Subgroups — C₇₃×#3100 #22607253

Order 9344.

492 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#480×C₇₃	C₂	1	2	yes	maximal, complemented
4	#474×C₇₃	C₂	1	2	yes	maximal, complemented
5	#459×C₇₃	C₂	1	2	yes	maximal, complemented
6	#447×C₇₃	C₂	1	2	yes	maximal, complemented
7	C₄×D₈×C₇₃	C₂	1	4	yes	maximal, complemented
8	#431×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₇₃	—	2	1	no	—
10	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₇₃	—	2	1	no	—
12	C₂×D₈×C₇₃	C₂²	1	4	yes	—
13	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic, complemented
14	#127×C₇₃	C₂²	1	1	yes	characteristic, complemented
15	#124×C₇₃	—	2	1	no	—
16	#121×C₇₃	—	2	1	no	—
17	#121×C₇₃	C₂²	1	2	yes	—
18	C₄×D₄×C₇₃	—	2	1	no	—
19	C₄×D₄×C₇₃	—	2	1	no	—
20	C₄×D₄×C₇₃	—	2	1	no	—
21	C₄×D₄×C₇₃	C₂²	1	2	yes	—
22	C₄×D₄×C₇₃	C₂²	1	2	yes	—
23	C₄×D₄×C₇₃	C₂²	1	2	yes	complemented
24	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
26	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
27	#107×C₇₃	C₂²	1	4	yes	complemented
28	C₄⋊C₈×C₇₃	C₂²	1	2	yes	complemented
29	#102×C₇₃	C₂²	1	4	yes	—
30	#102×C₇₃	C₂²	1	4	yes	complemented
31	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
32	C₄×C₈×C₇₃	C₂²	1	2	yes	complemented
33	C₂⁴×C₇₃	—	2	1	no	—
34	#41×C₇₃	—	4	4	no	—
35	C₂×Q₈×C₇₃	—	2	1	no	—
36	C₂×D₄×C₇₃	—	2	4	no	—
37	C₂×D₄×C₇₃	—	2	2	no	—
38	C₂×D₄×C₇₃	—	2	1	no	—
39	C₂×D₄×C₇₃	—	2	1	no	—
40	C₂×D₄×C₇₃	—	2	1	no	—
41	C₂×D₄×C₇₃	C₂³	1	2	yes	—
42	C₂×D₄×C₇₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₇₃	—	2	1	no	—
44	C₂²×C₄×C₇₃	—	2	2	no	—
45	C₂²×C₄×C₇₃	—	2	2	no	—
46	C₂²×C₄×C₇₃	—	2	2	no	—
47	C₂²×C₄×C₇₃	—	4	4	no	—
48	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
51	D₈×C₇₃	—	2	8	no	—
52	C₂×C₈×C₇₃	C₂³	1	4	yes	—
53	C₄⋊C₄×C₇₃	—	2	1	no	—
54	C₄⋊C₄×C₇₃	—	2	1	no	—
55	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
56	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₇₃	—	2	2	no	—
58	C₂²⋊C₄×C₇₃	—	2	2	no	—
59	C₂²⋊C₄×C₇₃	—	2	2	no	—
60	C₂²⋊C₄×C₇₃	—	2	2	no	—
61	C₄²×C₇₃	D₄	1	1	yes	characteristic
62	C₄²×C₇₃	D₄	1	1	yes	characteristic
63	C₄²×C₇₃	C₂³	1	1	yes	characteristic
64	C₄²×C₇₃	C₂³	1	1	yes	characteristic
65	C₂³×C₇₃	—	2	4	no	—
66	C₂³×C₇₃	—	2	2	no	—
67	C₂³×C₇₃	—	2	2	no	—
68	C₂³×C₇₃	—	4	2	no	—
69	C₂³×C₇₃	—	4	2	no	—
70	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₇₃	—	4	2	no	—
72	D₄×C₇₃	—	2	4	no	—
73	D₄×C₇₃	—	2	2	no	—
74	D₄×C₇₃	—	2	4	no	—
75	D₄×C₇₃	—	4	2	no	—
76	D₄×C₇₃	—	4	4	no	—
77	D₄×C₇₃	—	4	2	no	—
78	D₄×C₇₃	#41	1	4	yes	complemented
79	C₂×C₄×C₇₃	—	2	2	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	—	4	1	no	—
