Subgroups — #2177×D₇₃ #33239283

Order 18688.

798 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₄×D₈×D₇₃	C₂	1	2	yes	maximal, complemented
3	#22611808	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611807	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611806	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611805	C₂	1	2	yes	maximal, complemented
7	#22611804	C₂	1	2	yes	maximal, complemented
8	#367×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#357×D₇₃	C₂	1	2	yes	maximal, complemented
10	#346×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₇₃	C₂²	1	2	yes	—
13	C₄×D₄×D₇₃	—	2	2	no	—
14	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
15	C₂×#14556532	C₂²	1	2	yes	—
16	D₁₆×D₇₃	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
19	C₄×D₅₈₄	C₂²	1	2	yes	complemented
20	D₈×Dic₇₃	C₂²	1	2	yes	complemented
21	#17988669	C₂²	1	2	yes	complemented
22	C₄×#11924013	—	2	2	no	—
23	#17988667	—	2	2	no	—
24	#107×D₇₃	C₂²	1	2	yes	—
25	#102×D₇₃	—	2	2	no	—
26	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
27	#17988415	C₂²	1	1	yes	characteristic, complemented
28	#17988414	C₂²	1	2	yes	complemented
29	#17988389	C₂²	1	2	yes	complemented
30	#17988388	C₂²	1	2	yes	complemented
31	#17988387	C₂²	1	2	yes	complemented
32	#17988363	C₂²	1	1	yes	characteristic, complemented
33	#17988362	C₂²	1	1	yes	characteristic, complemented
34	#17988362	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₇₃	C₂²	1	2	yes	complemented
37	#367×C₇₃	C₂²	1	1	yes	characteristic, complemented
38	#357×C₇₃	C₂²	1	2	yes	complemented
39	#346×C₇₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₇₃	—	2	2	no	—
41	C₂²×C₄×D₇₃	—	4	2	no	—
42	C₂×D₅₈₄	C₂³	1	2	yes	—
43	C₂×#11924013	—	2	2	no	—
44	D₈×D₇₃	—	2	4	no	—
45	D₈×D₇₃	C₂×C₄	1	8	yes	complemented
46	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
47	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
48	C₄×D₂₉₂	—	2	2	no	—
49	D₄×Dic₇₃	—	2	2	no	—
50	#14556553	—	2	2	no	—
51	C₄×C₂²⋊D₇₃	—	4	2	no	—
52	#14556551	—	4	2	no	—
53	C₄⋊C₄×D₇₃	—	2	2	no	—
54	C₂²⋊C₄×D₇₃	—	4	2	no	—
55	C₄²×D₇₃	D₄	1	1	yes	characteristic
56	D₁₁₆₈	C₂×C₄	1	4	yes	complemented
57	#14556532	C₂×C₄	1	8	yes	complemented
58	C₂×#11924003	C₂³	1	1	yes	characteristic
59	C₂×#11924003	C₂³	1	1	yes	characteristic
60	C₁₆×D₇₃	—	2	2	no	—
61	C₁₆×D₇₃	C₂×C₄	1	4	yes	complemented
62	#14556525	C₂³	1	2	yes	—
63	#14556524	—	2	2	no	—
64	#14556513	—	2	2	no	—
65	#14556512	—	2	2	no	—
66	#14556511	—	2	2	no	—
67	C₄×#10596831	D₄	1	1	yes	characteristic
68	C₈×Dic₇₃	D₄	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₇₃	C₂³	1	2	yes	—
71	C₄×D₄×C₇₃	—	2	2	no	—
72	D₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
75	#107×C₇₃	C₂³	1	2	yes	—
76	#102×C₇₃	—	2	2	no	—
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₂³×D₇₃	—	4	2	no	—
79	C₂×D₂₉₂	—	2	2	no	—
80	C₂×C₂²⋊D₇₃	—	4	2	no	—
81	D₄×D₇₃	—	2	8	no	—
82	D₄×D₇₃	—	4	4	no	—
83	C₂²×Dic₇₃	—	4	2	no	—
