Subgroups — C₂²×C₄×C₄⋊Dic₇₃ #33244162

Order 18688.

5758 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	C₂³×C₄⋊Dic₇₃	C₂	1	2	yes	maximal
2	C₂³×C₄×Dic₇₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₄⋊Dic₇₃	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14556498	C₂	1	2	yes	maximal
5	C₂³×C₄²×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₇₃	C₂²	1	4	yes	—
7	C₂²×C₄⋊Dic₇₃	C₄	1	32	yes	direct product factor
8	C₂²×C₄⋊Dic₇₃	C₂²	1	24	yes	—
9	C₂²×C₄×Dic₇₃	—	2	16	no	—
10	C₂²×C₄×Dic₇₃	C₂²	1	24	yes	—
11	C₄×C₄⋊Dic₇₃	C₂²	1	64	yes	direct product factor
12	C₂×#14556498	C₂²	1	24	yes	—
13	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₇₃	C₄	1	16	yes	complemented
17	C₂²×C₄²×C₇₃	C₂²	1	12	yes	—
18	C₂³×Dic₇₃	—	2	16	no	—
19	C₂³×Dic₇₃	—	2	16	no	—
20	C₂³×Dic₇₃	C₂×C₄	1	32	yes	—
21	C₂³×Dic₇₃	C₂³	1	24	yes	—
22	C₂×C₄⋊Dic₇₃	C₂×C₄	1	192	yes	direct product factor
23	C₂×C₄⋊Dic₇₃	C₂³	1	32	yes	—
24	C₂×C₄×Dic₇₃	—	2	96	no	—
25	C₂×C₄×Dic₇₃	C₂³	1	32	yes	—
26	#14556498	C₂³	1	32	yes	—
27	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₇₃	—	2	8	no	—
29	C₂³×C₄×C₇₃	C₂×C₄	1	8	yes	—
30	C₂³×C₄×C₇₃	C₂×C₄	1	16	yes	—
31	C₂³×C₄×C₇₃	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₇₃	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₇₃	D₄	1	8	yes	—
34	C₂³×C₄×C₇₃	Q₈	1	8	yes	—
35	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
36	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
37	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
38	C₂×C₄²×C₇₃	C₂×C₄	1	96	yes	complemented
39	C₂×C₄²×C₇₃	C₂³	1	16	yes	—
40	C₂²×Dic₇₃	—	2	128	no	—
41	C₂²×Dic₇₃	—	2	48	no	—
42	C₂²×Dic₇₃	—	2	48	no	—
43	C₂²×Dic₇₃	C₂²×C₄	1	96	yes	—
44	C₂²×Dic₇₃	C₂⁴	1	16	yes	—
45	C₄⋊Dic₇₃	C₂²×C₄	1	128	yes	direct product factor
46	C₄×Dic₇₃	—	2	64	no	—
47	C₂⁴×C₇₃	C₂²×C₄	1	8	yes	—
48	C₂⁴×C₇₃	C₂²×C₄	1	4	yes	—
49	C₂⁴×C₇₃	C₂×D₄	1	4	yes	—
50	C₂⁴×C₇₃	C₂×Q₈	1	4	yes	—
51	C₂⁴×C₇₃	#41	1	8	yes	—
52	C₂⁴×C₇₃	C₂⁴	1	3	yes	—
53	C₂²×C₄×C₇₃	—	2	32	no	—
54	C₂²×C₄×C₇₃	—	2	24	no	—
55	C₂²×C₄×C₇₃	C₄²	1	64	yes	complemented
56	C₂²×C₄×C₇₃	C₄⋊C₄	1	64	yes	complemented
57	C₂²×C₄×C₇₃	C₂²×C₄	1	48	yes	—
58	C₂²×C₄×C₇₃	C₂²×C₄	1	24	yes	—
59	C₂²×C₄×C₇₃	C₂²×C₄	1	24	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	24	yes	—
61	C₂²×C₄×C₇₃	C₂×D₄	1	24	yes	—
62	C₂²×C₄×C₇₃	C₂×Q₈	1	24	yes	—
63	C₂²×C₄×C₇₃	C₂⁴	1	4	yes	—
64	C₂²×C₄×C₇₃	C₂⁴	1	4	yes	—
