Subgroups — #33056737

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 268 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	#22608796	C₂	1	8	yes	maximal, complemented
3	#22608687	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608382	C₂	1	2	yes	maximal, complemented
5	C₂×#17988309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₃×#3169	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556598	—	2	2	no	—
8	#17988665	—	2	2	no	—
9	#17988665	C₂²	1	4	yes	—
10	D₄×#10596831	—	2	2	no	—
11	D₄×#10596831	C₂²	1	4	yes	—
12	#17988611	C₂²	1	2	yes	—
13	#17988554	—	2	2	no	—
14	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
15	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
16	#17988338	C₂²	1	4	yes	—
17	D₄⋊#10596831	C₂²	1	8	yes	—
18	#17988309	C₂²	1	4	yes	—
19	#532×C₇₃	C₂²	1	2	yes	—
20	#500×C₇₃	C₄	1	1	yes	characteristic
21	#493×C₇₃	C₄	1	1	yes	characteristic
22	#486×C₇₃	C₄	1	2	yes	—
23	#466×C₇₃	C₄	1	4	yes	—
24	C₄×D₈×C₇₃	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	#14556598	—	4	8	no	—
27	C₂×#11924012	—	2	4	no	—
28	C₂×#11924012	—	2	2	no	—
29	C₂²×#10596831	—	2	4	no	—
30	C₂²×#10596831	—	2	1	no	—
31	C₂²×#10596831	D₄	1	2	yes	—
32	C₄⋊#10596831	—	2	2	no	—
33	C₄⋊#10596831	C₂³	1	4	yes	—
34	C₂²⋊#10596831	—	2	4	no	—
35	C₂²⋊#10596831	—	2	4	no	—
36	C₄×#10596831	—	2	1	no	—
37	C₄×#10596831	D₄	1	1	yes	characteristic
38	C₄×#10596831	D₄	1	1	yes	characteristic
39	C₄×#10596831	C₂³	1	2	yes	—
40	C₄×#10596831	C₂³	1	2	yes	—
41	C₂×#41×C₇₃	—	2	2	no	—
42	C₂×D₈×C₇₃	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
44	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
45	#127×C₇₃	C₂×C₄	1	2	yes	—
46	#124×C₇₃	—	2	2	no	—
47	#121×C₇₃	—	2	2	no	—
48	#121×C₇₃	C₂×C₄	1	4	yes	—
49	C₄×D₄×C₇₃	—	2	2	no	—
50	C₄×D₄×C₇₃	C₂³	1	4	yes	—
51	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
52	#107×C₇₃	C₂×C₄	1	4	yes	—
53	#102×C₇₃	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₇₃	C₂³	1	2	yes	—
57	#11924012	—	4	8	no	—
58	#11924012	—	4	4	no	—
59	C₂×#10596831	—	2	4	no	—
60	C₂×#10596831	—	2	4	no	—
61	C₂×#10596831	—	2	2	no	—
62	C₂×#10596831	—	2	4	no	—
63	C₂×#10596831	—	4	8	no	—
64	C₂×#10596831	C₂×D₄	1	2	yes	—
65	C₂×#10596831	C₂×D₄	1	2	yes	—
66	#41×C₇₃	—	4	4	no	—
67	#41×C₇₃	—	4	4	no	—
68	C₂×Q₈×C₇₃	—	2	2	no	—
69	C₂×D₄×C₇₃	—	2	4	no	—
70	C₂×D₄×C₇₃	—	2	2	no	—
71	C₂×D₄×C₇₃	—	2	2	no	—
72	C₂×D₄×C₇₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇₃	—	2	4	no	—
74	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
77	D₈×C₇₃	—	4	4	no	—
78	C₂×C₈×C₇₃	—	4	2	no	—
79	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
81	C₄⋊C₄×C₇₃	—	2	2	no	—
82	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₇₃	—	2	4	no	—
84	C₂²⋊C₄×C₇₃	—	2	4	no	—
85	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇₃	#41	1	1	yes	characteristic
88	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
89	C₂³×C₇₃	—	2	4	no	—
90	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
91	Q₈×C₇₃	—	4	4	no	—
92	D₄×C₇₃	—	2	8	no	—
93	D₄×C₇₃	—	4	8	no	—
94	D₄×C₇₃	—	4	4	no	—
95	D₄×C₇₃	—	4	4	no	—
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	4	no	—
99	C₂×C₄×C₇₃	—	4	4	no	—
100	C₂×C₄×C₇₃	—	4	4	no	—
101	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
105	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
108	C₈×C₇₃	—	2	2	no	—
109	C₈×C₇₃	—	2	2	no	—
110	#10596831	—	2	4	no	—
111	#10596831	—	2	2	no	—
112	#10596831	—	2	2	no	—
113	#10596831	—	4	4	no	—
114	C₂²×C₇₃	—	2	1	no	—
115	C₂²×C₇₃	—	2	1	no	—
116	C₂²×C₇₃	—	4	4	no	—
117	C₂²×C₇₃	—	4	4	no	—
118	C₂²×C₇₃	—	4	4	no	—
119	C₂²×C₇₃	#442	1	2	yes	—
120	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₇₃	—	2	1	no	—
122	C₄×C₇₃	—	2	1	no	—
123	C₄×C₇₃	—	4	4	no	—
124	C₄×C₇₃	#442	1	2	yes	—
125	C₄×C₇₃	#443	1	2	yes	—
126	C₄×C₇₃	#568	1	2	yes	—
127	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
128	#17981	—	73	1	no	maximal, Sylow
129	C₂×C₇₃	—	2	2	no	—
