Subgroups — C₂²×#541×C₇₃ #32972085

Order 18688.

7110 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#541×C₇₃	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#126×C₇₃	C₂	1	4	yes	maximal
4	C₂²×#125×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#125×C₇₃	C₂	1	4	yes	maximal, complemented
6	C₂²×C₂²⋊Q₈×C₇₃	C₂	1	2	yes	maximal
7	C₂²×C₄×Q₈×C₇₃	C₂	1	2	yes	maximal, complemented
8	C₂²×#117×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₇₃	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
11	#541×C₇₃	C₂²	1	256	yes	direct product factor
12	C₂×C₄⋊Q₈×C₇₃	C₂²	1	24	yes	complemented
13	C₂×#126×C₇₃	C₂²	1	96	yes	—
14	C₂×#125×C₇₃	C₂²	1	24	yes	complemented
15	C₂×#125×C₇₃	C₂²	1	96	yes	complemented
16	C₂×C₂²⋊Q₈×C₇₃	C₂²	1	48	yes	—
17	C₂×C₄×Q₈×C₇₃	C₂²	1	48	yes	complemented
18	C₂×#117×C₇₃	C₂²	1	24	yes	—
19	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
22	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
24	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄×C₇₃	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₇₃	C₂²	1	4	yes	—
27	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₇₃	C₂²	1	4	yes	—
30	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
31	C₂²×Q₈×C₇₃	—	2	16	no	—
32	C₂²×Q₈×C₇₃	C₂³	1	24	yes	—
33	C₂³×C₄×C₇₃	—	2	8	no	—
34	C₂³×C₄×C₇₃	C₂³	1	12	yes	—
35	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
36	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
38	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₇₃	C₂³	1	64	yes	complemented
42	#126×C₇₃	C₂³	1	256	yes	—
43	#125×C₇₃	C₂³	1	64	yes	complemented
44	#125×C₇₃	C₂³	1	256	yes	complemented
45	C₂²⋊Q₈×C₇₃	C₂³	1	128	yes	—
46	C₄×Q₈×C₇₃	C₂³	1	128	yes	complemented
47	#117×C₇₃	C₂³	1	64	yes	—
48	C₂×C₄⋊C₄×C₇₃	C₂³	1	24	yes	—
49	C₂×C₄⋊C₄×C₇₃	C₂³	1	24	yes	—
50	C₂×C₄⋊C₄×C₇₃	C₂³	1	48	yes	—
51	C₂×C₄⋊C₄×C₇₃	C₂³	1	48	yes	—
52	C₂×C₄⋊C₄×C₇₃	C₂³	1	48	yes	—
53	C₂×C₄⋊C₄×C₇₃	C₂³	1	48	yes	—
54	C₂×C₂²⋊C₄×C₇₃	C₂³	1	24	yes	—
55	C₂×C₂²⋊C₄×C₇₃	C₂³	1	48	yes	—
56	C₂×C₄²×C₇₃	C₂³	1	12	yes	—
57	C₂×C₄²×C₇₃	C₂³	1	12	yes	—
58	C₂×C₄²×C₇₃	C₂³	1	48	yes	—
59	C₂⁴×C₇₃	—	2	4	no	—
60	C₂⁴×C₇₃	—	2	8	no	—
61	C₂⁴×C₇₃	C₂⁴	1	6	yes	—
62	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
63	C₂×Q₈×C₇₃	—	2	96	no	—
64	C₂×Q₈×C₇₃	C₂⁴	1	32	yes	—
65	C₂²×C₄×C₇₃	—	2	4	no	—
66	C₂²×C₄×C₇₃	—	2	16	no	—
67	C₂²×C₄×C₇₃	—	2	8	no	—
68	C₂²×C₄×C₇₃	—	2	8	no	—
69	C₂²×C₄×C₇₃	—	2	16	no	—
70	C₂²×C₄×C₇₃	—	2	48	no	—
71	C₂²×C₄×C₇₃	#41	1	8	yes	—
72	C₂²×C₄×C₇₃	C₂⁴	1	12	yes	—
73	C₂²×C₄×C₇₃	C₂⁴	1	12	yes	—
74	C₂²×C₄×C₇₃	C₂⁴	1	24	yes	—
