Subgroups — C₂²×#509×C₇₃ #32972033

Order 18688.

2334 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	C₂²×#135×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#509×C₇₃	C₂	1	24	yes	maximal, direct product factor
4	C₂²×SD₃₂×C₇₃	C₂	1	2	yes	maximal, complemented
5	C₂²×D₁₆×C₇₃	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₇₃	—	2	1	no	—
8	C₂³×D₄×C₇₃	—	2	1	no	—
9	C₂×#135×C₇₃	—	2	6	no	—
10	C₂×#135×C₇₃	C₂²	1	12	yes	complemented
11	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
12	C₂²×SD₁₆×C₇₃	—	2	1	no	—
13	C₂²×D₈×C₇₃	—	2	1	no	—
14	C₂²×D₈×C₇₃	—	2	6	no	—
15	C₂²×D₈×C₇₃	C₂²	1	2	yes	—
16	C₂²×D₈×C₇₃	C₂²	1	12	yes	complemented
17	C₂²×D₈×C₇₃	C₂²	1	1	yes	characteristic, complemented
18	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#509×C₇₃	C₂²	1	64	yes	direct product factor
20	C₂×SD₃₂×C₇₃	C₂²	1	24	yes	complemented
21	C₂×D₁₆×C₇₃	C₂²	1	24	yes	complemented
22	C₂×M₃₂×C₇₃	C₂²	1	12	yes	complemented
23	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	complemented
24	C₂⁵×C₇₃	—	4	1	no	—
25	C₂×#41×C₇₃	—	2	12	no	—
26	C₂×#41×C₇₃	—	4	6	no	—
27	C₂²×Q₈×C₇₃	—	2	1	no	—
28	C₂²×D₄×C₇₃	—	2	2	no	—
29	C₂²×D₄×C₇₃	—	2	12	no	—
30	C₂²×D₄×C₇₃	—	2	1	no	—
31	C₂²×D₄×C₇₃	—	4	1	no	—
32	C₂²×D₄×C₇₃	—	4	6	no	—
33	C₂²×D₄×C₇₃	—	4	1	no	—
34	C₂³×C₄×C₇₃	—	4	1	no	—
35	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
36	#135×C₇₃	—	2	24	no	—
37	#135×C₇₃	C₂³	1	16	yes	complemented
38	C₂×Q₁₆×C₇₃	—	2	3	no	—
39	C₂×Q₁₆×C₇₃	C₂³	1	6	yes	complemented
40	C₂×SD₁₆×C₇₃	—	2	12	no	—
41	C₂×D₈×C₇₃	—	2	6	no	—
42	C₂×D₈×C₇₃	—	2	12	no	—
43	C₂×D₈×C₇₃	—	2	24	no	—
44	C₂×D₈×C₇₃	—	2	3	no	—
45	C₂×D₈×C₇₃	C₂³	1	12	yes	—
46	C₂×D₈×C₇₃	C₂³	1	16	yes	complemented
47	C₂×D₈×C₇₃	C₂³	1	6	yes	complemented
48	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	D₄	1	6	yes	—
50	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
52	SD₃₂×C₇₃	C₂³	1	32	yes	complemented
53	D₁₆×C₇₃	C₂³	1	32	yes	complemented
54	M₃₂×C₇₃	C₂³	1	16	yes	complemented
55	C₂×C₁₆×C₇₃	C₂³	1	12	yes	complemented
56	C₂⁴×C₇₃	—	4	12	no	—
57	C₂⁴×C₇₃	—	4	1	no	—
58	C₂⁴×C₇₃	—	4	2	no	—
59	C₂⁴×C₇₃	—	8	8	no	—
60	C₂⁴×C₇₃	D₈	1	1	yes	characteristic