65	C₂²×C₄×C₇₃	C₂⁴	1	4	yes	—
66	C₄²×C₇₃	C₂²×C₄	1	64	yes	complemented
67	C₂³×C₇₃	C₂×C₄²	1	16	yes	—
68	C₂³×C₇₃	C₂×C₄⋊C₄	1	16	yes	—
69	C₂³×C₇₃	#117	1	16	yes	—
70	C₂³×C₇₃	C₄×D₄	1	32	yes	—
71	C₂³×C₇₃	C₄×Q₈	1	32	yes	—
72	C₂³×C₇₃	C₂³×C₄	1	12	yes	—
73	C₂³×C₇₃	C₂³×C₄	1	6	yes	—
74	C₂³×C₇₃	C₂²×D₄	1	6	yes	—
75	C₂³×C₇₃	C₂²×Q₈	1	6	yes	—
76	C₂³×C₇₃	C₂×#41	1	12	yes	—
77	C₂³×C₇₃	C₂⁵	1	1	yes	characteristic
78	C₂×C₄×C₇₃	—	2	48	no	—
79	C₂×C₄×C₇₃	—	2	8	no	—
80	C₂×C₄×C₇₃	C₂×C₄²	1	96	yes	complemented
81	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	96	yes	complemented
82	C₂×C₄×C₇₃	C₂³×C₄	1	16	yes	—
83	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
84	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
85	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₇₃	C₂²×D₄	1	8	yes	—
87	C₂×C₄×C₇₃	C₂²×Q₈	1	8	yes	—
88	C₂×Dic₇₃	—	2	192	no	—
89	C₂×Dic₇₃	—	2	16	no	—
90	C₂×Dic₇₃	—	2	16	no	—
91	C₂×Dic₇₃	C₂³×C₄	1	32	yes	—
92	C₂²×C₇₃	C₄×C₄⋊C₄	1	64	yes	complemented
93	C₂²×C₇₃	C₂²×C₄²	1	12	yes	—
94	C₂²×C₇₃	C₂²×C₄⋊C₄	1	12	yes	—
95	C₂²×C₇₃	C₂×#117	1	12	yes	—
96	C₂²×C₇₃	C₂×C₄×D₄	1	24	yes	—
97	C₂²×C₇₃	C₂×C₄×Q₈	1	24	yes	—
98	C₂²×C₇₃	C₂⁴×C₄	1	2	yes	—
99	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
100	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
101	C₂²×C₇₃	C₂³×Q₈	1	1	yes	characteristic
102	C₂²×C₇₃	C₂²×#41	1	2	yes	—
103	C₄×C₇₃	—	2	8	no	—
104	C₄×C₇₃	C₂²×C₄²	1	16	yes	complemented
105	C₄×C₇₃	C₂²×C₄⋊C₄	1	16	yes	complemented
106	Dic₇₃	—	2	32	no	—
107	C₂²×C₄×C₄⋊C₄	—	73	1	no	maximal, Sylow
108	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	24	yes	complemented
109	C₂×C₇₃	C₂³×C₄²	1	1	yes	characteristic
110	C₂×C₇₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₇₃	C₂²×#117	1	1	yes	characteristic
112	C₂×C₇₃	C₂²×C₄×D₄	1	2	yes	—
113	C₂×C₇₃	C₂²×C₄×Q₈	1	2	yes	—
114	C₂³×C₄⋊C₄	—	73	2	no	—
115	C₂³×C₄²	—	73	2	no	—
116	C₂³×C₄²	D₇₃	1	1	yes	characteristic
117	C₂×C₄×C₄⋊C₄	—	73	24	no	—
118	C₂²×#95	—	73	2	no	—
119	C₇₃	C₂²×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
120	C₂⁴×C₄	—	73	4	no	—
121	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
122	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
123	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
124	C₂²×C₄⋊C₄	—	73	32	no	—
125	C₂²×C₄⋊C₄	—	73	24	no	—
126	C₂²×C₄²	—	73	24	no	—
127	C₂²×C₄²	—	146	16	no	—
