Subgroups — C₂×#22608216 #33051564

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988638	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#22608216	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988308	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×#10596831	C₂	1	2	yes	maximal, complemented
6	C₂×#322×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₇₃	C₄	1	2	yes	—
8	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
9	#17988638	C₄	1	8	yes	—
10	#17988638	C₂²	1	8	yes	—
11	C₂×#14556520	C₄	1	4	yes	—
12	C₂×#14556503	C₄	1	4	yes	—
13	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₂²	1	2	yes	—
15	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
16	C₄×C₈×D₇₃	C₄	1	4	yes	—
17	C₄×C₈×D₇₃	C₂²	1	4	yes	—
18	#17988308	C₂²	1	4	yes	complemented
19	C₈×#10596831	C₂²	1	8	yes	complemented
20	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#322×C₇₃	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₇₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10596831	C₂×C₄	1	4	yes	—
26	C₂²×#10596831	C₂×C₄	1	2	yes	—
27	C₂×C₈×D₇₃	C₈	1	16	yes	complemented
28	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
31	C₄²×D₇₃	C₂×C₄	1	4	yes	—
32	C₄²×D₇₃	C₂×C₄	1	2	yes	—
33	C₄²×D₇₃	C₂³	1	2	yes	—
34	#14556520	C₂×C₄	1	16	yes	—
35	#14556503	C₂×C₄	1	16	yes	—
36	C₄×#10596831	C₂×C₄	1	4	yes	—
37	C₄×#10596831	C₂×C₄	1	2	yes	—
38	C₄×#10596831	C₂³	1	4	yes	—
39	C₄×#10596831	C₂³	1	2	yes	—
40	C₈×Dic₇₃	C₂×C₄	1	8	yes	—
41	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
45	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₇₃	C₂³	1	4	yes	—
47	C₄×C₈×C₇₃	C₂³	1	2	yes	—
48	C₂³×D₇₃	C₄²	1	1	yes	characteristic
49	C₂²×Dic₇₃	C₄²	1	2	yes	—
50	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
51	C₂×C₄×D₇₃	—	2	8	no	—
52	C₂×C₄×D₇₃	C₄²	1	4	yes	—
53	C₂×C₄×D₇₃	C₄²	1	2	yes	—
54	C₂×C₄×D₇₃	C₂×C₈	1	8	yes	—
55	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
56	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
57	C₂×#10596831	—	2	8	no	—
58	C₂×#10596831	C₂×C₈	1	8	yes	—
59	C₂×#10596831	C₂²×C₄	1	8	yes	—
60	C₂×#10596831	C₂²×C₄	1	4	yes	—
61	C₈×D₇₃	C₂×C₈	1	32	yes	complemented
62	C₄×Dic₇₃	C₄²	1	2	yes	—
63	C₄×Dic₇₃	C₄²	1	4	yes	—
64	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
65	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₇₃	—	2	8	no	—
69	C₂×C₈×C₇₃	C₂×C₈	1	8	yes	—
70	C₂×C₈×C₇₃	C₂²×C₄	1	8	yes	—
71	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
72	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
75	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
76	C₂²×D₇₃	—	2	4	no	—
77	C₂²×D₇₃	C₄×C₈	1	4	yes	—