75	C₂²×C₄×C₇₃	C₂⁴	1	24	yes	—
76	C₂²×C₄×C₇₃	C₂⁴	1	6	yes	—
77	C₂²×C₄×C₇₃	C₂⁴	1	16	yes	—
78	C₂²×C₄×C₇₃	C₂⁴	1	6	yes	—
79	C₄⋊C₄×C₇₃	C₂⁴	1	32	yes	—
80	C₄⋊C₄×C₇₃	C₂⁴	1	32	yes	—
81	C₄⋊C₄×C₇₃	C₂⁴	1	64	yes	—
82	C₄⋊C₄×C₇₃	C₂⁴	1	64	yes	—
83	C₄⋊C₄×C₇₃	C₂⁴	1	64	yes	—
84	C₄⋊C₄×C₇₃	C₂⁴	1	64	yes	—
85	C₂²⋊C₄×C₇₃	C₂⁴	1	32	yes	—
86	C₂²⋊C₄×C₇₃	C₂⁴	1	64	yes	—
87	C₄²×C₇₃	C₂⁴	1	16	yes	—
88	C₄²×C₇₃	C₂⁴	1	16	yes	—
89	C₄²×C₇₃	C₂⁴	1	64	yes	—
90	C₂³×C₇₃	—	2	12	no	—
91	C₂³×C₇₃	—	2	24	no	—
92	C₂³×C₇₃	—	4	16	no	—
93	C₂³×C₇₃	C₂×#41	1	4	yes	—
94	C₂³×C₇₃	2₋¹⁺⁴	1	8	yes	—
95	C₂³×C₇₃	C₂⁵	1	4	yes	—
96	C₂³×C₇₃	C₂⁵	1	3	yes	—
97	Q₈×C₇₃	—	2	64	no	—
98	C₂×C₄×C₇₃	—	2	12	no	—
99	C₂×C₄×C₇₃	—	2	48	no	—
100	C₂×C₄×C₇₃	—	2	24	no	—
101	C₂×C₄×C₇₃	—	2	24	no	—
102	C₂×C₄×C₇₃	—	2	48	no	—
103	C₂×C₄×C₇₃	—	2	32	no	—
104	C₂×C₄×C₇₃	C₂×#41	1	24	yes	—
105	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
106	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
107	C₂×C₄×C₇₃	C₂⁵	1	16	yes	—
108	C₂×C₄×C₇₃	C₂⁵	1	16	yes	—
109	C₂×C₄×C₇₃	C₂⁵	1	4	yes	—
110	C₂×C₄×C₇₃	C₂⁵	1	4	yes	—
111	C₂²×C₇₃	—	2	4	no	—
112	C₂²×C₇₃	—	2	8	no	—
113	C₂²×C₇₃	—	4	24	no	—
114	C₂²×C₇₃	#541	1	16	yes	direct product factor
115	C₂²×C₇₃	C₂²×#41	1	6	yes	—
116	C₂²×C₇₃	C₂×2₋¹⁺⁴	1	12	yes	—
117	C₂²×C₇₃	C₂⁶	1	1	yes	characteristic
118	C₄×C₇₃	—	2	4	no	—
119	C₄×C₇₃	—	2	16	no	—
120	C₄×C₇₃	—	2	8	no	—
121	C₄×C₇₃	—	2	8	no	—
122	C₄×C₇₃	—	2	16	no	—
123	C₄×C₇₃	C₂²×#41	1	8	yes	—
124	C₂²×#541	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₇₃	—	4	4	no	—
126	C₂×C₇₃	C₂×#541	1	12	yes	direct product factor
127	C₂×C₇₃	C₂³×#41	1	1	yes	characteristic
128	C₂×C₇₃	C₂²×2₋¹⁺⁴	1	2	yes	—
129	C₂×#541	C₂×C₇₃	1	48	yes	direct product factor
130	C₂²×C₄⋊Q₈	C₂×C₇₃	1	1	yes	characteristic, complemented
131	C₂²×#126	C₂×C₇₃	1	4	yes	—
132	C₂²×#125	C₂×C₇₃	1	4	yes	complemented
133	C₂²×#125	C₂×C₇₃	1	1	yes	characteristic, complemented
134	C₂²×C₂²⋊Q₈	C₂×C₇₃	1	2	yes	—
135	C₂²×C₄×Q₈	C₂×C₇₃	1	2	yes	complemented
136	C₂²×#117	C₂×C₇₃	1	1	yes	characteristic
137	C₇₃	C₂²×#541	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³×Q₈	C₂²×C₇₃	1	2	yes	—
139	C₂⁴×C₄	C₂²×C₇₃	1	1	yes	characteristic
140	#541	C₂²×C₇₃	1	256	yes	direct product factor
141	C₂×C₄⋊Q₈	C₂²×C₇₃	1	24	yes	complemented
142	C₂×#126	C₂²×C₇₃	1	96	yes	—
143	C₂×#125	C₂²×C₇₃	1	24	yes	complemented
144	C₂×#125	C₂²×C₇₃	1	96	yes	complemented
145	C₂×C₂²⋊Q₈	C₂²×C₇₃	1	48	yes	—
146	C₂×C₄×Q₈	C₂²×C₇₃	1	48	yes	complemented
147	C₂×#117	C₂²×C₇₃	1	24	yes	—
148	C₂²×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