130	C₂×C₇₃	—	8	4	no	—
131	C₂×C₇₃	#2932	1	1	yes	characteristic
132	C₂×C₇₃	C₂×#442	1	1	yes	characteristic
133	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
134	#3169	D₇₃	1	1	yes	characteristic
135	#2931	—	73	2	no	—
136	#1586	—	73	8	no	—
137	#1566	—	73	1	no	—
138	#1492	—	73	2	no	—
139	C₂×#322	—	73	1	no	—
140	C₇₃	#17981	1	1	yes	characteristic, complemented, Sylow
141	C₂×#131	—	146	2	no	—
142	#532	C₂×D₇₃	1	2	yes	—
143	#500	Dic₇₃	1	1	yes	characteristic
144	#493	Dic₇₃	1	1	yes	characteristic
145	#486	Dic₇₃	1	2	yes	—
146	#466	Dic₇₃	1	4	yes	—
147	C₄×D₈	C₂×D₇₃	1	4	yes	—
148	#436	—	73	4	no	—
149	#436	—	146	2	no	—
150	C₈×D₄	—	73	4	no	—
151	C₈×D₄	—	146	2	no	—
152	#423	—	73	2	no	—
153	#408	—	146	2	no	—
154	C₂×C₄×C₈	—	73	1	no	—
155	C₂×C₄×C₈	—	73	1	no	—
156	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
157	#334	—	73	4	no	—
158	D₄⋊C₈	—	73	8	no	—
159	#322	—	73	4	no	—
160	C₂×#41	—	2	2	no	—
161	C₂×D₈	C₂×Dic₇₃	1	4	yes	—
162	#131	—	292	8	no	—
163	C₂×M₁₆	—	146	4	no	—
164	C₂×M₁₆	—	146	2	no	—
165	C₂²×C₈	—	73	2	no	—
166	C₂²×C₈	—	146	4	no	—
167	C₂²×C₈	—	146	1	no	—
168	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
169	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
170	#127	C₂×Dic₇₃	1	2	yes	—
171	#124	—	2	2	no	—
172	#121	—	2	2	no	—
173	#121	C₂×Dic₇₃	1	4	yes	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	C₂²×D₇₃	1	4	yes	—
176	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
177	#107	C₂×Dic₇₃	1	4	yes	—
178	C₄⋊C₈	—	73	4	no	—
179	C₄⋊C₈	—	146	2	no	—
180	#102	C₂×Dic₇₃	1	8	yes	—
181	C₂²⋊C₈	—	146	4	no	—
182	C₂²⋊C₈	—	146	4	no	—
183	C₄×C₈	—	73	1	no	—
184	C₄×C₈	—	73	1	no	—
185	C₄×C₈	—	73	2	no	—
186	C₄×C₈	—	73	2	no	—
187	C₄×C₈	—	146	1	no	—
188	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
189	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₇₃	1	2	yes	—
191	#41	—	4	4	no	—
192	#41	—	4	4	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	C₂²×Dic₇₃	1	4	yes	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
200	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
201	C₂²×C₄	#11924027	1	1	yes	characteristic
202	D₈	—	4	4	no	—
203	M₁₆	—	292	4	no	—
204	M₁₆	—	292	8	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	292	8	no	—
207	C₂×C₈	—	73	2	no	—
208	C₂×C₈	—	73	2	no	—
209	C₂×C₈	—	146	4	no	—
210	C₂×C₈	—	146	4	no	—
211	C₂×C₈	—	146	4	no	—
212	C₂×C₈	—	146	2	no	—
213	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
214	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
220	C₄²	D₄×D₇₃	1	1	yes	characteristic
221	C₄²	#11924027	1	1	yes	characteristic
222	C₄²	C₂³×D₇₃	1	1	yes	characteristic
223	C₂³	—	2	4	no	—
224	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
225	Q₈	—	4	4	no	—
226	D₄	—	2	8	no	—
227	D₄	—	4	8	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
238	C₂×C₄	#14556615	1	4	yes	—
239	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
240	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
241	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	292	4	no	—
245	C₈	—	146	4	no	—
246	C₈	—	146	2	no	—
247	C₈	—	146	2	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	#17988699	1	2	yes	—
254	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	#17988699	1	2	yes	—
259	C₄	#17988704	1	2	yes	—
260	C₄	#17989354	1	2	yes	—
261	C₄	C₂×#14556615	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	4	no	—
264	C₂	#22618649	1	1	yes	characteristic
265	C₂	C₂×#17988699	1	1	yes	characteristic
266	C₂	C₂×#17988704	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	#17981
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. #22608796	C₂	semidirect
3. #22608687	C₂	semidirect
4. #22608382	C₂	semidirect
5. C₂×#17988309	C₂	semidirect
140. C₇₃	#17981	semidirect

Maximal subgroups

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

subgroup	index
#22618643	2
#22608796	2
#22608687	2
#22608382	2
C₂×#17988309	2
C₇₃×#3169	2
#17981	73