Subgroups — C₂×C₇₃×#3415 #32972036

Order 18688.

2126 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₇₃	C₂	1	2	yes	maximal, complemented
2	C₇₃×#3415	C₂	1	16	yes	maximal, direct product factor
3	C₂×#509×C₇₃	C₂	1	6	yes	maximal, complemented
4	C₂×#508×C₇₃	C₂	1	3	yes	maximal, complemented
5	C₂²×D₁₆×C₇₃	C₂	1	3	yes	maximal, complemented
6	C₂×#504×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×H(5,𝔽₂)×C₇₃	—	2	2	no	—
8	#576×C₇₃	—	2	8	no	—
9	#576×C₇₃	C₂²	1	16	yes	complemented
10	C₂×AGL(1,ℤ/8)×C₇₃	—	2	6	no	—
11	C₂×#135×C₇₃	C₂²	1	6	yes	complemented
12	C₂²×D₈×C₇₃	C₂²	1	6	yes	complemented
13	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
14	#509×C₇₃	C₂²	1	48	yes	complemented
15	#508×C₇₃	C₂²	1	24	yes	complemented
16	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic, complemented
17	C₂×SD₃₂×C₇₃	C₂²	1	6	yes	complemented
18	C₂×D₁₆×C₇₃	C₂²	1	3	yes	complemented
19	C₂×D₁₆×C₇₃	C₂²	1	6	yes	complemented
20	C₂×D₁₆×C₇₃	C₂²	1	24	yes	complemented
21	#504×C₇₃	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₇₃	C₂²	1	3	yes	complemented
23	C₂²×C₁₆×C₇₃	C₂²	1	3	yes	complemented
24	H(5,𝔽₂)×C₇₃	—	2	16	no	—
25	H(5,𝔽₂)×C₇₃	—	4	8	no	—
26	C₂×#41×C₇₃	—	2	6	no	—
27	C₂×#41×C₇₃	—	4	2	no	—
28	C₂×#41×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₇₃	—	2	6	no	—
30	C₂²×D₄×C₇₃	—	4	6	no	—
31	AGL(1,ℤ/8)×C₇₃	—	2	48	no	—
32	#135×C₇₃	—	2	12	no	—
33	#135×C₇₃	C₂³	1	24	yes	complemented
34	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₇₃	—	2	6	no	—
36	C₂×D₈×C₇₃	—	2	6	no	—
37	C₂×D₈×C₇₃	—	2	12	no	—
38	C₂×D₈×C₇₃	C₂³	1	6	yes	—
39	C₂×D₈×C₇₃	C₂³	1	24	yes	complemented
40	#131×C₇₃	D₄	1	4	yes	—
41	#131×C₇₃	C₂³	1	4	yes	—
42	C₂×M₁₆×C₇₃	D₄	1	3	yes	—
43	C₂²×C₈×C₇₃	C₂³	1	3	yes	—
44	Q₃₂×C₇₃	C₂³	1	4	yes	complemented
45	SD₃₂×C₇₃	C₂³	1	24	yes	complemented
46	D₁₆×C₇₃	C₂³	1	12	yes	complemented
47	D₁₆×C₇₃	C₂³	1	24	yes	complemented
48	M₃₂×C₇₃	C₂³	1	12	yes	complemented
49	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₇₃	C₂³	1	3	yes	—
51	C₂×C₁₆×C₇₃	C₂³	1	12	yes	complemented
52	C₂⁴×C₇₃	—	4	6	no	—
53	#41×C₇₃	—	2	24	no	—
54	#41×C₇₃	—	4	16	no	—
55	#41×C₇₃	—	4	12	no	—
56	#41×C₇₃	D₈	1	4	yes	—
57	#41×C₇₃	C₂×D₄	1	4	yes	—
58	C₂×Q₈×C₇₃	—	2	2	no	—
59	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
60	C₂×D₄×C₇₃	—	2	6	no	—
61	C₂×D₄×C₇₃	—	2	24	no	—
62	C₂×D₄×C₇₃	—	4	6	no	—
63	C₂×D₄×C₇₃	—	4	48	no	—
64	C₂×D₄×C₇₃	—	4	6	no	—
65	C₂×D₄×C₇₃	—	4	12	no	—
66	C₂×D₄×C₇₃	D₈	1	3	yes	—