61	#41×C₇₃	—	2	16	no	—
62	#41×C₇₃	—	4	24	no	—
63	C₂×Q₈×C₇₃	—	2	6	no	—
64	C₂×Q₈×C₇₃	—	4	3	no	—
65	C₂×D₄×C₇₃	—	2	12	no	—
66	C₂×D₄×C₇₃	—	2	16	no	—
67	C₂×D₄×C₇₃	—	2	6	no	—
68	C₂×D₄×C₇₃	—	4	12	no	—
69	C₂×D₄×C₇₃	—	4	12	no	—
70	C₂×D₄×C₇₃	—	4	6	no	—
71	C₂×D₄×C₇₃	—	4	24	no	—
72	C₂×D₄×C₇₃	—	4	3	no	—
73	C₂²×C₄×C₇₃	—	4	1	no	—
74	C₂²×C₄×C₇₃	—	4	12	no	—
75	C₂²×C₄×C₇₃	D₈	1	6	yes	—
76	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₇₃	—	2	6	no	—
80	Q₁₆×C₇₃	C₂⁴	1	4	yes	complemented
81	SD₁₆×C₇₃	—	2	16	no	—
82	D₈×C₇₃	—	2	12	no	—
83	D₈×C₇₃	—	2	16	no	—
84	D₈×C₇₃	—	2	6	no	—
85	D₈×C₇₃	C₂⁴	1	8	yes	—
86	D₈×C₇₃	C₂⁴	1	4	yes	complemented
87	C₂×C₈×C₇₃	C₂×D₄	1	3	yes	—
88	C₂×C₈×C₇₃	C₂×D₄	1	6	yes	—
89	C₂×C₈×C₇₃	C₂×D₄	1	12	yes	—
90	C₂×C₈×C₇₃	C₂⁴	1	3	yes	—
91	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
92	C₁₆×C₇₃	C₂⁴	1	8	yes	complemented
93	C₂³×C₇₃	—	2	4	no	—
94	C₂³×C₇₃	—	4	6	no	—
95	C₂³×C₇₃	—	4	16	no	—
96	C₂³×C₇₃	—	4	12	no	—
97	C₂³×C₇₃	—	8	48	no	—
98	C₂³×C₇₃	—	8	4	no	—
99	C₂³×C₇₃	—	8	8	no	—
100	C₂³×C₇₃	C₂×D₈	1	6	yes	—
101	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
102	Q₈×C₇₃	—	2	4	no	—
103	Q₈×C₇₃	—	4	6	no	—
104	D₄×C₇₃	—	2	8	no	—
105	D₄×C₇₃	—	2	4	no	—
106	D₄×C₇₃	—	4	16	no	—
107	D₄×C₇₃	—	4	16	no	—
108	D₄×C₇₃	—	4	12	no	—
109	D₄×C₇₃	—	4	6	no	—
110	C₂×C₄×C₇₃	—	4	6	no	—
111	C₂×C₄×C₇₃	—	4	16	no	—
112	C₂×C₄×C₇₃	C₂×D₈	1	6	yes	—
113	C₂×C₄×C₇₃	C₂×D₈	1	3	yes	—
114	C₂×C₄×C₇₃	C₂×D₈	1	12	yes	—
115	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
116	C₂×C₄×C₇₃	C₂²×D₄	1	4	yes	—
117	C₈×C₇₃	C₂²×D₄	1	3	yes	—
118	C₈×C₇₃	C₂²×D₄	1	4	yes	—
119	C₈×C₇₃	C₂⁵	1	1	yes	characteristic
120	C₂²×C₇₃	—	2	12	no	—
121	C₂²×C₇₃	—	4	4	no	—
122	C₂²×C₇₃	—	4	8	no	—
123	C₂²×C₇₃	—	8	32	no	—
124	C₂²×C₇₃	—	8	12	no	—
125	C₂²×C₇₃	—	8	24	no	—
126	C₂²×C₇₃	#509	1	4	yes	direct product factor
127	C₂²×C₇₃	C₂²×D₈	1	4	yes	—
128	C₂²×C₇₃	C₂²×D₈	1	3	yes	—
129	C₄×C₇₃	—	4	4	no	—
130	C₄×C₇₃	C₂²×D₈	1	4	yes	—
131	C₄×C₇₃	C₂²×D₈	1	3	yes	—
132	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
133	C₂²×#509	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₇₃	—	2	4	no	—
135	C₂×C₇₃	—	8	4	no	—
136	C₂×C₇₃	—	8	8	no	—
