Subgroups — #32991806

Order 18688.

950 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988453	C₂	1	2	yes	maximal
2	C₂×#17988350	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₇₃	C₂²	1	4	yes	—
5	#17988453	—	2	4	no	—
6	#17988453	C₄	1	8	yes	complemented
7	C₂×#14556498	—	2	4	no	—
8	#17988350	—	2	8	no	—
9	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₃	—	2	4	no	—
12	C₂×C₄×C₈×C₇₃	C₄	1	4	yes	complemented
13	C₂³×Dic₇₃	—	2	4	no	—
14	C₂×C₄⋊Dic₇₃	—	2	8	no	—
15	C₂×C₄⋊Dic₇₃	—	2	4	no	—
16	C₂×C₄⋊Dic₇₃	—	2	8	no	—
17	C₂×C₄⋊Dic₇₃	C₂×C₄	1	8	yes	—
18	#14556498	—	2	16	no	—
19	C₂³×C₄×C₇₃	D₄	1	2	yes	—
20	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	—	2	4	no	—
22	C₂²×C₈×C₇₃	—	2	4	no	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₇₃	D₄	1	4	yes	—
25	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇₃	D₄	1	4	yes	—
29	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
30	C₄×C₈×C₇₃	—	2	8	no	—
31	C₂²×Dic₇₃	—	2	8	no	—
32	C₂²×Dic₇₃	—	4	4	no	—
33	C₂²×Dic₇₃	—	4	4	no	—
34	C₂²×Dic₇₃	—	4	16	no	—
35	C₄⋊Dic₇₃	—	2	16	no	—
36	C₄⋊Dic₇₃	—	4	8	no	—
37	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	—	2	8	no	—
39	C₂²×C₄×C₇₃	C₂²⋊C₄	1	8	yes	—
40	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
47	C₂²×C₄×C₇₃	#41	1	4	yes	—
48	C₂×C₈×C₇₃	—	2	4	no	—
49	C₂×C₈×C₇₃	—	2	4	no	—
50	C₂×C₈×C₇₃	—	2	8	no	—
51	C₂×C₈×C₇₃	—	2	8	no	—
52	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
53	C₄²×C₇₃	C₄²	1	4	yes	—
54	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₇₃	C₄⋊C₄	1	8	yes	—
56	C₂³×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
57	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂³×C₇₃	#124	1	2	yes	—
59	C₂³×C₇₃	#127	1	1	yes	characteristic
60	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
61	C₂³×C₇₃	#135	1	8	yes	—
62	C₂×C₄×C₇₃	—	2	16	no	—
63	C₂×C₄×C₇₃	—	2	8	no	—
64	C₂×C₄×C₇₃	#95	1	8	yes	—
65	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₇₃	#117	1	2	yes	—
70	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
71	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
72	C₂×C₄×C₇₃	#121	1	4	yes	—
73	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
74	C₂×C₄×C₇₃	#125	1	2	yes	—
75	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
76	C₂×Dic₇₃	—	4	8	no	—
77	C₂×Dic₇₃	—	4	16	no	—
78	C₂×Dic₇₃	—	8	8	no	—
79	C₂×Dic₇₃	—	8	8	no	—
80	C₈×C₇₃	—	2	8	no	—
81	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
82	C₂²×C₇₃	#382	1	1	yes	characteristic
83	C₂²×C₇₃	#384	1	1	yes	characteristic
84	C₂²×C₇₃	#390	1	2	yes	—
85	C₂²×C₇₃	#391	1	2	yes	—
86	C₂²×C₇₃	#416	1	8	yes	—
87	C₂²×C₇₃	#427	1	4	yes	—
88	C₂²×C₇₃	#488	1	8	yes	—
89	C₂²×C₇₃	#495	1	4	yes	—
90	C₂²×C₇₃	#499	1	4	yes	—
91	C₄×C₇₃	—	2	8	no	—
92	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
93	C₄×C₇₃	#380	1	2	yes	—
94	C₄×C₇₃	#389	1	4	yes	—
95	Dic₇₃	—	8	8	no	—
96	#10842	—	73	1	no	maximal, Sylow
97	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
98	C₂×C₇₃	#1735	1	2	yes	—
99	C₂×C₇₃	#1835	1	2	yes	—
100	C₂×C₇₃	#1846	1	2	yes	—
101	C₂×C₇₃	#1958	1	4	yes	—
102	C₂×C₇₃	#1991	1	4	yes	—
103	C₂²×C₄×C₈	D₇₃	1	1	yes	characteristic
104	C₂×#382	—	73	2	no	—
105	C₂×#340	—	73	4	no	—
106	C₇₃	#10842	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₈	C₂×D₇₃	1	2	yes	—
108	C₂²×C₄⋊C₄	—	73	4	no	—
