Subgroups — C₇₃×#23474 #32932878

Order 18688.

1164 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#410×C₇₃	C₂	1	3	yes	maximal
3	C₇₃×#2032	C₂	1	8	yes	maximal, complemented
4	C₇₃×#1740	C₂	1	3	yes	maximal, complemented
5	C₂²×#41×C₇₃	C₂²	1	1	yes	characteristic
6	#544×C₇₃	—	2	6	no	—
7	#543×C₇₃	C₂²	1	4	yes	—
8	#533×C₇₃	—	2	6	no	—
9	C₂×#117×C₇₃	C₂²	1	3	yes	—
10	C₂×#117×C₇₃	C₂²	1	3	yes	—
11	#410×C₇₃	—	2	6	no	—
12	#410×C₇₃	—	2	6	no	—
13	C₂×#99×C₇₃	C₂²	1	12	yes	—
14	#342×C₇₃	C₂²	1	12	yes	—
15	C₂×#41×C₇₃	—	2	3	no	—
16	C₂×#41×C₇₃	—	2	3	no	—
17	C₂×#41×C₇₃	—	2	3	no	—
18	C₂×#41×C₇₃	—	2	3	no	—
19	C₂²×Q₈×C₇₃	Q₈	1	1	yes	characteristic, complemented
20	C₂²×D₄×C₇₃	Q₈	1	1	yes	characteristic, complemented
21	C₂²×D₄×C₇₃	C₂³	1	2	yes	—
22	C₂³×C₄×C₇₃	Q₈	1	2	yes	complemented
23	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈×C₇₃	—	2	12	no	—
25	#125×C₇₃	—	2	12	no	—
26	C₂²⋊Q₈×C₇₃	—	2	24	no	—
27	C₄×Q₈×C₇₃	—	2	8	no	—
28	#117×C₇₃	—	2	3	no	—
29	#117×C₇₃	—	2	3	no	—
30	#117×C₇₃	—	2	3	no	—
31	#117×C₇₃	—	2	3	no	—
32	#117×C₇₃	—	2	3	no	—
33	#117×C₇₃	—	2	3	no	—
34	#117×C₇₃	D₄	1	6	yes	—
35	#117×C₇₃	D₄	1	6	yes	—
36	C₂×C₄⋊C₄×C₇₃	—	2	3	no	—
37	C₂×C₄⋊C₄×C₇₃	—	2	3	no	—
38	C₂×C₂²⋊C₄×C₇₃	C₂³	1	6	yes	—
39	C₂×C₂²⋊C₄×C₇₃	C₂³	1	6	yes	—
40	C₂×C₄²×C₇₃	—	2	3	no	—
41	C₂×C₄²×C₇₃	—	2	3	no	—
42	#99×C₇₃	—	2	24	no	—
43	C₂⁴×C₇₃	C₂×Q₈	1	2	yes	—
44	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
45	#41×C₇₃	—	4	3	no	—
46	#41×C₇₃	—	4	1	no	—
47	#41×C₇₃	—	4	3	no	—
48	#41×C₇₃	—	4	3	no	—
49	#41×C₇₃	—	4	3	no	—
50	#41×C₇₃	—	4	3	no	—
51	C₂×Q₈×C₇₃	—	2	8	no	—
52	C₂×Q₈×C₇₃	—	2	3	no	—
53	C₂×Q₈×C₇₃	—	2	3	no	—
54	C₂×D₄×C₇₃	—	2	6	no	—
55	C₂×D₄×C₇₃	—	2	6	no	—
56	C₂×D₄×C₇₃	—	2	3	no	—
57	C₂×D₄×C₇₃	—	2	3	no	—
58	C₂²×C₄×C₇₃	—	2	6	no	—
59	C₂²×C₄×C₇₃	—	2	6	no	—
60	C₂²×C₄×C₇₃	—	2	6	no	—
61	C₂²×C₄×C₇₃	—	2	6	no	—
62	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	#41	1	3	yes	—
68	C₂²×C₄×C₇₃	#41	1	3	yes	—
69	C₄⋊C₄×C₇₃	—	2	24	no	—
70	C₄⋊C₄×C₇₃	—	2	6	no	—
71	C₄⋊C₄×C₇₃	—	2	6	no	—
72	C₄⋊C₄×C₇₃	—	4	3	no	—
73	C₄⋊C₄×C₇₃	—	4	3	no	—
