Subgroups — C₇₃×C₂²⋊#338 #32920122

Order 18688.

1294 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#407×C₇₃	C₂	1	2	yes	maximal
3	C₂×#338×C₇₃	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₇₃	—	2	4	no	—
6	C₂²×M₁₆×C₇₃	—	2	8	no	—
7	C₂²×M₁₆×C₇₃	C₄	1	2	yes	—
8	C₂²×M₁₆×C₇₃	C₂²	1	4	yes	—
9	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
10	C₂³×C₈×C₇₃	C₄	1	2	yes	—
11	#407×C₇₃	—	2	2	no	—
12	#407×C₇₃	—	2	2	no	—
13	#407×C₇₃	—	2	4	no	—
14	C₂×C₂²⋊C₈×C₇₃	C₄	1	8	yes	—
15	#338×C₇₃	—	2	8	no	—
16	C₂⁵×C₇₃	Q₈	1	1	yes	characteristic
17	C₂³×C₄×C₇₃	—	2	4	no	—
18	C₂³×C₄×C₇₃	—	2	4	no	—
19	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₇₃	D₄	1	4	yes	—
25	C₂³×C₄×C₇₃	Q₈	1	1	yes	characteristic
26	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₇₃	—	2	32	no	—
28	C₂×M₁₆×C₇₃	—	2	4	no	—
29	C₂×M₁₆×C₇₃	—	2	4	no	—
30	C₂×M₁₆×C₇₃	—	2	4	no	—
31	C₂×M₁₆×C₇₃	—	2	4	no	—
32	C₂×M₁₆×C₇₃	—	2	4	no	—
33	C₂×M₁₆×C₇₃	—	2	8	no	—
34	C₂×M₁₆×C₇₃	—	2	8	no	—
35	C₂×M₁₆×C₇₃	—	2	8	no	—
36	C₂×M₁₆×C₇₃	D₄	1	8	yes	—
37	C₂²×C₈×C₇₃	—	2	4	no	—
38	C₂²×C₈×C₇₃	—	2	8	no	—
39	C₂²×C₈×C₇₃	C₂×C₄	1	8	yes	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
41	C₂²⋊C₈×C₇₃	—	2	8	no	—
42	C₂²⋊C₈×C₇₃	—	2	8	no	—
43	C₂⁴×C₇₃	—	2	4	no	—
44	C₂⁴×C₇₃	—	4	4	no	—
45	C₂⁴×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂⁴×C₇₃	C₄⋊C₄	1	4	yes	—
47	C₂⁴×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	—	2	4	no	—
50	C₂²×C₄×C₇₃	—	2	2	no	—
51	C₂²×C₄×C₇₃	—	2	2	no	—
52	C₂²×C₄×C₇₃	—	2	2	no	—
53	C₂²×C₄×C₇₃	—	2	2	no	—
54	C₂²×C₄×C₇₃	—	2	16	no	—
55	C₂²×C₄×C₇₃	—	2	8	no	—
56	C₂²×C₄×C₇₃	—	2	4	no	—
57	C₂²×C₄×C₇₃	—	2	4	no	—
58	C₂²×C₄×C₇₃	—	2	8	no	—
59	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₇₃	#41	1	4	yes	—
74	M₁₆×C₇₃	—	2	16	no	—
75	M₁₆×C₇₃	—	2	16	no	—
76	M₁₆×C₇₃	—	4	4	no	—
77	M₁₆×C₇₃	—	4	8	no	—
78	M₁₆×C₇₃	—	4	4	no	—
79	C₂×C₈×C₇₃	—	2	16	no	—
80	C₂×C₈×C₇₃	—	2	4	no	—
81	C₂×C₈×C₇₃	—	2	4	no	—
82	C₂×C₈×C₇₃	—	2	8	no	—
83	C₂×C₈×C₇₃	—	2	16	no	—
84	C₂×C₈×C₇₃	C₂²⋊C₄	1	8	yes	—
85	C₂³×C₇₃	—	2	4	no	—
86	C₂³×C₇₃	—	2	2	no	—
87	C₂³×C₇₃	—	2	8	no	—
88	C₂³×C₇₃	—	2	2	no	—
89	C₂³×C₇₃	—	2	8	no	—
90	C₂³×C₇₃	—	4	4	no	—
91	C₂³×C₇₃	—	4	2	no	—
92	C₂³×C₇₃	—	4	2	no	—
93	C₂³×C₇₃	—	4	16	no	—
94	C₂³×C₇₃	#95	1	2	yes	—
95	C₂³×C₇₃	#95	1	2	yes	—
96	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₇₃	C₂²⋊Q₈	1	4	yes	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	2	8	no	—
102	C₂×C₄×C₇₃	—	2	4	no	—
103	C₂×C₄×C₇₃	—	2	4	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	—	2	8	no	—
109	C₂×C₄×C₇₃	—	2	8	no	—
110	C₂×C₄×C₇₃	#95	1	2	yes	—
