Subgroups — #33001018

Order 18688.

514 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988541	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988366	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988366	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988362	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608009	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11924011	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₂×#14556548	C₄	1	1	yes	characteristic
11	C₂²×#11924003	C₂²	1	1	yes	characteristic
12	C₂²×#11924003	C₂²	1	1	yes	characteristic
13	#17988541	—	2	2	no	—
14	#17988541	—	2	2	no	—
15	C₂×#14556502	C₄	1	1	yes	characteristic
16	C₂×#14556501	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	#17988366	C₄	1	4	yes	complemented
19	#17988366	C₄	1	4	yes	complemented
20	#17988362	—	2	2	no	—
21	C₁₆×Dic₇₃	—	2	2	no	—
22	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#11924011	—	2	2	no	—
27	C₂×#11924011	—	2	2	no	—
28	C₂×#11924011	C₈	1	4	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₇₃	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
37	#14556548	—	2	2	no	—
38	#14556548	—	2	2	no	—
39	C₂×#11924007	C₈	1	2	yes	—
40	C₂×#11924005	C₈	1	2	yes	—
41	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×#11924003	—	2	1	no	—
43	C₂×#11924003	—	2	1	no	—
44	C₂×#11924003	—	2	1	no	—
45	C₂×#11924003	—	2	1	no	—
46	C₂×#11924003	C₂×C₄	1	2	yes	—
47	C₂×#11924003	C₂×C₄	1	2	yes	—
48	#14556502	—	2	2	no	—
49	#14556501	—	2	2	no	—
50	C₈×Dic₇₃	D₄	1	2	yes	—
51	C₈×Dic₇₃	D₄	1	2	yes	—
52	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₇₃	—	2	1	no	—
56	C₂×C₁₆×C₇₃	—	2	1	no	—
57	C₂×C₁₆×C₇₃	—	2	1	no	—
58	C₂×C₁₆×C₇₃	—	2	1	no	—
59	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
60	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₇₃	—	2	1	no	—
62	C₄×C₈×C₇₃	—	2	1	no	—
63	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₇₃	—	2	1	no	—
67	C₂×C₄×D₇₃	—	2	1	no	—
68	C₂×C₄×D₇₃	—	2	1	no	—
69	C₂×C₄×D₇₃	—	2	1	no	—
70	C₂×C₄×D₇₃	C₄²	1	2	yes	—
71	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
72	#11924011	—	2	8	no	—
73	C₂×#10596831	—	2	1	no	—
74	C₂×#10596831	—	2	1	no	—
75	C₂×#10596831	C₄²	1	2	yes	—
76	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
77	C₂×#10596831	C₂×C₈	1	2	yes	—
78	C₂×#10596831	M₁₆	1	2	yes	—
79	C₈×D₇₃	—	2	2	no	—
80	C₈×D₇₃	—	2	2	no	—
81	C₈×D₇₃	—	2	2	no	—
82	C₈×D₇₃	—	2	2	no	—
83	#11924007	—	2	4	no	—
84	#11924005	—	2	4	no	—
85	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
86	C₄×Dic₇₃	M₁₆	1	2	yes	—
87	#11924003	—	2	2	no	—
88	#11924003	—	2	2	no	—
89	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	—	2	1	no	—
92	C₂×C₈×C₇₃	—	2	1	no	—
93	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
94	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
100	C₄²×C₇₃	—	2	1	no	—
101	C₄²×C₇₃	—	2	1	no	—
102	C₁₆×C₇₃	—	2	2	no	—
103	C₁₆×C₇₃	—	2	2	no	—
104	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
105	C₂²×D₇₃	—	2	1	no	—
106	C₂²×D₇₃	—	2	1	no	—
107	C₂²×D₇₃	—	2	2	no	—
108	C₂²×D₇₃	—	2	2	no	—