83	C₂×C₄×C₇₃	—	4	2	no	—
84	C₂×C₄×C₇₃	—	4	4	no	—
85	C₂×C₄×C₇₃	—	4	4	no	—
86	C₂×C₄×C₇₃	—	4	4	no	—
87	C₂×C₄×C₇₃	D₈	1	4	yes	complemented
88	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₇₃	—	2	4	no	—
95	C₂²×C₇₃	—	2	1	no	—
96	C₂²×C₇₃	—	2	1	no	—
97	C₂²×C₇₃	—	2	4	no	—
98	C₂²×C₇₃	—	4	8	no	—
99	C₂²×C₇₃	—	4	2	no	—
100	C₂²×C₇₃	—	4	2	no	—
101	C₂²×C₇₃	—	4	2	no	—
102	C₂²×C₇₃	—	4	2	no	—
103	C₂²×C₇₃	—	4	2	no	—
104	C₂²×C₇₃	C₂×D₈	1	2	yes	—
105	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₇₃	—	2	1	no	—
107	C₄×C₇₃	—	2	1	no	—
108	C₄×C₇₃	—	2	1	no	—
109	C₄×C₇₃	—	4	2	no	—
110	C₄×C₇₃	—	4	2	no	—
111	C₄×C₇₃	C₂×D₈	1	2	yes	—
112	C₄×C₇₃	C₂×D₈	1	2	yes	—
113	C₄×C₇₃	C₂×#41	1	2	yes	—
114	C₂×C₇₃	—	2	2	no	—
115	C₂×C₇₃	—	4	4	no	—
116	C₂×C₇₃	—	8	2	no	—
117	C₂×C₇₃	#525	1	1	yes	characteristic
118	C₂×C₇₃	C₂²×D₈	1	1	yes	characteristic
119	C₂×C₇₃	#575	1	1	yes	characteristic
120	#3100	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₇₃	#3100	1	1	yes	characteristic, direct product factor, Sylow
122	#532	C₂×C₇₃	1	1	yes	characteristic, complemented
123	C₂×C₄×D₄	C₂×C₇₃	1	1	yes	characteristic, complemented
124	#480	C₂×C₇₃	1	2	yes	complemented
125	#474	C₂×C₇₃	1	2	yes	complemented
126	#459	C₂×C₇₃	1	2	yes	complemented
127	#447	C₂×C₇₃	1	2	yes	complemented
128	C₄×D₈	C₂×C₇₃	1	4	yes	complemented
129	#431	C₂×C₇₃	1	1	yes	characteristic, complemented
130	C₂×#41	—	2	1	no	—
131	C₂²×D₄	C₂²×C₇₃	1	1	yes	characteristic
132	C₂³×C₄	—	2	1	no	—
133	C₂×D₈	C₂²×C₇₃	1	4	yes	—
134	C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic, complemented
135	#127	C₂²×C₇₃	1	1	yes	characteristic, complemented
136	#124	—	2	1	no	—
137	#121	—	2	1	no	—
138	#121	C₂²×C₇₃	1	2	yes	—
139	C₄×D₄	—	2	1	no	—
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	C₂²×C₇₃	1	2	yes	—
143	C₄×D₄	C₂²×C₇₃	1	2	yes	—
144	C₄×D₄	C₂²×C₇₃	1	2	yes	complemented
145	C₂×C₄⋊C₄	C₂²×C₇₃	1	1	yes	characteristic
146	C₂×C₂²⋊C₄	—	2	1	no	—
147	C₂×C₄²	C₂²×C₇₃	1	1	yes	characteristic
148	#107	C₂²×C₇₃	1	4	yes	complemented
149	C₄⋊C₈	C₂²×C₇₃	1	2	yes	complemented
150	#102	C₂²×C₇₃	1	4	yes	—
151	#102	C₂²×C₇₃	1	4	yes	complemented
152	C₂²⋊C₈	C₂²×C₇₃	1	2	yes	—
153	C₄×C₈	C₂²×C₇₃	1	2	yes	complemented
154	C₂⁴	—	2	1	no	—
155	#41	—	4	4	no	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	C₂³×C₇₃	1	2	yes	—
163	C₂×D₄	C₂³×C₇₃	1	2	yes	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
170	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
172	D₈	—	2	8	no	—
173	C₂×C₈	C₂³×C₇₃	1	4	yes	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