78	C₂²×D₇₃	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
82	C₂×C₄×C₇₃	#131	1	8	yes	—
83	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
85	C₂×Dic₇₃	—	2	4	no	—
86	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
87	C₂×Dic₇₃	C₂×C₄²	1	4	yes	—
88	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
89	C₄×D₇₃	—	2	8	no	—
90	C₄×D₇₃	—	2	8	no	—
91	C₄×D₇₃	C₄×C₈	1	8	yes	—
92	C₄×D₇₃	C₂²×C₈	1	8	yes	—
93	C₈×C₇₃	—	2	8	no	—
94	C₈×C₇₃	C₂²×C₈	1	8	yes	—
95	#10596831	—	2	8	no	—
96	#10596831	C₂²×C₈	1	8	yes	—
97	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
98	C₂²×C₇₃	#405	1	4	yes	—
99	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
100	C₂×D₇₃	—	2	4	no	—
101	C₂×D₇₃	—	2	8	no	—
102	C₂×D₇₃	C₂×C₄×C₈	1	4	yes	—
103	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
104	C₄×C₇₃	#405	1	4	yes	—
105	C₄×C₇₃	C₂³×C₈	1	2	yes	—
106	C₄×C₇₃	C₂×#131	1	4	yes	—
107	Dic₇₃	—	2	4	no	—
108	Dic₇₃	C₂×C₄×C₈	1	4	yes	—
109	C₂×#1450	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	#1450	1	4	yes	complemented
111	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
112	C₂×C₇₃	C₂×#405	1	2	yes	—
113	D₇₃	—	2	4	no	—
114	C₂×#431	—	73	2	no	—
115	C₂²×C₄×C₈	—	73	1	no	—
116	#1450	—	73	8	no	—
117	C₂×#322	—	73	1	no	—
118	C₂×#322	D₇₃	1	1	yes	characteristic, complemented
119	C₂×C₈²	—	73	2	no	—
120	C₇₃	C₂×#1450	1	1	yes	characteristic, complemented, Sylow
121	C₂³×C₈	—	73	2	no	—
122	C₂²×C₄²	—	73	1	no	—
123	#431	—	73	8	no	—
124	#431	—	73	8	no	—
125	C₂×C₄⋊C₈	—	73	4	no	—
126	C₂×C₂²⋊C₈	—	73	4	no	—
127	C₂×C₄×C₈	—	73	2	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₂×C₄×C₈	—	73	2	no	—
130	C₂×C₄×C₈	—	73	4	no	—
131	C₂×C₄×C₈	—	73	4	no	—
132	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
134	#322	—	73	4	no	—
135	#322	C₂×D₇₃	1	4	yes	complemented
136	C₈²	—	73	8	no	—
137	C₂³×C₄	—	73	1	no	—
138	C₂³×C₄	—	73	2	no	—
139	C₂²×C₈	—	73	2	no	—
140	C₂²×C₈	—	73	16	no	—
141	C₂²×C₈	—	73	8	no	—
142	C₂²×C₈	—	73	4	no	—
143	C₂²×C₈	C₄×D₇₃	1	4	yes	—
144	C₂²×C₈	C₄×D₇₃	1	2	yes	—
145	C₂×C₄²	—	73	1	no	—
146	C₂×C₄²	—	73	2	no	—
147	C₂×C₄²	—	73	2	no	—
148	C₂×C₄²	—	73	4	no	—
149	C₂×C₄²	—	73	2	no	—
150	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
151	C₄⋊C₈	—	73	16	no	—
152	C₂²⋊C₈	—	73	16	no	—
153	C₄×C₈	—	73	8	no	—
154	C₄×C₈	—	73	2	no	—
155	C₄×C₈	—	73	4	no	—
156	C₄×C₈	—	73	2	no	—
157	C₄×C₈	—	73	4	no	—
158	C₄×C₈	C₄×D₇₃	1	2	yes	—
159	C₄×C₈	C₄×D₇₃	1	4	yes	—
160	C₄×C₈	C₂²×D₇₃	1	4	yes	—
161	C₄×C₈	C₂²×D₇₃	1	2	yes	—
162	C₂⁴	—	73	1	no	—
163	C₂²×C₄	—	73	1	no	—
164	C₂²×C₄	—	73	2	no	—
165	C₂²×C₄	—	73	8	no	—
166	C₂²×C₄	—	73	2	no	—
167	C₂²×C₄	—	73	2	no	—
168	C₂²×C₄	—	73	4	no	—