137	C₂×C₇₃	C₂×#509	1	6	yes	direct product factor
138	C₂×C₇₃	C₂³×D₈	1	1	yes	characteristic
139	C₂²×#135	C₂×C₇₃	1	1	yes	characteristic, complemented
140	C₂³×D₈	C₂×C₇₃	1	1	yes	characteristic, complemented
141	C₂×#509	C₂×C₇₃	1	24	yes	direct product factor
142	C₂²×SD₃₂	C₂×C₇₃	1	2	yes	complemented
143	C₂²×D₁₆	C₂×C₇₃	1	2	yes	complemented
144	C₂²×M₃₂	C₂×C₇₃	1	1	yes	characteristic, complemented
145	C₇₃	C₂²×#509	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×#41	—	2	1	no	—
147	C₂³×D₄	—	2	1	no	—
148	C₂×#135	—	2	6	no	—
149	C₂×#135	C₂²×C₇₃	1	12	yes	complemented
150	C₂²×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic, complemented
151	C₂²×SD₁₆	—	2	1	no	—
152	C₂²×D₈	—	2	1	no	—
153	C₂²×D₈	—	2	6	no	—
154	C₂²×D₈	C₂²×C₇₃	1	2	yes	—
155	C₂²×D₈	C₂²×C₇₃	1	12	yes	complemented
156	C₂²×D₈	C₂²×C₇₃	1	1	yes	characteristic, complemented
157	C₂³×C₈	C₂²×C₇₃	1	1	yes	characteristic
158	#509	C₂²×C₇₃	1	64	yes	direct product factor
159	C₂×SD₃₂	C₂²×C₇₃	1	24	yes	complemented
160	C₂×D₁₆	C₂²×C₇₃	1	24	yes	complemented
161	C₂×M₃₂	C₂²×C₇₃	1	12	yes	complemented
162	C₂²×C₁₆	C₂²×C₇₃	1	2	yes	complemented
163	C₂⁵	—	4	1	no	—
164	C₂×#41	—	2	12	no	—
165	C₂×#41	—	4	6	no	—
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	2	12	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	4	1	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂²×D₄	—	4	6	no	—
173	C₂³×C₄	—	4	1	no	—
174	C₂³×C₄	D₄×C₇₃	1	1	yes	characteristic
175	#135	—	2	24	no	—
176	#135	C₂³×C₇₃	1	16	yes	complemented
177	C₂×Q₁₆	—	2	3	no	—
178	C₂×Q₁₆	C₂³×C₇₃	1	6	yes	complemented
179	C₂×SD₁₆	—	2	12	no	—
180	C₂×D₈	—	2	6	no	—
181	C₂×D₈	—	2	12	no	—
182	C₂×D₈	—	2	24	no	—
183	C₂×D₈	—	2	3	no	—
184	C₂×D₈	C₂³×C₇₃	1	12	yes	—
185	C₂×D₈	C₂³×C₇₃	1	16	yes	complemented
186	C₂×D₈	C₂³×C₇₃	1	6	yes	complemented
187	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
188	C₂²×C₈	D₄×C₇₃	1	6	yes	—
189	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₇₃	1	6	yes	—
191	SD₃₂	C₂³×C₇₃	1	32	yes	complemented
192	D₁₆	C₂³×C₇₃	1	32	yes	complemented
193	M₃₂	C₂³×C₇₃	1	16	yes	complemented
194	C₂×C₁₆	C₂³×C₇₃	1	12	yes	complemented
195	C₂⁴	—	4	1	no	—
196	C₂⁴	—	4	12	no	—
