Subgroups — #33002501

Order 18688.

404 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988417	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988416	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17988366	C₂	1	1	yes	maximal, characteristic
5	#22609966	C₂	1	1	yes	maximal, characteristic
6	#22608024	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₇₃	—	2	1	no	—
9	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₂²×#11924003	C₂²	1	1	yes	characteristic
11	C₂×#14556527	C₂²	1	1	yes	characteristic
12	C₂×#14556527	C₂²	1	1	yes	characteristic
13	C₂×#14556526	—	2	1	no	—
14	#108×D₇₃	—	2	2	no	—
15	#108×D₇₃	—	2	2	no	—
16	#17988550	—	2	1	no	—
17	#17988417	—	2	2	no	—
18	#17988416	—	2	2	no	—
19	#17988366	C₄	1	2	yes	—
20	#17988366	C₄	1	2	yes	—
21	#17988351	—	2	1	no	—
22	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
26	C₂×#11924012	—	2	1	no	—
27	C₂×#11924012	—	2	1	no	—
28	C₂×#11924011	—	4	1	no	—
29	M₁₆×D₇₃	—	2	4	no	—
30	M₁₆×D₇₃	—	4	2	no	—
31	C₂²×#10596831	D₄	1	1	yes	characteristic
32	C₂×C₈×D₇₃	—	2	1	no	—
33	C₂×C₈×D₇₃	—	2	1	no	—
34	C₂×C₈×D₇₃	—	2	1	no	—
35	C₂×C₈×D₇₃	—	2	1	no	—
36	C₂×C₈×D₇₃	—	4	1	no	—
37	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
40	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
41	C₂×#11924003	—	2	1	no	—
42	C₂×#11924003	—	2	1	no	—
43	C₂×#11924003	C₂×C₄	1	2	yes	—
44	#14556527	—	2	1	no	—
45	#14556527	—	2	1	no	—
46	#14556527	—	2	1	no	—
47	#14556527	D₄	1	2	yes	—
48	#14556526	—	2	4	no	—
49	#14556503	—	4	1	no	—
50	#14556503	—	4	1	no	—
51	C₂×M₁₆×C₇₃	—	2	1	no	—
52	C₂×M₁₆×C₇₃	—	2	1	no	—
53	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₇₃	—	2	1	no	—
55	C₂×C₁₆×C₇₃	—	2	1	no	—
56	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
57	#108×C₇₃	—	2	1	no	—
58	#108×C₇₃	—	2	1	no	—
59	#108×C₇₃	—	2	1	no	—
60	#108×C₇₃	D₄	1	2	yes	—
61	C₂³×D₇₃	Q₁₆	1	1	yes	characteristic
62	C₂²×Dic₇₃	Q₁₆	1	1	yes	characteristic
63	C₂×C₄×D₇₃	—	2	1	no	—
64	C₂×C₄×D₇₃	—	2	1	no	—
65	C₂×C₄×D₇₃	—	2	1	no	—
66	C₂×C₄×D₇₃	—	2	1	no	—
67	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
70	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
71	#11924012	—	2	2	no	—
72	#11924012	—	4	1	no	—
73	#11924012	—	4	1	no	—
74	#11924012	—	4	1	no	—
75	#11924011	—	4	4	no	—
76	C₂×#10596831	—	2	1	no	—
77	C₂×#10596831	—	2	1	no	—
78	C₂×#10596831	—	4	1	no	—
79	C₂×#10596831	—	4	1	no	—
80	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
81	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
82	C₈×D₇₃	—	2	1	no	—
83	C₈×D₇₃	—	2	1	no	—
84	C₈×D₇₃	—	2	1	no	—
85	C₈×D₇₃	—	2	1	no	—
86	C₈×D₇₃	—	2	1	no	—
87	C₈×D₇₃	—	2	1	no	—
88	C₈×D₇₃	—	2	1	no	—
89	C₈×D₇₃	—	2	1	no	—
90	C₈×D₇₃	—	4	4	no	—
91	#11924003	—	2	2	no	—
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	M₁₆×C₇₃	—	2	2	no	—
94	M₁₆×C₇₃	—	4	1	no	—
95	M₁₆×C₇₃	—	4	1	no	—
96	M₁₆×C₇₃	—	4	1	no	—
97	C₂×C₈×C₇₃	—	4	1	no	—
98	C₂×C₈×C₇₃	—	4	1	no	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
105	C₁₆×C₇₃	—	2	2	no	—
106	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₂²×D₇₃	—	2	1	no	—
108	C₂²×D₇₃	—	2	1	no	—
109	C₂²×D₇₃	—	2	2	no	—
110	C₂²×D₇₃	—	4	1	no	—
111	C₂²×D₇₃	#107	1	1	yes	characteristic
112	C₂²×D₇₃	#107	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
115	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
