Subgroups — #543×Dic₇₃ #33245139

Order 18688.

2602 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612666	C₂	1	12	yes	maximal, complemented
2	#22612545	C₂	1	3	yes	maximal
3	C₂²⋊Q₈×Dic₇₃	C₂	1	12	yes	maximal, complemented
4	C₂×C₂²⋊C₄×Dic₇₃	C₂	1	3	yes	maximal
5	C₂×#543×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₇₃	C₂²	1	1	yes	characteristic
7	C₂×Q₈×Dic₇₃	C₂²	1	4	yes	complemented
8	C₂²×C₂²⋊Dic₇₃	C₂²	1	3	yes	—
9	C₂²×C₄×Dic₇₃	C₂²	1	6	yes	—
10	#17988494	C₂²	1	12	yes	complemented
11	#17988486	C₂²	1	12	yes	complemented
12	#17988473	C₂²	1	24	yes	complemented
13	#17988462	C₂²	1	24	yes	—
14	#17988447	C₂²	1	12	yes	—
15	C₄⋊C₄×Dic₇₃	C₂²	1	12	yes	complemented
16	C₂²⋊C₄×Dic₇₃	C₂²	1	12	yes	—
17	C₂²⋊C₄×Dic₇₃	C₂²	1	12	yes	—
18	C₂×#14556498	C₂²	1	6	yes	—
19	#543×C₇₃	C₄	1	16	yes	complemented
20	C₂×C₂²⋊Q₈×C₇₃	C₂²	1	12	yes	—
21	C₂²×C₂²⋊C₄×C₇₃	C₂²	1	3	yes	—
22	C₂³×Dic₇₃	—	4	4	no	—
23	C₂³×Dic₇₃	Q₈	1	8	yes	complemented
24	C₂³×Dic₇₃	C₂³	1	3	yes	—
25	C₂³×Dic₇₃	C₂³	1	6	yes	—
26	C₂³×Dic₇₃	C₂³	1	3	yes	—
27	Q₈×Dic₇₃	—	4	8	no	—
28	C₂×C₄⋊Dic₇₃	C₂³	1	12	yes	—
29	C₂×C₄⋊Dic₇₃	C₂³	1	12	yes	—
30	C₂×C₂²⋊Dic₇₃	—	2	12	no	—
31	C₂×C₂²⋊Dic₇₃	C₂³	1	12	yes	—
32	C₂×C₄×Dic₇₃	—	4	12	no	—
33	C₂×C₄×Dic₇₃	C₂³	1	12	yes	—
34	C₂×C₄×Dic₇₃	C₂³	1	12	yes	—
35	#14556498	C₂³	1	24	yes	—
36	#14556498	C₂³	1	24	yes	—
37	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
38	C₂²×Q₈×C₇₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
40	C₂²⋊Q₈×C₇₃	C₂×C₄	1	96	yes	complemented
41	C₂×C₄⋊C₄×C₇₃	C₂³	1	12	yes	—
42	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	24	yes	—
43	C₂×C₂²⋊C₄×C₇₃	C₂³	1	12	yes	—
44	C₂²×Dic₇₃	—	2	12	no	—
45	C₂²×Dic₇₃	—	2	12	no	—
46	C₂²×Dic₇₃	—	4	16	no	—
47	C₂²×Dic₇₃	—	4	6	no	—
48	C₂²×Dic₇₃	—	4	6	no	—
49	C₂²×Dic₇₃	—	4	6	no	—
50	C₂²×Dic₇₃	C₂×Q₈	1	12	yes	—
51	C₂²×Dic₇₃	C₂×Q₈	1	12	yes	—
52	C₂²×Dic₇₃	C₂⁴	1	12	yes	—
53	C₂²×Dic₇₃	C₂⁴	1	3	yes	—
54	C₂²×Dic₇₃	C₂⁴	1	1	yes	characteristic
55	C₄⋊Dic₇₃	—	4	12	no	—
56	C₄⋊Dic₇₃	—	4	12	no	—
57	C₂²⋊Dic₇₃	—	2	24	no	—
58	C₄×Dic₇₃	—	2	24	no	—
59	C₄×Dic₇₃	—	4	12	no	—
60	C₂⁴×C₇₃	—	4	2	no	—
61	C₂⁴×C₇₃	—	4	2	no	—