197	C₂⁴	—	4	2	no	—
198	C₂⁴	—	8	8	no	—
199	C₂⁴	D₈×C₇₃	1	1	yes	characteristic
200	#41	—	2	16	no	—
201	#41	—	4	24	no	—
202	C₂×Q₈	—	2	6	no	—
203	C₂×Q₈	—	4	3	no	—
204	C₂×D₄	—	2	12	no	—
205	C₂×D₄	—	2	16	no	—
206	C₂×D₄	—	2	6	no	—
207	C₂×D₄	—	4	12	no	—
208	C₂×D₄	—	4	12	no	—
209	C₂×D₄	—	4	6	no	—
210	C₂×D₄	—	4	24	no	—
211	C₂×D₄	—	4	3	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	12	no	—
214	C₂²×C₄	D₈×C₇₃	1	1	yes	characteristic
215	C₂²×C₄	D₈×C₇₃	1	6	yes	—
216	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₇₃	1	6	yes	—
218	Q₁₆	—	2	6	no	—
219	Q₁₆	C₂⁴×C₇₃	1	4	yes	complemented
220	SD₁₆	—	2	16	no	—
221	D₈	—	2	12	no	—
222	D₈	—	2	16	no	—
223	D₈	—	2	6	no	—
224	D₈	C₂⁴×C₇₃	1	8	yes	—
225	D₈	C₂⁴×C₇₃	1	4	yes	complemented
226	C₂×C₈	C₂×D₄×C₇₃	1	3	yes	—
227	C₂×C₈	C₂×D₄×C₇₃	1	6	yes	—
228	C₂×C₈	C₂×D₄×C₇₃	1	12	yes	—
229	C₂×C₈	C₂⁴×C₇₃	1	3	yes	—
230	C₂×C₈	C₂⁴×C₇₃	1	4	yes	—
231	C₁₆	C₂⁴×C₇₃	1	8	yes	complemented
232	C₂³	—	2	4	no	—
233	C₂³	—	4	6	no	—
234	C₂³	—	4	16	no	—
235	C₂³	—	4	12	no	—
236	C₂³	—	8	48	no	—
237	C₂³	—	8	8	no	—
238	C₂³	—	8	4	no	—
239	C₂³	C₂×D₈×C₇₃	1	6	yes	—
240	C₂³	C₂×D₈×C₇₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	4	6	no	—
243	D₄	—	2	8	no	—
244	D₄	—	2	4	no	—
245	D₄	—	4	16	no	—
246	D₄	—	4	16	no	—
247	D₄	—	4	12	no	—
248	D₄	—	4	6	no	—
249	C₂×C₄	—	4	6	no	—
250	C₂×C₄	—	4	16	no	—
251	C₂×C₄	C₂×D₈×C₇₃	1	6	yes	—
252	C₂×C₄	C₂×D₈×C₇₃	1	3	yes	—
253	C₂×C₄	C₂×D₈×C₇₃	1	12	yes	—
254	C₂×C₄	C₂²×D₄×C₇₃	1	3	yes	—
255	C₂×C₄	C₂²×D₄×C₇₃	1	4	yes	—
256	C₈	C₂²×D₄×C₇₃	1	3	yes	—
257	C₈	C₂²×D₄×C₇₃	1	4	yes	—
258	C₈	C₂⁵×C₇₃	1	1	yes	characteristic
259	C₂²	—	2	12	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	8	32	no	—
263	C₂²	—	8	24	no	—
264	C₂²	—	8	12	no	—
265	C₂²	#509×C₇₃	1	4	yes	direct product factor
266	C₂²	C₂²×D₈×C₇₃	1	4	yes	—
267	C₂²	C₂²×D₈×C₇₃	1	3	yes	—
268	C₄	—	4	4	no	—
269	C₄	C₂²×D₈×C₇₃	1	4	yes	—
270	C₄	C₂²×D₈×C₇₃	1	3	yes	—
271	C₄	C₂³×D₄×C₇₃	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	—	8	8	no	—
274	C₂	—	8	4	no	—
275	C₂	C₂×#509×C₇₃	1	6	yes	direct product factor
