Subgroups — C₂³×C₄×D₂₉₂ #33992550

Order 18688.

19998 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₇₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×D₂₉₂	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#11924005	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄²×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₃	C₂²	1	2	yes	—
8	C₂³×D₂₉₂	C₄	1	32	yes	direct product factor
9	C₂³×D₂₉₂	C₂²	1	28	yes	—
10	C₂⁴×Dic₇₃	C₂²	1	2	yes	—
11	C₂³×C₄×D₇₃	—	2	32	no	—
12	C₂³×C₄×D₇₃	C₂²	1	56	yes	complemented
13	C₂×C₄×D₂₉₂	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₇₃	C₂²	1	28	yes	complemented
15	C₂²×#11924005	C₂²	1	56	yes	complemented
16	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₇₃	C₂²	1	28	yes	complemented
20	C₂⁴×D₇₃	—	2	16	no	—
21	C₂⁴×D₇₃	—	2	16	no	—
22	C₂⁴×D₇₃	C₂×C₄	1	32	yes	complemented
23	C₂⁴×D₇₃	C₂³	1	28	yes	—
24	C₂²×D₂₉₂	C₂×C₄	1	448	yes	direct product factor
25	C₂²×D₂₉₂	C₂³	1	112	yes	—
26	C₂³×Dic₇₃	—	2	16	no	—
27	C₂³×Dic₇₃	C₂³	1	28	yes	—
28	C₂²×C₄×D₇₃	—	2	448	no	—
29	C₂²×C₄×D₇₃	C₂³	1	224	yes	complemented
30	C₄×D₂₉₂	C₂³	1	512	yes	direct product factor
31	C₂×C₄⋊Dic₇₃	C₂³	1	112	yes	complemented
32	C₂×#11924005	C₂³	1	224	yes	complemented
33	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₇₃	—	2	8	no	—
35	C₂³×C₄×C₇₃	C₂×C₄	1	16	yes	complemented
36	C₂³×C₄×C₇₃	D₄	1	16	yes	complemented
37	C₂³×C₄×C₇₃	C₂³	1	14	yes	—
38	C₂³×C₄×C₇₃	C₂³	1	14	yes	—
39	C₂³×C₄×C₇₃	C₂³	1	14	yes	—
40	C₂×C₄²×C₇₃	C₂³	1	112	yes	complemented
41	C₂³×D₇₃	—	2	256	no	—
42	C₂³×D₇₃	—	2	112	no	—
43	C₂³×D₇₃	—	2	112	no	—
44	C₂³×D₇₃	C₂²×C₄	1	224	yes	complemented
45	C₂³×D₇₃	C₂⁴	1	56	yes	—
46	C₂×D₂₉₂	C₂²×C₄	1	896	yes	direct product factor
47	C₂×D₂₉₂	C₂⁴	1	64	yes	—
48	C₂²×Dic₇₃	—	2	112	no	—
49	C₂²×Dic₇₃	C₂⁴	1	56	yes	—
50	C₂×C₄×D₇₃	—	2	896	no	—
51	C₂×C₄×D₇₃	C₂⁴	1	128	yes	complemented
52	C₄⋊Dic₇₃	C₂⁴	1	64	yes	complemented
53	#11924005	C₂⁴	1	128	yes	complemented
54	C₂⁴×C₇₃	C₂²×C₄	1	8	yes	—
55	C₂⁴×C₇₃	C₂×D₄	1	8	yes	—
56	C₂⁴×C₇₃	#41	1	8	yes	—
57	C₂⁴×C₇₃	C₂⁴	1	7	yes	—
58	C₂²×C₄×C₇₃	—	2	56	no	—
59	C₂²×C₄×C₇₃	C₂²×C₄	1	112	yes	complemented
60	C₂²×C₄×C₇₃	C₂×D₄	1	112	yes	complemented
61	C₂²×C₄×C₇₃	C₂⁴	1	28	yes	—
62	C₂²×C₄×C₇₃	C₂⁴	1	28	yes	—
63	C₂²×C₄×C₇₃	C₂⁴	1	28	yes	—
64	C₄²×C₇₃	C₂⁴	1	64	yes	complemented