62	C₂⁴×C₇₃	C₂²×C₄	1	4	yes	—
63	C₂⁴×C₇₃	C₂×Q₈	1	4	yes	—
64	C₂⁴×C₇₃	#41	1	4	yes	—
65	C₂⁴×C₇₃	C₂⁴	1	3	yes	—
66	C₂×Q₈×C₇₃	—	4	4	no	—
67	C₂×Q₈×C₇₃	—	4	4	no	—
68	C₂×Q₈×C₇₃	C₂²×C₄	1	16	yes	complemented
69	C₂²×C₄×C₇₃	—	4	6	no	—
70	C₂²×C₄×C₇₃	—	4	6	no	—
71	C₂²×C₄×C₇₃	C₂²×C₄	1	24	yes	—
72	C₂²×C₄×C₇₃	C₂⁴	1	12	yes	—
73	C₄⋊C₄×C₇₃	C₂²×C₄	1	48	yes	complemented
74	C₂²⋊C₄×C₇₃	C₂²×C₄	1	48	yes	—
75	C₂³×C₇₃	—	2	6	no	—
76	C₂³×C₇₃	—	2	6	no	—
77	C₂³×C₇₃	—	4	8	no	—
78	C₂³×C₇₃	—	4	4	no	—
79	C₂³×C₇₃	—	4	4	no	—
80	C₂³×C₇₃	—	4	8	no	—
81	C₂³×C₇₃	C₄×Q₈	1	16	yes	complemented
82	C₂³×C₇₃	C₂³×C₄	1	6	yes	—
83	C₂³×C₇₃	C₂²×Q₈	1	6	yes	—
84	C₂³×C₇₃	C₂×#41	1	6	yes	—
85	C₂³×C₇₃	C₂⁵	1	1	yes	characteristic
86	Q₈×C₇₃	—	4	16	no	—
87	C₂×C₄×C₇₃	—	2	12	no	—
88	C₂×C₄×C₇₃	—	2	12	no	—
89	C₂×C₄×C₇₃	—	4	24	no	—
90	C₂×C₄×C₇₃	C₂³×C₄	1	24	yes	—
91	C₂×Dic₇₃	—	2	6	no	—
92	C₂×Dic₇₃	—	2	24	no	—
93	C₂×Dic₇₃	—	2	12	no	—
94	C₂×Dic₇₃	—	2	12	no	—
95	C₂×Dic₇₃	—	2	12	no	—
96	C₂×Dic₇₃	—	4	12	no	—
97	C₂×Dic₇₃	—	4	24	no	—
98	C₂×Dic₇₃	C₂²×Q₈	1	6	yes	—
99	C₂×Dic₇₃	C₂²×Q₈	1	2	yes	—
100	C₂×Dic₇₃	H(5,𝔽₂)	1	4	yes	—
101	C₂²×C₇₃	—	2	12	no	—
102	C₂²×C₇₃	—	2	6	no	—
103	C₂²×C₇₃	—	2	6	no	—
104	C₂²×C₇₃	—	2	12	no	—
105	C₂²×C₇₃	—	4	16	no	—
106	C₂²×C₇₃	C₂×C₄×Q₈	1	12	yes	—
107	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
108	C₂²×C₇₃	C₂³×Q₈	1	1	yes	characteristic
109	C₂²×C₇₃	C₂²×#41	1	1	yes	characteristic
110	C₂²×C₇₃	C₂×H(5,𝔽₂)	1	2	yes	—
111	C₂²×C₇₃	#585	1	2	yes	—
112	C₄×C₇₃	—	2	24	no	—
113	Dic₇₃	—	2	6	no	—
114	Dic₇₃	—	4	12	no	—
115	Dic₇₃	#543	1	4	yes	direct product factor
116	C₄×#543	—	73	1	no	maximal, Sylow
117	C₂×C₇₃	—	2	12	no	—
118	C₂×C₇₃	C₂²×C₄×Q₈	1	1	yes	characteristic
119	C₂×C₇₃	C₄×H(5,𝔽₂)	1	2	yes	—
120	C₂×C₇₃	C₂×#543	1	1	yes	characteristic
121	C₂×C₇₃	#3458	1	1	yes	characteristic
122	C₂×C₇₃	#3459	1	2	yes	—
123	C₂×#543	D₇₃	1	1	yes	characteristic
124	#2329	—	73	12	no	—
125	#2315	—	73	3	no	—
126	C₄×C₂²⋊Q₈	—	73	12	no	—
127	C₂×C₄×C₂²⋊C₄	—	73	3	no	—
128	C₇₃	C₄×#543	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴×C₄	—	73	1	no	—
130	#543	Dic₇₃	1	16	yes	direct product factor