67	C₂²×C₄×C₇₃	—	4	6	no	—
68	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
69	Q₁₆×C₇₃	—	2	2	no	—
70	Q₁₆×C₇₃	C₂⁴	1	4	yes	—
71	SD₁₆×C₇₃	—	2	24	no	—
72	D₈×C₇₃	—	2	6	no	—
73	D₈×C₇₃	—	2	24	no	—
74	D₈×C₇₃	C₂⁴	1	12	yes	—
75	M₁₆×C₇₃	C₂×D₄	1	12	yes	—
76	C₂×C₈×C₇₃	C₂×D₄	1	3	yes	—
77	C₂×C₈×C₇₃	C₂×D₄	1	6	yes	—
78	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₇₃	C₂⁴	1	6	yes	—
80	C₁₆×C₇₃	C₂⁴	1	2	yes	—
81	C₁₆×C₇₃	C₂⁴	1	6	yes	—
82	C₂³×C₇₃	—	4	6	no	—
83	C₂³×C₇₃	—	4	24	no	—
84	C₂³×C₇₃	—	16	12	no	—
85	C₂³×C₇₃	C₂×D₈	1	3	yes	—
86	Q₈×C₇₃	—	2	4	no	—
87	Q₈×C₇₃	—	4	2	no	—
88	Q₈×C₇₃	C₂×D₈	1	4	yes	—
89	D₄×C₇₃	—	2	12	no	—
90	D₄×C₇₃	—	4	24	no	—
91	D₄×C₇₃	—	4	24	no	—
92	D₄×C₇₃	—	4	6	no	—
93	D₄×C₇₃	C₂×D₈	1	12	yes	—
94	C₂×C₄×C₇₃	—	4	2	no	—
95	C₂×C₄×C₇₃	—	4	24	no	—
96	C₂×C₄×C₇₃	C₂×D₈	1	6	yes	—
97	C₂×C₄×C₇₃	C₂×D₈	1	3	yes	—
98	C₂×C₄×C₇₃	C₂²×D₄	1	6	yes	—
99	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₇₃	C₂²×D₄	1	6	yes	—
102	C₈×C₇₃	C₂⁵	1	1	yes	characteristic
103	C₂²×C₇₃	—	2	6	no	—
104	C₂²×C₇₃	—	4	12	no	—
105	C₂²×C₇₃	—	8	12	no	—
106	C₂²×C₇₃	—	16	24	no	—
107	C₂²×C₇₃	C₂²×D₈	1	6	yes	—
108	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
109	C₄×C₇₃	—	4	4	no	—
110	C₄×C₇₃	C₂²×D₈	1	6	yes	—
111	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
112	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
113	C₂×#3415	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₇₃	—	2	6	no	—
115	C₂×C₇₃	—	8	12	no	—
116	C₂×C₇₃	#3415	1	2	yes	direct product factor
117	C₂×C₇₃	C₂³×D₈	1	1	yes	characteristic
118	C₂×#576	C₂×C₇₃	1	2	yes	complemented
119	#3415	C₂×C₇₃	1	16	yes	direct product factor
120	C₂×#509	C₂×C₇₃	1	6	yes	complemented
121	C₂×#508	C₂×C₇₃	1	3	yes	complemented
122	C₂²×D₁₆	C₂×C₇₃	1	3	yes	complemented
123	C₂×#504	C₂×C₇₃	1	1	yes	characteristic, complemented
124	C₇₃	C₂×#3415	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×H(5,𝔽₂)	—	2	2	no	—
126	#576	—	2	8	no	—
127	#576	C₂²×C₇₃	1	16	yes	complemented
128	C₂×AGL(1,ℤ/8)	—	2	6	no	—
129	C₂×#135	C₂²×C₇₃	1	6	yes	complemented
130	C₂²×D₈	C₂²×C₇₃	1	6	yes	complemented
131	C₂×#131	C₂²×C₇₃	1	1	yes	characteristic
132	#509	C₂²×C₇₃	1	48	yes	complemented
133	#508	C₂²×C₇₃	1	24	yes	complemented
134	C₂×Q₃₂	C₂²×C₇₃	1	1	yes	characteristic, complemented
135	C₂×SD₃₂	C₂²×C₇₃	1	6	yes	complemented
136	C₂×D₁₆	C₂²×C₇₃	1	3	yes	complemented
137	C₂×D₁₆	C₂²×C₇₃	1	6	yes	complemented
138	C₂×D₁₆	C₂²×C₇₃	1	24	yes	complemented
139	#504	C₂²×C₇₃	1	8	yes	complemented
