Subgroups — C₂×#22609617 #33083310

Order 18688.

1614 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556502	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556501	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₇₃	C₂	1	2	yes	maximal
4	C₂×#17988433	C₂	1	1	yes	maximal, characteristic
5	#22609617	C₂	1	8	yes	maximal, direct product factor
6	C₂²×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10596831	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₇₃	C₂²	1	2	yes	—
9	C₂×#14556502	C₄	1	8	yes	complemented
10	C₂×#14556502	C₂²	1	4	yes	—
11	C₂×#14556501	C₄	1	16	yes	complemented
12	C₂×#14556501	C₂²	1	8	yes	—
13	C₂×C₈×Dic₇₃	C₄	1	16	yes	complemented
14	C₂×C₈×Dic₇₃	C₂²	1	8	yes	—
15	#17988433	C₄	1	4	yes	—
16	#17988433	C₂²	1	4	yes	—
17	C₂×#14556498	C₄	1	4	yes	—
18	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₃	C₄	1	8	yes	complemented
22	C₂×C₄×C₈×C₇₃	C₂²	1	4	yes	—
23	C₂³×Dic₇₃	C₂×C₄	1	4	yes	—
24	C₂²×#10596831	C₂×C₄	1	8	yes	—
25	C₂²×#10596831	C₂×C₄	1	8	yes	—
26	C₂²×#10596831	C₂×C₄	1	8	yes	—
27	C₂²×#10596831	C₂³	1	4	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	8	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	8	yes	—
31	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
32	#14556502	C₂×C₄	1	16	yes	complemented
33	#14556501	C₂×C₄	1	32	yes	complemented
34	C₈×Dic₇₃	C₂×C₄	1	32	yes	complemented
35	#14556498	C₂×C₄	1	16	yes	—
36	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	C₂×C₄	1	8	yes	—
39	C₂²×C₈×C₇₃	C₂×C₄	1	8	yes	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
48	C₄×C₈×C₇₃	C₂×C₄	1	16	yes	complemented
49	C₂²×Dic₇₃	—	2	16	no	—
50	C₂²×Dic₇₃	C₄²	1	16	yes	—
51	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
52	C₂×#10596831	C₄²	1	32	yes	—
53	C₂×#10596831	C₂²×C₄	1	8	yes	—
54	C₂×#10596831	C₂²×C₄	1	8	yes	—
55	C₂×#10596831	C₂²×C₄	1	8	yes	—
56	C₄×Dic₇₃	C₄²	1	16	yes	—
57	C₄×Dic₇₃	C₂²×C₄	1	8	yes	—
58	C₄×Dic₇₃	C₂²×C₄	1	8	yes	—
59	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	—	2	8	no	—
61	C₂²×C₄×C₇₃	C₄²	1	8	yes	—
62	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₄²	1	16	yes	—
70	C₂×C₈×C₇₃	C₄²	1	16	yes	—
71	C₂×C₈×C₇₃	C₂²×C₄	1	8	yes	—
72	C₂×C₈×C₇₃	C₂²×C₄	1	8	yes	—
73	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
75	C₄²×C₇₃	C₄²	1	8	yes	—
76	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
77	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
78	C₂³×C₇₃	C₂×C₄²	1	4	yes	—
79	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
80	C₂³×C₇₃	#131	1	8	yes	—
81	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	—	2	8	no	—
83	C₂×C₄×C₇₃	—	2	16	no	—
84	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
85	C₂×C₄×C₇₃	C₂×C₄²	1	8	yes	—
86	C₂×C₄×C₇₃	C₂×C₄²	1	8	yes	—
87	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
90	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
92	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
93	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×Dic₇₃	—	2	16	no	—
95	C₂×Dic₇₃	—	2	32	no	—
96	C₂×Dic₇₃	C₂×C₄²	1	16	yes	—
97	C₈×C₇₃	C₂×C₄²	1	8	yes	—
98	C₈×C₇₃	C₂×C₄²	1	8	yes	—
99	#10596831	C₂×C₄²	1	16	yes	—
100	C₂²×C₇₃	C₄³	1	4	yes	—
101	C₂²×C₇₃	#405	1	16	yes	—
102	C₂²×C₇₃	#405	1	8	yes	—
103	C₂²×C₇₃	C₂²×C₄²	1	2	yes	—
104	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
105	C₂²×C₇₃	C₂×#131	1	4	yes	—
106	C₄×C₇₃	—	2	8	no	—
107	C₄×C₇₃	C₄³	1	4	yes	—
108	C₄×C₇₃	C₂²×C₄²	1	2	yes	—
109	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
110	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
111	Dic₇₃	—	2	16	no	—
112	C₂×#1726	—	73	1	no	maximal, Sylow
113	C₂×C₇₃	#1726	1	8	yes	complemented
114	C₂×C₇₃	C₂×C₄³	1	1	yes	characteristic
115	C₂×C₇₃	C₂×#405	1	4	yes	—
116	C₂×C₇₃	C₂×#405	1	2	yes	—
117	C₂²×#97	—	73	1	no	—
118	C₂²×#97	—	73	2	no	—
119	C₂²×C₄×C₈	—	73	2	no	—
120	C₂²×C₄×C₈	D₇₃	1	1	yes	characteristic
121	C₂×#376	—	73	1	no	—
122	#1726	—	73	8	no	—
123	C₇₃	C₂×#1726	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₈	—	73	2	no	—
125	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
126	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
127	C₂²×C₄²	—	73	2	no	—
128	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
129	C₂×#97	—	73	16	no	—
130	C₂×#97	—	73	8	no	—
131	C₂×#97	—	73	8	no	—
132	C₂×#97	—	73	4	no	—
133	C₂×C₄×C₈	—	73	16	no	—
134	C₂×C₄×C₈	—	73	8	no	—
135	C₂×C₄×C₈	Dic₇₃	1	8	yes	complemented
