Subgroups — #33084045

Order 18688.

1142 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988625	C₂	1	2	yes	maximal
2	C₄×#14556550	C₂	1	2	yes	maximal
3	#22609828	C₂	1	8	yes	maximal, complemented
4	#22609648	C₂	1	2	yes	maximal
5	C₂²×#97×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556550	C₂²	1	4	yes	—
7	#17988625	C₄	1	16	yes	complemented
8	C₂×#14556525	C₂²	1	4	yes	—
9	C₂×#14556523	C₂²	1	4	yes	—
10	C₄×C₄⋊Dic₇₃	C₂²	1	8	yes	—
11	C₄×C₂²⋊Dic₇₃	—	2	4	no	—
12	#17988433	—	2	2	no	—
13	C₄²×Dic₇₃	—	2	2	no	—
14	#17988350	C₂²	1	8	yes	—
15	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₇₃	C₄	1	8	yes	complemented
19	C₂×#97×C₇₃	C₂²	1	4	yes	—
20	#14556550	—	2	8	no	—
21	#14556550	C₂×C₄	1	16	yes	—
22	C₂×C₄⋊Dic₇₃	C₂³	1	8	yes	—
23	C₂×C₂²⋊Dic₇₃	—	2	4	no	—
24	C₂×C₄×Dic₇₃	—	2	4	no	—
25	C₂×C₄×Dic₇₃	—	2	8	no	—
26	#14556525	C₂×C₄	1	16	yes	complemented
27	#14556523	C₂×C₄	1	16	yes	complemented
28	#14556498	—	2	8	no	—
29	C₂³×C₄×C₇₃	D₄	1	2	yes	—
30	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	—	2	4	no	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₇₃	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
37	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇₃	D₄	1	2	yes	—
39	C₂×C₄²×C₇₃	Q₈	1	4	yes	—
40	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
41	#97×C₇₃	C₂×C₄	1	16	yes	—
42	C₂²×Dic₇₃	—	2	8	no	—
43	C₄⋊Dic₇₃	—	2	8	no	—
44	C₄⋊Dic₇₃	C₂²×C₄	1	16	yes	—
45	C₂²⋊Dic₇₃	—	2	16	no	—
46	C₄×Dic₇₃	—	2	16	no	—
47	C₄×Dic₇₃	—	4	16	no	—
48	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	—	4	4	no	—
50	C₂²×C₄×C₇₃	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₇₃	#41	1	4	yes	—
57	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₇₃	—	2	8	no	—
59	C₂×C₈×C₇₃	—	2	4	no	—
60	C₂×C₈×C₇₃	C₄²	1	16	yes	complemented
61	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
65	C₄²×C₇₃	C₄⋊C₄	1	8	yes	—
66	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
68	C₂³×C₇₃	—	2	4	no	—
69	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₂³×C₇₃	C₄×D₄	1	4	yes	—
71	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	—	2	8	no	—
73	C₂×C₄×C₇₃	—	4	8	no	—
74	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
75	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
76	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₇₃	#117	1	4	yes	—
78	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
79	C₂×C₄×C₇₃	C₄×Q₈	1	8	yes	—
80	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
81	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
84	C₂×Dic₇₃	—	2	16	no	—
85	C₂×Dic₇₃	—	4	16	no	—
86	C₈×C₇₃	—	2	4	no	—
87	C₈×C₇₃	C₂×C₄²	1	8	yes	—
88	C₂²×C₇₃	—	2	4	no	—
89	C₂²×C₇₃	—	2	8	no	—
90	C₂²×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
91	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₇₃	C₂×C₄×D₄	1	2	yes	—
93	C₂²×C₇₃	#575	1	4	yes	—
94	C₄×C₇₃	—	2	8	no	—
95	C₄×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
96	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
97	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
98	C₄×C₇₃	C₂×C₄×Q₈	1	2	yes	—
99	Dic₇₃	—	4	16	no	—
100	#20451	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	—	2	4	no	—
102	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₇₃	#2911	1	2	yes	—
104	C₂×C₇₃	#2943	1	4	yes	—
105	C₂×#427	—	73	2	no	—
106	C₂²×#97	D₇₃	1	1	yes	characteristic
107	C₄×#117	—	73	2	no	—
108	#1776	—	73	8	no	—
109	#1735	—	73	2	no	—
110	C₇₃	#20451	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
112	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
113	C₂×#117	—	73	4	no	—
114	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
115	#427	—	73	16	no	—