84	C₂×C₄×D₇₃	—	4	2	no	—
85	C₂×C₄×D₇₃	—	4	2	no	—
86	C₂×C₄×D₇₃	—	4	2	no	—
87	C₂×C₄×D₇₃	—	8	8	no	—
88	C₂×C₄×D₇₃	D₈	1	2	yes	—
89	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
90	D₅₈₄	—	2	2	no	—
91	D₅₈₄	C₂²×C₄	1	4	yes	complemented
92	#11924013	—	2	8	no	—
93	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
94	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
95	C₈×D₇₃	—	2	2	no	—
96	C₈×D₇₃	C₂²×C₄	1	2	yes	—
97	C₈×D₇₃	#41	1	2	yes	—
98	C₄⋊Dic₇₃	—	2	2	no	—
99	#11924007	—	4	2	no	—
100	C₂²⋊Dic₇₃	—	4	2	no	—
101	#11924005	—	4	2	no	—
102	#11924005	—	4	2	no	—
103	C₄×Dic₇₃	D₈	1	2	yes	—
104	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
105	#11924003	—	2	1	no	—
106	#11924003	C₂²×C₄	1	2	yes	complemented
107	C₂×D₄×C₇₃	—	2	2	no	—
108	C₂²×C₄×C₇₃	—	4	2	no	—
109	D₈×C₇₃	—	2	2	no	—
110	D₈×C₇₃	C₂²×C₄	1	4	yes	complemented
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₇₃	—	2	2	no	—
114	C₂²⋊C₄×C₇₃	—	4	2	no	—
115	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₇₃	—	2	1	no	—
117	C₁₆×C₇₃	C₂²×C₄	1	2	yes	complemented
118	C₂³×C₇₃	—	4	2	no	—
119	C₂²×D₇₃	—	4	2	no	—
120	C₂²×D₇₃	—	4	8	no	—
121	C₂²×D₇₃	—	8	4	no	—
122	C₂²×D₇₃	—	8	4	no	—
123	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
124	D₄×C₇₃	—	2	4	no	—
125	D₄×C₇₃	—	4	2	no	—
126	C₂×C₄×C₇₃	—	4	2	no	—
127	C₂×C₄×C₇₃	—	8	4	no	—
128	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₇₃	—	4	8	no	—
131	C₂×Dic₇₃	—	4	2	no	—
132	C₂×Dic₇₃	—	8	4	no	—
133	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
134	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
135	D₂₉₂	—	2	4	no	—
136	D₂₉₂	—	4	2	no	—
137	C₄×D₇₃	—	2	4	no	—
138	C₄×D₇₃	—	8	4	no	—
139	C₄×D₇₃	—	8	4	no	—
140	C₄×D₇₃	—	8	4	no	—
141	C₄×D₇₃	C₄×D₄	1	2	yes	—
142	C₄×D₇₃	#135	1	2	yes	—
143	C₈×C₇₃	—	2	1	no	—
144	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
145	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
146	#10596831	—	2	1	no	—
147	#10596831	C₄×D₄	1	2	yes	—
148	C₂²×C₇₃	—	4	4	no	—
149	C₂²×C₇₃	—	8	2	no	—
150	C₂²×C₇₃	—	8	2	no	—
151	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₇₃	—	4	4	no	—
153	C₂×D₇₃	—	8	2	no	—
154	C₂×D₇₃	—	8	2	no	—
155	C₂×D₇₃	—	8	16	no	—
156	C₂×D₇₃	C₄×D₈	1	2	yes	—
157	C₂×D₇₃	#509	1	4	yes	—
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	8	2	no	—
160	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
162	Dic₇₃	—	2	2	no	—
163	Dic₇₃	—	8	2	no	—
164	Dic₇₃	C₄×D₈	1	2	yes	—
165	C₂×#2177	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	8	4	no	—
167	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
168	C₂×C₇₃	C₂×#509	1	2	yes	—
169	D₇₃	—	8	4	no	—
170	D₇₃	#2177	1	4	yes	direct product factor
171	C₂²×D₁₆	—	73	1	no	—
172	C₂×C₄×D₈	—	73	2	no	—
173	#2177	—	73	1	no	—
174	#2177	—	73	2	no	—
175	#2177	—	73	1	no	—
176	#2177	—	73	2	no	—
177	#2177	—	73	1	no	—
178	#2177	D₇₃	1	1	yes	characteristic, direct product factor
179	C₂×#367	—	73	1	no	—
180	C₂×#357	—	73	2	no	—