169	C₂²×C₄	—	73	4	no	—
170	C₂²×C₄	—	146	8	no	—
171	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
173	C₂×C₈	—	2	8	no	—
174	C₂×C₈	—	73	8	no	—
175	C₂×C₈	—	73	4	no	—
176	C₂×C₈	—	73	32	no	—
177	C₂×C₈	—	73	8	no	—
178	C₂×C₈	—	146	8	no	—
179	C₂×C₈	C₈×D₇₃	1	8	yes	—
180	C₂×C₈	C₂×C₄×D₇₃	1	8	yes	—
181	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
182	C₄²	—	73	4	no	—
183	C₄²	—	73	2	no	—
184	C₄²	—	73	2	no	—
185	C₄²	—	73	4	no	—
186	C₄²	C₂×C₄×D₇₃	1	2	yes	—
187	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
188	C₄²	C₂³×D₇₃	1	1	yes	characteristic
189	C₂³	—	73	2	no	—
190	C₂³	—	73	4	no	—
191	C₂³	—	146	4	no	—
192	C₂³	C₄²×D₇₃	1	1	yes	characteristic
193	C₂×C₄	—	73	2	no	—
194	C₂×C₄	—	73	4	no	—
195	C₂×C₄	—	73	8	no	—
196	C₂×C₄	—	73	8	no	—
197	C₂×C₄	—	73	4	no	—
198	C₂×C₄	—	146	4	no	—
199	C₂×C₄	—	146	8	no	—
200	C₂×C₄	—	146	8	no	—
201	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
202	C₂×C₄	C₄²×D₇₃	1	2	yes	—
203	C₂×C₄	C₂×C₈×D₇₃	1	4	yes	—
204	C₂×C₄	#14556599	1	8	yes	—
205	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
207	C₈	—	2	8	no	—
208	C₈	—	73	8	no	—
209	C₈	—	146	8	no	—
210	C₈	C₂×C₈×D₇₃	1	8	yes	—
211	C₂²	—	73	4	no	—
212	C₂²	—	146	8	no	—
213	C₂²	—	146	4	no	—
214	C₂²	C₄×C₈×D₇₃	1	2	yes	—
215	C₂²	#17988544	1	4	yes	—
216	C₂²	C₂×C₄²×D₇₃	1	1	yes	characteristic
217	C₄	—	73	4	no	—
218	C₄	—	146	4	no	—
219	C₄	C₄×C₈×D₇₃	1	2	yes	—
220	C₄	#17988544	1	4	yes	—
221	C₄	C₂²×C₈×D₇₃	1	2	yes	—
222	C₄	C₂×#14556599	1	4	yes	—
223	C₂	—	146	4	no	—
224	C₂	#22608216	1	4	yes	direct product factor
225	C₂	C₂×C₄×C₈×D₇₃	1	1	yes	characteristic
226	C₂	C₂×#17988544	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₃
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#322×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₄²
prime radical	C₂³×D₇₃	C₄²
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×#322	D₇₃
hypocenter	C₇₃	C₂×#1450
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #22608216	C₂	direct (N × H)
4. C₂×#17988308	C₂	semidirect
5. C₂×C₈×#10596831	C₂	semidirect
6. C₂×#322×C₇₃	C₂	semidirect
18. #17988308	C₂²	semidirect
19. C₈×#10596831	C₂²	semidirect
22. #322×C₇₃	C₂²	semidirect
27. C₂×C₈×D₇₃	C₈	semidirect
61. C₈×D₇₃	C₂×C₈	semidirect
110. C₂×C₇₃	#1450	semidirect
118. C₂×#322	D₇₃	semidirect
120. C₇₃	C₂×#1450	semidirect
135. #322	C₂×D₇₃	semidirect
224. C₂	#22608216	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×#17988638	2
C₂×C₄×C₈×D₇₃	2
#22608216	2
C₂×#17988308	2
C₂×C₈×#10596831	2
C₂×#322×C₇₃	2
C₂×#1450	73

Subgroups — C2×#22608216 #33051564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22608216 #33051564