Subgroups — #427×Dic₇₃ #33084036

Order 18688.

1182 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988625	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#117×Dic₇₃	C₂	1	2	yes	maximal
4	#107×Dic₇₃	C₂	1	4	yes	maximal, complemented
5	#106×Dic₇₃	C₂	1	4	yes	maximal, complemented
6	#22609647	C₂	1	2	yes	maximal
7	C₂×#427×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×#10596831	C₂²	1	1	yes	characteristic
9	C₂×#14556550	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
11	#17988625	C₄	1	8	yes	complemented
12	C₂×#14556525	C₂²	1	2	yes	—
13	C₂×#14556523	C₂²	1	2	yes	—
14	C₂×C₈×Dic₇₃	C₄	1	8	yes	complemented
15	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₇₃	C₂²	1	8	yes	—
18	C₂²⋊C₄×Dic₇₃	—	2	4	no	—
19	#17988433	—	2	2	no	—
20	C₄²×Dic₇₃	—	2	2	no	—
21	#17988349	C₂²	1	8	yes	—
22	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×#117×C₇₃	C₂²	1	2	yes	—
24	#427×C₇₃	C₄	1	8	yes	complemented
25	C₂×#107×C₇₃	C₂²	1	2	yes	—
26	C₂×#106×C₇₃	C₂²	1	2	yes	—
27	C₂³×Dic₇₃	D₄	1	2	yes	—
28	C₂²×#10596831	—	2	2	no	—
29	C₂²×#10596831	—	2	2	no	—
30	C₂²×#10596831	C₂×C₄	1	4	yes	—
31	C₂²×#10596831	C₂³	1	2	yes	—
32	#14556550	—	2	4	no	—
33	#14556550	C₂×C₄	1	8	yes	—
34	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
36	C₂×C₄×Dic₇₃	—	2	8	no	—
37	C₂×C₄×Dic₇₃	—	2	2	no	—
38	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₇₃	Q₈	1	4	yes	—
42	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
43	#14556525	C₂×C₄	1	8	yes	complemented
44	#14556523	C₂×C₄	1	8	yes	complemented
45	C₈×Dic₇₃	C₂×C₄	1	8	yes	—
46	C₈×Dic₇₃	C₂×C₄	1	8	yes	—
47	#14556498	—	2	8	no	—
48	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	#117×C₇₃	—	2	4	no	—
55	#117×C₇₃	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
57	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
58	C₂×C₄²×C₇₃	—	2	2	no	—
59	#107×C₇₃	C₂×C₄	1	8	yes	complemented
60	#106×C₇₃	C₂×C₄	1	8	yes	complemented
61	C₂²×Dic₇₃	—	2	8	no	—
62	C₂²×Dic₇₃	—	2	4	no	—
63	C₂²×Dic₇₃	C₄⋊C₄	1	8	yes	—
64	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
65	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
66	C₂×#10596831	—	2	8	no	—
67	C₂×#10596831	—	2	2	no	—
68	C₂×#10596831	—	2	2	no	—
69	C₂×#10596831	C₂²×C₄	1	4	yes	—
70	C₄⋊Dic₇₃	—	2	4	no	—
71	C₄⋊Dic₇₃	C₂²×C₄	1	8	yes	—
72	C₂²⋊Dic₇₃	—	2	8	no	—
73	C₄×Dic₇₃	—	2	8	no	—
74	C₄×Dic₇₃	—	4	16	no	—
75	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
76	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
77	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
78	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
79	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	—	2	4	no	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₇₃	#41	1	2	yes	—
87	C₂²×C₄×C₇₃	#41	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₇₃	C₄²	1	16	yes	complemented
90	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
96	C₄⋊C₄×C₇₃	—	2	4	no	—
97	C₄⋊C₄×C₇₃	C₂²×C₄	1	8	yes	—
98	C₂²⋊C₄×C₇₃	—	2	8	no	—
99	C₄²×C₇₃	—	2	8	no	—
100	C₂³×C₇₃	—	2	4	no	—
101	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂³×C₇₃	C₄×D₄	1	2	yes	—
103	C₂³×C₇₃	C₄×D₄	1	2	yes	—
104	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	—	2	8	no	—
106	C₂×C₄×C₇₃	—	4	8	no	—
107	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