109	C₂²×D₇₃	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	—	2	2	no	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂×Dic₇₃	—	2	1	no	—
120	C₂×Dic₇₃	—	2	1	no	—
121	C₂×Dic₇₃	—	2	2	no	—
122	C₂×Dic₇₃	—	2	2	no	—
123	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
124	C₂×Dic₇₃	C₂²⋊C₈	1	4	yes	—
125	C₄×D₇₃	—	2	2	no	—
126	C₄×D₇₃	—	2	2	no	—
127	C₄×D₇₃	—	2	2	no	—
128	C₄×D₇₃	—	2	2	no	—
129	C₈×C₇₃	—	2	2	no	—
130	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
131	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
132	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
133	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
134	#10596831	—	2	2	no	—
135	#10596831	C₂²⋊C₈	1	4	yes	—
136	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
137	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₇₃	#431	1	1	yes	characteristic
139	C₂²×C₇₃	#504	1	2	yes	—
140	C₂²×C₇₃	#504	1	2	yes	—
141	C₂×D₇₃	—	2	2	no	—
142	C₂×D₇₃	—	2	2	no	—
143	C₂×D₇₃	—	2	2	no	—
144	C₂×D₇₃	—	4	2	no	—
145	C₂×D₇₃	—	4	2	no	—
146	C₄×C₇₃	—	4	2	no	—
147	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
149	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₇₃	#431	1	1	yes	characteristic
151	Dic₇₃	—	2	4	no	—
152	Dic₇₃	—	2	2	no	—
153	#12600	—	73	1	no	maximal, Sylow
154	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
155	C₂×C₇₃	#2108	1	1	yes	characteristic
156	C₂×C₇₃	#2108	1	1	yes	characteristic
157	C₂×C₇₃	#2113	1	2	yes	—
158	C₂×C₇₃	#2164	1	2	yes	—
159	D₇₃	—	4	2	no	—
160	C₂×#405	—	73	1	no	—
161	C₂×C₂²⋊C₁₆	—	73	1	no	—
162	C₂×C₂²⋊C₁₆	—	73	1	no	—
163	C₂×#346	—	73	1	no	—
164	C₂×C₄×C₁₆	—	73	1	no	—
165	#1374	—	73	1	no	—
166	#1374	D₇₃	1	1	yes	characteristic, complemented
167	C₇₃	#12600	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	—	73	1	no	—
169	C₂³×C₈	—	73	1	no	—
170	C₂×#117	—	73	1	no	—
171	C₂²×C₁₆	—	73	1	no	—
172	C₂²×C₁₆	—	73	1	no	—
173	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
174	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
175	#405	—	146	2	no	—
176	#405	—	146	2	no	—
177	C₂×#97	—	73	1	no	—
178	C₂×#97	—	73	1	no	—
179	C₂×C₄×C₈	—	73	1	no	—
180	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
181	C₂²⋊C₁₆	—	73	4	no	—
182	C₂²⋊C₁₆	—	73	4	no	—
183	#346	—	146	2	no	—
184	C₄×C₁₆	—	146	2	no	—
185	C₂³×C₄	—	73	1	no	—
186	C₂×M₁₆	—	73	4	no	—
187	C₂×M₁₆	—	146	2	no	—
188	C₂×M₁₆	—	146	2	no	—
189	C₂²×C₈	—	73	1	no	—
190	C₂²×C₈	—	73	1	no	—
191	C₂²×C₈	—	73	2	no	—
192	C₂²×C₈	—	73	2	no	—
193	C₂²×C₈	—	146	1	no	—
194	C₂²×C₈	—	146	1	no	—
195	C₂²×C₈	—	146	1	no	—
196	C₂²×C₈	—	146	1	no	—
197	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
199	#117	—	146	2	no	—
200	#117	—	146	2	no	—
201	C₂×C₄⋊C₄	—	73	2	no	—
202	C₂×C₂²⋊C₄	—	73	2	no	—
203	C₂×C₄²	—	73	1	no	—
204	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	73	2	no	—
210	C₂×C₁₆	—	73	2	no	—
211	C₂×C₁₆	—	146	1	no	—
212	C₂×C₁₆	—	146	1	no	—
213	C₂×C₁₆	—	146	1	no	—
214	C₂×C₁₆	—	146	1	no	—
215	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
216	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
217	#97	—	146	2	no	—
218	#97	—	146	2	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	73	2	no	—
