Subgroups — C₂²×#17989643 #34032678

Order 18688.

2334 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556666	C₂	1	2	yes	maximal, complemented
2	C₂×#17989643	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14556665	C₂	1	2	yes	maximal, complemented
4	C₂²×#14556664	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556666	C₂²	1	24	yes	complemented
7	#17989643	C₂²	1	64	yes	direct product factor
8	C₂×#14556665	C₂²	1	24	yes	complemented
9	C₂×#14556664	C₂²	1	24	yes	complemented
10	C₂²×#11924031	C₂²	1	4	yes	complemented
11	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
12	C₂³×#10596831	C₂²	1	1	yes	characteristic
13	C₂²×C₈×D₇₃	C₄	1	12	yes	—
14	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
15	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
16	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₇₃	C₂²	1	12	yes	—
18	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
20	#14556666	C₂³	1	32	yes	complemented
21	#14556665	C₂³	1	32	yes	complemented
22	#14556664	C₂³	1	32	yes	complemented
23	C₂×#11924031	C₂³	1	24	yes	complemented
24	C₂⁴×D₇₃	C₈	1	1	yes	characteristic
25	C₂³×Dic₇₃	C₈	1	1	yes	characteristic
26	C₂²×C₄×D₇₃	C₈	1	12	yes	—
27	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
28	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
29	C₂²×C₄×D₇₃	C₂×C₄	1	12	yes	—
30	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10596831	C₂×C₄	1	6	yes	—
33	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₂²×#10596831	C₂³	1	1	yes	characteristic
35	C₂²×#10596831	C₂³	1	6	yes	—
36	C₂×C₈×D₇₃	C₂×C₄	1	6	yes	—
37	C₂×C₈×D₇₃	C₂×C₄	1	48	yes	—
38	C₂×C₈×D₇₃	C₂×C₄	1	12	yes	—
39	C₂×C₈×D₇₃	C₂×C₄	1	12	yes	—
40	C₂×C₈×D₇₃	C₂×C₄	1	6	yes	—
41	C₂×C₈×D₇₃	C₂³	1	6	yes	—
42	C₂×C₈×D₇₃	C₂³	1	16	yes	—
43	C₂×C₈×D₇₃	C₂³	1	6	yes	—
44	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	C₂×C₄	1	12	yes	—
46	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
48	#11924031	C₂⁴	1	16	yes	complemented
49	C₂³×D₇₃	—	2	8	no	—
50	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂³×D₇₃	C₂×C₈	1	12	yes	—
52	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₂×C₈	1	12	yes	—
54	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₄×D₇₃	C₂×C₈	1	12	yes	—
57	C₂×C₄×D₇₃	C₂×C₈	1	6	yes	—
58	C₂×C₄×D₇₃	C₂×C₈	1	48	yes	—
59	C₂×C₄×D₇₃	C₂×C₈	1	6	yes	—