74	C₂²⋊C₄×C₇₃	—	2	6	no	—
75	C₂²⋊C₄×C₇₃	—	2	6	no	—
76	C₂²⋊C₄×C₇₃	—	2	6	no	—
77	C₂²⋊C₄×C₇₃	C₂×D₄	1	12	yes	—
78	C₄²×C₇₃	—	2	6	no	—
79	C₄²×C₇₃	—	2	6	no	—
80	C₄²×C₇₃	—	4	3	no	—
81	C₄²×C₇₃	—	4	3	no	—
82	C₂³×C₇₃	—	2	6	no	—
83	C₂³×C₇₃	—	8	1	no	—
84	C₂³×C₇₃	—	8	2	no	—
85	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
86	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
87	C₂³×C₇₃	C₂×#41	1	3	yes	—
88	Q₈×C₇₃	—	4	3	no	—
89	Q₈×C₇₃	—	4	1	no	—
90	Q₈×C₇₃	—	8	8	no	—
91	D₄×C₇₃	—	4	2	no	—
92	D₄×C₇₃	—	4	6	no	—
93	D₄×C₇₃	—	4	3	no	—
94	D₄×C₇₃	—	4	1	no	—
95	C₂×C₄×C₇₃	—	2	12	no	—
96	C₂×C₄×C₇₃	—	2	6	no	—
97	C₂×C₄×C₇₃	—	4	6	no	—
98	C₂×C₄×C₇₃	—	4	2	no	—
99	C₂×C₄×C₇₃	—	4	6	no	—
100	C₂×C₄×C₇₃	—	8	6	no	—
101	C₂×C₄×C₇₃	—	8	6	no	—
102	C₂×C₄×C₇₃	#120	1	3	yes	—
103	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂²⋊Q₈	1	3	yes	—
105	C₂×C₄×C₇₃	C₂²⋊Q₈	1	3	yes	—
106	C₂×C₄×C₇₃	C₂²⋊Q₈	1	3	yes	—
107	C₂×C₄×C₇₃	C₂²⋊Q₈	1	3	yes	—
108	C₂×C₄×C₇₃	#124	1	3	yes	—
109	C₂×C₄×C₇₃	#124	1	3	yes	—
110	C₂×C₄×C₇₃	#124	1	3	yes	—
111	C₂×C₄×C₇₃	#124	1	3	yes	—
112	C₂²×C₇₃	—	4	3	no	—
113	C₂²×C₇₃	—	4	3	no	—
114	C₂²×C₇₃	—	4	1	no	—
115	C₂²×C₇₃	—	4	2	no	—
116	C₂²×C₇₃	—	8	6	no	—
117	C₂²×C₇₃	C₂×#120	1	1	yes	characteristic
118	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	3	yes	—
119	C₂²×C₇₃	C₂×#124	1	3	yes	—
120	C₄×C₇₃	—	4	12	no	—
121	C₄×C₇₃	—	4	2	no	—
122	C₄×C₇₃	—	8	6	no	—
123	C₄×C₇₃	#393	1	3	yes	—
124	C₄×C₇₃	#393	1	3	yes	—
125	C₄×C₇₃	#393	1	1	yes	characteristic
126	C₄×C₇₃	#393	1	1	yes	characteristic
127	#23474	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₇₃	—	2	1	no	—
129	C₂×C₇₃	—	2	3	no	—
130	C₂×C₇₃	—	2	3	no	—
131	C₂×C₇₃	—	8	2	no	—
132	C₂×C₇₃	C₂×#393	1	1	yes	characteristic
133	#3458	C₂×C₇₃	1	1	yes	characteristic, complemented
134	C₂×#410	C₂×C₇₃	1	3	yes	—
135	#2032	C₂×C₇₃	1	8	yes	complemented
136	#1740	C₂×C₇₃	1	3	yes	complemented
137	C₇₃	#23474	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×#41	C₂²×C₇₃	1	1	yes	characteristic
139	#544	—	2	6	no	—
140	#543	C₂²×C₇₃	1	4	yes	—
