Subgroups — #32985825

Order 18688.

562 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988934	C₂	1	1	yes	maximal, characteristic
2	#22610891	C₂	1	1	yes	maximal, characteristic, complemented
3	#22610736	C₂	1	1	yes	maximal, characteristic
4	#22610736	C₂	1	1	yes	maximal, characteristic
5	#22610732	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988308	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556591	C₂²	1	1	yes	characteristic
9	C₂×#14556588	C₂²	1	1	yes	characteristic
10	#17988934	—	2	2	no	—
11	#17988934	C₄	1	4	yes	complemented
12	C₂×#14556514	—	2	2	no	—
13	C₂×#14556513	—	2	2	no	—
14	C₂×#14556511	—	2	2	no	—
15	C₂×#14556511	—	2	2	no	—
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
18	#17988349	—	2	2	no	—
19	#17988308	—	2	2	no	—
20	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	#384×C₇₃	C₂²	1	1	yes	characteristic
22	#384×C₇₃	C₂²	1	1	yes	characteristic
23	#340×C₇₃	—	2	2	no	—
24	C₂²×Dic₁₄₆	—	2	1	no	—
25	C₂²×D₂₉₂	—	2	1	no	—
26	C₂²×D₂₉₂	—	2	1	no	—
27	C₂²×D₂₉₂	—	2	1	no	—
28	C₂²×#10596831	D₄	1	2	yes	—
29	C₂²×#10596831	D₄	1	2	yes	—
30	#14556591	—	2	1	no	—
31	#14556591	—	2	1	no	—
32	#14556591	—	2	1	no	—
33	#14556591	C₂×C₄	1	2	yes	—
34	#14556588	—	2	1	no	—
35	#14556588	—	2	1	no	—
36	#14556588	—	2	1	no	—
37	#14556588	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊Dic₇₃	—	2	1	no	—
39	C₂×C₄⋊Dic₇₃	—	2	1	no	—
40	#14556514	—	4	4	no	—
41	#14556513	—	2	8	no	—
42	#14556511	—	4	4	no	—
43	#14556511	—	4	4	no	—
44	C₄×#10596831	—	2	1	no	—
45	C₄×#10596831	—	2	1	no	—
46	C₄×#10596831	—	2	1	no	—
47	C₄×#10596831	—	2	1	no	—
48	C₂²×C₈×C₇₃	D₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₇₃	D₄	1	2	yes	—
58	C₂³×D₇₃	—	4	1	no	—
59	C₂×Dic₁₄₆	—	2	2	no	—
60	C₂×Dic₁₄₆	—	4	2	no	—
61	C₂×Dic₁₄₆	—	4	2	no	—
62	C₂×Dic₁₄₆	—	4	1	no	—
63	C₂×D₂₉₂	—	2	2	no	—
64	C₂×D₂₉₂	—	2	2	no	—
65	C₂×D₂₉₂	—	2	2	no	—
66	C₂×D₂₉₂	—	4	2	no	—
67	C₂×D₂₉₂	—	4	2	no	—
68	C₂×D₂₉₂	—	4	2	no	—
69	C₂×D₂₉₂	—	4	2	no	—
70	C₂×D₂₉₂	—	4	2	no	—
71	C₂×D₂₉₂	—	4	2	no	—
72	C₂×D₂₉₂	—	4	1	no	—
73	C₂×D₂₉₂	—	4	1	no	—
74	C₂×D₂₉₂	—	4	1	no	—
75	C₂²×Dic₇₃	—	4	1	no	—
76	C₂×#10596831	—	2	2	no	—
77	C₂×#10596831	—	2	2	no	—
78	C₂×#10596831	—	2	2	no	—
79	C₂×#10596831	—	2	2	no	—
80	C₂×#10596831	—	2	2	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₄⋊Dic₇₃	—	2	2	no	—
83	C₄⋊Dic₇₃	—	2	2	no	—
84	C₄⋊Dic₇₃	—	4	1	no	—
85	C₄⋊Dic₇₃	—	4	1	no	—
86	C₂²×C₄×C₇₃	—	4	1	no	—
87	C₂²×C₄×C₇₃	—	4	1	no	—