177	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₄²	D₄×C₇₃	1	1	yes	characteristic
183	C₄²	D₄×C₇₃	1	1	yes	characteristic
184	C₄²	C₂³×C₇₃	1	1	yes	characteristic
185	C₄²	C₂³×C₇₃	1	1	yes	characteristic
186	C₂³	—	2	4	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	4	2	no	—
190	C₂³	—	4	2	no	—
191	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	D₄	—	2	4	no	—
194	D₄	—	2	2	no	—
195	D₄	—	2	4	no	—
196	D₄	—	4	2	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	2	no	—
199	D₄	#41×C₇₃	1	4	yes	complemented
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	D₈×C₇₃	1	4	yes	complemented
209	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
211	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	4	8	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	C₂×D₈×C₇₃	1	2	yes	—
226	C₂²	C₂²×D₄×C₇₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	C₂×D₈×C₇₃	1	2	yes	—
233	C₄	C₂×D₈×C₇₃	1	2	yes	—
234	C₄	C₂×#41×C₇₃	1	2	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	4	4	no	—
237	C₂	—	8	2	no	—
238	C₂	#525×C₇₃	1	1	yes	characteristic
239	C₂	C₂²×D₈×C₇₃	1	1	yes	characteristic
240	C₂	#575×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₃
socle	C₂²×C₇₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #532×C₇₃	C₂	semidirect
2. C₂×C₄×D₄×C₇₃	C₂	semidirect
3. #480×C₇₃	C₂	semidirect
4. #474×C₇₃	C₂	semidirect
5. #459×C₇₃	C₂	semidirect
6. #447×C₇₃	C₂	semidirect
7. C₄×D₈×C₇₃	C₂	semidirect
8. #431×C₇₃	C₂	semidirect
13. C₄⋊Q₈×C₇₃	C₂²	semidirect
14. #127×C₇₃	C₂²	semidirect
23. C₄×D₄×C₇₃	C₂²	semidirect
27. #107×C₇₃	C₂²	semidirect
28. C₄⋊C₈×C₇₃	C₂²	semidirect
30. #102×C₇₃	C₂²	semidirect
32. C₄×C₈×C₇₃	C₂²	semidirect
78. D₄×C₇₃	#41	semidirect
87. C₂×C₄×C₇₃	D₈	semidirect
120. #3100	C₇₃	direct (N × H)
121. C₇₃	#3100	direct (N × H)
122. #532	C₂×C₇₃	semidirect
123. C₂×C₄×D₄	C₂×C₇₃	semidirect
124. #480	C₂×C₇₃	semidirect
125. #474	C₂×C₇₃	semidirect
126. #459	C₂×C₇₃	semidirect
127. #447	C₂×C₇₃	semidirect
128. C₄×D₈	C₂×C₇₃	semidirect
129. #431	C₂×C₇₃	semidirect
134. C₄⋊Q₈	C₂²×C₇₃	semidirect
135. #127	C₂²×C₇₃	semidirect
144. C₄×D₄	C₂²×C₇₃	semidirect
148. #107	C₂²×C₇₃	semidirect
149. C₄⋊C₈	C₂²×C₇₃	semidirect
151. #102	C₂²×C₇₃	semidirect
153. C₄×C₈	C₂²×C₇₃	semidirect
199. D₄	#41×C₇₃	semidirect
208. C₂×C₄	D₈×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
#532×C₇₃	2
C₂×C₄×D₄×C₇₃	2
#480×C₇₃	2
#474×C₇₃	2
#459×C₇₃	2
#447×C₇₃	2
C₄×D₈×C₇₃	2
#431×C₇₃	2
#3100	73

Subgroups — C73×#3100 #22607253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#3100 #22607253