65	C₂³×C₇₃	C₄×D₄	1	64	yes	complemented
66	C₂³×C₇₃	C₂³×C₄	1	28	yes	—
67	C₂³×C₇₃	C₂²×D₄	1	28	yes	—
68	C₂³×C₇₃	C₂×#41	1	28	yes	—
69	C₂³×C₇₃	C₂⁵	1	7	yes	—
70	C₂²×D₇₃	—	2	896	no	—
71	C₂²×D₇₃	—	2	112	no	—
72	C₂²×D₇₃	—	2	112	no	—
73	C₂²×D₇₃	C₂³×C₄	1	224	yes	complemented
74	C₂²×D₇₃	C₂⁵	1	16	yes	—
75	C₂×C₄×C₇₃	—	2	56	no	—
76	C₂×C₄×C₇₃	C₂³×C₄	1	112	yes	complemented
77	C₂×C₄×C₇₃	C₂²×D₄	1	112	yes	complemented
78	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
79	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
80	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
81	C₂×Dic₇₃	—	2	112	no	—
82	C₂×Dic₇₃	C₂⁵	1	16	yes	—
83	D₂₉₂	C₂³×C₄	1	256	yes	direct product factor
84	C₄×D₇₃	—	2	256	no	—
85	C₂²×C₇₃	C₂×C₄×D₄	1	112	yes	complemented
86	C₂²×C₇₃	C₂⁴×C₄	1	14	yes	—
87	C₂²×C₇₃	C₂³×D₄	1	14	yes	—
88	C₂²×C₇₃	C₂²×#41	1	14	yes	—
89	C₂²×C₇₃	C₂⁶	1	1	yes	characteristic
90	C₂×D₇₃	—	2	16	no	—
91	C₂×D₇₃	—	2	16	no	—
92	C₂×D₇₃	—	2	448	no	—
93	C₂×D₇₃	C₂⁴×C₄	1	32	yes	complemented
94	C₄×C₇₃	—	2	8	no	—
95	C₄×C₇₃	C₂⁴×C₄	1	16	yes	complemented
96	C₄×C₇₃	C₂³×D₄	1	16	yes	complemented
97	Dic₇₃	—	2	16	no	—
98	C₂³×C₄×D₄	—	73	1	no	maximal, Sylow
99	C₂×C₇₃	C₂²×C₄×D₄	1	28	yes	complemented
100	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
101	C₂×C₇₃	C₂⁴×D₄	1	1	yes	characteristic
102	C₂×C₇₃	C₂³×#41	1	1	yes	characteristic
103	D₇₃	—	2	32	no	—
104	C₂⁴×D₄	—	73	1	no	—
105	C₂⁵×C₄	—	73	2	no	—
106	C₂²×C₄×D₄	—	73	56	no	—
107	C₂³×C₄⋊C₄	—	73	1	no	—
108	C₂³×C₂²⋊C₄	—	73	2	no	—
109	C₂³×C₄²	D₇₃	1	1	yes	characteristic, complemented
110	C₇₃	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
111	C₂⁶	—	73	2	no	—
112	C₂³×D₄	—	73	32	no	—
113	C₂³×D₄	—	73	28	no	—
114	C₂⁴×C₄	—	73	2	no	—
115	C₂⁴×C₄	—	73	56	no	—
116	C₂⁴×C₄	—	146	32	no	—
117	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
118	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
119	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
120	C₂×C₄×D₄	—	73	448	no	—
121	C₂²×C₄⋊C₄	—	73	28	no	—
122	C₂²×C₂²⋊C₄	—	73	56	no	—
123	C₂²×C₄²	C₂×D₇₃	1	28	yes	complemented
124	C₂⁵	—	73	32	no	—
125	C₂⁵	—	73	28	no	—
126	C₂⁵	—	146	16	no	—
127	C₂⁵	—	146	16	no	—
128	C₂⁵	C₂²×D₇₃	1	1	yes	characteristic
129	C₂²×D₄	—	73	448	no	—
130	C₂²×D₄	—	73	112	no	—
131	C₂³×C₄	—	2	8	no	—
132	C₂³×C₄	—	73	28	no	—