60	C₂×C₄×D₇₃	C₂×C₈	1	12	yes	—
61	C₂×C₄×D₇₃	C₂²×C₄	1	16	yes	—
62	C₂×C₄×D₇₃	C₂²×C₄	1	6	yes	—
63	C₂×C₄×D₇₃	C₂²×C₄	1	6	yes	—
64	C₂×#10596831	C₂²×C₄	1	3	yes	—
65	C₂×#10596831	C₂²×C₄	1	12	yes	—
66	C₂×#10596831	C₂²×C₄	1	6	yes	—
67	C₂×#10596831	C₂⁴	1	3	yes	—
68	C₂×#10596831	C₂⁴	1	4	yes	—
69	C₈×D₇₃	C₂²×C₄	1	12	yes	—
70	C₈×D₇₃	C₂²×C₄	1	16	yes	—
71	C₈×D₇₃	C₂²×C₄	1	16	yes	—
72	C₈×D₇₃	C₂²×C₄	1	12	yes	—
73	C₈×D₇₃	C₂⁴	1	4	yes	—
74	C₈×D₇₃	C₂⁴	1	4	yes	—
75	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×C₈	1	6	yes	—
77	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₃	C₂²×C₄	1	16	yes	—
81	C₂×C₈×C₇₃	C₂²×C₄	1	6	yes	—
82	C₂×C₈×C₇₃	C₂²×C₄	1	6	yes	—
83	C₂³×C₇₃	—	2	4	no	—
84	C₂³×C₇₃	C₂²×C₈	1	6	yes	—
85	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
86	C₂²×D₇₃	—	2	4	no	—
87	C₂²×D₇₃	—	2	48	no	—
88	C₂²×D₇₃	M₃₂	1	8	yes	complemented
89	C₂²×D₇₃	C₂²×C₈	1	6	yes	—
90	C₂²×D₇₃	C₂²×C₈	1	16	yes	—
91	C₂²×D₇₃	C₂²×C₈	1	6	yes	—
92	C₂×C₄×C₇₃	C₂²×C₈	1	12	yes	—
93	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
94	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
95	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
97	C₂×Dic₇₃	C₂²×C₈	1	16	yes	—
98	C₂×Dic₇₃	C₂²×C₈	1	6	yes	—
99	C₂×Dic₇₃	C₂²×C₈	1	6	yes	—
100	C₄×D₇₃	C₂²×C₈	1	16	yes	—
101	C₄×D₇₃	C₂²×C₈	1	12	yes	—
102	C₄×D₇₃	C₂²×C₈	1	12	yes	—
103	C₄×D₇₃	C₂²×C₈	1	16	yes	—
104	C₄×D₇₃	C₂³×C₄	1	4	yes	—
105	C₄×D₇₃	C₂³×C₄	1	4	yes	—
106	C₈×C₇₃	C₂³×C₄	1	4	yes	—
107	C₈×C₇₃	C₂³×C₄	1	4	yes	—
108	#10596831	C₂³×C₄	1	3	yes	—
109	#10596831	C₂³×C₄	1	4	yes	—
110	#10596831	C₂⁵	1	1	yes	characteristic
111	C₂²×C₇₃	—	2	12	no	—
112	C₂²×C₇₃	C₂×M₃₂	1	4	yes	complemented
113	C₂²×C₇₃	C₂³×C₈	1	4	yes	—
114	C₂²×C₇₃	C₂³×C₈	1	3	yes	—
115	C₂×D₇₃	—	2	12	no	—
116	C₂×D₇₃	—	2	32	no	—
117	C₂×D₇₃	C₂×M₃₂	1	24	yes	complemented
118	C₂×D₇₃	C₂³×C₈	1	4	yes	—
119	C₂×D₇₃	C₂³×C₈	1	4	yes	—
120	C₄×C₇₃	C₂³×C₈	1	3	yes	—
121	C₄×C₇₃	C₂³×C₈	1	4	yes	—
122	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
123	Dic₇₃	C₂³×C₈	1	4	yes	—
124	Dic₇₃	C₂³×C₈	1	4	yes	—
125	C₂³×M₃₂	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	—	2	4	no	—
127	C₂×C₇₃	C₂²×M₃₂	1	6	yes	complemented
128	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
129	D₇₃	—	2	4	no	—
130	D₇₃	C₂²×M₃₂	1	8	yes	complemented
131	C₂⁴×C₈	—	73	1	no	—
132	C₂²×M₃₂	—	73	2	no	—
