Subgroups — #33056550

Order 18688.

734 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618643	C₂	1	2	yes	maximal, complemented
2	#22608845	C₂	1	4	yes	maximal, complemented
3	#22608755	C₂	1	4	yes	maximal, complemented
4	#22608687	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608382	C₂	1	2	yes	maximal, complemented
6	C₂×#17988308	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#3169	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556598	—	2	2	no	—
9	#17988665	—	2	2	no	—
10	#17988665	C₂²	1	4	yes	—
11	D₄×#10596831	—	2	2	no	—
12	D₄×#10596831	C₂²	1	4	yes	—
13	#17988611	C₂²	1	2	yes	—
14	#17988554	—	2	2	no	—
15	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	#17988338	C₂²	1	4	yes	—
18	D₄⋊#10596831	C₂²	1	8	yes	—
19	#17988308	C₂²	1	2	yes	—
20	#17988308	C₂²	1	2	yes	—
21	#532×C₇₃	C₂²	1	2	yes	—
22	#500×C₇₃	C₄	1	1	yes	characteristic
23	#493×C₇₃	C₄	1	1	yes	characteristic
24	#486×C₇₃	C₄	1	2	yes	—
25	#466×C₇₃	C₄	1	4	yes	—
26	C₄×D₈×C₇₃	C₂²	1	4	yes	—
27	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
28	#14556598	—	4	8	no	—
29	C₂×#11924012	—	2	4	no	—
30	C₂×#11924012	—	2	2	no	—
31	C₂²×#10596831	—	2	4	no	—
32	C₂²×#10596831	—	2	1	no	—
33	C₂²×#10596831	D₄	1	2	yes	—
34	C₄⋊#10596831	—	2	2	no	—
35	C₄⋊#10596831	C₂³	1	4	yes	—
36	C₂²⋊#10596831	—	2	4	no	—
37	C₂²⋊#10596831	—	2	4	no	—
38	C₄×#10596831	—	2	1	no	—
39	C₄×#10596831	D₄	1	2	yes	—
40	C₄×#10596831	C₂³	1	2	yes	—
41	C₄×#10596831	C₂³	1	1	yes	characteristic
42	C₄×#10596831	C₂³	1	1	yes	characteristic
43	C₂×#41×C₇₃	—	2	2	no	—
44	C₂×D₈×C₇₃	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
47	#127×C₇₃	C₂×C₄	1	2	yes	—
48	#124×C₇₃	—	2	2	no	—
49	#121×C₇₃	—	2	2	no	—
50	#121×C₇₃	C₂×C₄	1	4	yes	—
51	C₄×D₄×C₇₃	—	2	2	no	—
52	C₄×D₄×C₇₃	C₂³	1	4	yes	—
53	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
54	#107×C₇₃	C₂×C₄	1	4	yes	—
55	#102×C₇₃	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₃	C₂³	1	2	yes	—
59	#11924012	—	4	8	no	—
60	#11924012	—	4	4	no	—
61	C₂×#10596831	—	2	4	no	—
62	C₂×#10596831	—	2	4	no	—
63	C₂×#10596831	—	2	2	no	—
64	C₂×#10596831	—	4	8	no	—
65	C₂×#10596831	—	4	2	no	—
66	C₂×#10596831	C₂×D₄	1	2	yes	—
67	C₂×#10596831	C₂×D₄	1	2	yes	—
68	#41×C₇₃	—	4	4	no	—
69	#41×C₇₃	—	4	4	no	—
70	C₂×Q₈×C₇₃	—	2	2	no	—
71	C₂×D₄×C₇₃	—	2	4	no	—
72	C₂×D₄×C₇₃	—	2	2	no	—
73	C₂×D₄×C₇₃	—	2	2	no	—
74	C₂×D₄×C₇₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₇₃	—	2	4	no	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
79	D₈×C₇₃	—	2	8	no	—
80	C₂×C₈×C₇₃	—	2	4	no	—
81	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₇₃	—	2	2	no	—
84	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×C₇₃	—	2	4	no	—
86	C₂²⋊C₄×C₇₃	—	2	4	no	—
87	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇₃	#41	1	1	yes	characteristic
90	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
91	C₂³×C₇₃	—	2	4	no	—
92	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
93	Q₈×C₇₃	—	4	4	no	—
94	D₄×C₇₃	—	2	8	no	—
95	D₄×C₇₃	—	4	8	no	—
96	D₄×C₇₃	—	4	4	no	—
97	D₄×C₇₃	—	4	4	no	—
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	—	4	4	no	—
102	C₂×C₄×C₇₃	—	4	4	no	—
103	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
110	C₈×C₇₃	—	2	2	no	—
111	C₈×C₇₃	#131	1	4	yes	complemented
112	#10596831	—	2	2	no	—
113	#10596831	—	2	2	no	—
114	#10596831	—	4	2	no	—
115	#10596831	—	4	4	no	—
116	C₂²×C₇₃	—	2	1	no	—
117	C₂²×C₇₃	—	2	1	no	—
118	C₂²×C₇₃	—	4	4	no	—
119	C₂²×C₇₃	—	4	4	no	—
120	C₂²×C₇₃	—	4	4	no	—
121	C₂²×C₇₃	#442	1	2	yes	—
122	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₇₃	—	2	1	no	—
124	C₄×C₇₃	—	2	1	no	—
125	C₄×C₇₃	—	4	4	no	—
126	C₄×C₇₃	#442	1	2	yes	—
127	C₄×C₇₃	#444	1	2	yes	—
128	C₄×C₇₃	C₂×#131	1	2	yes	—
129	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	#17962	—	73	1	no	maximal, Sylow
131	C₂×C₇₃	—	2	2	no	—
132	C₂×C₇₃	—	8	4	no	—
133	C₂×C₇₃	#2931	1	1	yes	characteristic
134	C₂×C₇₃	C₂×#442	1	1	yes	characteristic
135	C₂×C₇₃	C₂×#444	1	1	yes	characteristic