119	C₂×Dic₇₃	—	2	1	no	—
120	C₂×Dic₇₃	—	2	1	no	—
121	C₂×Dic₇₃	#107	1	1	yes	characteristic
122	C₂×Dic₇₃	#107	1	1	yes	characteristic
123	C₄×D₇₃	—	2	1	no	—
124	C₄×D₇₃	—	2	1	no	—
125	C₄×D₇₃	—	2	1	no	—
126	C₄×D₇₃	—	2	1	no	—
127	C₄×D₇₃	—	2	1	no	—
128	C₄×D₇₃	—	2	1	no	—
129	C₄×D₇₃	—	2	1	no	—
130	C₄×D₇₃	—	2	1	no	—
131	C₈×C₇₃	—	4	2	no	—
132	C₈×C₇₃	—	8	1	no	—
133	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
134	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
135	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
137	#10596831	—	2	1	no	—
138	#10596831	—	2	1	no	—
139	#10596831	—	4	2	no	—
140	#10596831	—	8	1	no	—
141	C₂²×C₇₃	—	2	1	no	—
142	C₂²×C₇₃	—	2	1	no	—
143	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
144	C₂²×C₇₃	#451	1	1	yes	characteristic
145	C₂²×C₇₃	#484	1	1	yes	characteristic
146	C₂×D₇₃	—	2	1	no	—
147	C₂×D₇₃	—	2	1	no	—
148	C₂×D₇₃	—	2	1	no	—
149	C₂×D₇₃	—	2	1	no	—
150	C₂×D₇₃	—	2	1	no	—
151	C₂×D₇₃	—	4	2	no	—
152	C₂×D₇₃	—	4	2	no	—
153	C₂×D₇₃	#368	1	2	yes	complemented
154	C₄×C₇₃	#385	1	1	yes	characteristic
155	C₄×C₇₃	#428	1	1	yes	characteristic
156	C₄×C₇₃	#447	1	1	yes	characteristic
157	C₄×C₇₃	#480	1	1	yes	characteristic
158	Dic₇₃	—	2	1	no	—
159	Dic₇₃	—	2	1	no	—
160	#12922	—	73	1	no	maximal, Sylow
161	C₂×C₇₃	—	2	1	no	—
162	C₂×C₇₃	—	2	1	no	—
163	C₂×C₇₃	#1852	1	1	yes	characteristic
164	C₂×C₇₃	C₂×#368	1	1	yes	characteristic
165	C₂×C₇₃	#2161	1	1	yes	characteristic
166	D₇₃	—	2	2	no	—
167	D₇₃	—	4	1	no	—
168	C₂²×#108	—	73	1	no	—
169	C₂×#368	—	73	1	no	—
170	C₂×#368	—	73	1	no	—
171	C₂×C₂²⋊C₁₆	—	73	1	no	—
172	#1810	—	73	1	no	—
173	#1381	—	73	1	no	—
174	#1381	D₇₃	1	1	yes	characteristic, complemented
175	C₇₃	#12922	1	1	yes	characteristic, complemented, Sylow
176	C₂²×M₁₆	—	146	1	no	—
177	C₂³×C₈	—	73	1	no	—
178	C₂²×C₁₆	—	73	1	no	—
179	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
180	C₂×#108	—	73	1	no	—
181	C₂×#108	—	73	1	no	—
182	C₂×#108	—	146	2	no	—
183	C₂×#108	—	146	1	no	—
184	C₂×#108	—	146	2	no	—
185	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
186	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
187	#407	—	146	1	no	—
188	#368	—	146	2	no	—
189	#368	—	146	2	no	—
190	C₂²⋊C₁₆	—	73	2	no	—
191	C₂²⋊C₁₆	—	73	2	no	—
192	#341	—	146	1	no	—
193	C₂³×C₄	—	73	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	292	1	no	—
197	C₂×M₁₆	—	292	2	no	—
198	C₂×M₁₆	—	146	4	no	—
199	C₂×M₁₆	—	146	1	no	—
200	C₂×M₁₆	—	146	1	no	—
201	C₂²×C₈	—	292	1	no	—
202	C₂²×C₈	—	73	1	no	—
203	C₂²×C₈	—	73	1	no	—
204	C₂²×C₈	—	73	1	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₂²×C₈	—	146	1	no	—
208	C₂²×C₈	—	146	1	no	—
209	C₂²×C₈	—	146	1	no	—
210	C₂²×C₈	—	146	1	no	—
211	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	73	2	no	—
215	C₂×C₁₆	—	146	1	no	—
216	C₂×C₁₆	—	146	1	no	—
217	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	2	1	no	—
221	#108	—	73	2	no	—
222	#108	—	146	4	no	—
223	#108	—	146	1	no	—
224	#108	—	146	1	no	—
225	#108	—	146	1	no	—
226	#108	D₂₉₂	1	2	yes	—
227	C₂²⋊C₈	—	292	1	no	—
228	C₂²⋊C₈	—	292	1	no	—
229	C₂⁴	—	73	1	no	—
230	C₂²×C₄	—	73	1	no	—
231	C₂²×C₄	—	73	1	no	—
232	C₂²×C₄	—	73	1	no	—
233	C₂²×C₄	—	73	1	no	—
234	C₂²×C₄	—	73	1	no	—
235	C₂²×C₄	—	146	1	no	—
236	C₂²×C₄	—	146	1	no	—
237	C₂²×C₄	—	146	1	no	—
238	C₂²×C₄	—	146	1	no	—