111	C₂×C₄×C₇₃	#95	1	2	yes	—
112	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₇₃	#117	1	2	yes	—
117	C₂×C₄×C₇₃	C₄×D₄	1	8	yes	—
118	C₂×C₄×C₇₃	#120	1	4	yes	—
119	C₂×C₄×C₇₃	#121	1	4	yes	—
120	C₂×C₄×C₇₃	#123	1	4	yes	—
121	C₈×C₇₃	—	2	8	no	—
122	C₈×C₇₃	—	4	8	no	—
123	C₂²×C₇₃	—	2	8	no	—
124	C₂²×C₇₃	—	2	2	no	—
125	C₂²×C₇₃	—	2	2	no	—
126	C₂²×C₇₃	—	2	2	no	—
127	C₂²×C₇₃	—	2	4	no	—
128	C₂²×C₇₃	—	2	2	no	—
129	C₂²×C₇₃	—	2	4	no	—
130	C₂²×C₇₃	—	4	4	no	—
131	C₂²×C₇₃	—	4	16	no	—
132	C₂²×C₇₃	—	4	4	no	—
133	C₂²×C₇₃	#338	1	4	yes	complemented
134	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
135	C₂²×C₇₃	#380	1	2	yes	—
136	C₂²×C₇₃	#385	1	4	yes	—
137	C₄×C₇₃	—	2	2	no	—
138	C₄×C₇₃	—	2	2	no	—
139	C₄×C₇₃	C₄×C₂²⋊C₄	1	2	yes	—
140	C₄×C₇₃	#379	1	1	yes	characteristic
141	C₄×C₇₃	#381	1	1	yes	characteristic
142	C₄×C₇₃	#386	1	4	yes	—
143	C₂²⋊#338	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₇₃	—	2	4	no	—
145	C₂×C₇₃	—	2	2	no	—
146	C₂×C₇₃	—	2	4	no	—
147	C₂×C₇₃	—	4	2	no	—
148	C₂×C₇₃	C₂²⋊#95	1	1	yes	characteristic
149	C₂×C₇₃	C₂×#338	1	2	yes	—
150	C₂³×M₁₆	C₂×C₇₃	1	1	yes	characteristic
151	C₂×#407	C₂×C₇₃	1	2	yes	—
152	C₂×#338	C₂×C₇₃	1	4	yes	complemented
153	C₇₃	C₂²⋊#338	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴×C₄	C₂²×C₇₃	1	1	yes	characteristic
155	C₂²×M₁₆	—	2	4	no	—
156	C₂²×M₁₆	—	2	8	no	—
157	C₂²×M₁₆	C₄×C₇₃	1	2	yes	—
158	C₂²×M₁₆	C₂²×C₇₃	1	4	yes	—
159	C₂²×M₁₆	C₂²×C₇₃	1	2	yes	—
160	C₂³×C₈	C₄×C₇₃	1	2	yes	—
161	#407	—	2	4	no	—
162	#407	—	2	2	no	—
163	#407	—	2	2	no	—
164	C₂×C₂²⋊C₈	C₄×C₇₃	1	8	yes	—
165	#338	—	2	8	no	—
166	C₂⁵	Q₈×C₇₃	1	1	yes	characteristic
167	C₂³×C₄	—	2	4	no	—
168	C₂³×C₄	—	2	4	no	—
169	C₂³×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
170	C₂³×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₇₃	1	4	yes	—
172	C₂³×C₄	D₄×C₇₃	1	4	yes	—
173	C₂³×C₄	D₄×C₇₃	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₇₃	1	1	yes	characteristic
175	C₂³×C₄	Q₈×C₇₃	1	1	yes	characteristic
176	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
177	C₂×M₁₆	—	2	32	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	8	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	2	8	no	—
185	C₂×M₁₆	—	2	8	no	—
186	C₂×M₁₆	D₄×C₇₃	1	8	yes	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	—	2	8	no	—
189	C₂²×C₈	C₂×C₄×C₇₃	1	8	yes	—
190	C₂²×C₈	C₂×C₄×C₇₃	1	4	yes	—
191	C₂²⋊C₈	—	2	8	no	—
192	C₂²⋊C₈	—	2	8	no	—
193	C₂⁴	—	2	4	no	—
194	C₂⁴	—	4	4	no	—
195	C₂⁴	C₄⋊C₄×C₇₃	1	1	yes	characteristic
196	C₂⁴	C₄⋊C₄×C₇₃	1	4	yes	—
197	C₂⁴	C₄⋊C₄×C₇₃	1	1	yes	characteristic