181	C₂×#346	—	73	1	no	—
182	C₇₃	C₂×#2177	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₈	—	73	2	no	—
184	C₂×C₄×D₄	—	146	2	no	—
185	C₂×D₁₆	—	73	2	no	—
186	C₂×D₁₆	—	73	8	no	—
187	C₂×D₁₆	—	73	1	no	—
188	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
189	C₂²×C₁₆	—	73	1	no	—
190	C₂²×C₁₆	—	73	1	no	—
191	C₄×D₈	—	73	2	no	—
192	C₄×D₈	—	73	2	no	—
193	C₄×D₈	—	73	2	no	—
194	C₄×D₈	—	146	2	no	—
195	C₄×D₈	—	146	2	no	—
196	C₄×D₈	C₂×D₇₃	1	2	yes	complemented
197	C₂×#107	—	73	2	no	—
198	C₂×#102	—	146	2	no	—
199	C₂×C₄×C₈	—	73	1	no	—
200	#367	—	73	1	no	—
201	#367	—	73	2	no	—
202	#367	C₂×D₇₃	1	1	yes	characteristic, complemented
203	#357	—	73	2	no	—
204	#357	—	73	2	no	—
205	#357	—	73	2	no	—
206	#357	C₂×D₇₃	1	2	yes	complemented
207	#346	—	73	1	no	—
208	#346	—	73	1	no	—
209	#346	—	73	1	no	—
210	#346	C₂×D₇₃	1	1	yes	characteristic, complemented
211	C₂²×D₄	—	146	2	no	—
212	C₂³×C₄	—	292	2	no	—
213	C₂×D₈	—	73	8	no	—
214	C₂×D₈	—	73	2	no	—
215	C₂×D₈	—	146	4	no	—
216	C₂×D₈	—	146	2	no	—
217	C₂×D₈	C₂²×D₇₃	1	2	yes	—
218	C₂²×C₈	—	73	1	no	—
219	C₂²×C₈	—	73	1	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	292	2	no	—
222	C₄×D₄	—	292	2	no	—
223	C₄×D₄	—	146	2	no	—
224	C₄×D₄	—	146	2	no	—
225	C₄×D₄	—	146	2	no	—
226	C₂×C₄⋊C₄	—	146	2	no	—
227	C₂×C₂²⋊C₄	—	292	2	no	—
228	C₂×C₄²	—	73	1	no	—
229	D₁₆	—	73	8	no	—
230	D₁₆	—	73	4	no	—
231	D₁₆	C₄×D₇₃	1	4	yes	complemented
232	C₂×C₁₆	—	73	1	no	—
233	C₂×C₁₆	—	73	4	no	—
234	C₂×C₁₆	—	73	1	no	—
235	C₂×C₁₆	—	146	2	no	—
236	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
238	#107	—	73	2	no	—
239	#107	—	146	2	no	—
240	#107	C₂²×D₇₃	1	2	yes	—
241	#102	—	2	2	no	—
242	#102	—	146	2	no	—
243	#102	—	146	2	no	—
244	#102	—	146	2	no	—
245	C₄×C₈	—	73	1	no	—
246	C₄×C₈	—	73	1	no	—
247	C₄×C₈	—	73	1	no	—
248	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
249	C₂⁴	—	292	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	292	4	no	—
252	C₂×D₄	—	292	2	no	—
253	C₂×D₄	—	146	8	no	—
254	C₂×D₄	—	146	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	292	2	no	—
257	C₂²×C₄	—	292	2	no	—
258	C₂²×C₄	—	292	2	no	—
259	C₂²×C₄	—	292	2	no	—
260	C₂²×C₄	—	584	8	no	—
261	C₂²×C₄	—	73	1	no	—
262	C₂²×C₄	—	73	2	no	—
263	D₈	—	2	2	no	—
264	D₈	—	73	4	no	—
265	D₈	—	146	8	no	—
266	D₈	—	146	2	no	—
267	D₈	C₂×C₄×D₇₃	1	4	yes	complemented
268	C₂×C₈	—	73	1	no	—
269	C₂×C₈	—	73	2	no	—
270	C₂×C₈	—	73	2	no	—
271	C₂×C₈	—	73	1	no	—
272	C₂×C₈	—	146	2	no	—
273	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	292	2	no	—
277	C₄⋊C₄	—	146	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	292	2	no	—
280	C₂²⋊C₄	—	292	2	no	—
281	C₂²⋊C₄	—	292	2	no	—
282	C₄²	—	73	2	no	—
283	C₄²	—	73	1	no	—
284	C₄²	D₄×D₇₃	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	73	2	no	—
287	C₁₆	—	146	1	no	—
288	C₁₆	C₂×C₄×D₇₃	1	2	yes	complemented
289	C₂³	—	4	2	no	—
290	C₂³	—	292	8	no	—
291	C₂³	—	292	2	no	—
292	C₂³	—	584	4	no	—
293	C₂³	—	584	4	no	—
294	C₂³	—	73	1	no	—
