Subgroups — #32985039

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618309	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988335	C₂	1	2	yes	maximal
4	C₂×#17988332	C₂	1	2	yes	maximal
5	C₇₃×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	1	yes	characteristic
7	C₂×#14556591	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
9	C₂×C₄×#10596831	—	2	1	no	—
10	#17988493	—	2	1	no	—
11	#17988493	—	2	1	no	—
12	#17988493	C₄	1	4	yes	—
13	#17988453	—	2	2	no	—
14	#17988342	—	2	1	no	—
15	#17988335	—	2	4	no	—
16	#17988332	—	2	4	no	—
17	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
18	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
19	C₄³×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₄₆	—	2	1	no	—
21	C₂²×Dic₁₄₆	C₂×C₄	1	2	yes	—
22	C₂²×#10596831	—	2	2	no	—
23	#14556591	—	2	2	no	—
24	#14556591	—	2	2	no	—
25	#14556591	—	2	2	no	—
26	#14556591	—	2	2	no	—
27	C₂×C₄⋊Dic₇₃	—	2	1	no	—
28	C₂×C₄⋊Dic₇₃	—	2	1	no	—
29	C₂×C₄⋊Dic₇₃	—	2	1	no	—
30	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	C₄⋊#10596831	—	2	2	no	—
32	C₄⋊#10596831	D₄	1	4	yes	—
33	C₄×#10596831	—	4	2	no	—
34	#14556498	—	2	4	no	—
35	#14556498	—	2	4	no	—
36	C₂²×C₈×C₇₃	—	2	2	no	—
37	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₃	D₄	1	2	yes	—
41	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₃	—	2	2	no	—
43	C₄⋊C₈×C₇₃	—	2	2	no	—
44	C₄⋊C₈×C₇₃	—	4	4	no	—
45	C₂×Dic₁₄₆	—	2	4	no	—
46	C₂×Dic₁₄₆	—	4	2	no	—
47	C₂×Dic₁₄₆	—	4	2	no	—
48	C₂×Dic₁₄₆	—	4	1	no	—
49	C₂×Dic₁₄₆	—	4	1	no	—
50	C₂×Dic₁₄₆	—	4	4	no	—
51	C₂²×Dic₇₃	—	2	2	no	—
52	C₂²×Dic₇₃	—	2	2	no	—
53	C₂×#10596831	—	2	4	no	—
54	C₂×#10596831	—	4	2	no	—
55	C₂×#10596831	—	4	2	no	—
56	C₄⋊Dic₇₃	—	2	4	no	—
57	C₄⋊Dic₇₃	—	4	2	no	—
58	C₄⋊Dic₇₃	—	4	1	no	—
59	C₄⋊Dic₇₃	—	4	1	no	—
60	C₄⋊Dic₇₃	—	4	1	no	—
61	C₄⋊Dic₇₃	—	4	1	no	—
62	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₇₃	—	4	2	no	—
68	C₂×C₈×C₇₃	—	4	4	no	—
69	C₄²×C₇₃	—	2	2	no	—
70	C₄²×C₇₃	—	2	1	no	—
71	C₄²×C₇₃	—	2	1	no	—
72	C₄²×C₇₃	—	4	2	no	—
73	C₄²×C₇₃	SD₁₆	1	4	yes	—
74	C₄²×C₇₃	Q₁₆	1	4	yes	—
75	C₄²×C₇₃	C₂×D₄	1	2	yes	—
76	C₄²×C₇₃	#41	1	2	yes	—
77	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	—	2	2	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	—	2	2	no	—
83	C₂×C₄×C₇₃	#103	1	4	yes	—
84	C₂×C₄×C₇₃	#103	1	4	yes	—
85	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₇₃	#121	1	2	yes	—
87	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₇₃	#135	1	2	yes	—
92	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
94	C₂×Dic₇₃	—	4	2	no	—
95	C₂×Dic₇₃	—	4	2	no	—
96	C₂×Dic₇₃	—	4	4	no	—
97	C₂×Dic₇₃	—	4	2	no	—
98	C₂×Dic₇₃	—	4	2	no	—
99	Dic₁₄₆	—	8	2	no	—
100	Dic₁₄₆	—	8	2	no	—
101	Dic₁₄₆	—	8	4	no	—
102	C₈×C₇₃	—	8	2	no	—
103	#10596831	—	4	4	no	—
104	C₂²×C₇₃	#390	1	1	yes	characteristic
105	C₂²×C₇₃	#413	1	2	yes	—
106	C₂²×C₇₃	C₂×#103	1	2	yes	—
107	C₂²×C₇₃	#416	1	1	yes	characteristic
108	C₂²×C₇₃	#417	1	1	yes	characteristic
109	C₄×C₇₃	—	2	4	no	—
110	C₄×C₇₃	—	2	4	no	—
111	C₄×C₇₃	—	2	2	no	—
112	C₄×C₇₃	#463	1	2	yes	—
113	C₄×C₇₃	#464	1	2	yes	—
114	C₄×C₇₃	#487	1	2	yes	—
115	C₄×C₇₃	#492	1	1	yes	characteristic
116	C₄×C₇₃	#494	1	1	yes	characteristic
117	Dic₇₃	—	8	2	no	—
118	Dic₇₃	—	8	2	no	—
119	#9697	—	73	1	no	maximal, Sylow
120	C₂×C₇₃	#1542	1	2	yes	—
121	C₂×C₇₃	#1550	1	2	yes	—
122	C₂×C₇₃	#1956	1	1	yes	characteristic
123	C₂×C₇₃	#1967	1	1	yes	characteristic
124	C₂×C₇₃	#1983	1	1	yes	characteristic
125	#2862	—	73	1	no	—
126	#1842	D₇₃	1	1	yes	characteristic
127	C₄×C₄⋊C₈	—	73	1	no	—
128	C₂×#333	—	73	2	no	—