116	C₂×#107	—	73	4	no	—
117	C₂×#106	—	73	4	no	—
118	C₂×#97	Dic₇₃	1	8	yes	complemented
119	C₂×#97	C₂×D₇₃	1	4	yes	—
120	C₄×C₄⋊C₄	—	73	8	no	—
121	C₄×C₂²⋊C₄	—	146	4	no	—
122	#376	—	146	2	no	—
123	C₄³	—	146	2	no	—
124	#340	—	73	8	no	—
125	C₂³×C₄	D₂₉₂	1	2	yes	—
126	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
127	C₂²×C₈	—	2	4	no	—
128	C₂²×C₈	C₄×D₇₃	1	8	yes	—
129	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
130	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
131	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
132	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
133	#117	—	73	16	no	—
134	#117	—	146	8	no	—
135	C₂×C₄⋊C₄	—	73	8	no	—
136	C₂×C₂²⋊C₄	—	146	4	no	—
137	C₂×C₄²	—	146	8	no	—
138	C₂×C₄²	—	146	4	no	—
139	C₂×C₄²	Dic₁₄₆	1	4	yes	—
140	C₂×C₄²	D₂₉₂	1	2	yes	—
141	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
142	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
143	#107	—	73	16	no	—
144	#106	—	73	16	no	—
145	#97	C₂×Dic₇₃	1	16	yes	—
146	#95	—	146	8	no	—
147	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	—	146	8	no	—
150	C₂²×C₄	C₄⋊Dic₇₃	1	8	yes	—
151	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
152	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
153	C₂²×C₄	C₂×D₂₉₂	1	4	yes	—
154	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
156	C₂²×C₄	#11924028	1	4	yes	—
157	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	C₄×Dic₇₃	1	16	yes	complemented
161	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
162	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
163	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
164	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
165	C₄⋊C₄	—	73	16	no	—
166	C₄⋊C₄	—	146	8	no	—
167	C₂²⋊C₄	—	146	16	no	—
168	C₄²	—	292	16	no	—
169	C₄²	—	146	16	no	—
170	C₄²	C₄⋊Dic₇₃	1	8	yes	—
171	C₄²	C₂²×Dic₇₃	1	4	yes	—
172	C₄²	C₂×Dic₁₄₆	1	4	yes	—
173	C₂³	—	2	4	no	—
174	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
175	C₂³	C₄×D₂₉₂	1	4	yes	—
176	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	292	16	no	—
180	C₂×C₄	—	146	16	no	—
181	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₇₃	1	8	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
184	C₂×C₄	#14556550	1	4	yes	—
185	C₂×C₄	C₄×D₂₉₂	1	4	yes	—
186	C₂×C₄	C₄×Dic₁₄₆	1	8	yes	—
187	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
188	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
190	C₂×C₄	C₂×#11924028	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	C₂×C₄×Dic₇₃	1	8	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	4	no	—
195	C₂²	C₄×C₄⋊Dic₇₃	1	4	yes	—
196	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
197	C₂²	C₂×C₄×D₂₉₂	1	2	yes	—
198	C₂²	#17989387	1	4	yes	—
199	C₄	—	2	8	no	—
200	C₄	—	292	16	no	—
201	C₄	C₄×C₄⋊Dic₇₃	1	4	yes	—
202	C₄	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
203	C₄	C₂×#14556550	1	1	yes	characteristic
204	C₄	C₂×C₄×Dic₁₄₆	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂×C₄×C₄⋊Dic₇₃	1	1	yes	characteristic
207	C₂	#22618521	1	2	yes	—
208	C₂	#22618733	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂²×#97×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂²×#97	D₇₃
hypocenter	C₇₃	#20451
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #22609828	C₂	semidirect
7. #17988625	C₄	semidirect
18. C₂×#97×C₇₃	C₄	semidirect
26. #14556525	C₂×C₄	semidirect
27. #14556523	C₂×C₄	semidirect
60. C₂×C₈×C₇₃	C₄²	semidirect
110. C₇₃	#20451	semidirect
118. C₂×#97	Dic₇₃	semidirect
160. C₂×C₈	C₄×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988625	2
C₄×#14556550	2
#22609828	2
#22609648	2
C₂²×#97×C₇₃	2
#20451	73