239	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	1	no	—
244	M₁₆	—	292	4	no	—
245	M₁₆	—	292	1	no	—
246	M₁₆	—	292	1	no	—
247	M₁₆	—	292	1	no	—
248	M₁₆	—	146	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	292	4	no	—
252	C₂×C₈	—	292	1	no	—
253	C₂×C₈	—	292	1	no	—
254	C₂×C₈	—	73	1	no	—
255	C₂×C₈	—	73	1	no	—
256	C₂×C₈	—	146	1	no	—
257	C₂×C₈	—	146	1	no	—
258	C₂×C₈	—	146	1	no	—
259	C₂×C₈	—	146	1	no	—
260	C₂×C₈	—	146	1	no	—
261	C₂×C₈	—	146	1	no	—
262	C₂×C₈	—	146	1	no	—
263	C₂×C₈	—	146	1	no	—
264	C₂×C₈	—	146	1	no	—
265	C₂×C₈	—	146	1	no	—
266	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
267	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
269	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
270	C₂×C₈	#11924027	1	1	yes	characteristic
271	C₂×C₈	#11924028	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	—	146	2	no	—
274	C₂³	—	292	1	no	—
275	C₂³	—	73	1	no	—
276	C₂³	—	73	1	no	—
277	C₂³	—	146	2	no	—
278	C₂³	—	146	1	no	—
279	C₂³	—	146	1	no	—
280	C₂³	Q₁₆×D₇₃	1	1	yes	characteristic
281	C₂×C₄	—	73	1	no	—
282	C₂×C₄	—	73	1	no	—
283	C₂×C₄	—	146	1	no	—
284	C₂×C₄	—	146	1	no	—
285	C₂×C₄	—	146	1	no	—
286	C₂×C₄	—	146	1	no	—
287	C₂×C₄	—	146	1	no	—
288	C₂×C₄	—	146	1	no	—
289	C₂×C₄	—	146	1	no	—
290	C₂×C₄	—	146	1	no	—
291	C₂×C₄	—	146	1	no	—
292	C₂×C₄	—	146	1	no	—
293	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
294	C₂×C₄	#14556561	1	1	yes	characteristic
295	C₂×C₄	#14556573	1	1	yes	characteristic
296	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
297	C₂×C₄	#14556618	1	1	yes	characteristic
298	C₂×C₄	#14556620	1	1	yes	characteristic
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	—	292	2	no	—
302	C₈	—	584	1	no	—
303	C₈	—	146	1	no	—
304	C₈	—	146	1	no	—
305	C₈	#14556551	1	1	yes	characteristic
306	C₈	C₄×D₂₉₂	1	1	yes	characteristic
307	C₈	#14556569	1	1	yes	characteristic
308	C₈	#14556571	1	1	yes	characteristic
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	292	2	no	—
312	C₂²	—	292	2	no	—
313	C₂²	—	73	2	no	—
314	C₂²	—	146	1	no	—
315	C₂²	—	146	1	no	—
316	C₂²	—	146	1	no	—
317	C₂²	—	146	1	no	—
318	C₂²	—	146	1	no	—
319	C₂²	#107×D₇₃	1	1	yes	characteristic
320	C₂²	#17988744	1	1	yes	characteristic
321	C₂²	#17988921	1	1	yes	characteristic
322	C₄	—	146	1	no	—
323	C₄	—	146	1	no	—
324	C₄	#17988461	1	1	yes	characteristic
325	C₄	#17988626	1	1	yes	characteristic
326	C₄	#17988722	1	1	yes	characteristic
327	C₄	#17988899	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	292	1	no	—
331	C₂	—	146	2	no	—
332	C₂	#22610120	1	1	yes	characteristic
333	C₂	#368×D₇₃	1	1	yes	characteristic
334	C₂	#22611724	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1381	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂³×D₇₃	Q₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1381	D₇₃
hypocenter	C₇₃	#12922
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

Complements

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

N	H ≅ G/N	type
2. C₂×#17988417	C₂	semidirect
3. C₂×#17988416	C₂	semidirect
6. #22608024	C₂	semidirect
7. C₇₃×#1381	C₂	semidirect
153. C₂×D₇₃	#368	semidirect
174. #1381	D₇₃	semidirect
175. C₇₃	#12922	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108×D₇₃	2
C₂×#17988417	2
C₂×#17988416	2
C₂×#17988366	2
#22609966	2
#22608024	2
C₇₃×#1381	2
#12922	73