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	—	2	2	no	—
92	C₂×C₈×C₇₃	—	2	2	no	—
93	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
94	C₄⋊C₄×C₇₃	—	4	2	no	—
95	C₄⋊C₄×C₇₃	—	4	2	no	—
96	C₄⋊C₄×C₇₃	—	4	2	no	—
97	C₄⋊C₄×C₇₃	—	4	2	no	—
98	C₄⋊C₄×C₇₃	—	4	2	no	—
99	C₄⋊C₄×C₇₃	—	4	2	no	—
100	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₃	#41	1	1	yes	characteristic
103	C₄²×C₇₃	#41	1	1	yes	characteristic
104	C₂³×C₇₃	#127	1	1	yes	characteristic
105	C₂²×D₇₃	—	4	2	no	—
106	C₂²×D₇₃	—	8	1	no	—
107	C₂²×D₇₃	—	8	1	no	—
108	C₂²×D₇₃	—	8	1	no	—
109	C₂²×D₇₃	—	8	1	no	—
110	C₂²×D₇₃	—	8	1	no	—
111	C₂²×D₇₃	—	8	1	no	—
112	C₂×C₄×C₇₃	—	8	1	no	—
113	C₂×C₄×C₇₃	—	8	1	no	—
114	C₂×C₄×C₇₃	—	8	1	no	—
115	C₂×C₄×C₇₃	—	8	1	no	—
116	C₂×C₄×C₇₃	—	8	1	no	—
117	C₂×C₄×C₇₃	—	8	1	no	—
118	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
124	C₂×C₄×C₇₃	#135	1	2	yes	—
125	C₂×C₄×C₇₃	#135	1	2	yes	—
126	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₇₃	#137	1	2	yes	—
130	C₂×Dic₇₃	—	4	2	no	—
131	C₂×Dic₇₃	—	8	1	no	—
132	C₂×Dic₇₃	—	8	1	no	—
133	D₂₉₂	—	4	4	no	—
134	D₂₉₂	—	4	4	no	—
135	D₂₉₂	—	4	4	no	—
136	D₂₉₂	—	8	2	no	—
137	D₂₉₂	—	8	2	no	—
138	D₂₉₂	—	8	2	no	—
139	Dic₁₄₆	—	4	4	no	—
140	Dic₁₄₆	—	8	2	no	—
141	C₈×C₇₃	—	2	4	no	—
142	#10596831	—	4	2	no	—
143	#10596831	—	4	2	no	—
144	C₂²×C₇₃	#390	1	1	yes	characteristic
145	C₂²×C₇₃	#495	1	1	yes	characteristic
146	C₂²×C₇₃	#495	1	1	yes	characteristic
147	C₂²×C₇₃	#496	1	1	yes	characteristic
148	C₂²×C₇₃	#496	1	1	yes	characteristic
149	C₂²×C₇₃	#496	1	1	yes	characteristic
150	C₂²×C₇₃	#497	1	1	yes	characteristic
151	C₂×D₇₃	—	8	2	no	—
152	C₂×D₇₃	—	8	2	no	—
153	C₂×D₇₃	—	8	2	no	—
154	C₂×D₇₃	—	16	1	no	—
155	C₂×D₇₃	—	16	1	no	—
156	C₂×D₇₃	—	16	1	no	—
157	C₂×D₇₃	—	16	1	no	—
158	C₄×C₇₃	—	16	1	no	—
159	C₄×C₇₃	—	16	1	no	—
160	C₄×C₇₃	#416	1	2	yes	—
161	C₄×C₇₃	#440	1	2	yes	—
162	C₄×C₇₃	#442	1	2	yes	—
163	C₄×C₇₃	#468	1	2	yes	—
164	C₄×C₇₃	#469	1	2	yes	—
165	C₄×C₇₃	#488	1	2	yes	—
166	Dic₇₃	—	8	2	no	—
167	#9838	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	#1717	1	1	yes	characteristic
169	C₂×C₇₃	#1717	1	1	yes	characteristic
170	C₂×C₇₃	#1719	1	1	yes	characteristic
171	C₂×C₇₃	#1720	1	1	yes	characteristic
172	C₂×C₇₃	#1958	1	1	yes	characteristic
173	C₂×C₇₃	#1972	1	1	yes	characteristic
174	C₂×C₇₃	#1973	1	1	yes	characteristic
175	D₇₃	—	16	2	no	—
176	C₂×#486	—	73	1	no	—
177	#1990	—	73	1	no	—
178	#1957	—	73	1	no	—
179	#1957	—	73	1	no	—
180	#1956	—	73	1	no	—