140	C₂×M₃₂	C₂²×C₇₃	1	3	yes	complemented
141	C₂²×C₁₆	C₂²×C₇₃	1	3	yes	complemented
142	H(5,𝔽₂)	—	2	16	no	—
143	H(5,𝔽₂)	—	4	8	no	—
144	C₂×#41	—	2	6	no	—
145	C₂×#41	—	4	2	no	—
146	C₂×#41	D₄×C₇₃	1	1	yes	characteristic
147	C₂²×D₄	—	2	6	no	—
148	C₂²×D₄	—	4	6	no	—
149	AGL(1,ℤ/8)	—	2	48	no	—
150	#135	—	2	12	no	—
151	#135	C₂³×C₇₃	1	24	yes	complemented
152	C₂×Q₁₆	C₂³×C₇₃	1	2	yes	—
153	C₂×SD₁₆	—	2	6	no	—
154	C₂×D₈	—	2	6	no	—
155	C₂×D₈	—	2	12	no	—
156	C₂×D₈	C₂³×C₇₃	1	6	yes	—
157	C₂×D₈	C₂³×C₇₃	1	24	yes	complemented
158	#131	D₄×C₇₃	1	4	yes	—
159	#131	C₂³×C₇₃	1	4	yes	—
160	C₂×M₁₆	D₄×C₇₃	1	3	yes	—
161	C₂²×C₈	C₂³×C₇₃	1	3	yes	—
162	Q₃₂	C₂³×C₇₃	1	4	yes	complemented
163	SD₃₂	C₂³×C₇₃	1	24	yes	complemented
164	D₁₆	C₂³×C₇₃	1	12	yes	complemented
165	D₁₆	C₂³×C₇₃	1	24	yes	complemented
166	M₃₂	C₂³×C₇₃	1	12	yes	complemented
167	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
168	C₂×C₁₆	C₂³×C₇₃	1	3	yes	—
169	C₂×C₁₆	C₂³×C₇₃	1	12	yes	complemented
170	C₂⁴	—	4	6	no	—
171	#41	—	2	24	no	—
172	#41	—	4	16	no	—
173	#41	—	4	12	no	—
174	#41	D₈×C₇₃	1	4	yes	—
175	#41	C₂×D₄×C₇₃	1	4	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	D₈×C₇₃	1	1	yes	characteristic
178	C₂×D₄	—	2	6	no	—
179	C₂×D₄	—	2	24	no	—
180	C₂×D₄	—	4	6	no	—
181	C₂×D₄	—	4	48	no	—
182	C₂×D₄	—	4	6	no	—
183	C₂×D₄	—	4	12	no	—
184	C₂×D₄	D₈×C₇₃	1	3	yes	—
185	C₂²×C₄	—	4	6	no	—
186	C₂²×C₄	C₂×D₄×C₇₃	1	3	yes	—
187	Q₁₆	—	2	2	no	—
188	Q₁₆	C₂⁴×C₇₃	1	4	yes	—
189	SD₁₆	—	2	24	no	—
190	D₈	—	2	6	no	—
191	D₈	—	2	24	no	—
192	D₈	C₂⁴×C₇₃	1	12	yes	—
193	M₁₆	C₂×D₄×C₇₃	1	12	yes	—
194	C₂×C₈	C₂×D₄×C₇₃	1	3	yes	—
195	C₂×C₈	C₂×D₄×C₇₃	1	6	yes	—
196	C₂×C₈	C₂⁴×C₇₃	1	1	yes	characteristic
197	C₂×C₈	C₂⁴×C₇₃	1	6	yes	—
198	C₁₆	C₂⁴×C₇₃	1	2	yes	—
199	C₁₆	C₂⁴×C₇₃	1	6	yes	—
200	C₂³	—	4	6	no	—
201	C₂³	—	4	24	no	—
202	C₂³	—	16	12	no	—
203	C₂³	C₂×D₈×C₇₃	1	3	yes	—
204	Q₈	—	2	4	no	—
205	Q₈	—	4	2	no	—
206	Q₈	C₂×D₈×C₇₃	1	4	yes	—
207	D₄	—	2	12	no	—
208	D₄	—	4	24	no	—
209	D₄	—	4	24	no	—
210	D₄	—	4	6	no	—
211	D₄	C₂×D₈×C₇₃	1	12	yes	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	24	no	—
214	C₂×C₄	C₂×D₈×C₇₃	1	3	yes	—
215	C₂×C₄	C₂×D₈×C₇₃	1	6	yes	—
216	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₄×C₇₃	1	6	yes	—
218	C₈	C₂²×D₄×C₇₃	1	1	yes	characteristic
219	C₈	C₂²×D₄×C₇₃	1	6	yes	—
220	C₈	C₂⁵×C₇₃	1	1	yes	characteristic
221	C₂²	—	2	6	no	—
222	C₂²	—	4	12	no	—
223	C₂²	—	8	12	no	—
224	C₂²	—	16	24	no	—
225	C₂²	C₂²×D₈×C₇₃	1	6	yes	—
226	C₂²	C₂²×D₈×C₇₃	1	1	yes	characteristic
