Subgroups — #32986002

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556591	C₂	1	1	yes	maximal, characteristic
2	#22609677	C₂	1	1	yes	maximal, characteristic
3	#22607770	C₂	1	4	yes	maximal
4	C₇₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556591	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₄₆	—	2	1	no	—
7	C₂×C₄×#10596831	C₂²	1	2	yes	—
8	#17988493	—	2	1	no	—
9	#17988485	—	2	1	no	—
10	#17988485	C₄	1	2	yes	—
11	#17988457	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₇₃	—	2	1	no	—
13	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
14	#17988342	—	2	1	no	—
15	#17988342	—	2	1	no	—
16	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
17	C₄³×C₇₃	C₂²	1	1	yes	characteristic
18	#336×C₇₃	—	2	1	no	—
19	#336×C₇₃	—	2	1	no	—
20	C₂²×Dic₁₄₆	—	2	1	no	—
21	C₂²×#10596831	—	2	2	no	—
22	#14556591	—	2	2	no	—
23	#14556591	C₈	1	4	yes	complemented
24	C₄×Dic₁₄₆	—	4	4	no	—
25	C₂×C₄⋊Dic₇₃	—	2	1	no	—
26	C₂×C₄⋊Dic₇₃	—	2	1	no	—
27	C₂×C₄⋊Dic₇₃	—	2	1	no	—
28	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×#11924007	—	2	2	no	—
31	C₂×C₄×Dic₇₃	—	2	2	no	—
32	C₄×#10596831	—	2	2	no	—
33	C₄×#10596831	Q₈	1	4	yes	complemented
34	#14556498	—	2	2	no	—
35	#14556498	—	2	2	no	—
36	C₂²×C₈×C₇₃	—	2	2	no	—
37	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₇₃	—	2	4	no	—
45	C₂×Dic₁₄₆	—	2	4	no	—
46	C₂×Dic₁₄₆	—	4	2	no	—
47	C₂×Dic₁₄₆	—	4	2	no	—
48	C₂²×Dic₇₃	—	2	2	no	—
49	C₂²×Dic₇₃	—	2	2	no	—
50	C₂×#10596831	—	2	4	no	—
51	C₂×#10596831	—	4	2	no	—
52	C₂×#10596831	—	4	2	no	—
53	C₄⋊Dic₇₃	—	2	4	no	—
54	C₄⋊Dic₇₃	—	2	2	no	—
55	C₄⋊Dic₇₃	—	2	4	no	—
56	C₄⋊Dic₇₃	—	4	2	no	—
57	C₄⋊Dic₇₃	—	4	2	no	—
58	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
59	#11924007	—	4	4	no	—
60	C₄×Dic₇₃	—	4	4	no	—
61	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	—	4	2	no	—
69	C₂×C₈×C₇₃	—	4	2	no	—
70	C₂×C₈×C₇₃	—	4	2	no	—
71	C₄²×C₇₃	—	2	2	no	—
72	C₄²×C₇₃	—	2	2	no	—
73	C₄²×C₇₃	—	2	2	no	—
74	C₄²×C₇₃	—	4	1	no	—
75	C₄²×C₇₃	—	4	1	no	—
76	C₄²×C₇₃	C₂×C₈	1	2	yes	—
77	C₄²×C₇₃	M₁₆	1	2	yes	—
78	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
79	C₄²×C₇₃	#41	1	2	yes	—
80	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	—	2	1	no	—
82	C₂×C₄×C₇₃	—	2	1	no	—
83	C₂×C₄×C₇₃	—	2	1	no	—
84	C₂×C₄×C₇₃	—	2	1	no	—
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	1	no	—
87	C₂×C₄×C₇₃	—	2	1	no	—
88	C₂×C₄×C₇₃	—	2	1	no	—
89	C₂×C₄×C₇₃	—	2	1	no	—
90	C₂×C₄×C₇₃	—	2	1	no	—
91	C₂×C₄×C₇₃	—	2	1	no	—