222	C₄×C₈	—	73	2	no	—
223	C₂⁴	—	73	1	no	—
224	C₂²×C₄	—	73	2	no	—
225	C₂²×C₄	—	73	1	no	—
226	C₂²×C₄	—	73	2	no	—
227	C₂²×C₄	—	73	2	no	—
228	C₂²×C₄	—	146	1	no	—
229	C₂²×C₄	—	146	1	no	—
230	C₂²×C₄	—	146	1	no	—
231	C₂²×C₄	—	146	1	no	—
232	C₂²×C₄	C₈×D₇₃	1	2	yes	—
233	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
234	M₁₆	—	146	8	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	73	2	no	—
238	C₂×C₈	—	73	2	no	—
239	C₂×C₈	—	73	2	no	—
240	C₂×C₈	—	73	2	no	—
241	C₂×C₈	—	146	1	no	—
242	C₂×C₈	—	146	1	no	—
243	C₂×C₈	—	146	2	no	—
244	C₂×C₈	—	146	2	no	—
245	C₂×C₈	—	146	2	no	—
246	C₂×C₈	—	146	2	no	—
247	C₂×C₈	C₈×D₇₃	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
250	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
251	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
252	C₂×C₈	#11924027	1	1	yes	characteristic
253	C₂×C₈	#11924029	1	1	yes	characteristic
254	C₄⋊C₄	—	146	4	no	—
255	C₂²⋊C₄	—	146	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	73	2	no	—
259	C₄²	—	73	2	no	—
260	C₁₆	—	2	2	no	—
261	C₁₆	—	2	2	no	—
262	C₁₆	—	146	2	no	—
263	C₁₆	—	146	2	no	—
264	C₂³	—	73	2	no	—
265	C₂³	—	146	1	no	—
266	C₂³	—	146	2	no	—
267	C₂³	—	146	2	no	—
268	C₂³	—	146	1	no	—
269	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	73	2	no	—
274	C₂×C₄	—	73	4	no	—
275	C₂×C₄	—	146	1	no	—
276	C₂×C₄	—	146	2	no	—
277	C₂×C₄	—	146	2	no	—
278	C₂×C₄	—	146	1	no	—
279	C₂×C₄	—	146	2	no	—
280	C₂×C₄	—	146	2	no	—
281	C₂×C₄	—	146	2	no	—
282	C₂×C₄	—	146	2	no	—
283	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
285	C₂×C₄	#14556549	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	73	4	no	—
291	C₈	—	146	2	no	—
292	C₈	#14556551	1	1	yes	characteristic
293	C₈	#14556551	1	1	yes	characteristic
294	C₈	#14556553	1	1	yes	characteristic
295	C₈	#14556553	1	1	yes	characteristic
296	C₂²	—	292	2	no	—
297	C₂²	—	292	2	no	—
298	C₂²	—	146	2	no	—
299	C₂²	—	146	2	no	—
300	C₂²	—	146	2	no	—
301	C₂²	C₄×C₈×D₇₃	1	1	yes	characteristic
302	C₂²	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
303	C₂²	#17988639	1	1	yes	characteristic
304	C₂²	#17989003	1	2	yes	—
305	C₂²	#17989005	1	2	yes	—
306	C₄	—	4	2	no	—
307	C₄	—	146	2	no	—
308	C₄	—	146	4	no	—
309	C₄	#17988434	1	1	yes	characteristic
310	C₄	#17988539	1	1	yes	characteristic
311	C₄	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
312	C₄	#17988639	1	1	yes	characteristic
313	C₂	—	292	2	no	—
314	C₂	#22609688	1	1	yes	characteristic
315	C₂	#22611526	1	1	yes	characteristic
316	C₂	#22611529	1	1	yes	characteristic
317	C₂	#22611543	1	2	yes	—
318	C₂	#22611737	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₇₃×#1374	C₂
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1374	D₇₃
hypocenter	C₇₃	#12600
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. C₂×#17988362	C₂	semidirect
5. C₂×C₁₆×Dic₇₃	C₂	semidirect
6. #22608009	C₂	semidirect
7. C₇₃×#1374	C₂	semidirect
18. #17988366	C₄	semidirect
19. #17988366	C₄	semidirect
166. #1374	D₇₃	semidirect
167. C₇₃	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988541	2
C₂×#17988366	2
C₂×#17988362	2
C₂×C₁₆×Dic₇₃	2
#22608009	2
C₇₃×#1374	2
#12600	73