Subgroups — C₇₃×#3188 #22607341

Order 9344.

472 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#525×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#121×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#484×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#482×C₇₃	C₂	1	2	yes	maximal, complemented
5	#481×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#466×C₇₃	C₂	1	2	yes	maximal, complemented
7	#465×C₇₃	C₂	1	2	yes	maximal, complemented
8	#450×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#449×C₇₃	C₂	1	2	yes	maximal, complemented
10	#447×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₇₃	—	2	1	no	—
14	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₇₃	C₂²	1	2	yes	complemented
17	C₂×D₈×C₇₃	C₂²	1	2	yes	complemented
18	C₂²×C₈×C₇₃	C₂²	1	2	yes	complemented
19	#123×C₇₃	—	2	1	no	—
20	C₂²⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic, complemented
22	#121×C₇₃	—	2	1	no	—
23	#121×C₇₃	—	2	1	no	—
24	#121×C₇₃	C₂²	1	1	yes	characteristic
25	#121×C₇₃	C₂²	1	2	yes	—
26	#121×C₇₃	C₂²	1	2	yes	—
27	#121×C₇₃	C₂²	1	1	yes	characteristic, complemented
28	#120×C₇₃	—	2	1	no	—
29	C₄×D₄×C₇₃	—	2	1	no	—
30	C₄×D₄×C₇₃	—	2	1	no	—
31	#117×C₇₃	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
34	#107×C₇₃	C₂²	1	2	yes	complemented
35	#106×C₇₃	C₂²	1	2	yes	complemented
36	#103×C₇₃	C₂²	1	2	yes	complemented
37	#102×C₇₃	C₂²	1	2	yes	complemented
38	#102×C₇₃	C₂²	1	2	yes	complemented
39	#102×C₇₃	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
42	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	complemented
43	C₂⁴×C₇₃	—	2	1	no	—
44	C₂⁴×C₇₃	D₄	1	1	yes	characteristic
45	#41×C₇₃	—	4	2	no	—
46	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₇₃	—	2	1	no	—
48	C₂×D₄×C₇₃	—	2	1	no	—
49	C₂×D₄×C₇₃	—	2	1	no	—
50	C₂×D₄×C₇₃	—	2	2	no	—
51	C₂×D₄×C₇₃	—	2	2	no	—
52	C₂×D₄×C₇₃	—	2	1	no	—
53	C₂×D₄×C₇₃	—	4	2	no	—
54	C₂×D₄×C₇₃	—	4	4	no	—
55	C₂×D₄×C₇₃	C₂³	1	2	yes	—
56	C₂×D₄×C₇₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	—	2	1	no	—
58	C₂²×C₄×C₇₃	—	2	1	no	—
59	C₂²×C₄×C₇₃	—	4	2	no	—
60	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
66	SD₁₆×C₇₃	—	4	2	no	—
67	D₈×C₇₃	—	4	2	no	—
68	C₂×C₈×C₇₃	—	4	2	no	—
69	C₂×C₈×C₇₃	C₂³	1	2	yes	—
70	C₂×C₈×C₇₃	C₂³	1	2	yes	—
71	C₄⋊C₄×C₇₃	—	2	1	no	—
72	C₄⋊C₄×C₇₃	—	2	1	no	—
73	C₄⋊C₄×C₇₃	—	2	1	no	—
74	C₄⋊C₄×C₇₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₇₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₇₃	—	2	1	no	—
78	C₂²⋊C₄×C₇₃	—	2	1	no	—
79	C₂²⋊C₄×C₇₃	—	2	1	no	—
80	C₂²⋊C₄×C₇₃	—	2	2	no	—
81	C₂²⋊C₄×C₇₃	—	2	2	no	—
82	C₂²⋊C₄×C₇₃	—	2	1	no	—
83	C₂²⋊C₄×C₇₃	—	2	1	no	—
84	C₄²×C₇₃	—	2	1	no	—
85	C₂³×C₇₃	—	2	2	no	—
86	C₂³×C₇₃	—	2	2	no	—
87	C₂³×C₇₃	—	2	1	no	—
88	C₂³×C₇₃	—	2	2	no	—
89	C₂³×C₇₃	—	4	1	no	—
90	C₂³×C₇₃	—	4	2	no	—
91	C₂³×C₇₃	—	4	2	no	—
92	C₂³×C₇₃	—	4	2	no	—
93	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
96	Q₈×C₇₃	—	2	2	no	—
97	D₄×C₇₃	—	2	2	no	—
98	D₄×C₇₃	—	2	2	no	—
99	D₄×C₇₃	—	2	2	no	—
100	D₄×C₇₃	—	4	2	no	—
101	D₄×C₇₃	—	4	4	no	—
102	D₄×C₇₃	—	4	2	no	—
103	D₄×C₇₃	—	4	2	no	—
104	D₄×C₇₃	—	4	2	no	—
105	D₄×C₇₃	—	4	4	no	—
106	C₂×C₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	2	1	no	—
110	C₂×C₄×C₇₃	—	2	1	no	—
111	C₂×C₄×C₇₃	—	2	1	no	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	—	4	1	no	—