92	C₂×C₄×C₇₃	—	2	1	no	—
93	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇₃	#131	1	2	yes	—
102	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₇₃	#137	1	2	yes	—
104	C₂×Dic₇₃	—	2	4	no	—
105	C₂×Dic₇₃	—	4	2	no	—
106	C₂×Dic₇₃	—	4	2	no	—
107	C₂×Dic₇₃	—	4	2	no	—
108	C₂×Dic₇₃	—	4	2	no	—
109	C₂×Dic₇₃	—	4	2	no	—
110	Dic₁₄₆	—	4	8	no	—
111	C₈×C₇₃	—	8	2	no	—
112	#10596831	—	4	4	no	—
113	C₂²×C₇₃	#391	1	1	yes	characteristic
114	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₇₃	#408	1	1	yes	characteristic
116	C₂²×C₇₃	#418	1	1	yes	characteristic
117	C₂²×C₇₃	#419	1	1	yes	characteristic
118	C₂²×C₇₃	#421	1	1	yes	characteristic
119	C₂²×C₇₃	#421	1	1	yes	characteristic
120	C₄×C₇₃	—	2	2	no	—
121	C₄×C₇₃	—	2	2	no	—
122	C₄×C₇₃	—	2	2	no	—
123	C₄×C₇₃	—	4	1	no	—
124	C₄×C₇₃	—	4	1	no	—
125	C₄×C₇₃	C₈×Q₈	1	2	yes	—
126	C₄×C₇₃	#446	1	2	yes	—
127	C₄×C₇₃	#489	1	2	yes	—
128	C₄×C₇₃	#501	1	2	yes	—
129	Dic₇₃	—	4	4	no	—
130	Dic₇₃	—	8	2	no	—
131	#9873	—	73	1	no	maximal, Sylow
132	C₂×C₇₃	#1478	1	1	yes	characteristic
133	C₂×C₇₃	#1479	1	1	yes	characteristic
134	C₂×C₇₃	#1496	1	1	yes	characteristic
135	C₂×C₇₃	#1497	1	1	yes	characteristic
136	C₂×C₇₃	#1984	1	1	yes	characteristic
137	C₂×C₇₃	#1993	1	1	yes	characteristic
138	C₂×C₇₃	#1995	1	1	yes	characteristic
139	C₄×C₄⋊Q₈	—	73	1	no	—
140	#1831	D₇₃	1	1	yes	characteristic
141	#1744	—	73	1	no	—
142	#1279	—	73	4	no	—
143	C₇₃	#9873	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄⋊Q₈	—	73	1	no	—
145	C₂×C₄×Q₈	—	146	1	no	—
146	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
147	C₂×C₄×C₈	—	73	2	no	—
148	#391	—	146	1	no	—
149	#389	—	73	2	no	—
150	#389	—	146	1	no	—
151	#384	—	73	1	no	—
152	C₄×C₄⋊C₄	—	73	2	no	—
153	C₄×C₄⋊C₄	—	146	1	no	—
154	C₄³	C₂×D₇₃	1	1	yes	characteristic
155	#336	—	2	1	no	—
156	#336	—	2	1	no	—
157	#336	—	146	1	no	—
158	#336	—	146	1	no	—
159	C₂²×Q₈	—	146	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	146	2	no	—
162	C₄⋊Q₈	—	73	4	no	—
163	C₄⋊Q₈	—	146	2	no	—
164	C₄×Q₈	—	292	4	no	—
165	C₂×C₄⋊C₄	—	73	2	no	—
166	C₂×C₄⋊C₄	—	73	2	no	—
167	C₂×C₄⋊C₄	—	146	1	no	—
168	C₂×C₄⋊C₄	—	146	2	no	—
169	C₂×C₄⋊C₄	—	146	1	no	—
170	C₂×C₄⋊C₄	—	146	1	no	—
171	C₂×C₄²	—	146	2	no	—
172	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
174	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
175	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
179	C₄⋊C₈	—	2	4	no	—
180	C₄×C₈	—	73	4	no	—
181	C₄×C₈	—	146	2	no	—
182	#95	—	146	2	no	—
183	#95	—	146	2	no	—