108	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₇₃	#117	1	4	yes	—
110	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₇₃	—	2	4	no	—
121	C₂×Dic₇₃	—	2	16	no	—
122	C₂×Dic₇₃	—	2	8	no	—
123	C₂×Dic₇₃	—	4	8	no	—
124	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×Dic₇₃	#135	1	8	yes	—
127	C₈×C₇₃	C₂×C₄²	1	4	yes	—
128	C₈×C₇₃	C₂×C₄²	1	4	yes	—
129	#10596831	—	2	4	no	—
130	C₂²×C₇₃	—	2	4	no	—
131	C₂²×C₇₃	—	2	4	no	—
132	C₂²×C₇₃	—	2	4	no	—
133	C₂²×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
134	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₇₃	C₂×#135	1	4	yes	—
138	C₄×C₇₃	—	4	8	no	—
139	C₄×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
140	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
141	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
142	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
144	Dic₇₃	—	2	4	no	—
145	Dic₇₃	—	4	8	no	—
146	Dic₇₃	#427	1	8	yes	direct product factor
147	C₄×#427	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	2	4	no	—
149	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₇₃	C₂×#427	1	2	yes	—
151	C₂×C₇₃	C₄×#135	1	2	yes	—
152	C₂×C₇₃	C₄×#135	1	2	yes	—
153	C₂×#427	—	73	1	no	—
154	C₂×#427	D₇₃	1	1	yes	characteristic
155	C₂²×C₄×C₈	—	73	1	no	—
156	C₄×#117	—	73	2	no	—
157	C₄×#107	—	73	4	no	—
158	C₄×#106	—	73	4	no	—
159	#1735	—	73	2	no	—
160	C₇₃	C₄×#427	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₈	—	73	1	no	—
162	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
163	C₂×#117	—	73	2	no	—
164	C₂×#117	C₂×D₇₃	1	2	yes	—
165	C₂²×C₄²	—	73	1	no	—
166	#427	—	73	8	no	—
167	#427	Dic₇₃	1	8	yes	direct product factor
168	C₂×#107	—	73	2	no	—
169	C₂×#107	C₂×D₇₃	1	2	yes	—
170	C₂×#106	—	73	2	no	—
171	C₂×#106	C₂×D₇₃	1	2	yes	—
172	C₂×C₄×C₈	—	73	8	no	—
173	C₂×C₄×C₈	—	73	2	no	—
174	C₂×C₄×C₈	—	73	2	no	—
175	C₄×C₄⋊C₄	—	73	8	no	—
176	C₄×C₂²⋊C₄	—	146	4	no	—
177	#376	—	146	2	no	—
178	C₄³	—	146	2	no	—
179	#340	—	73	8	no	—
180	C₂³×C₄	—	73	2	no	—
181	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
182	C₂²×C₈	—	73	4	no	—
183	C₂²×C₈	—	73	2	no	—
184	C₂²×C₈	—	146	2	no	—
185	C₂²×C₈	—	146	2	no	—
186	C₂²×C₈	C₄×D₇₃	1	4	yes	—
187	C₂²×C₈	C₄×D₇₃	1	4	yes	—
188	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
189	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
191	#117	—	2	4	no	—
192	#117	—	73	8	no	—
193	#117	—	146	4	no	—
194	#117	C₂×Dic₇₃	1	8	yes	—
195	C₂×C₄⋊C₄	—	73	4	no	—
196	C₂×C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	146	2	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	73	4	no	—
201	C₂×C₄²	—	73	1	no	—
202	C₂×C₄²	—	73	1	no	—
203	C₂×C₄²	—	73	4	no	—
204	C₂×C₄²	—	73	2	no	—
205	C₂×C₄²	—	146	8	no	—
206	C₂×C₄²	—	146	2	no	—
207	#107	—	73	8	no	—
208	#107	C₂×Dic₇₃	1	8	yes	complemented
209	#106	—	73	8	no	—
210	#106	C₂×Dic₇₃	1	8	yes	complemented
211	C₄×C₈	—	73	8	no	—
212	C₄×C₈	—	73	8	no	—
213	#95	—	146	8	no	—
214	C₂⁴	D₄×D₇₃	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	73	2	no	—
217	C₂²×C₄	—	73	8	no	—
218	C₂²×C₄	—	73	2	no	—
219	C₂²×C₄	—	146	4	no	—
220	C₂²×C₄	—	146	8	no	—
221	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
224	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
225	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