128	C₂²×C₄²	Dic₇₃	1	16	yes	complemented
129	C₂²×C₄²	C₂×D₇₃	1	12	yes	—
130	C₄×C₄⋊C₄	—	73	64	no	—
131	C₂×#95	—	73	24	no	—
132	C₂⁵	C₂²×D₇₃	1	1	yes	characteristic
133	C₂³×C₄	—	2	8	no	—
134	C₂³×C₄	—	73	32	no	—
135	C₂³×C₄	—	73	24	no	—
136	C₂³×C₄	—	146	16	no	—
137	C₂³×C₄	—	146	16	no	—
138	C₂³×C₄	Dic₁₄₆	1	8	yes	—
139	C₂³×C₄	C₄×D₇₃	1	16	yes	—
140	C₂³×C₄	D₂₉₂	1	8	yes	—
141	C₂³×C₄	C₂×Dic₇₃	1	8	yes	—
142	C₂³×C₄	C₂×Dic₇₃	1	8	yes	—
143	C₂³×C₄	C₂×Dic₇₃	1	8	yes	—
144	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
145	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
146	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
147	C₂×C₄⋊C₄	—	73	192	no	—
148	C₂×C₄⋊C₄	—	73	32	no	—
149	C₂×C₄²	—	73	32	no	—
150	C₂×C₄²	—	146	96	no	—
151	C₂×C₄²	C₂×Dic₇₃	1	96	yes	complemented
152	C₂×C₄²	C₂²×D₇₃	1	16	yes	—
153	#95	—	73	32	no	—
154	C₂⁴	C₂×C₄×D₇₃	1	8	yes	—
155	C₂⁴	C₂²×Dic₇₃	1	4	yes	—
156	C₂⁴	C₂×D₂₉₂	1	4	yes	—
157	C₂⁴	C₂×Dic₁₄₆	1	4	yes	—
158	C₂⁴	#11924028	1	8	yes	—
159	C₂⁴	C₂³×D₇₃	1	3	yes	—
160	C₂²×C₄	—	2	32	no	—
161	C₂²×C₄	—	2	24	no	—
162	C₂²×C₄	—	73	96	no	—
163	C₂²×C₄	—	73	16	no	—
164	C₂²×C₄	—	146	128	no	—
165	C₂²×C₄	—	146	48	no	—
166	C₂²×C₄	—	146	48	no	—
167	C₂²×C₄	C₄×Dic₇₃	1	64	yes	complemented
168	C₂²×C₄	C₄⋊Dic₇₃	1	64	yes	direct product factor
169	C₂²×C₄	C₂×C₄×D₇₃	1	48	yes	—
170	C₂²×C₄	C₂²×Dic₇₃	1	24	yes	—
171	C₂²×C₄	C₂²×Dic₇₃	1	24	yes	—
172	C₂²×C₄	C₂²×Dic₇₃	1	24	yes	—
173	C₂²×C₄	C₂×D₂₉₂	1	24	yes	—
174	C₂²×C₄	C₂×Dic₁₄₆	1	24	yes	—
175	C₂²×C₄	C₂³×D₇₃	1	4	yes	—
176	C₂²×C₄	C₂³×D₇₃	1	4	yes	—
177	C₂²×C₄	C₂³×D₇₃	1	4	yes	—
178	C₄⋊C₄	—	73	128	no	—
179	C₄²	—	146	64	no	—
180	C₄²	C₂²×Dic₇₃	1	64	yes	complemented
181	C₂³	C₂×C₄×Dic₇₃	1	16	yes	—
182	C₂³	C₂×C₄⋊Dic₇₃	1	16	yes	—
183	C₂³	#14556550	1	16	yes	—
184	C₂³	C₄×D₂₉₂	1	32	yes	—
185	C₂³	C₄×Dic₁₄₆	1	32	yes	—
186	C₂³	C₂²×C₄×D₇₃	1	12	yes	—
187	C₂³	C₂³×Dic₇₃	1	6	yes	—
188	C₂³	C₂²×D₂₉₂	1	6	yes	—
189	C₂³	C₂²×Dic₁₄₆	1	6	yes	—
190	C₂³	C₂×#11924028	1	12	yes	—
191	C₂³	C₂⁴×D₇₃	1	1	yes	characteristic
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	48	no	—
194	C₂×C₄	—	73	32	no	—
195	C₂×C₄	—	146	16	no	—
196	C₂×C₄	—	146	16	no	—
197	C₂×C₄	—	146	192	no	—
198	C₂×C₄	C₂×C₄×Dic₇₃	1	96	yes	complemented
199	C₂×C₄	C₂×C₄⋊Dic₇₃	1	96	yes	direct product factor
200	C₂×C₄	C₂²×C₄×D₇₃	1	16	yes	—
201	C₂×C₄	C₂³×Dic₇₃	1	8	yes	—
202	C₂×C₄	C₂³×Dic₇₃	1	8	yes	—