114	C₂×C₄×C₇₃	—	4	1	no	—
115	C₂×C₄×C₇₃	—	4	1	no	—
116	C₂×C₄×C₇₃	—	4	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
122	C₈×C₇₃	—	2	2	no	—
123	C₈×C₇₃	—	2	2	no	—
124	C₂²×C₇₃	—	2	1	no	—
125	C₂²×C₇₃	—	2	1	no	—
126	C₂²×C₇₃	—	2	1	no	—
127	C₂²×C₇₃	—	2	1	no	—
128	C₂²×C₇₃	—	2	1	no	—
129	C₂²×C₇₃	—	2	1	no	—
130	C₂²×C₇₃	—	2	1	no	—
131	C₂²×C₇₃	—	4	4	no	—
132	C₂²×C₇₃	—	4	1	no	—
133	C₂²×C₇₃	—	4	2	no	—
134	C₂²×C₇₃	—	4	1	no	—
135	C₂²×C₇₃	—	4	2	no	—
136	C₂²×C₇₃	—	4	2	no	—
137	C₂²×C₇₃	—	4	1	no	—
138	C₂²×C₇₃	—	8	4	no	—
139	C₂²×C₇₃	#135	1	2	yes	—
140	C₂²×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
141	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₇₃	—	2	1	no	—
143	C₄×C₇₃	—	2	1	no	—
144	C₄×C₇₃	—	2	1	no	—
145	C₄×C₇₃	—	4	1	no	—
146	C₄×C₇₃	—	4	1	no	—
147	C₄×C₇₃	—	4	2	no	—
148	C₄×C₇₃	—	4	1	no	—
149	C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
150	C₂×C₇₃	—	2	2	no	—
151	C₂×C₇₃	—	2	2	no	—
152	C₂×C₇₃	—	4	1	no	—
153	C₂×C₇₃	—	8	1	no	—
154	C₂×C₇₃	—	8	2	no	—
155	C₂×C₇₃	#534	1	1	yes	characteristic
156	C₂×C₇₃	C₂×#135	1	1	yes	characteristic
157	C₂×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	#3188	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₇₃	#3188	1	1	yes	characteristic, direct product factor, Sylow
160	#525	C₂×C₇₃	1	1	yes	characteristic, complemented
161	C₂×#121	C₂×C₇₃	1	1	yes	characteristic, complemented
162	#484	C₂×C₇₃	1	1	yes	characteristic, complemented
163	#482	C₂×C₇₃	1	2	yes	complemented
164	#481	C₂×C₇₃	1	1	yes	characteristic, complemented
165	#466	C₂×C₇₃	1	2	yes	complemented
166	#465	C₂×C₇₃	1	2	yes	complemented
167	#450	C₂×C₇₃	1	1	yes	characteristic, complemented
168	#449	C₂×C₇₃	1	2	yes	complemented
169	#447	C₂×C₇₃	1	1	yes	characteristic, complemented
170	C₂×C₂²⋊C₈	C₂×C₇₃	1	1	yes	characteristic, complemented
171	C₂×#41	C₂²×C₇₃	1	1	yes	characteristic, complemented
172	C₂²×D₄	—	2	1	no	—
173	C₂²×D₄	C₂²×C₇₃	1	1	yes	characteristic
174	C₂³×C₄	C₂²×C₇₃	1	1	yes	characteristic
175	C₂×SD₁₆	C₂²×C₇₃	1	2	yes	complemented
176	C₂×D₈	C₂²×C₇₃	1	2	yes	complemented
177	C₂²×C₈	C₂²×C₇₃	1	2	yes	complemented
178	#123	—	2	1	no	—
179	C₂²⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
180	C₂²⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic, complemented
181	#121	—	2	1	no	—
182	#121	—	2	1	no	—
183	#121	C₂²×C₇₃	1	2	yes	—
184	#121	C₂²×C₇₃	1	2	yes	—
185	#121	C₂²×C₇₃	1	1	yes	characteristic
186	#121	C₂²×C₇₃	1	1	yes	characteristic, complemented
187	#120	—	2	1	no	—
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	—	2	1	no	—
190	#117	C₂²×C₇₃	1	1	yes	characteristic, complemented
191	C₂×C₄⋊C₄	C₂²×C₇₃	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	#107	C₂²×C₇₃	1	2	yes	complemented
194	#106	C₂²×C₇₃	1	2	yes	complemented
195	#103	C₂²×C₇₃	1	2	yes	complemented
196	#102	C₂²×C₇₃	1	2	yes	complemented
197	#102	C₂²×C₇₃	1	2	yes	complemented
198	#102	C₂²×C₇₃	1	2	yes	complemented
199	C₂²⋊C₈	C₂²×C₇₃	1	1	yes	characteristic
200	C₂²⋊C₈	C₂²×C₇₃	1	1	yes	characteristic
201	C₂²⋊C₈	C₂²×C₇₃	1	2	yes	complemented
202	C₂⁴	—	2	1	no	—
203	C₂⁴	D₄×C₇₃	1	1	yes	characteristic
204	#41	—	4	2	no	—
205	C₂×Q₈	C₂³×C₇₃	1	1	yes	characteristic
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	C₂³×C₇₃	1	2	yes	—
