yes	complemented
145	C₇₃	#20718	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×#41	C₂²×C₇₃	1	1	yes	characteristic
147	C₄×#41	—	2	8	no	—
148	C₂×C₄×Q₈	C₄×C₇₃	1	2	yes	—
149	C₂×C₄×D₄	C₄×C₇₃	1	4	yes	—
150	C₂×C₄×D₄	C₄×C₇₃	1	2	yes	—
151	C₂×#117	C₂²×C₇₃	1	2	yes	—
152	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
153	C₂²×C₄²	C₂²×C₇₃	1	2	yes	—
154	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
155	#431	—	2	8	no	—
156	C₂×C₄⋊C₈	C₂²×C₇₃	1	4	yes	—
157	C₂×C₂²⋊C₈	C₂²×C₇₃	1	8	yes	—
158	C₂×C₂²⋊C₈	C₂²×C₇₃	1	4	yes	—
159	C₂×C₄×C₈	C₂²×C₇₃	1	4	yes	—
160	#336	C₂²×C₇₃	1	8	yes	—
161	C₂×#41	—	2	4	no	—
162	C₂×#41	—	2	8	no	—
163	C₂²×Q₈	C₂×C₄×C₇₃	1	1	yes	characteristic
164	C₂²×D₄	C₂×C₄×C₇₃	1	1	yes	characteristic
165	C₂²×D₄	C₂×C₄×C₇₃	1	2	yes	—
166	C₂³×C₄	C₂×C₄×C₇₃	1	4	yes	—
167	C₂³×C₄	C₂×C₄×C₇₃	1	2	yes	—
168	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
169	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
170	C₂²×C₈	C₂³×C₇₃	1	8	yes	—
171	C₄×Q₈	—	2	8	no	—
172	C₄×D₄	—	2	16	no	—
173	C₄×D₄	—	2	8	no	—
174	#117	—	2	4	no	—
175	#117	—	2	8	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	4	yes	—
178	C₂×C₂²⋊C₄	C₂×C₄×C₇₃	1	2	yes	—
179	C₂×C₂²⋊C₄	C₂×C₄×C₇₃	1	4	yes	—
180	C₂×C₄²	—	2	8	no	—
181	C₂×C₄²	C₂×C₄×C₇₃	1	4	yes	—
182	C₂×C₄²	C₂×C₄×C₇₃	1	2	yes	—
183	C₂×C₄²	D₄×C₇₃	1	8	yes	—
184	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
185	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
187	C₄⋊C₈	—	2	8	no	—
188	C₂²⋊C₈	—	2	8	no	—
189	C₂²⋊C₈	—	2	16	no	—
190	C₄×C₈	—	2	8	no	—
191	C₂⁴	C₂²×C₄×C₇₃	1	1	yes	characteristic
192	C₂⁴	C₂²×C₄×C₇₃	1	2	yes	—
193	#41	—	2	32	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	16	no	—
207	C₂²×C₄	C₂²⋊C₄×C₇₃	1	8	yes	—
208	C₂²×C₄	C₂²×C₄×C₇₃	1	4	yes	—
209	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
214	C₂²×C₄	#41×C₇₃	1	8	yes	—
215	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
216	C₂×C₈	—	2	8	no	—
217	C₂×C₈	—	2	16	no	—
218	C₄⋊C₄	—	2	8	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₂²⋊C₄	—	2	8	no	—
222	C₄²	—	2	8	no	—
223	C₄²	—	2	4	no	—
224	C₄²	C₂²⋊C₄×C₇₃	1	8	yes	—
225	C₄²	C₂×D₄×C₇₃	1	4	yes	—
226	C₄²	C₂×D₄×C₇₃	1	4	yes	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	4	no	—
231	C₂³	—	2	8	no	—
232	C₂³	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
233	C₂³	#117×C₇₃	1	4	yes	—
234	C₂³	C₂³×C₄×C₇₃	1	1	yes	characteristic
235	Q₈	—	2	8	no	—
236	D₄	—	2	8	no	—
237	D₄	—	2	16	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	16	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	4	16	no	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	4	yes	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	4	yes	—
251	C₂×C₄	#117×C₇₃	1	4	yes	—
252	C₂×C₄	#131×C₇₃	1	8	yes	—
253	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
254	C₂×C₄	C₂×#41×C₇₃	1	4	yes	—
255	C₈	—	4	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	C₂²×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
264	C₂²	C₂×#117×C₇₃	1	2	yes	—
265	C₂²	C₂×#131×C₇₃	1	4	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	4	no	—
268	C₄	—	2	4	no	—
269	C₄	—	4	4	no	—
270	C₄	#408×C₇₃	1	8	yes	—
271	C₄	#525×C₇₃	1	4	yes	—
272	C₂	—	4	2	no	—
273	C₂	—	4	4	no	—
274	C₂	C₇₃×#2280	1	1	yes	characteristic
275	C₂	C₂×#408×C₇₃	1	2	yes	—
276	C₂	C₇₃×#2922	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₇₃	C₂³
commutator	C₂³	C₂³×C₄×C₇₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂²×#41×C₇₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#431×C₇₃	C₂	semidirect
3. C₇₃×#1818	C₂	semidirect
4. C₇₃×#1798	C₂	semidirect
6. C₄×C₂²⋊C₈×C₇₃	C₂	semidirect
133. #20718	C₇₃	direct (N × H)
140. C₂×#431	C₂×C₇₃	semidirect
141. #1818	C₂×C₇₃	semidirect
142. #1798	C₂×C₇₃	semidirect
144. C₄×C₂²⋊C₈	C₂×C₇₃	semidirect
145. C₇₃	#20718	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₇₃	2
C₂×#431×C₇₃	2
C₇₃×#1818	2
C₇₃×#1798	2
C₇₃×#1780	2
C₄×C₂²⋊C₈×C₇₃	2
#20718	73

Subgroups — C73×#20718 #32930122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#20718 #32930122