141	#533	—	2	6	no	—
142	C₂×#117	C₂²×C₇₃	1	3	yes	—
143	C₂×#117	C₂²×C₇₃	1	3	yes	—
144	#410	—	2	6	no	—
145	#410	—	2	6	no	—
146	C₂×#99	C₂²×C₇₃	1	12	yes	—
147	#342	C₂²×C₇₃	1	12	yes	—
148	C₂×#41	—	2	3	no	—
149	C₂×#41	—	2	3	no	—
150	C₂×#41	—	2	3	no	—
151	C₂×#41	—	2	3	no	—
152	C₂²×Q₈	Q₈×C₇₃	1	1	yes	characteristic, complemented
153	C₂²×D₄	Q₈×C₇₃	1	1	yes	characteristic, complemented
154	C₂²×D₄	C₂³×C₇₃	1	2	yes	—
155	C₂³×C₄	Q₈×C₇₃	1	2	yes	complemented
156	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
157	C₄⋊Q₈	—	2	12	no	—
158	#125	—	2	12	no	—
159	C₂²⋊Q₈	—	2	24	no	—
160	C₄×Q₈	—	2	8	no	—
161	#117	—	2	3	no	—
162	#117	—	2	3	no	—
163	#117	—	2	3	no	—
164	#117	—	2	3	no	—
165	#117	—	2	3	no	—
166	#117	—	2	3	no	—
167	#117	D₄×C₇₃	1	6	yes	—
168	#117	D₄×C₇₃	1	6	yes	—
169	C₂×C₄⋊C₄	—	2	3	no	—
170	C₂×C₄⋊C₄	—	2	3	no	—
171	C₂×C₂²⋊C₄	C₂³×C₇₃	1	6	yes	—
172	C₂×C₂²⋊C₄	C₂³×C₇₃	1	6	yes	—
173	C₂×C₄²	—	2	3	no	—
174	C₂×C₄²	—	2	3	no	—
175	#99	—	2	24	no	—
176	C₂⁴	C₂×Q₈×C₇₃	1	2	yes	—
177	C₂⁴	C₂⁴×C₇₃	1	1	yes	characteristic
178	#41	—	4	1	no	—
179	#41	—	4	3	no	—
180	#41	—	4	3	no	—
181	#41	—	4	3	no	—
182	#41	—	4	3	no	—
183	#41	—	4	3	no	—
184	C₂×Q₈	—	2	8	no	—
185	C₂×Q₈	—	2	3	no	—
186	C₂×Q₈	—	2	3	no	—
187	C₂×D₄	—	2	6	no	—
188	C₂×D₄	—	2	6	no	—
189	C₂×D₄	—	2	3	no	—
190	C₂×D₄	—	2	3	no	—
191	C₂²×C₄	—	2	6	no	—
192	C₂²×C₄	—	2	6	no	—
193	C₂²×C₄	—	2	6	no	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	C₂×D₄×C₇₃	1	3	yes	—
196	C₂²×C₄	C₂×D₄×C₇₃	1	3	yes	—
197	C₂²×C₄	C₂×Q₈×C₇₃	1	2	yes	—
198	C₂²×C₄	C₂×Q₈×C₇₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×Q₈×C₇₃	1	1	yes	characteristic
200	C₂²×C₄	#41×C₇₃	1	3	yes	—
201	C₂²×C₄	#41×C₇₃	1	3	yes	—
202	C₄⋊C₄	—	2	24	no	—
203	C₄⋊C₄	—	2	6	no	—
204	C₄⋊C₄	—	2	6	no	—
205	C₄⋊C₄	—	4	3	no	—
206	C₄⋊C₄	—	4	3	no	—
207	C₂²⋊C₄	—	2	6	no	—
208	C₂²⋊C₄	—	2	6	no	—
209	C₂²⋊C₄	—	2	6	no	—
210	C₂²⋊C₄	C₂×D₄×C₇₃	1	12	yes	—
211	C₄²	—	2	6	no	—
212	C₄²	—	2	6	no	—
213	C₄²	—	4	3	no	—
214	C₄²	—	4	3	no	—
215	C₂³	—	2	6	no	—
216	C₂³	—	8	1	no	—
217	C₂³	—	8	2	no	—
218	C₂³	C₂²×D₄×C₇₃	1	3	yes	—