181	#1834	D₇₃	1	1	yes	characteristic, complemented
182	C₂×#322	—	73	1	no	—
183	C₇₃	#9838	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄⋊Q₈	—	73	1	no	—
185	C₂×#127	—	73	1	no	—
186	#486	—	73	4	no	—
187	#486	—	146	2	no	—
188	C₂×#103	—	146	2	no	—
189	C₂×#102	—	146	2	no	—
190	C₂×#102	—	146	2	no	—
191	C₂×#102	—	146	2	no	—
192	C₂×C₄×C₈	—	73	1	no	—
193	C₂×C₄×C₈	—	73	1	no	—
194	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
195	#384	C₂×D₇₃	1	1	yes	characteristic
196	#384	C₂×D₇₃	1	1	yes	characteristic
197	#340	—	2	2	no	—
198	#340	—	146	2	no	—
199	#322	—	146	2	no	—
200	C₂²×Q₈	—	146	1	no	—
201	C₂²×D₄	—	146	1	no	—
202	C₂²×D₄	—	146	1	no	—
203	C₂²×D₄	—	146	1	no	—
204	C₂²×C₈	—	73	2	no	—
205	C₂²×C₈	—	73	2	no	—
206	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
207	C₄⋊Q₈	—	73	2	no	—
208	C₄⋊Q₈	—	146	1	no	—
209	C₄⋊Q₈	—	146	1	no	—
210	C₄⋊Q₈	—	146	1	no	—
211	#127	—	73	2	no	—
212	#127	—	146	1	no	—
213	#127	—	146	1	no	—
214	#127	—	146	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	146	1	no	—
222	C₂×C₄⋊C₄	—	146	1	no	—
223	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
224	#103	—	292	4	no	—
225	#102	—	292	4	no	—
226	#102	—	292	4	no	—
227	#102	—	146	8	no	—
228	C₄×C₈	—	146	1	no	—
229	C₄×C₈	—	146	1	no	—
230	C₄×C₈	—	146	1	no	—
231	C₄×C₈	—	146	1	no	—
232	C₄×C₈	C₄×D₇₃	1	2	yes	—
233	C₄×C₈	D₂₉₂	1	2	yes	—
234	C₂⁴	—	292	1	no	—
235	C₂×Q₈	—	292	1	no	—
236	C₂×Q₈	—	292	2	no	—
237	C₂×Q₈	—	292	2	no	—
238	C₂×Q₈	—	146	2	no	—
239	C₂×D₄	—	292	1	no	—
240	C₂×D₄	—	292	1	no	—
241	C₂×D₄	—	292	1	no	—
242	C₂×D₄	—	292	2	no	—
243	C₂×D₄	—	292	2	no	—
244	C₂×D₄	—	292	2	no	—
245	C₂×D₄	—	292	2	no	—
246	C₂×D₄	—	292	2	no	—
247	C₂×D₄	—	292	2	no	—
248	C₂×D₄	—	146	2	no	—
249	C₂×D₄	—	146	2	no	—
250	C₂×D₄	—	146	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	292	1	no	—
254	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	146	2	no	—
260	C₂×C₈	—	146	2	no	—
261	C₂×C₈	—	146	2	no	—
262	C₂×C₈	—	146	2	no	—
263	C₂×C₈	—	146	2	no	—
264	C₂×C₈	—	146	2	no	—
265	C₂×C₈	#11924005	1	4	yes	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	292	1	no	—
273	C₄⋊C₄	—	292	1	no	—
274	C₄⋊C₄	—	146	2	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
277	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
278	C₄²	#11924029	1	1	yes	characteristic
279	C₄²	#11924029	1	1	yes	characteristic
280	C₂³	—	292	2	no	—
281	C₂³	—	584	1	no	—
282	C₂³	—	584	1	no	—
283	C₂³	—	584	1	no	—
284	C₂³	—	584	1	no	—
285	C₂³	—	584	1	no	—