203	C₂×C₄	C₂³×Dic₇₃	1	8	yes	—
204	C₂×C₄	C₂²×D₂₉₂	1	8	yes	—
205	C₂×C₄	C₂²×Dic₁₄₆	1	8	yes	—
206	C₂²	C₄×C₄⋊Dic₇₃	1	64	yes	direct product factor
207	C₂²	C₂²×C₄×Dic₇₃	1	12	yes	—
208	C₂²	C₂²×C₄⋊Dic₇₃	1	12	yes	—
209	C₂²	C₂×#14556550	1	12	yes	—
210	C₂²	C₂×C₄×D₂₉₂	1	24	yes	—
211	C₂²	C₂×C₄×Dic₁₄₆	1	24	yes	—
212	C₂²	C₂³×C₄×D₇₃	1	2	yes	—
213	C₂²	C₂⁴×Dic₇₃	1	1	yes	characteristic
214	C₂²	C₂³×D₂₉₂	1	1	yes	characteristic
215	C₂²	C₂³×Dic₁₄₆	1	1	yes	characteristic
216	C₂²	C₂²×#11924028	1	2	yes	—
217	C₄	—	2	8	no	—
218	C₄	—	146	32	no	—
219	C₄	C₂²×C₄×Dic₇₃	1	16	yes	complemented
220	C₄	C₂²×C₄⋊Dic₇₃	1	16	yes	direct product factor
221	C₂	C₂×C₄×C₄⋊Dic₇₃	1	24	yes	direct product factor
222	C₂	C₂³×C₄×Dic₇₃	1	1	yes	characteristic
223	C₂	C₂³×C₄⋊Dic₇₃	1	1	yes	characteristic
224	C₂	C₂²×#14556550	1	1	yes	characteristic
225	C₂	C₂²×C₄×D₂₉₂	1	2	yes	—
226	C₂	C₂²×C₄×Dic₁₄₆	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₇₃
commutator	C₂×C₇₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₇₃
Fitting	C₂³×C₄²×C₇₃	C₂
socle, prime radical	C₂⁵×C₇₃	C₂³
cosocle	C₂³×C₇₃	C₂⁵
hypercenter	C₂³×C₄²	D₇₃
hypocenter	C₇₃	C₂²×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₄⋊Dic₇₃	C₂	direct (N × H)
7. C₂²×C₄⋊Dic₇₃	C₄	direct (N × H)
11. C₄×C₄⋊Dic₇₃	C₂²	direct (N × H)
16. C₂²×C₄²×C₇₃	C₄	semidirect
22. C₂×C₄⋊Dic₇₃	C₂×C₄	direct (N × H)
38. C₂×C₄²×C₇₃	C₂×C₄	semidirect
45. C₄⋊Dic₇₃	C₂²×C₄	direct (N × H)
55. C₂²×C₄×C₇₃	C₄²	semidirect
56. C₂²×C₄×C₇₃	C₄⋊C₄	semidirect
66. C₄²×C₇₃	C₂²×C₄	semidirect
80. C₂×C₄×C₇₃	C₂×C₄²	semidirect
81. C₂×C₄×C₇₃	C₂×C₄⋊C₄	semidirect
92. C₂²×C₇₃	C₄×C₄⋊C₄	semidirect
104. C₄×C₇₃	C₂²×C₄²	semidirect
105. C₄×C₇₃	C₂²×C₄⋊C₄	semidirect
108. C₂×C₇₃	C₂×C₄×C₄⋊C₄	semidirect
119. C₇₃	C₂²×C₄×C₄⋊C₄	semidirect
128. C₂²×C₄²	Dic₇₃	semidirect
151. C₂×C₄²	C₂×Dic₇₃	semidirect
167. C₂²×C₄	C₄×Dic₇₃	semidirect
168. C₂²×C₄	C₄⋊Dic₇₃	direct (N × H)
180. C₄²	C₂²×Dic₇₃	semidirect
198. C₂×C₄	C₂×C₄×Dic₇₃	semidirect
199. C₂×C₄	C₂×C₄⋊Dic₇₃	direct (N × H)
206. C₂²	C₄×C₄⋊Dic₇₃	direct (N × H)
219. C₄	C₂²×C₄×Dic₇₃	semidirect
220. C₄	C₂²×C₄⋊Dic₇₃	direct (N × H)
221. C₂	C₂×C₄×C₄⋊Dic₇₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₇₃	2
C₂³×C₄×Dic₇₃	2
C₂×C₄×C₄⋊Dic₇₃	2
C₂²×#14556498	2
C₂³×C₄²×C₇₃	2
C₂²×C₄×C₄⋊C₄	73

Subgroups — C22×C4×C4⋊Dic73 #33244162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₄⋊Dic₇₃ #33244162