227	C₄	—	4	4	no	—
228	C₄	C₂²×D₈×C₇₃	1	6	yes	—
229	C₄	C₂²×D₈×C₇₃	1	1	yes	characteristic
230	C₄	C₂³×D₄×C₇₃	1	1	yes	characteristic
231	C₂	—	2	6	no	—
232	C₂	—	8	12	no	—
233	C₂	C₇₃×#3415	1	2	yes	direct product factor
234	C₂	C₂³×D₈×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₇₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×#41×C₇₃ ◁ C₂×#131×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#576×C₇₃	C₂	semidirect
2. C₇₃×#3415	C₂	direct (N × H)
3. C₂×#509×C₇₃	C₂	semidirect
4. C₂×#508×C₇₃	C₂	semidirect
5. C₂²×D₁₆×C₇₃	C₂	semidirect
6. C₂×#504×C₇₃	C₂	semidirect
9. #576×C₇₃	C₂²	semidirect
11. C₂×#135×C₇₃	C₂²	semidirect
12. C₂²×D₈×C₇₃	C₂²	semidirect
14. #509×C₇₃	C₂²	semidirect
15. #508×C₇₃	C₂²	semidirect
16. C₂×Q₃₂×C₇₃	C₂²	semidirect
17. C₂×SD₃₂×C₇₃	C₂²	semidirect
18. C₂×D₁₆×C₇₃	C₂²	semidirect
19. C₂×D₁₆×C₇₃	C₂²	semidirect
20. C₂×D₁₆×C₇₃	C₂²	semidirect
21. #504×C₇₃	C₂²	semidirect
22. C₂×M₃₂×C₇₃	C₂²	semidirect
23. C₂²×C₁₆×C₇₃	C₂²	semidirect
33. #135×C₇₃	C₂³	semidirect
39. C₂×D₈×C₇₃	C₂³	semidirect
44. Q₃₂×C₇₃	C₂³	semidirect
45. SD₃₂×C₇₃	C₂³	semidirect
46. D₁₆×C₇₃	C₂³	semidirect
47. D₁₆×C₇₃	C₂³	semidirect
48. M₃₂×C₇₃	C₂³	semidirect
51. C₂×C₁₆×C₇₃	C₂³	semidirect
113. C₂×#3415	C₇₃	direct (N × H)
116. C₂×C₇₃	#3415	direct (N × H)
118. C₂×#576	C₂×C₇₃	semidirect
119. #3415	C₂×C₇₃	direct (N × H)
120. C₂×#509	C₂×C₇₃	semidirect
121. C₂×#508	C₂×C₇₃	semidirect
122. C₂²×D₁₆	C₂×C₇₃	semidirect
123. C₂×#504	C₂×C₇₃	semidirect
124. C₇₃	C₂×#3415	direct (N × H)
127. #576	C₂²×C₇₃	semidirect
129. C₂×#135	C₂²×C₇₃	semidirect
130. C₂²×D₈	C₂²×C₇₃	semidirect
132. #509	C₂²×C₇₃	semidirect
133. #508	C₂²×C₇₃	semidirect
134. C₂×Q₃₂	C₂²×C₇₃	semidirect
135. C₂×SD₃₂	C₂²×C₇₃	semidirect
136. C₂×D₁₆	C₂²×C₇₃	semidirect
137. C₂×D₁₆	C₂²×C₇₃	semidirect
138. C₂×D₁₆	C₂²×C₇₃	semidirect
139. #504	C₂²×C₇₃	semidirect
140. C₂×M₃₂	C₂²×C₇₃	semidirect
141. C₂²×C₁₆	C₂²×C₇₃	semidirect
151. #135	C₂³×C₇₃	semidirect
157. C₂×D₈	C₂³×C₇₃	semidirect
162. Q₃₂	C₂³×C₇₃	semidirect
163. SD₃₂	C₂³×C₇₃	semidirect
164. D₁₆	C₂³×C₇₃	semidirect
165. D₁₆	C₂³×C₇₃	semidirect
166. M₃₂	C₂³×C₇₃	semidirect
169. C₂×C₁₆	C₂³×C₇₃	semidirect
233. C₂	C₇₃×#3415	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×#576×C₇₃	2
C₇₃×#3415	2
C₂×#509×C₇₃	2
C₂×#508×C₇₃	2
C₂²×D₁₆×C₇₃	2
C₂×#504×C₇₃	2
C₂×#3415	73

Subgroups — C2×C73×#3415 #32972036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₃×#3415 #32972036