226	C₂²×C₄	#11924027	1	2	yes	—
227	C₂²×C₄	#11924029	1	2	yes	—
228	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
229	C₂×C₈	—	73	4	no	—
230	C₂×C₈	—	146	8	no	—
231	C₂×C₈	—	146	2	no	—
232	C₂×C₈	—	146	2	no	—
233	C₂×C₈	C₄×Dic₇₃	1	16	yes	complemented
234	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
238	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
239	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	73	8	no	—
242	C₄⋊C₄	—	146	4	no	—
243	C₄⋊C₄	C₂²×Dic₇₃	1	8	yes	—
244	C₂²⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	—	146	8	no	—
246	C₄²	—	2	8	no	—
247	C₄²	—	292	16	no	—
248	C₄²	—	73	4	no	—
249	C₄²	—	73	4	no	—
250	C₄²	—	73	4	no	—
251	C₄²	—	73	4	no	—
252	C₄²	—	146	8	no	—
253	C₂³	—	2	4	no	—
254	C₂³	C₄⋊C₄×D₇₃	1	2	yes	—
255	C₂³	#14556553	1	2	yes	—
256	C₂³	D₄×Dic₇₃	1	2	yes	—
257	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	292	8	no	—
261	C₂×C₄	—	73	4	no	—
262	C₂×C₄	—	73	8	no	—
263	C₂×C₄	—	73	4	no	—
264	C₂×C₄	—	146	8	no	—
265	C₂×C₄	—	146	16	no	—
266	C₂×C₄	—	146	4	no	—
267	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
268	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
269	C₂×C₄	#14556549	1	4	yes	—
270	C₂×C₄	#14556553	1	2	yes	—
271	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
272	C₂×C₄	#14556556	1	4	yes	—
273	C₂×C₄	Q₈×Dic₇₃	1	4	yes	—
274	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
277	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
279	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
280	C₈	—	146	4	no	—
281	C₈	C₂×C₄×Dic₇₃	1	4	yes	—
282	C₈	C₂×C₄×Dic₇₃	1	4	yes	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	C₄⋊C₄×Dic₇₃	1	4	yes	—
287	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
288	C₂²	C₂×#14556553	1	1	yes	characteristic
289	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
290	C₂²	#135×D₇₃	1	4	yes	—
291	C₄	—	4	8	no	—
292	C₄	—	292	8	no	—
293	C₄	—	73	8	no	—
294	C₄	—	146	4	no	—
295	C₄	C₄⋊C₄×Dic₇₃	1	4	yes	—
296	C₄	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
297	C₄	C₂×#14556549	1	1	yes	characteristic
298	C₄	C₂×#14556556	1	1	yes	characteristic
299	C₄	C₂×Q₈×Dic₇₃	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	C₂×C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
302	C₂	#427×D₇₃	1	2	yes	—
303	C₂	#22618702	1	2	yes	—
304	C₂	#135×Dic₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#427×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#427	D₇₃
hypocenter	C₇₃	C₄×#427
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #107×Dic₇₃	C₂	semidirect
5. #106×Dic₇₃	C₂	semidirect
11. #17988625	C₄	semidirect
14. C₂×C₈×Dic₇₃	C₄	semidirect
24. #427×C₇₃	C₄	semidirect
43. #14556525	C₂×C₄	semidirect
44. #14556523	C₂×C₄	semidirect
59. #107×C₇₃	C₂×C₄	semidirect
60. #106×C₇₃	C₂×C₄	semidirect
89. C₂×C₈×C₇₃	C₄²	semidirect
146. Dic₇₃	#427	direct (N × H)
160. C₇₃	C₄×#427	semidirect
167. #427	Dic₇₃	direct (N × H)
208. #107	C₂×Dic₇₃	semidirect
210. #106	C₂×Dic₇₃	semidirect
233. C₂×C₈	C₄×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988625	2
C₂²×C₈×Dic₇₃	2
#117×Dic₇₃	2
#107×Dic₇₃	2
#106×Dic₇₃	2
#22609647	2
C₂×#427×C₇₃	2
C₄×#427	73

Subgroups — #427×Dic73 #33084036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #427×Dic₇₃ #33084036