109	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
110	C₂×C₄×C₈	—	2	4	no	—
111	C₂×C₄×C₈	Dic₇₃	1	4	yes	complemented
112	#382	—	73	8	no	—
113	#382	—	146	4	no	—
114	C₂×#95	—	146	4	no	—
115	#340	—	146	8	no	—
116	C₂³×C₄	—	146	4	no	—
117	C₂³×C₄	D₂₉₂	1	2	yes	—
118	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	Dic₁₄₆	1	4	yes	—
122	C₂²×C₈	D₂₉₂	1	4	yes	—
123	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
124	C₂×C₄⋊C₄	—	73	8	no	—
125	C₂×C₄⋊C₄	—	146	4	no	—
126	C₂×C₄⋊C₄	—	146	8	no	—
127	C₂×C₄⋊C₄	—	146	8	no	—
128	C₂×C₄²	Dic₁₄₆	1	2	yes	—
129	C₂×C₄²	C₄×D₇₃	1	4	yes	—
130	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₇₃	1	4	yes	—
132	C₄×C₈	—	2	8	no	—
133	#95	—	146	16	no	—
134	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	292	4	no	—
137	C₂²×C₄	—	292	4	no	—
138	C₂²×C₄	—	292	16	no	—
139	C₂²×C₄	—	146	8	no	—
140	C₂²×C₄	#11924005	1	8	yes	—
141	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
142	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
143	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
145	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
146	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
148	C₂²×C₄	#11924028	1	4	yes	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	C₄⋊Dic₇₃	1	8	yes	—
154	C₄⋊C₄	—	292	8	no	—
155	C₄⋊C₄	—	146	16	no	—
156	C₄²	C₄×Dic₇₃	1	4	yes	—
157	C₄²	C₂²⋊Dic₇₃	1	4	yes	—
158	C₄²	#11924007	1	8	yes	—
159	C₂³	C₂×#11924005	1	2	yes	—
160	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
161	C₂³	#14556579	1	2	yes	—
162	C₂³	#14556588	1	1	yes	characteristic
163	C₂³	#14556591	1	1	yes	characteristic
164	C₂³	#14556613	1	8	yes	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	16	no	—
167	C₂×C₄	—	292	8	no	—
168	C₂×C₄	—	292	16	no	—
169	C₂×C₄	—	584	8	no	—
170	C₂×C₄	—	584	8	no	—
171	C₂×C₄	#14556498	1	8	yes	—
172	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
174	C₂×C₄	C₂×#11924007	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
176	C₂×C₄	#14556550	1	2	yes	—
177	C₂×C₄	C₄×D₂₉₂	1	4	yes	—
178	C₂×C₄	C₄×Dic₁₄₆	1	4	yes	—
179	C₂×C₄	#14556565	1	4	yes	—
180	C₂×C₄	#14556570	1	4	yes	—
181	C₂×C₄	#14556583	1	2	yes	—
182	C₂×C₄	#14556591	1	2	yes	—
183	C₈	—	2	8	no	—
184	C₂²	C₂×#14556498	1	1	yes	characteristic
185	C₂²	#17988453	1	1	yes	characteristic
186	C₂²	#17988457	1	1	yes	characteristic
187	C₂²	#17988489	1	2	yes	—
188	C₂²	#17988493	1	2	yes	—
189	C₂²	#17988584	1	8	yes	—
190	C₂²	#17988625	1	4	yes	—
191	C₂²	#17988942	1	8	yes	—
192	C₂²	#17988971	1	4	yes	—
193	C₂²	#17988989	1	4	yes	—
194	C₄	—	2	8	no	—
195	C₄	—	584	8	no	—
196	C₄	C₄×C₄⋊Dic₇₃	1	2	yes	—
197	C₄	#17988447	1	2	yes	—
198	C₄	#17988485	1	4	yes	—
199	C₂	#22608161	1	1	yes	characteristic
200	C₂	#22609648	1	2	yes	—
201	C₂	#22610062	1	2	yes	—
202	C₂	#22610095	1	2	yes	—
203	C₂	#22610742	1	4	yes	—
204	C₂	#22610896	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂²×C₄×C₈×C₇₃	C₂
socle, prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂²×C₄×C₈	D₇₃
hypocenter	C₇₃	#10842
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17988453	C₄	semidirect
12. C₂×C₄×C₈×C₇₃	C₄	semidirect
106. C₇₃	#10842	semidirect
111. C₂×C₄×C₈	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988453	2
C₂×#17988350	2
C₂²×C₄×C₈×C₇₃	2
#10842	73