Subgroups — #32990896

Order 18688.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610687	C₂	1	2	yes	maximal
2	#22610649	C₂	1	1	yes	maximal, characteristic
3	#22607814	C₂	1	2	yes	maximal
4	#22607766	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556524	—	2	2	no	—
7	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
9	C₂×#14556520	—	2	1	no	—
10	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	#17988349	—	2	2	no	—
12	#17988341	—	2	1	no	—
13	C₂×#107×C₇₃	—	2	1	no	—
14	C₂×#107×C₇₃	—	2	1	no	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#384×C₇₃	C₂²	1	2	yes	—
17	C₂²×#10596831	—	2	2	no	—
18	C₂²×#10596831	—	2	1	no	—
19	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	—	2	1	no	—
22	#14556524	—	4	4	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	C₄⋊#10596831	—	2	2	no	—
25	C₄⋊#10596831	Q₈	1	4	yes	—
26	#14556520	—	4	2	no	—
27	#14556498	—	2	1	no	—
28	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
33	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
34	#107×C₇₃	—	4	2	no	—
35	#107×C₇₃	—	4	2	no	—
36	C₄×C₈×C₇₃	—	2	2	no	—
37	C₂²×Dic₇₃	—	2	1	no	—
38	C₂²×Dic₇₃	—	2	1	no	—
39	C₂×#10596831	—	2	4	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	1	no	—
43	C₂×#10596831	—	4	1	no	—
44	C₂×#10596831	—	4	1	no	—
45	C₄⋊Dic₇₃	—	2	1	no	—
46	C₄⋊Dic₇₃	—	2	2	no	—
47	C₄⋊Dic₇₃	Q₁₆	1	2	yes	—
48	C₄×Dic₇₃	—	4	1	no	—
49	C₄×Dic₇₃	—	4	1	no	—
50	C₂²×C₄×C₇₃	—	4	2	no	—
51	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	D₈	1	2	yes	—
59	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
60	C₄⋊C₄×C₇₃	—	4	2	no	—
61	C₄⋊C₄×C₇₃	—	4	2	no	—
62	C₄⋊C₄×C₇₃	—	4	2	no	—
63	C₄⋊C₄×C₇₃	—	4	2	no	—
64	C₄⋊C₄×C₇₃	—	8	2	no	—
65	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
66	C₄²×C₇₃	#41	1	2	yes	—
67	C₂³×C₇₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	8	2	no	—
71	C₂×C₄×C₇₃	—	8	2	no	—
72	C₂×C₄×C₇₃	—	8	2	no	—
73	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#125	1	2	yes	—
76	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
81	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
84	C₂×Dic₇₃	—	2	2	no	—
85	C₂×Dic₇₃	—	4	1	no	—
86	C₂×Dic₇₃	—	4	1	no	—
87	C₂×Dic₇₃	—	4	1	no	—
88	C₂×Dic₇₃	—	4	1	no	—
89	C₂×Dic₇₃	—	4	1	no	—
90	C₈×C₇₃	—	2	2	no	—
91	C₈×C₇₃	—	2	2	no	—
92	#10596831	—	4	4	no	—
93	#10596831	—	8	1	no	—
94	C₂²×C₇₃	#398	1	1	yes	characteristic
95	C₂²×C₇₃	#459	1	1	yes	characteristic
96	C₂²×C₇₃	#462	1	1	yes	characteristic
97	C₂²×C₇₃	#467	1	1	yes	characteristic
98	C₂²×C₇₃	#469	1	1	yes	characteristic
99	C₂²×C₇₃	#472	1	1	yes	characteristic
100	C₂²×C₇₃	#473	1	1	yes	characteristic
101	C₄×C₇₃	—	2	2	no	—
102	C₄×C₇₃	—	16	2	no	—
103	C₄×C₇₃	#477	1	2	yes	—
104	C₄×C₇₃	#478	1	2	yes	—
105	C₄×C₇₃	#480	1	1	yes	characteristic
106	C₄×C₇₃	#484	1	1	yes	characteristic
107	C₄×C₇₃	#500	1	1	yes	characteristic
108	C₄×C₇₃	#501	1	1	yes	characteristic
109	Dic₇₃	—	4	2	no	—
110	Dic₇₃	—	8	1	no	—
111	#10650	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	#1669	1	1	yes	characteristic
113	C₂×C₇₃	#1676	1	1	yes	characteristic
114	C₂×C₇₃	#1687	1	1	yes	characteristic
115	C₂×C₇₃	#1694	1	1	yes	characteristic
116	C₂×C₇₃	#2083	1	1	yes	characteristic
117	C₂×C₇₃	#2087	1	1	yes	characteristic
118	C₂×C₇₃	#2089	1	1	yes	characteristic
119	#1947	—	73	2	no	—
120	#1939	—	73	1	no	—
121	#1845	D₇₃	1	1	yes	characteristic
122	#1296	—	73	2	no	—
123	#1277	—	73	1	no	—
124	C₇₃	#10650	1	1	yes	characteristic, complemented, Sylow