133	C₂³×C₄	—	73	224	no	—
134	C₂³×C₄	—	146	16	no	—
135	C₂³×C₄	—	146	448	no	—
136	C₂³×C₄	C₄×D₇₃	1	16	yes	complemented
137	C₂³×C₄	D₂₉₂	1	16	yes	direct product factor
138	C₂³×C₄	C₂²×D₇₃	1	14	yes	—
139	C₂³×C₄	C₂²×D₇₃	1	14	yes	—
140	C₂³×C₄	C₂²×D₇₃	1	14	yes	—
141	C₄×D₄	—	73	512	no	—
142	C₂×C₄⋊C₄	—	73	112	no	—
143	C₂×C₂²⋊C₄	—	73	224	no	—
144	C₂×C₄²	C₂²×D₇₃	1	112	yes	complemented
145	C₂⁴	—	73	224	no	—
146	C₂⁴	—	73	56	no	—
147	C₂⁴	—	146	256	no	—
148	C₂⁴	—	146	112	no	—
149	C₂⁴	—	146	112	no	—
150	C₂⁴	C₂×C₄×D₇₃	1	8	yes	—
151	C₂⁴	C₂×D₂₉₂	1	8	yes	—
152	C₂⁴	#11924028	1	8	yes	—
153	C₂⁴	C₂³×D₇₃	1	7	yes	—
154	C₂×D₄	—	73	896	no	—
155	C₂×D₄	—	73	64	no	—
156	C₂²×C₄	—	2	56	no	—
157	C₂²×C₄	—	73	56	no	—
158	C₂²×C₄	—	73	128	no	—
159	C₂²×C₄	—	146	112	no	—
160	C₂²×C₄	—	146	896	no	—
161	C₂²×C₄	C₂×C₄×D₇₃	1	112	yes	complemented
162	C₂²×C₄	C₂×D₂₉₂	1	112	yes	direct product factor
163	C₂²×C₄	C₂³×D₇₃	1	28	yes	—
164	C₂²×C₄	C₂³×D₇₃	1	28	yes	—
165	C₂²×C₄	C₂³×D₇₃	1	28	yes	—
166	C₄⋊C₄	—	73	64	no	—
167	C₂²⋊C₄	—	73	128	no	—
168	C₄²	C₂³×D₇₃	1	64	yes	complemented
169	C₂³	—	73	224	no	—
170	C₂³	—	73	16	no	—
171	C₂³	—	146	112	no	—
172	C₂³	—	146	112	no	—
173	C₂³	—	146	896	no	—
174	C₂³	C₄×D₂₉₂	1	64	yes	direct product factor
175	C₂³	C₂²×C₄×D₇₃	1	28	yes	—
176	C₂³	C₂²×D₂₉₂	1	28	yes	—
177	C₂³	C₂×#11924028	1	28	yes	—
178	C₂³	C₂⁴×D₇₃	1	7	yes	—
179	D₄	—	73	256	no	—
180	C₂×C₄	—	2	56	no	—
181	C₂×C₄	—	73	16	no	—
182	C₂×C₄	—	146	112	no	—
183	C₂×C₄	—	146	256	no	—
184	C₂×C₄	C₂²×C₄×D₇₃	1	112	yes	complemented
185	C₂×C₄	C₂²×D₂₉₂	1	112	yes	direct product factor
186	C₂×C₄	C₂⁴×D₇₃	1	8	yes	—
187	C₂×C₄	C₂⁴×D₇₃	1	8	yes	—
188	C₂×C₄	C₂⁴×D₇₃	1	8	yes	—
189	C₂²	—	73	32	no	—
190	C₂²	—	146	448	no	—
191	C₂²	—	146	16	no	—
192	C₂²	—	146	16	no	—
193	C₂²	C₂×C₄×D₂₉₂	1	112	yes	direct product factor
194	C₂²	C₂³×C₄×D₇₃	1	14	yes	—
195	C₂²	C₂³×D₂₉₂	1	14	yes	—
196	C₂²	C₂²×#11924028	1	14	yes	—
197	C₂²	C₂⁵×D₇₃	1	1	yes	characteristic
198	C₄	—	2	8	no	—
199	C₄	—	146	16	no	—
200	C₄	C₂³×C₄×D₇₃	1	16	yes	complemented
201	C₄	C₂³×D₂₉₂	1	16	yes	direct product factor
202	C₂	—	146	32	no	—
203	C₂	C₂²×C₄×D₂₉₂	1	28	yes	direct product factor
204	C₂	C₂⁴×C₄×D₇₃	1	1	yes	characteristic
205	C₂	C₂⁴×D₂₉₂	1	1	yes	characteristic
206	C₂	C₂³×#11924028	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₇₃