198	C₂⁴	C₂×Q₈×C₇₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	16	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	C₄²×C₇₃	1	2	yes	—
210	C₂²×C₄	C₄²×C₇₃	1	2	yes	—
211	C₂²×C₄	C₂²⋊C₄×C₇₃	1	4	yes	—
212	C₂²×C₄	C₂²⋊C₄×C₇₃	1	4	yes	—
213	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
215	C₂²×C₄	C₄⋊C₄×C₇₃	1	1	yes	characteristic
216	C₂²×C₄	C₄⋊C₄×C₇₃	1	1	yes	characteristic
217	C₂²×C₄	C₄⋊C₄×C₇₃	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
223	C₂²×C₄	#41×C₇₃	1	4	yes	—
224	M₁₆	—	2	16	no	—
225	M₁₆	—	2	16	no	—
226	M₁₆	—	4	4	no	—
227	M₁₆	—	4	8	no	—
228	M₁₆	—	4	4	no	—
229	C₂×C₈	—	2	16	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	8	no	—
233	C₂×C₈	—	2	16	no	—
234	C₂×C₈	C₂²⋊C₄×C₇₃	1	8	yes	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	2	8	no	—
240	C₂³	—	4	4	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	16	no	—
244	C₂³	#95×C₇₃	1	2	yes	—
245	C₂³	#95×C₇₃	1	2	yes	—
246	C₂³	C₂×C₄⋊C₄×C₇₃	1	1	yes	characteristic
247	C₂³	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
248	C₂³	C₂²⋊Q₈×C₇₃	1	4	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	#95×C₇₃	1	2	yes	—
261	C₂×C₄	#95×C₇₃	1	2	yes	—
262	C₂×C₄	C₂×C₄²×C₇₃	1	1	yes	characteristic
263	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
266	C₂×C₄	#117×C₇₃	1	2	yes	—
267	C₂×C₄	C₄×D₄×C₇₃	1	8	yes	—
268	C₂×C₄	#120×C₇₃	1	4	yes	—
269	C₂×C₄	#121×C₇₃	1	4	yes	—
270	C₂×C₄	#123×C₇₃	1	4	yes	—
271	C₈	—	2	8	no	—
272	C₈	—	4	8	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	16	no	—
282	C₂²	—	4	4	no	—
283	C₂²	#338×C₇₃	1	4	yes	complemented
284	C₂²	C₂×#95×C₇₃	1	1	yes	characteristic
285	C₂²	#380×C₇₃	1	2	yes	—
286	C₂²	#385×C₇₃	1	4	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	C₄×C₂²⋊C₄×C₇₃	1	2	yes	—
290	C₄	#379×C₇₃	1	1	yes	characteristic
291	C₄	#381×C₇₃	1	1	yes	characteristic
292	C₄	#386×C₇₃	1	4	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	2	4	no	—
295	C₂	—	2	4	no	—
296	C₂	—	4	2	no	—
297	C₂	C₇₃×C₂²⋊#95	1	1	yes	characteristic
298	C₂	C₂×#338×C₇₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₇₃
Frattini, cosocle	C₂³×C₄	C₂³×C₇₃
socle	C₂²×C₇₃	C₂×#95
prime radical	C₂⁵×C₇₃	Q₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#338×C₇₃	C₂	semidirect
133. C₂²×C₇₃	#338	semidirect
143. C₂²⋊#338	C₇₃	direct (N × H)
152. C₂×#338	C₂×C₇₃	semidirect
153. C₇₃	C₂²⋊#338	direct (N × H)
283. C₂²	#338×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₇₃	2
C₂×#407×C₇₃	2
C₂×#338×C₇₃	2
C₂²⋊#338	73

Subgroups — C73×C22⋊#338 #32920122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×C₂²⋊#338 #32920122