149	C₂²×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
150	C₂²×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
151	C₂²×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
152	C₂²×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
153	C₂²×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
154	C₂²×C₂²⋊C₄	C₂²×C₇₃	1	2	yes	—
155	C₂²×C₂²⋊C₄	C₂²×C₇₃	1	4	yes	—
156	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
157	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
158	C₂²×C₄²	C₂²×C₇₃	1	4	yes	—
159	C₂⁵	C₂³×C₇₃	1	1	yes	characteristic
160	C₂²×Q₈	—	2	16	no	—
161	C₂²×Q₈	C₂³×C₇₃	1	24	yes	—
162	C₂³×C₄	—	2	8	no	—
163	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
164	C₂³×C₄	C₂³×C₇₃	1	4	yes	—
165	C₂³×C₄	C₂³×C₇₃	1	4	yes	—
166	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
167	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
168	C₂³×C₄	C₂³×C₇₃	1	12	yes	—
169	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
170	C₄⋊Q₈	C₂³×C₇₃	1	64	yes	complemented
171	#126	C₂³×C₇₃	1	256	yes	—
172	#125	C₂³×C₇₃	1	64	yes	complemented
173	#125	C₂³×C₇₃	1	256	yes	complemented
174	C₂²⋊Q₈	C₂³×C₇₃	1	128	yes	—
175	C₄×Q₈	C₂³×C₇₃	1	128	yes	complemented
176	#117	C₂³×C₇₃	1	64	yes	—
177	C₂×C₄⋊C₄	C₂³×C₇₃	1	24	yes	—
178	C₂×C₄⋊C₄	C₂³×C₇₃	1	24	yes	—
179	C₂×C₄⋊C₄	C₂³×C₇₃	1	48	yes	—
180	C₂×C₄⋊C₄	C₂³×C₇₃	1	48	yes	—
181	C₂×C₄⋊C₄	C₂³×C₇₃	1	48	yes	—
182	C₂×C₄⋊C₄	C₂³×C₇₃	1	48	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₇₃	1	24	yes	—
184	C₂×C₂²⋊C₄	C₂³×C₇₃	1	48	yes	—
185	C₂×C₄²	C₂³×C₇₃	1	12	yes	—
186	C₂×C₄²	C₂³×C₇₃	1	12	yes	—
187	C₂×C₄²	C₂³×C₇₃	1	48	yes	—
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	2	8	no	—
190	C₂⁴	C₂⁴×C₇₃	1	6	yes	—
191	C₂⁴	C₂⁴×C₇₃	1	1	yes	characteristic
192	C₂×Q₈	—	2	96	no	—
193	C₂×Q₈	C₂⁴×C₇₃	1	32	yes	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	16	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	16	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	48	no	—
200	C₂²×C₄	#41×C₇₃	1	8	yes	—
201	C₂²×C₄	C₂⁴×C₇₃	1	12	yes	—
202	C₂²×C₄	C₂⁴×C₇₃	1	12	yes	—
203	C₂²×C₄	C₂⁴×C₇₃	1	24	yes	—
204	C₂²×C₄	C₂⁴×C₇₃	1	24	yes	—
205	C₂²×C₄	C₂⁴×C₇₃	1	6	yes	—
206	C₂²×C₄	C₂⁴×C₇₃	1	16	yes	—
207	C₂²×C₄	C₂⁴×C₇₃	1	6	yes	—
208	C₄⋊C₄	C₂⁴×C₇₃	1	32	yes	—
209	C₄⋊C₄	C₂⁴×C₇₃	1	32	yes	—
210	C₄⋊C₄	C₂⁴×C₇₃	1	64	yes	—
211	C₄⋊C₄	C₂⁴×C₇₃	1	64	yes	—
212	C₄⋊C₄	C₂⁴×C₇₃	1	64	yes	—
213	C₄⋊C₄	C₂⁴×C₇₃	1	64	yes	—
214	C₂²⋊C₄	C₂⁴×C₇₃	1	32	yes	—
215	C₂²⋊C₄	C₂⁴×C₇₃	1	64	yes	—
216	C₄²	C₂⁴×C₇₃	1	16	yes	—
217	C₄²	C₂⁴×C₇₃	1	16	yes	—
218	C₄²	C₂⁴×C₇₃	1	64	yes	—