286	C₂³	—	584	1	no	—
287	C₂³	#14556587	1	1	yes	characteristic
288	Q₈	—	292	4	no	—
289	Q₈	—	584	2	no	—
290	D₄	—	292	4	no	—
291	D₄	—	292	4	no	—
292	D₄	—	292	4	no	—
293	D₄	—	584	2	no	—
294	D₄	—	584	2	no	—
295	D₄	—	584	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	292	2	no	—
303	C₂×C₄	—	584	1	no	—
304	C₂×C₄	—	584	1	no	—
305	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
306	C₂×C₄	#14556553	1	1	yes	characteristic
307	C₂×C₄	#14556553	1	1	yes	characteristic
308	C₂×C₄	#14556566	1	1	yes	characteristic
309	C₂×C₄	#14556566	1	1	yes	characteristic
310	C₂×C₄	#14556580	1	1	yes	characteristic
311	C₂×C₄	#14556611	1	2	yes	—
312	C₂×C₄	#14556611	1	2	yes	—
313	C₂×C₄	#14556615	1	2	yes	—
314	C₂×C₄	#14556618	1	2	yes	—
315	C₂×C₄	#14556618	1	2	yes	—
316	C₂×C₄	#14556623	1	2	yes	—
317	C₈	—	2	4	no	—
318	C₈	—	292	2	no	—
319	C₈	—	292	2	no	—
320	C₂²	—	584	2	no	—
321	C₂²	—	584	2	no	—
322	C₂²	—	584	2	no	—
323	C₂²	—	1168	1	no	—
324	C₂²	—	1168	1	no	—
325	C₂²	—	1168	1	no	—
326	C₂²	—	1168	1	no	—
327	C₂²	#17988488	1	1	yes	characteristic
328	C₂²	#17988969	1	1	yes	characteristic
329	C₂²	#17988969	1	1	yes	characteristic
330	C₂²	#17988973	1	1	yes	characteristic
331	C₂²	#17988973	1	1	yes	characteristic
332	C₂²	#17988975	1	1	yes	characteristic
333	C₂²	#17988980	1	1	yes	characteristic
334	C₄	—	16	1	no	—
335	C₄	—	16	1	no	—
336	C₄	—	584	2	no	—
337	C₄	#17988581	1	2	yes	—
338	C₄	#17988686	1	2	yes	—
339	C₄	#17988698	1	2	yes	—
340	C₄	#17988834	1	2	yes	—
341	C₄	#17988841	1	2	yes	—
342	C₄	#17988944	1	2	yes	—
343	C₂	—	1168	2	no	—
344	C₂	#22609567	1	1	yes	characteristic
345	C₂	#22609571	1	1	yes	characteristic
346	C₂	#22609585	1	1	yes	characteristic
347	C₂	#22609588	1	1	yes	characteristic
348	C₂	#22610740	1	1	yes	characteristic
349	C₂	#22610805	1	1	yes	characteristic
350	C₂	#22610810	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556587
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1834	C₂
socle	C₂³×C₇₃	#127
prime radical	C₂×#14556588	C₂²
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1834	D₇₃
hypocenter	C₇₃	#9838
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
2. #22610891	C₂	semidirect
5. #22610732	C₂	semidirect
6. C₂×#17988308	C₂	semidirect
7. C₇₃×#1834	C₂	semidirect
11. #17988934	C₄	semidirect
181. #1834	D₇₃	semidirect
183. C₇₃	#9838	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988934	2
#22610891	2
#22610736	2
#22610732	2
C₂×#17988308	2
C₇₃×#1834	2
#9838	73