129	C₂×#332	—	73	2	no	—
130	C₇₃	#9697	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄⋊Q₈	—	73	1	no	—
132	C₂×C₄⋊Q₈	—	73	1	no	—
133	C₂×C₄⋊C₈	—	2	2	no	—
134	C₂×C₄⋊C₈	—	73	2	no	—
135	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
136	C₂×C₄×C₈	—	146	1	no	—
137	#391	—	73	4	no	—
138	#391	—	146	1	no	—
139	#391	—	146	1	no	—
140	#382	—	146	2	no	—
141	C₄³	C₂×D₇₃	1	1	yes	characteristic
142	#336	—	146	1	no	—
143	#333	—	146	4	no	—
144	#332	—	146	4	no	—
145	C₂²×Q₈	—	73	2	no	—
146	C₂²×Q₈	—	146	1	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	146	2	no	—
149	C₄⋊Q₈	—	146	2	no	—
150	C₄⋊Q₈	—	146	2	no	—
151	C₄⋊Q₈	—	146	2	no	—
152	C₄⋊Q₈	—	146	2	no	—
153	C₂×C₄⋊C₄	—	73	2	no	—
154	C₂×C₄⋊C₄	—	146	1	no	—
155	C₂×C₄⋊C₄	—	146	1	no	—
156	C₂×C₄⋊C₄	—	146	1	no	—
157	C₂×C₄²	C₄×D₇₃	1	2	yes	—
158	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
159	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
161	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	4	4	no	—
165	C₄⋊C₈	—	73	4	no	—
166	C₄⋊C₈	—	146	2	no	—
167	C₄×C₈	—	292	2	no	—
168	#95	—	146	4	no	—
169	#95	—	146	4	no	—
170	C₂×Q₈	—	292	4	no	—
171	C₂×Q₈	—	292	2	no	—
172	C₂×Q₈	—	292	2	no	—
173	C₂×Q₈	—	292	1	no	—
174	C₂×Q₈	—	292	1	no	—
175	C₂×Q₈	—	146	4	no	—
176	C₂²×C₄	—	146	2	no	—
177	C₂²×C₄	—	146	2	no	—
178	C₂²×C₄	#11924005	1	2	yes	—
179	C₂²×C₄	#11924005	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	—	292	2	no	—
186	C₂×C₈	—	292	2	no	—
187	C₂×C₈	—	146	4	no	—
188	C₄⋊C₄	—	292	2	no	—
189	C₄⋊C₄	—	292	1	no	—
190	C₄⋊C₄	—	292	1	no	—
191	C₄⋊C₄	—	292	1	no	—
192	C₄⋊C₄	—	292	1	no	—
193	C₄⋊C₄	—	146	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	4	2	no	—
198	C₄²	#11924016	1	4	yes	—
199	C₄²	#11924018	1	4	yes	—
200	C₄²	D₄×D₇₃	1	2	yes	—
201	C₄²	#11924029	1	2	yes	—
202	C₂³	C₂×#11924005	1	1	yes	characteristic
203	Q₈	—	584	4	no	—
204	Q₈	—	584	2	no	—
205	Q₈	—	584	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	292	2	no	—
212	C₂×C₄	—	292	2	no	—
213	C₂×C₄	—	292	4	no	—
214	C₂×C₄	—	292	2	no	—
215	C₂×C₄	—	292	2	no	—
216	C₂×C₄	#14556514	1	4	yes	—
217	C₂×C₄	#14556514	1	4	yes	—
218	C₂×C₄	#14556553	1	2	yes	—
219	C₂×C₄	#14556566	1	2	yes	—
220	C₂×C₄	#14556580	1	1	yes	characteristic
221	C₂×C₄	#14556587	1	1	yes	characteristic
222	C₂×C₄	C₂×#11924016	1	2	yes	—
223	C₂×C₄	C₂×#11924018	1	2	yes	—
224	C₂×C₄	#14556613	1	2	yes	—
225	C₂×C₄	#14556619	1	1	yes	characteristic
226	C₂×C₄	#14556624	1	1	yes	characteristic
227	C₈	—	8	2	no	—
228	C₈	—	292	4	no	—
229	C₂²	#17988488	1	1	yes	characteristic
230	C₂²	#17988569	1	2	yes	—
231	C₂²	C₂×#14556514	1	2	yes	—
232	C₂²	#17988584	1	1	yes	characteristic
233	C₂²	#17988589	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	584	2	no	—
238	C₄	—	584	2	no	—
239	C₄	#17988803	1	2	yes	—
240	C₄	#17988810	1	2	yes	—
241	C₄	#17988939	1	2	yes	—
242	C₄	#17988959	1	1	yes	characteristic
243	C₄	#17988966	1	1	yes	characteristic
244	C₂	#22608589	1	2	yes	—
245	C₂	#22608622	1	2	yes	—
246	C₂	#22610732	1	1	yes	characteristic
247	C₂	#22610778	1	1	yes	characteristic
248	C₂	#22610861	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11924005
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1842	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1842	D₇₃
hypocenter	C₇₃	#9697
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
130. C₇₃	#9697	semidirect

Maximal subgroups

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

subgroup	index
#22618309	2
C₄×C₄⋊#10596831	2
C₂×#17988335	2
C₂×#17988332	2
C₇₃×#1842	2
#9697	73