276	C₂	C₂³×D₈×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₇₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ C₂³×C₈×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#135×C₇₃	C₂	semidirect
2. C₂³×D₈×C₇₃	C₂	semidirect
3. C₂×#509×C₇₃	C₂	direct (N × H)
4. C₂²×SD₃₂×C₇₃	C₂	semidirect
5. C₂²×D₁₆×C₇₃	C₂	semidirect
6. C₂²×M₃₂×C₇₃	C₂	semidirect
10. C₂×#135×C₇₃	C₂²	semidirect
11. C₂²×Q₁₆×C₇₃	C₂²	semidirect
16. C₂²×D₈×C₇₃	C₂²	semidirect
17. C₂²×D₈×C₇₃	C₂²	semidirect
19. #509×C₇₃	C₂²	direct (N × H)
20. C₂×SD₃₂×C₇₃	C₂²	semidirect
21. C₂×D₁₆×C₇₃	C₂²	semidirect
22. C₂×M₃₂×C₇₃	C₂²	semidirect
23. C₂²×C₁₆×C₇₃	C₂²	semidirect
37. #135×C₇₃	C₂³	semidirect
39. C₂×Q₁₆×C₇₃	C₂³	semidirect
46. C₂×D₈×C₇₃	C₂³	semidirect
47. C₂×D₈×C₇₃	C₂³	semidirect
52. SD₃₂×C₇₃	C₂³	semidirect
53. D₁₆×C₇₃	C₂³	semidirect
54. M₃₂×C₇₃	C₂³	semidirect
55. C₂×C₁₆×C₇₃	C₂³	semidirect
80. Q₁₆×C₇₃	C₂⁴	semidirect
86. D₈×C₇₃	C₂⁴	semidirect
92. C₁₆×C₇₃	C₂⁴	semidirect
126. C₂²×C₇₃	#509	direct (N × H)
133. C₂²×#509	C₇₃	direct (N × H)
137. C₂×C₇₃	C₂×#509	direct (N × H)
139. C₂²×#135	C₂×C₇₃	semidirect
140. C₂³×D₈	C₂×C₇₃	semidirect
141. C₂×#509	C₂×C₇₃	direct (N × H)
142. C₂²×SD₃₂	C₂×C₇₃	semidirect
143. C₂²×D₁₆	C₂×C₇₃	semidirect
144. C₂²×M₃₂	C₂×C₇₃	semidirect
145. C₇₃	C₂²×#509	direct (N × H)
149. C₂×#135	C₂²×C₇₃	semidirect
150. C₂²×Q₁₆	C₂²×C₇₃	semidirect
155. C₂²×D₈	C₂²×C₇₃	semidirect
156. C₂²×D₈	C₂²×C₇₃	semidirect
158. #509	C₂²×C₇₃	direct (N × H)
159. C₂×SD₃₂	C₂²×C₇₃	semidirect
160. C₂×D₁₆	C₂²×C₇₃	semidirect
161. C₂×M₃₂	C₂²×C₇₃	semidirect
162. C₂²×C₁₆	C₂²×C₇₃	semidirect
176. #135	C₂³×C₇₃	semidirect
178. C₂×Q₁₆	C₂³×C₇₃	semidirect
185. C₂×D₈	C₂³×C₇₃	semidirect
186. C₂×D₈	C₂³×C₇₃	semidirect
191. SD₃₂	C₂³×C₇₃	semidirect
192. D₁₆	C₂³×C₇₃	semidirect
193. M₃₂	C₂³×C₇₃	semidirect
194. C₂×C₁₆	C₂³×C₇₃	semidirect
219. Q₁₆	C₂⁴×C₇₃	semidirect
225. D₈	C₂⁴×C₇₃	semidirect
231. C₁₆	C₂⁴×C₇₃	semidirect
265. C₂²	#509×C₇₃	direct (N × H)
275. C₂	C₂×#509×C₇₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#135×C₇₃	2
C₂³×D₈×C₇₃	2
C₂×#509×C₇₃	2
C₂²×SD₃₂×C₇₃	2
C₂²×D₁₆×C₇₃	2
C₂²×M₃₂×C₇₃	2
C₂²×#509	73

Subgroups — C22×#509×C73 #32972033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#509×C₇₃ #32972033