131	C₂×C₂²⋊Q₈	C₂×D₇₃	1	12	yes	—
132	C₂×C₄×Q₈	—	73	4	no	—
133	C₂²×C₂²⋊C₄	—	73	3	no	—
134	C₂²×C₂²⋊C₄	C₂×D₇₃	1	3	yes	—
135	C₂²×C₄²	—	73	6	no	—
136	#391	—	73	12	no	—
137	#389	—	73	12	no	—
138	#387	—	73	24	no	—
139	#385	—	73	24	no	—
140	#380	—	73	12	no	—
141	C₄×C₄⋊C₄	—	73	12	no	—
142	C₄×C₂²⋊C₄	—	73	12	no	—
143	C₄×C₂²⋊C₄	—	73	12	no	—
144	C₂×#95	—	73	6	no	—
145	C₂⁵	C₂²×D₇₃	1	1	yes	characteristic
146	C₂²×Q₈	C₂²×D₇₃	1	4	yes	—
147	C₂³×C₄	—	292	4	no	—
148	C₂³×C₄	—	73	8	no	—
149	C₂³×C₄	—	73	3	no	—
150	C₂³×C₄	—	73	6	no	—
151	C₂³×C₄	—	73	3	no	—
152	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
153	C₂²⋊Q₈	C₂×Dic₇₃	1	96	yes	complemented
154	C₄×Q₈	—	292	8	no	—
155	C₂×C₄⋊C₄	—	73	12	no	—
156	C₂×C₄⋊C₄	—	73	12	no	—
157	C₂×C₄⋊C₄	C₂²×D₇₃	1	12	yes	—
158	C₂×C₂²⋊C₄	—	73	12	no	—
159	C₂×C₂²⋊C₄	—	146	12	no	—
160	C₂×C₂²⋊C₄	C₂×Dic₇₃	1	24	yes	—
161	C₂×C₂²⋊C₄	C₂²×D₇₃	1	12	yes	—
162	C₂×C₄²	—	292	12	no	—
163	C₂×C₄²	—	73	12	no	—
164	C₂×C₄²	—	73	12	no	—
165	#95	—	73	24	no	—
166	#95	—	73	24	no	—
167	C₂⁴	—	4	2	no	—
168	C₂⁴	—	4	2	no	—
169	C₂⁴	C₂²×Dic₇₃	1	4	yes	—
170	C₂⁴	Q₈×D₇₃	1	4	yes	—
171	C₂⁴	#11924029	1	4	yes	—
172	C₂⁴	C₂³×D₇₃	1	3	yes	—
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	C₂²×Dic₇₃	1	16	yes	complemented
176	C₂²×C₄	—	4	6	no	—
177	C₂²×C₄	—	4	6	no	—
178	C₂²×C₄	—	292	6	no	—
179	C₂²×C₄	—	292	6	no	—
180	C₂²×C₄	—	292	6	no	—
181	C₂²×C₄	—	292	16	no	—
182	C₂²×C₄	—	73	12	no	—
183	C₂²×C₄	—	73	1	no	—
184	C₂²×C₄	—	73	12	no	—
185	C₂²×C₄	—	73	3	no	—
186	C₂²×C₄	—	73	12	no	—
187	C₂²×C₄	—	146	12	no	—
188	C₂²×C₄	—	146	12	no	—
189	C₂²×C₄	C₂²×Dic₇₃	1	24	yes	—
190	C₂²×C₄	C₂³×D₇₃	1	12	yes	—
191	C₄⋊C₄	—	292	12	no	—
192	C₄⋊C₄	—	292	12	no	—
193	C₄⋊C₄	C₂²×Dic₇₃	1	48	yes	complemented
194	C₂²⋊C₄	—	146	24	no	—
195	C₂²⋊C₄	C₂²×Dic₇₃	1	48	yes	—
196	C₄²	—	292	12	no	—
197	C₄²	—	146	24	no	—
198	C₂³	—	2	6	no	—
199	C₂³	—	2	6	no	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	8	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	8	no	—
204	C₂³	Q₈×Dic₇₃	1	16	yes	complemented
205	C₂³	C₂³×Dic₇₃	1	6	yes	—
206	C₂³	C₂×Q₈×D₇₃	1	6	yes	—
207	C₂³	C₂×#11924029	1	6	yes	—