133	C₂²×M₃₂	—	73	24	no	—
134	C₂²×M₃₂	—	73	2	no	—
135	C₂³×C₁₆	—	73	2	no	—
136	C₇₃	C₂³×M₃₂	1	1	yes	characteristic, complemented, Sylow
137	C₂⁴×C₄	—	73	1	no	—
138	C₂³×C₈	—	73	1	no	—
139	C₂³×C₈	—	73	1	no	—
140	C₂³×C₈	—	73	12	no	—
141	C₂³×C₈	—	73	12	no	—
142	C₂³×C₈	—	73	1	no	—
143	C₂³×C₈	—	73	1	no	—
144	C₂³×C₈	—	73	1	no	—
145	C₂³×C₈	#64408	1	1	yes	characteristic
146	C₂×M₃₂	—	73	24	no	—
147	C₂×M₃₂	—	73	64	no	—
148	C₂×M₃₂	—	73	24	no	—
149	C₂²×C₁₆	—	73	4	no	—
150	C₂²×C₁₆	—	73	24	no	—
151	C₂⁵	—	73	1	no	—
152	C₂³×C₄	—	73	1	no	—
153	C₂³×C₄	—	73	1	no	—
154	C₂³×C₄	—	73	12	no	—
155	C₂³×C₄	—	73	1	no	—
156	C₂³×C₄	—	73	12	no	—
157	C₂³×C₄	—	73	1	no	—
158	C₂³×C₄	—	73	1	no	—
159	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
160	C₂²×C₈	—	73	6	no	—
161	C₂²×C₈	—	73	1	no	—
162	C₂²×C₈	—	73	1	no	—
163	C₂²×C₈	—	73	6	no	—
164	C₂²×C₈	—	73	6	no	—
165	C₂²×C₈	—	73	48	no	—
166	C₂²×C₈	—	73	12	no	—
167	C₂²×C₈	—	73	12	no	—
168	C₂²×C₈	—	73	6	no	—
169	C₂²×C₈	—	73	6	no	—
170	C₂²×C₈	—	73	6	no	—
171	C₂²×C₈	—	73	16	no	—
172	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
173	C₂²×C₈	C₂×#64408	1	12	yes	—
174	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
175	M₃₂	—	73	32	no	—
176	M₃₂	—	73	32	no	—
177	C₂×C₁₆	—	73	24	no	—
178	C₂×C₁₆	—	73	32	no	—
179	C₂⁴	—	73	1	no	—
180	C₂⁴	—	73	12	no	—
181	C₂⁴	—	73	1	no	—
182	C₂⁴	—	146	8	no	—
183	C₂⁴	#11924036	1	1	yes	characteristic
184	C₂²×C₄	—	73	12	no	—
185	C₂²×C₄	—	73	1	no	—
186	C₂²×C₄	—	73	1	no	—
187	C₂²×C₄	—	73	12	no	—
188	C₂²×C₄	—	73	6	no	—
189	C₂²×C₄	—	73	48	no	—
190	C₂²×C₄	—	73	12	no	—
191	C₂²×C₄	—	73	6	no	—
192	C₂²×C₄	—	73	6	no	—
193	C₂²×C₄	—	73	16	no	—
194	C₂²×C₄	—	73	6	no	—
195	C₂²×C₄	#11924036	1	6	yes	—
196	C₂²×C₄	#11924036	1	1	yes	characteristic
197	C₂²×C₄	C₂²×#64408	1	6	yes	—
198	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
199	C₂×C₈	—	73	3	no	—
200	C₂×C₈	—	73	12	no	—
201	C₂×C₈	—	73	6	no	—
202	C₂×C₈	—	73	3	no	—
203	C₂×C₈	—	73	4	no	—
204	C₂×C₈	—	73	12	no	—
205	C₂×C₈	—	73	12	no	—
206	C₂×C₈	—	73	16	no	—
207	C₂×C₈	—	73	16	no	—
208	C₂×C₈	—	73	4	no	—
209	C₂×C₈	—	73	4	no	—
210	C₂×C₈	C₂²×#64408	1	16	yes	—
211	C₂×C₈	C₂²×#64408	1	6	yes	—
212	C₂×C₈	C₂²×#64408	1	6	yes	—
213	C₁₆	—	73	16	no	—
214	C₂³	—	2	4	no	—
215	C₂³	—	73	8	no	—
216	C₂³	—	73	6	no	—
217	C₂³	—	73	16	no	—
218	C₂³	—	73	6	no	—