295	D₄	—	2	4	no	—
296	D₄	—	4	2	no	—
297	D₄	—	292	8	no	—
298	D₄	—	292	2	no	—
299	D₄	—	146	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	292	2	no	—
303	C₂×C₄	—	584	4	no	—
304	C₂×C₄	—	584	4	no	—
305	C₂×C₄	—	584	4	no	—
306	C₂×C₄	—	584	4	no	—
307	C₂×C₄	—	73	1	no	—
308	C₂×C₄	—	73	2	no	—
309	C₂×C₄	—	73	2	no	—
310	C₂×C₄	—	73	2	no	—
311	C₂×C₄	—	146	4	no	—
312	C₂×C₄	D₈×D₇₃	1	2	yes	—
313	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	73	2	no	—
316	C₈	—	146	1	no	—
317	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
318	C₈	#41×D₇₃	1	1	yes	characteristic
319	C₂²	—	4	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	292	4	no	—
323	C₂²	—	584	16	no	—
324	C₂²	—	584	2	no	—
325	C₂²	—	584	2	no	—
326	C₂²	—	73	2	no	—
327	C₂²	—	73	4	no	—
328	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	584	2	no	—
332	C₄	—	73	2	no	—
333	C₄	—	146	2	no	—
334	C₄	C₄×D₄×D₇₃	1	1	yes	characteristic
335	C₄	#135×D₇₃	1	1	yes	characteristic
336	C₂	—	8	4	no	—
337	C₂	—	584	4	no	—
338	C₂	—	73	4	no	—
339	C₂	C₄×D₈×D₇₃	1	1	yes	characteristic
340	C₂	#509×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2177	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×D₁₆×D₇₃	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2177	D₇₃
hypocenter	C₇₃	C₂×#2177
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. C₄×D₈×D₇₃	C₂	semidirect
3. #22611808	C₂	semidirect
4. #22611807	C₂	semidirect
5. #22611806	C₂	semidirect
6. #22611805	C₂	semidirect
7. #22611804	C₂	semidirect
8. #367×D₇₃	C₂	semidirect
9. #357×D₇₃	C₂	semidirect
10. #346×D₇₃	C₂	semidirect
11. C₇₃×#2177	C₂	semidirect
16. D₁₆×D₇₃	C₄	semidirect
19. C₄×D₅₈₄	C₂²	semidirect
20. D₈×Dic₇₃	C₂²	semidirect
21. #17988669	C₂²	semidirect
27. #17988415	C₂²	semidirect
28. #17988414	C₂²	semidirect
29. #17988389	C₂²	semidirect
30. #17988388	C₂²	semidirect
31. #17988387	C₂²	semidirect
32. #17988363	C₂²	semidirect
33. #17988362	C₂²	semidirect
34. #17988362	C₂²	semidirect
36. C₄×D₈×C₇₃	C₂²	semidirect
37. #367×C₇₃	C₂²	semidirect
38. #357×C₇₃	C₂²	semidirect
39. #346×C₇₃	C₂²	semidirect
45. D₈×D₇₃	C₂×C₄	semidirect
56. D₁₁₆₈	C₂×C₄	semidirect
57. #14556532	C₂×C₄	semidirect
61. C₁₆×D₇₃	C₂×C₄	semidirect
72. D₁₆×C₇₃	C₂×C₄	semidirect
91. D₅₈₄	C₂²×C₄	semidirect
106. #11924003	C₂²×C₄	semidirect
110. D₈×C₇₃	C₂²×C₄	semidirect
117. C₁₆×C₇₃	C₂²×C₄	semidirect
170. D₇₃	#2177	direct (N × H)
178. #2177	D₇₃	direct (N × H)
182. C₇₃	C₂×#2177	semidirect
196. C₄×D₈	C₂×D₇₃	semidirect
202. #367	C₂×D₇₃	semidirect
206. #357	C₂×D₇₃	semidirect
210. #346	C₂×D₇₃	semidirect
231. D₁₆	C₄×D₇₃	semidirect
267. D₈	C₂×C₄×D₇₃	semidirect
288. C₁₆	C₂×C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₇₃	2
C₄×D₈×D₇₃	2
#22611808	2
#22611807	2
#22611806	2
#22611805	2
#22611804	2
#367×D₇₃	2
#357×D₇₃	2
#346×D₇₃	2
C₇₃×#2177	2
C₂×#2177	73

Subgroups — #2177×D73 #33239283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2177×D₇₃ #33239283