215	C₂×D₄	C₂³×C₇₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
220	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
221	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
223	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
224	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
225	SD₁₆	—	4	2	no	—
226	D₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	C₂³×C₇₃	1	2	yes	—
229	C₂×C₈	C₂³×C₇₃	1	2	yes	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	C₂³×C₇₃	1	1	yes	characteristic
234	C₄⋊C₄	C₂³×C₇₃	1	1	yes	characteristic
235	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
253	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
254	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	2	2	no	—
258	D₄	—	2	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	4	no	—
264	D₄	—	4	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
280	C₂×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	2	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	8	4	no	—
298	C₂²	#135×C₇₃	1	2	yes	—
299	C₂²	AGL(1,ℤ/8)×C₇₃	1	2	yes	—
300	C₂²	C₂²×D₄×C₇₃	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	1	no	—
308	C₄	H(5,𝔽₂)×C₇₃	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	2	2	no	—
311	C₂	—	4	1	no	—
312	C₂	—	8	2	no	—
313	C₂	—	8	1	no	—
314	C₂	#534×C₇₃	1	1	yes	characteristic
315	C₂	C₂×#135×C₇₃	1	1	yes	characteristic
316	C₂	C₂×AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #525×C₇₃	C₂	semidirect
2. C₂×#121×C₇₃	C₂	semidirect
3. #484×C₇₃	C₂	semidirect
4. #482×C₇₃	C₂	semidirect
5. #481×C₇₃	C₂	semidirect
6. #466×C₇₃	C₂	semidirect
7. #465×C₇₃	C₂	semidirect
8. #450×C₇₃	C₂	semidirect
9. #449×C₇₃	C₂	semidirect
10. #447×C₇₃	C₂	semidirect
11. C₂×C₂²⋊C₈×C₇₃	C₂	semidirect
12. C₂×#41×C₇₃	C₂²	semidirect
16. C₂×SD₁₆×C₇₃	C₂²	semidirect
17. C₂×D₈×C₇₃	C₂²	semidirect
18. C₂²×C₈×C₇₃	C₂²	semidirect
21. C₂²⋊Q₈×C₇₃	C₂²	semidirect
27. #121×C₇₃	C₂²	semidirect
31. #117×C₇₃	C₂²	semidirect
34. #107×C₇₃	C₂²	semidirect
35. #106×C₇₃	C₂²	semidirect
36. #103×C₇₃	C₂²	semidirect
37. #102×C₇₃	C₂²	semidirect
38. #102×C₇₃	C₂²	semidirect
39. #102×C₇₃	C₂²	semidirect
42. C₂²⋊C₈×C₇₃	C₂²	semidirect
158. #3188	C₇₃	direct (N × H)
159. C₇₃	#3188	direct (N × H)
160. #525	C₂×C₇₃	semidirect
161. C₂×#121	C₂×C₇₃	semidirect
162. #484	C₂×C₇₃	semidirect
163. #482	C₂×C₇₃	semidirect
164. #481	C₂×C₇₃	semidirect
165. #466	C₂×C₇₃	semidirect
166. #465	C₂×C₇₃	semidirect
167. #450	C₂×C₇₃	semidirect
168. #449	C₂×C₇₃	semidirect
169. #447	C₂×C₇₃	semidirect
170. C₂×C₂²⋊C₈	C₂×C₇₃	semidirect
171. C₂×#41	C₂²×C₇₃	semidirect
175. C₂×SD₁₆	C₂²×C₇₃	semidirect
176. C₂×D₈	C₂²×C₇₃	semidirect
177. C₂²×C₈	C₂²×C₇₃	semidirect
180. C₂²⋊Q₈	C₂²×C₇₃	semidirect
186. #121	C₂²×C₇₃	semidirect
190. #117	C₂²×C₇₃	semidirect
193. #107	C₂²×C₇₃	semidirect
194. #106	C₂²×C₇₃	semidirect
195. #103	C₂²×C₇₃	semidirect
196. #102	C₂²×C₇₃	semidirect
197. #102	C₂²×C₇₃	semidirect
198. #102	C₂²×C₇₃	semidirect
201. C₂²⋊C₈	C₂²×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
#525×C₇₃	2
C₂×#121×C₇₃	2
#484×C₇₃	2
#482×C₇₃	2
#481×C₇₃	2
#466×C₇₃	2
#465×C₇₃	2
#450×C₇₃	2
#449×C₇₃	2
#447×C₇₃	2
C₂×C₂²⋊C₈×C₇₃	2
#3188	73

Subgroups — C73×#3188 #22607341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#3188 #22607341