136	C₂×C₄×C₈	C₂×D₇₃	1	4	yes	—
137	#376	—	73	4	no	—
138	#376	—	73	4	no	—
139	C₂×#95	—	73	4	no	—
140	C₂³×C₄	—	73	4	no	—
141	C₂³×C₄	C₄×D₇₃	1	2	yes	—
142	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
143	C₂²×C₈	—	73	8	no	—
144	C₂²×C₈	—	73	8	no	—
145	C₂²×C₈	—	73	8	no	—
146	C₂²×C₈	—	73	4	no	—
147	C₂²×C₈	C₄×D₇₃	1	8	yes	—
148	C₂²×C₈	C₄×D₇₃	1	8	yes	—
149	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
150	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
151	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
152	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
153	C₂×C₄²	—	73	4	no	—
154	C₂×C₄²	—	73	8	no	—
155	C₂×C₄²	—	73	8	no	—
156	C₂×C₄²	—	73	4	no	—
157	C₂×C₄²	C₄×D₇₃	1	2	yes	—
158	C₂×C₄²	C₄×D₇₃	1	4	yes	—
159	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
160	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
161	#97	—	73	32	no	—
162	#97	—	73	16	no	—
163	C₄×C₈	—	73	32	no	—
164	C₄×C₈	C₂×Dic₇₃	1	16	yes	complemented
165	#95	—	73	16	no	—
166	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	73	16	no	—
169	C₂²×C₄	—	73	8	no	—
170	C₂²×C₄	—	146	16	no	—
171	C₂²×C₄	C₄×Dic₇₃	1	8	yes	—
172	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
173	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
175	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
176	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
177	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
178	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
179	C₂×C₈	—	73	32	no	—
180	C₂×C₈	—	73	8	no	—
181	C₂×C₈	—	73	8	no	—
182	C₂×C₈	—	73	8	no	—
183	C₂×C₈	C₄×Dic₇₃	1	16	yes	—
184	C₂×C₈	C₄×Dic₇₃	1	16	yes	—
185	C₂×C₈	C₂×C₄×D₇₃	1	8	yes	—
186	C₂×C₈	C₂×C₄×D₇₃	1	8	yes	—
187	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
188	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
189	C₄²	—	73	16	no	—
190	C₄²	—	73	8	no	—
191	C₄²	—	73	8	no	—
192	C₄²	C₄×Dic₇₃	1	8	yes	—
193	C₄²	C₂×C₄×D₇₃	1	4	yes	—
194	C₄²	C₂²×Dic₇₃	1	4	yes	—
195	C₂³	C₄²×D₇₃	1	4	yes	—
196	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
197	C₂³	#14556597	1	8	yes	—
198	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
199	C₂×C₄	—	2	16	no	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	73	16	no	—
202	C₂×C₄	—	146	32	no	—
203	C₂×C₄	—	146	16	no	—
204	C₂×C₄	C₄²×D₇₃	1	4	yes	—
205	C₂×C₄	C₄²×D₇₃	1	8	yes	—
206	C₂×C₄	C₂×C₄×Dic₇₃	1	8	yes	—
207	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
208	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
209	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
210	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
212	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
213	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
214	C₈	—	73	16	no	—
215	C₈	C₂×C₄×Dic₇₃	1	8	yes	—
216	C₈	C₂×C₄×Dic₇₃	1	8	yes	—
217	C₂²	C₄²×Dic₇₃	1	4	yes	—
218	C₂²	#17988541	1	16	yes	—
219	C₂²	#17988542	1	8	yes	—
220	C₂²	C₂×C₄²×D₇₃	1	2	yes	—
221	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
222	C₂²	C₂×#14556597	1	4	yes	—
223	C₄	—	2	8	no	—
224	C₄	—	146	16	no	—
225	C₄	C₄²×Dic₇₃	1	4	yes	—
226	C₄	C₂×C₄²×D₇₃	1	2	yes	—
227	C₄	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
228	C₄	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
229	C₂	#22609617	1	8	yes	direct product factor
230	C₂	C₂×C₄²×Dic₇₃	1	1	yes	characteristic
231	C₂	C₂×#17988541	1	4	yes	—
232	C₂	C₂×#17988542	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₇₃
commutator	C₂×C₇₃	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂²×C₄×C₈×C₇₃	C₂
socle, prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₃
hypocenter	C₇₃	C₂×#1726
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
5. #22609617	C₂	direct (N × H)
9. C₂×#14556502	C₄	semidirect
11. C₂×#14556501	C₄	semidirect
13. C₂×C₈×Dic₇₃	C₄	semidirect
21. C₂×C₄×C₈×C₇₃	C₄	semidirect
32. #14556502	C₂×C₄	semidirect
33. #14556501	C₂×C₄	semidirect
34. C₈×Dic₇₃	C₂×C₄	semidirect
48. C₄×C₈×C₇₃	C₂×C₄	semidirect
113. C₂×C₇₃	#1726	semidirect
123. C₇₃	C₂×#1726	semidirect
135. C₂×C₄×C₈	Dic₇₃	semidirect
164. C₄×C₈	C₂×Dic₇₃	semidirect
229. C₂	#22609617	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×#14556502	2
C₂²×#14556501	2
C₂²×C₈×Dic₇₃	2
C₂×#17988433	2
#22609617	2
C₂²×C₄×C₈×C₇₃	2
C₂×#1726	73

Subgroups — C2×#22609617 #33083310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22609617 #33083310