219	C₂³	—	2	12	no	—
220	C₂³	—	2	24	no	—
221	C₂³	—	4	16	no	—
222	C₂³	C₂×#41×C₇₃	1	4	yes	—
223	C₂³	2₋¹⁺⁴×C₇₃	1	8	yes	—
224	C₂³	C₂⁵×C₇₃	1	4	yes	—
225	C₂³	C₂⁵×C₇₃	1	3	yes	—
226	Q₈	—	2	64	no	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	48	no	—
229	C₂×C₄	—	2	24	no	—
230	C₂×C₄	—	2	24	no	—
231	C₂×C₄	—	2	48	no	—
232	C₂×C₄	—	2	32	no	—
233	C₂×C₄	C₂×#41×C₇₃	1	24	yes	—
234	C₂×C₄	C₂⁵×C₇₃	1	8	yes	—
235	C₂×C₄	C₂⁵×C₇₃	1	8	yes	—
236	C₂×C₄	C₂⁵×C₇₃	1	16	yes	—
237	C₂×C₄	C₂⁵×C₇₃	1	16	yes	—
238	C₂×C₄	C₂⁵×C₇₃	1	4	yes	—
239	C₂×C₄	C₂⁵×C₇₃	1	4	yes	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	24	no	—
243	C₂²	#541×C₇₃	1	16	yes	direct product factor
244	C₂²	C₂²×#41×C₇₃	1	6	yes	—
245	C₂²	C₂×2₋¹⁺⁴×C₇₃	1	12	yes	—
246	C₂²	C₂⁶×C₇₃	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	16	no	—
249	C₄	—	2	8	no	—
250	C₄	—	2	8	no	—
251	C₄	—	2	16	no	—
252	C₄	C₂²×#41×C₇₃	1	8	yes	—
253	C₂	—	4	4	no	—
254	C₂	C₂×#541×C₇₃	1	12	yes	direct product factor
255	C₂	C₂³×#41×C₇₃	1	1	yes	characteristic
256	C₂	C₂²×2₋¹⁺⁴×C₇₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇₃
prime radical	C₂⁵×C₇₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#541×C₇₃	C₂	direct (N × H)
2. C₂²×C₄⋊Q₈×C₇₃	C₂	semidirect
4. C₂²×#125×C₇₃	C₂	semidirect
5. C₂²×#125×C₇₃	C₂	semidirect
7. C₂²×C₄×Q₈×C₇₃	C₂	semidirect
11. #541×C₇₃	C₂²	direct (N × H)
12. C₂×C₄⋊Q₈×C₇₃	C₂²	semidirect
14. C₂×#125×C₇₃	C₂²	semidirect
15. C₂×#125×C₇₃	C₂²	semidirect
17. C₂×C₄×Q₈×C₇₃	C₂²	semidirect
41. C₄⋊Q₈×C₇₃	C₂³	semidirect
43. #125×C₇₃	C₂³	semidirect
44. #125×C₇₃	C₂³	semidirect
46. C₄×Q₈×C₇₃	C₂³	semidirect
114. C₂²×C₇₃	#541	direct (N × H)
124. C₂²×#541	C₇₃	direct (N × H)
126. C₂×C₇₃	C₂×#541	direct (N × H)
129. C₂×#541	C₂×C₇₃	direct (N × H)
130. C₂²×C₄⋊Q₈	C₂×C₇₃	semidirect
132. C₂²×#125	C₂×C₇₃	semidirect
133. C₂²×#125	C₂×C₇₃	semidirect
135. C₂²×C₄×Q₈	C₂×C₇₃	semidirect
137. C₇₃	C₂²×#541	direct (N × H)
140. #541	C₂²×C₇₃	direct (N × H)
141. C₂×C₄⋊Q₈	C₂²×C₇₃	semidirect
143. C₂×#125	C₂²×C₇₃	semidirect
144. C₂×#125	C₂²×C₇₃	semidirect
146. C₂×C₄×Q₈	C₂²×C₇₃	semidirect
170. C₄⋊Q₈	C₂³×C₇₃	semidirect
172. #125	C₂³×C₇₃	semidirect
173. #125	C₂³×C₇₃	semidirect
175. C₄×Q₈	C₂³×C₇₃	semidirect
243. C₂²	#541×C₇₃	direct (N × H)
254. C₂	C₂×#541×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
C₂×#541×C₇₃	2
C₂²×C₄⋊Q₈×C₇₃	2
C₂²×#126×C₇₃	2
C₂²×#125×C₇₃	2
C₂²×C₂²⋊Q₈×C₇₃	2
C₂²×C₄×Q₈×C₇₃	2
C₂²×#117×C₇₃	2
C₂²×#541	73

Subgroups — C22×#541×C73 #32972085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#541×C₇₃ #32972085