219	C₂³	C₂²×Q₈×C₇₃	1	1	yes	characteristic
220	C₂³	C₂×#41×C₇₃	1	3	yes	—
221	Q₈	—	4	1	no	—
222	Q₈	—	4	3	no	—
223	Q₈	—	8	8	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	6	no	—
226	D₄	—	4	1	no	—
227	D₄	—	4	3	no	—
228	C₂×C₄	—	2	12	no	—
229	C₂×C₄	—	2	6	no	—
230	C₂×C₄	—	4	6	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	6	no	—
233	C₂×C₄	—	8	6	no	—
234	C₂×C₄	—	8	6	no	—
235	C₂×C₄	#120×C₇₃	1	1	yes	characteristic
236	C₂×C₄	#120×C₇₃	1	3	yes	—
237	C₂×C₄	C₂²⋊Q₈×C₇₃	1	3	yes	—
238	C₂×C₄	C₂²⋊Q₈×C₇₃	1	3	yes	—
239	C₂×C₄	C₂²⋊Q₈×C₇₃	1	3	yes	—
240	C₂×C₄	C₂²⋊Q₈×C₇₃	1	3	yes	—
241	C₂×C₄	#124×C₇₃	1	3	yes	—
242	C₂×C₄	#124×C₇₃	1	3	yes	—
243	C₂×C₄	#124×C₇₃	1	3	yes	—
244	C₂×C₄	#124×C₇₃	1	3	yes	—
245	C₂²	—	4	3	no	—
246	C₂²	—	4	3	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	8	6	no	—
250	C₂²	C₂×#120×C₇₃	1	1	yes	characteristic
251	C₂²	C₂×C₂²⋊Q₈×C₇₃	1	3	yes	—
252	C₂²	C₂×#124×C₇₃	1	3	yes	—
253	C₄	—	4	12	no	—
254	C₄	—	4	2	no	—
255	C₄	—	8	6	no	—
256	C₄	#393×C₇₃	1	3	yes	—
257	C₄	#393×C₇₃	1	1	yes	characteristic
258	C₄	#393×C₇₃	1	1	yes	characteristic
259	C₄	#393×C₇₃	1	3	yes	—
260	C₂	—	2	1	no	—
261	C₂	—	2	3	no	—
262	C₂	—	2	3	no	—
263	C₂	—	8	2	no	—
264	C₂	C₂×#393×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇₃	#393
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇₃
socle	C₂×C₇₃	C₂×#393
prime radical	C₂²×#41×C₇₃	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₃×#3458	C₂	semidirect
3. C₇₃×#2032	C₂	semidirect
4. C₇₃×#1740	C₂	semidirect
19. C₂²×Q₈×C₇₃	Q₈	semidirect
20. C₂²×D₄×C₇₃	Q₈	semidirect
22. C₂³×C₄×C₇₃	Q₈	semidirect
127. #23474	C₇₃	direct (N × H)
133. #3458	C₂×C₇₃	semidirect
135. #2032	C₂×C₇₃	semidirect
136. #1740	C₂×C₇₃	semidirect
137. C₇₃	#23474	direct (N × H)
152. C₂²×Q₈	Q₈×C₇₃	semidirect
153. C₂²×D₄	Q₈×C₇₃	semidirect
155. C₂³×C₄	Q₈×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₇₃×#3458	2
C₂×#410×C₇₃	2
C₇₃×#2032	2
C₇₃×#1740	2
#23474	73

Subgroups — C73×#23474 #32932878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#23474 #32932878