commutator	C₂×C₇₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₇₃
Fitting	C₂³×C₄²×C₇₃	C₂
socle	C₂⁵×C₇₃	C₂³
prime radical	C₂⁴×D₂₉₂	C₂
cosocle	C₂²×C₇₃	C₂⁶
hypercenter	C₂³×C₄²	D₇₃
hypocenter	C₇₃	C₂³×C₄×D₄
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
2. C₂⁴×C₄×D₇₃	C₂	semidirect
3. C₂²×C₄×D₂₉₂	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₇₃	C₂	semidirect
5. C₂³×#11924005	C₂	semidirect
6. C₂³×C₄²×C₇₃	C₂	semidirect
8. C₂³×D₂₉₂	C₄	direct (N × H)
12. C₂³×C₄×D₇₃	C₂²	semidirect
13. C₂×C₄×D₂₉₂	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₇₃	C₂²	semidirect
15. C₂²×#11924005	C₂²	semidirect
19. C₂²×C₄²×C₇₃	C₂²	semidirect
22. C₂⁴×D₇₃	C₂×C₄	semidirect
24. C₂²×D₂₉₂	C₂×C₄	direct (N × H)
29. C₂²×C₄×D₇₃	C₂³	semidirect
30. C₄×D₂₉₂	C₂³	direct (N × H)
31. C₂×C₄⋊Dic₇₃	C₂³	semidirect
32. C₂×#11924005	C₂³	semidirect
35. C₂³×C₄×C₇₃	C₂×C₄	semidirect
36. C₂³×C₄×C₇₃	D₄	semidirect
40. C₂×C₄²×C₇₃	C₂³	semidirect
44. C₂³×D₇₃	C₂²×C₄	semidirect
46. C₂×D₂₉₂	C₂²×C₄	direct (N × H)
51. C₂×C₄×D₇₃	C₂⁴	semidirect
52. C₄⋊Dic₇₃	C₂⁴	semidirect
53. #11924005	C₂⁴	semidirect
59. C₂²×C₄×C₇₃	C₂²×C₄	semidirect
60. C₂²×C₄×C₇₃	C₂×D₄	semidirect
64. C₄²×C₇₃	C₂⁴	semidirect
65. C₂³×C₇₃	C₄×D₄	semidirect
73. C₂²×D₇₃	C₂³×C₄	semidirect
76. C₂×C₄×C₇₃	C₂³×C₄	semidirect
77. C₂×C₄×C₇₃	C₂²×D₄	semidirect
83. D₂₉₂	C₂³×C₄	direct (N × H)
85. C₂²×C₇₃	C₂×C₄×D₄	semidirect
93. C₂×D₇₃	C₂⁴×C₄	semidirect
95. C₄×C₇₃	C₂⁴×C₄	semidirect
96. C₄×C₇₃	C₂³×D₄	semidirect
99. C₂×C₇₃	C₂²×C₄×D₄	semidirect
109. C₂³×C₄²	D₇₃	semidirect
110. C₇₃	C₂³×C₄×D₄	semidirect
123. C₂²×C₄²	C₂×D₇₃	semidirect
136. C₂³×C₄	C₄×D₇₃	semidirect
137. C₂³×C₄	D₂₉₂	direct (N × H)
144. C₂×C₄²	C₂²×D₇₃	semidirect
161. C₂²×C₄	C₂×C₄×D₇₃	semidirect
162. C₂²×C₄	C₂×D₂₉₂	direct (N × H)
168. C₄²	C₂³×D₇₃	semidirect
174. C₂³	C₄×D₂₉₂	direct (N × H)
184. C₂×C₄	C₂²×C₄×D₇₃	semidirect
185. C₂×C₄	C₂²×D₂₉₂	direct (N × H)
193. C₂²	C₂×C₄×D₂₉₂	direct (N × H)
200. C₄	C₂³×C₄×D₇₃	semidirect
201. C₄	C₂³×D₂₉₂	direct (N × H)
203. C₂	C₂²×C₄×D₂₉₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₉₂	2
C₂⁴×C₄×D₇₃	2
C₂²×C₄×D₂₉₂	2
C₂³×C₄⋊Dic₇₃	2
C₂³×#11924005	2
C₂³×C₄²×C₇₃	2
C₂³×C₄×D₄	73

Subgroups — C23×C4×D292 #33992550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×D₂₉₂ #33992550