219	C₂³	—	146	48	no	—
220	C₂³	—	146	4	no	—
221	C₂³	C₂×#11924036	1	6	yes	—
222	C₂³	C₂×#11924036	1	1	yes	characteristic
223	C₂×C₄	—	73	16	no	—
224	C₂×C₄	—	73	6	no	—
225	C₂×C₄	—	73	6	no	—
226	C₂×C₄	—	73	16	no	—
227	C₂×C₄	—	73	12	no	—
228	C₂×C₄	—	73	12	no	—
229	C₂×C₄	—	73	16	no	—
230	C₂×C₄	—	73	4	no	—
231	C₂×C₄	—	73	4	no	—
232	C₂×C₄	C₂×#11924036	1	12	yes	—
233	C₂×C₄	C₂×#11924036	1	6	yes	—
234	C₂×C₄	C₂×#11924036	1	3	yes	—
235	C₂×C₄	C₂³×#64408	1	4	yes	—
236	C₂×C₄	C₂³×#64408	1	3	yes	—
237	C₈	—	73	1	no	—
238	C₈	—	73	3	no	—
239	C₈	—	73	4	no	—
240	C₈	C₂³×#64408	1	4	yes	—
241	C₈	C₂³×#64408	1	4	yes	—
242	C₂²	—	2	12	no	—
243	C₂²	—	73	24	no	—
244	C₂²	—	73	4	no	—
245	C₂²	—	73	4	no	—
246	C₂²	—	146	32	no	—
247	C₂²	—	146	12	no	—
248	C₂²	#17989643	1	4	yes	direct product factor
249	C₂²	C₂²×#11924036	1	4	yes	—
250	C₂²	C₂²×#11924036	1	3	yes	—
251	C₄	—	73	4	no	—
252	C₄	—	73	4	no	—
253	C₄	C₂²×#11924036	1	3	yes	—
254	C₄	C₂²×#11924036	1	4	yes	—
255	C₄	C₂⁴×#64408	1	1	yes	characteristic
256	C₂	—	2	4	no	—
257	C₂	—	73	8	no	—
258	C₂	—	146	4	no	—
259	C₂	C₂×#17989643	1	6	yes	direct product factor
260	C₂	C₂³×#11924036	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×#64408
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂⁴×D₇₃	C₈
cosocle	#10596831	C₂⁵
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	C₂³×M₃₂
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14556666	C₂	semidirect
2. C₂×#17989643	C₂	direct (N × H)
3. C₂²×#14556665	C₂	semidirect
4. C₂²×#14556664	C₂	semidirect
6. C₂×#14556666	C₂²	semidirect
7. #17989643	C₂²	direct (N × H)
8. C₂×#14556665	C₂²	semidirect
9. C₂×#14556664	C₂²	semidirect
10. C₂²×#11924031	C₂²	semidirect
20. #14556666	C₂³	semidirect
21. #14556665	C₂³	semidirect
22. #14556664	C₂³	semidirect
23. C₂×#11924031	C₂³	semidirect
48. #11924031	C₂⁴	semidirect
88. C₂²×D₇₃	M₃₂	semidirect
112. C₂²×C₇₃	C₂×M₃₂	semidirect
117. C₂×D₇₃	C₂×M₃₂	semidirect
127. C₂×C₇₃	C₂²×M₃₂	semidirect
130. D₇₃	C₂²×M₃₂	semidirect
136. C₇₃	C₂³×M₃₂	semidirect
248. C₂²	#17989643	direct (N × H)
259. C₂	C₂×#17989643	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×#14556666	2
C₂×#17989643	2
C₂²×#14556665	2
C₂²×#14556664	2
C₂³×C₈×D₇₃	2
C₂³×M₃₂	73

Subgroups — C22×#17989643 #34032678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989643 #34032678