125	C₂×#107	—	2	1	no	—
126	C₂×#107	—	2	1	no	—
127	C₂×#107	—	146	2	no	—
128	C₂×C₄⋊C₈	—	73	2	no	—
129	C₂×C₄⋊C₈	—	73	1	no	—
130	C₂×C₄⋊C₈	—	146	1	no	—
131	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
132	#384	C₂×D₇₃	1	2	yes	—
133	C₄×C₄⋊C₄	—	73	1	no	—
134	#340	—	146	2	no	—
135	#336	—	146	1	no	—
136	C₂²×C₈	—	146	1	no	—
137	C₂²×C₈	—	146	2	no	—
138	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	73	1	no	—
144	C₂×C₄⋊C₄	—	73	1	no	—
145	C₂×C₄²	—	146	1	no	—
146	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
147	#107	—	4	2	no	—
148	#107	—	4	2	no	—
149	#107	—	292	4	no	—
150	C₄⋊C₈	—	292	2	no	—
151	C₄⋊C₈	—	73	4	no	—
152	C₄⋊C₈	—	146	2	no	—
153	C₄⋊C₈	—	146	2	no	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	146	1	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	146	1	no	—
158	C₂²×C₄	—	146	1	no	—
159	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
161	C₂²×C₄	#11924027	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	292	2	no	—
167	C₂×C₈	—	292	2	no	—
168	C₂×C₈	—	292	1	no	—
169	C₂×C₈	—	292	1	no	—
170	C₂×C₈	—	292	1	no	—
171	C₂×C₈	—	146	4	no	—
172	C₂×C₈	#11924013	1	2	yes	—
173	C₂×C₈	#11924018	1	2	yes	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	8	2	no	—
179	C₄⋊C₄	—	73	2	no	—
180	C₄⋊C₄	—	146	2	no	—
181	C₄⋊C₄	—	146	1	no	—
182	C₄²	—	292	1	no	—
183	C₄²	—	292	1	no	—
184	C₄²	Q₈×D₇₃	1	2	yes	—
185	C₄²	#11924028	1	2	yes	—
186	C₂³	#14556562	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	292	1	no	—
193	C₂×C₄	—	292	1	no	—
194	C₂×C₄	—	292	1	no	—
195	C₂×C₄	—	292	1	no	—
196	C₂×C₄	—	292	1	no	—
197	C₂×C₄	—	146	2	no	—
198	C₂×C₄	#14556569	1	2	yes	—
199	C₂×C₄	#14556574	1	1	yes	characteristic
200	C₂×C₄	#14556581	1	2	yes	—
201	C₂×C₄	#14556589	1	1	yes	characteristic
202	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₇₃	1	1	yes	characteristic
204	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
205	C₂×C₄	#14556614	1	1	yes	characteristic
206	C₂×C₄	#14556615	1	2	yes	—
207	C₂×C₄	#14556617	1	1	yes	characteristic
208	C₂×C₄	#14556622	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	292	4	no	—
212	C₈	—	584	1	no	—
213	C₂²	#17988518	1	1	yes	characteristic
214	C₂²	#17988779	1	1	yes	characteristic
215	C₂²	#17988794	1	1	yes	characteristic
216	C₂²	#17988825	1	1	yes	characteristic
217	C₂²	#17988837	1	1	yes	characteristic
218	C₂²	#17988854	1	1	yes	characteristic
219	C₂²	#17988859	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	16	2	no	—
222	C₄	—	292	2	no	—
223	C₄	—	584	1	no	—
224	C₄	#17988881	1	2	yes	—
225	C₄	#17988890	1	2	yes	—
226	C₄	#17988900	1	1	yes	characteristic
227	C₄	#17988922	1	1	yes	characteristic
228	C₄	#17988990	1	1	yes	characteristic
229	C₄	#17988993	1	1	yes	characteristic
230	C₂	#22609293	1	1	yes	characteristic
231	C₂	#22609332	1	1	yes	characteristic
232	C₂	#22609407	1	1	yes	characteristic
233	C₂	#22609446	1	1	yes	characteristic
234	C₂	#22611403	1	1	yes	characteristic
235	C₂	#22611425	1	1	yes	characteristic
236	C₂	#22611437	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556562
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1845	C₂
socle, prime radical	C₂³×C₇₃	#121
hypercenter	#1845	D₇₃
hypocenter	C₇₃	#10650
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
124. C₇₃	#10650	semidirect

Maximal subgroups

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

subgroup	index
#22610687	2
#22610649	2
#22607814	2
#22607766	2
C₇₃×#1845	2
#10650	73