136	#3169	D₇₃	1	1	yes	characteristic
137	#2931	—	73	2	no	—
138	#1594	—	73	4	no	—
139	#1580	—	73	4	no	—
140	#1566	—	73	1	no	—
141	#1492	—	73	2	no	—
142	C₂×#322	—	73	1	no	—
143	C₇₃	#17962	1	1	yes	characteristic, complemented, Sylow
144	C₂×#131	—	146	2	no	—
145	#532	C₂×D₇₃	1	2	yes	—
146	#500	Dic₇₃	1	1	yes	characteristic
147	#493	Dic₇₃	1	1	yes	characteristic
148	#486	Dic₇₃	1	2	yes	—
149	#466	Dic₇₃	1	4	yes	—
150	C₄×D₈	C₂×D₇₃	1	4	yes	—
151	#436	—	73	4	no	—
152	#436	—	146	2	no	—
153	C₈×D₄	—	73	4	no	—
154	C₈×D₄	—	146	2	no	—
155	#423	—	73	2	no	—
156	#408	—	146	2	no	—
157	C₂×C₄×C₈	—	73	1	no	—
158	C₂×C₄×C₈	—	73	1	no	—
159	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
160	#334	—	73	4	no	—
161	D₄⋊C₈	—	73	8	no	—
162	#322	—	73	2	no	—
163	#322	—	73	2	no	—
164	C₂×#41	—	2	2	no	—
165	C₂×D₈	C₂×Dic₇₃	1	4	yes	—
166	#131	—	292	8	no	—
167	C₂×M₁₆	—	146	4	no	—
168	C₂×M₁₆	—	146	2	no	—
169	C₂²×C₈	—	73	2	no	—
170	C₂²×C₈	—	146	4	no	—
171	C₂²×C₈	—	146	1	no	—
172	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
173	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
174	#127	C₂×Dic₇₃	1	2	yes	—
175	#124	—	2	2	no	—
176	#121	—	2	2	no	—
177	#121	C₂×Dic₇₃	1	4	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂²×D₇₃	1	4	yes	—
180	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
181	#107	C₂×Dic₇₃	1	4	yes	—
182	C₄⋊C₈	—	73	4	no	—
183	C₄⋊C₈	—	146	2	no	—
184	#102	C₂×Dic₇₃	1	8	yes	—
185	C₂²⋊C₈	—	146	4	no	—
186	C₂²⋊C₈	—	146	4	no	—
187	C₄×C₈	—	73	2	no	—
188	C₄×C₈	—	73	1	no	—
189	C₄×C₈	—	73	1	no	—
190	C₄×C₈	—	73	2	no	—
191	C₄×C₈	—	146	1	no	—
192	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
193	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₇₃	1	2	yes	—
195	#41	—	4	4	no	—
196	#41	—	4	4	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂²×Dic₇₃	1	4	yes	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
204	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
205	C₂²×C₄	#11924027	1	1	yes	characteristic
206	D₈	—	2	8	no	—
207	M₁₆	—	292	4	no	—
208	M₁₆	—	292	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	292	2	no	—
211	C₂×C₈	—	292	8	no	—
212	C₂×C₈	—	73	2	no	—
213	C₂×C₈	—	73	2	no	—
214	C₂×C₈	—	146	4	no	—
215	C₂×C₈	—	146	2	no	—
216	C₂×C₈	—	146	4	no	—
217	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
218	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
224	C₄²	D₄×D₇₃	1	1	yes	characteristic
225	C₄²	#11924027	1	1	yes	characteristic
226	C₄²	C₂³×D₇₃	1	1	yes	characteristic
227	C₂³	—	2	4	no	—
228	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
240	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
241	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
242	C₂×C₄	#14556615	1	4	yes	—
243	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	292	2	no	—
248	C₈	—	292	4	no	—
249	C₈	—	146	2	no	—
250	C₈	—	146	2	no	—
251	C₈	#14556598	1	4	yes	complemented
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	#17988699	1	2	yes	—
258	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	4	no	—
262	C₄	#17988699	1	2	yes	—
263	C₄	#17988709	1	2	yes	—
264	C₄	C₂×#14556598	1	2	yes	—
265	C₄	C₂×#14556615	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	4	no	—
268	C₂	#22618643	1	1	yes	characteristic
269	C₂	C₂×#17988699	1	1	yes	characteristic
270	C₂	C₂×#17988709	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting, prime radical	C₇₃×#3169	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#3169	D₇₃
hypocenter	C₇₃	#17962
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #22618643	C₂	semidirect
2. #22608845	C₂	semidirect
3. #22608755	C₂	semidirect
4. #22608687	C₂	semidirect
5. #22608382	C₂	semidirect
6. C₂×#17988308	C₂	semidirect
111. C₈×C₇₃	#131	semidirect
143. C₇₃	#17962	semidirect
251. C₈	#14556598	semidirect

Maximal subgroups

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

subgroup	index
#22618643	2
#22608845	2
#22608755	2
#22608687	2
#22608382	2
C₂×#17988308	2
C₇₃×#3169	2
#17962	73