208	C₂³	C₂⁴×D₇₃	1	1	yes	characteristic
209	Q₈	—	4	16	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	4	24	no	—
213	C₂×C₄	—	292	24	no	—
214	C₂×C₄	—	292	12	no	—
215	C₂×C₄	—	73	2	no	—
216	C₂×C₄	—	73	4	no	—
217	C₂×C₄	—	73	6	no	—
218	C₂×C₄	—	146	12	no	—
219	C₂×C₄	—	146	12	no	—
220	C₂×C₄	—	146	12	no	—
221	C₂×C₄	—	146	24	no	—
222	C₂×C₄	—	146	6	no	—
223	C₂×C₄	C₂³×Dic₇₃	1	24	yes	—
224	C₂²	—	2	6	no	—
225	C₂²	—	2	6	no	—
226	C₂²	—	2	12	no	—
227	C₂²	—	2	12	no	—
228	C₂²	—	4	16	no	—
229	C₂²	C₂×Q₈×Dic₇₃	1	12	yes	—
230	C₂²	C₂⁴×Dic₇₃	1	1	yes	characteristic
231	C₂²	C₂²×Q₈×D₇₃	1	1	yes	characteristic
232	C₂²	C₂²×#11924029	1	1	yes	characteristic
233	C₂²	H(5,𝔽₂)×D₇₃	1	2	yes	—
234	C₂²	#17989422	1	2	yes	—
235	C₄	—	2	24	no	—
236	C₄	—	292	12	no	—
237	C₄	—	73	4	no	—
238	C₄	—	146	6	no	—
239	C₂	—	2	12	no	—
240	C₂	C₂²×Q₈×Dic₇₃	1	1	yes	characteristic
241	C₂	H(5,𝔽₂)×Dic₇₃	1	2	yes	—
242	C₂	#543×D₇₃	1	1	yes	characteristic
243	C₂	#22623333	1	1	yes	characteristic
244	C₂	#22623343	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇₃
commutator	C₂²×C₇₃	C₂⁴×C₄
Fitting	C₂×#543×C₇₃	C₂
socle, cosocle	C₂³×C₇₃	C₂⁵
prime radical	C₂⁵×C₇₃	C₂³
hypercenter	C₂×#543	D₇₃
hypocenter	C₇₃	C₄×#543
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
1. #22612666	C₂	semidirect
3. C₂²⋊Q₈×Dic₇₃	C₂	semidirect
7. C₂×Q₈×Dic₇₃	C₂²	semidirect
10. #17988494	C₂²	semidirect
11. #17988486	C₂²	semidirect
12. #17988473	C₂²	semidirect
15. C₄⋊C₄×Dic₇₃	C₂²	semidirect
19. #543×C₇₃	C₄	semidirect
23. C₂³×Dic₇₃	Q₈	semidirect
40. C₂²⋊Q₈×C₇₃	C₂×C₄	semidirect
68. C₂×Q₈×C₇₃	C₂²×C₄	semidirect
73. C₄⋊C₄×C₇₃	C₂²×C₄	semidirect
81. C₂³×C₇₃	C₄×Q₈	semidirect
115. Dic₇₃	#543	direct (N × H)
128. C₇₃	C₄×#543	semidirect
130. #543	Dic₇₃	direct (N × H)
153. C₂²⋊Q₈	C₂×Dic₇₃	semidirect
175. C₂×Q₈	C₂²×Dic₇₃	semidirect
193. C₄⋊C₄	C₂²×Dic₇₃	semidirect
204. C₂³	Q₈×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612666	2
#22612545	2
C₂²⋊Q₈×Dic₇₃	2
C₂×C₂²⋊C₄×Dic₇₃	2
C₂×#543×C₇₃	2
C₄×#543	73

Subgroups — #543×Dic73 #33245139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #543×Dic₇₃ #33245139