184	C₂×Q₈	—	292	2	no	—
185	C₂×Q₈	—	292	2	no	—
186	C₂×Q₈	—	146	4	no	—
187	C₂²×C₄	—	146	2	no	—
188	C₂²×C₄	—	146	2	no	—
189	C₂²×C₄	#11924005	1	1	yes	characteristic
190	C₂²×C₄	#11924005	1	1	yes	characteristic
191	C₂²×C₄	#11924005	1	1	yes	characteristic
192	C₂²×C₄	#11924005	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	292	2	no	—
200	C₂×C₈	—	292	2	no	—
201	C₂×C₈	—	146	4	no	—
202	C₄⋊C₄	—	292	4	no	—
203	C₄⋊C₄	—	292	2	no	—
204	C₄⋊C₄	—	292	2	no	—
205	C₄⋊C₄	—	73	4	no	—
206	C₄⋊C₄	—	146	4	no	—
207	C₄⋊C₄	—	146	4	no	—
208	C₄⋊C₄	—	146	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	292	4	no	—
215	C₄²	C₈×D₇₃	1	2	yes	—
216	C₄²	#11924011	1	2	yes	—
217	C₄²	Q₈×D₇₃	1	2	yes	—
218	C₄²	#11924027	1	2	yes	—
219	C₂³	C₂×#11924005	1	1	yes	characteristic
220	Q₈	—	292	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	292	2	no	—
234	C₂×C₄	—	292	2	no	—
235	C₂×C₄	—	292	2	no	—
236	C₂×C₄	—	292	2	no	—
237	C₂×C₄	—	292	2	no	—
238	C₂×C₄	—	146	4	no	—
239	C₂×C₄	#14556503	1	2	yes	—
240	C₂×C₄	#14556503	1	2	yes	—
241	C₂×C₄	#14556556	1	2	yes	—
242	C₂×C₄	#14556571	1	2	yes	—
243	C₂×C₄	#14556578	1	1	yes	characteristic
244	C₂×C₄	#14556589	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
247	C₂×C₄	#14556599	1	2	yes	—
248	C₂×C₄	#14556616	1	2	yes	—
249	C₂×C₄	#14556621	1	2	yes	—
250	C₈	—	8	2	no	—
251	C₈	—	292	4	no	—
252	C₂²	#17988492	1	1	yes	characteristic
253	C₂²	C₂×#14556503	1	1	yes	characteristic
254	C₂²	#17988555	1	1	yes	characteristic
255	C₂²	#17988591	1	1	yes	characteristic
256	C₂²	#17988595	1	1	yes	characteristic
257	C₂²	#17988603	1	1	yes	characteristic
258	C₂²	#17988603	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	292	4	no	—
265	C₄	—	584	2	no	—
266	C₄	#17988711	1	2	yes	—
267	C₄	#17988714	1	2	yes	—
268	C₄	#17988946	1	2	yes	—
269	C₄	#17988997	1	2	yes	—
270	C₂	#22608321	1	1	yes	characteristic
271	C₂	#22608325	1	1	yes	characteristic
272	C₂	#22608399	1	1	yes	characteristic
273	C₂	#22608403	1	1	yes	characteristic
274	C₂	#22610866	1	1	yes	characteristic
275	C₂	#22610903	1	1	yes	characteristic
276	C₂	#22610911	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1831	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1831	D₇₃
hypocenter	C₇₃	#9873
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
23. #14556591	C₈	semidirect
33. C₄×#10596831	Q₈	semidirect
143. C₇₃	#9873	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556591	2
#22609677	2
#22607770	2
C₇₃×#1831	2
#9873	73