Subgroups — #33084579

Order 18688.

998 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988710	C₂	1	2	yes	maximal
2	C₄×#14556613	C₂	1	1	yes	maximal, characteristic
3	#22618356	C₂	1	2	yes	maximal, complemented
4	C₄×#14556527	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609815	C₂	1	2	yes	maximal, complemented
6	#22609768	C₂	1	4	yes	maximal, complemented
7	#22609742	C₂	1	2	yes	maximal, complemented
8	C₇₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₂²	1	1	yes	characteristic
10	C₂×#14556597	C₂²	1	4	yes	—
11	C₄×#11924028	C₂²	1	2	yes	—
12	#17988710	C₄	1	16	yes	complemented
13	C₄×Dic₂₉₂	C₄	1	2	yes	complemented
14	C₄×#11924017	C₄	1	4	yes	complemented
15	C₄×D₅₈₄	C₄	1	2	yes	complemented
16	C₂×#14556527	C₂²	1	2	yes	—
17	#17988625	C₂²	1	1	yes	characteristic
18	C₂×#14556519	C₂²	1	4	yes	—
19	#17988584	C₂²	1	2	yes	—
20	#17988544	—	2	4	no	—
21	#17988542	C₂²	1	4	yes	—
22	C₄×#11924012	C₂²	1	2	yes	—
23	#17988539	—	2	2	no	—
24	#97×D₇₃	—	2	2	no	—
25	#17988352	C₂²	1	2	yes	—
26	#17988347	C₂²	1	4	yes	—
27	C₂×#97×C₇₃	C₂²	1	2	yes	—
28	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	#376×C₇₃	C₂²	1	1	yes	characteristic
32	C₂×#11924028	C₂³	1	2	yes	—
33	#14556613	C₂×C₄	1	8	yes	—
34	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
35	C₂×#11924017	C₂×C₄	1	2	yes	—
36	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
37	#14556597	—	2	8	no	—
38	#14556597	C₂×C₄	1	16	yes	complemented
39	C₂×#11924012	C₂³	1	4	yes	—
40	C₂×#11924011	—	2	4	no	—
41	C₂×C₈×D₇₃	—	2	4	no	—
42	C₄×Dic₁₄₆	C₂×C₄	1	4	yes	—
43	C₄×D₂₉₂	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₇₃	—	2	4	no	—
45	#14556550	C₂³	1	2	yes	—
46	#14556548	—	2	2	no	—
47	C₄²×D₇₃	—	2	2	no	—
48	#14556527	C₂×C₄	1	8	yes	complemented
49	#14556525	C₂×C₄	1	2	yes	—
50	#14556523	C₂×C₄	1	2	yes	—
51	#14556519	C₂×C₄	1	16	yes	complemented
52	#14556516	C₂×C₄	1	4	yes	—
53	#14556513	C₂×C₄	1	4	yes	—
54	#14556502	—	2	2	no	—
55	#14556501	—	2	4	no	—
56	C₄×#10596831	—	2	2	no	—
57	C₈×Dic₇₃	—	2	4	no	—
58	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
63	#97×C₇₃	C₂×C₄	1	8	yes	complemented
64	#97×C₇₃	D₄	1	2	yes	—
65	#97×C₇₃	D₄	1	2	yes	—
66	C₄×C₈×C₇₃	—	2	4	no	—
67	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
69	#95×C₇₃	—	2	2	no	—
70	#11924028	—	2	4	no	—
71	#11924028	C₂²×C₄	1	8	yes	—
72	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
73	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
74	C₂×C₂²⋊D₇₃	—	2	2	no	—
75	C₂×C₄×D₇₃	—	2	2	no	—
76	C₂×C₄×D₇₃	—	2	4	no	—
77	Dic₂₉₂	C₄²	1	4	yes	—
78	#11924017	C₄²	1	8	yes	—
79	D₅₈₄	C₄²	1	4	yes	—
80	#11924012	—	2	4	no	—
81	#11924012	C₂²×C₄	1	8	yes	—
82	#11924011	—	2	16	no	—
83	C₂×#10596831	—	2	4	no	—
84	C₈×D₇₃	—	2	16	no	—
85	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
86	#11924007	—	2	4	no	—
87	C₂²⋊Dic₇₃	—	2	2	no	—
88	#11924005	—	2	4	no	—
89	C₄×Dic₇₃	—	2	4	no	—
90	C₄×Dic₇₃	—	2	2	no	—
91	C₂²×C₄×C₇₃	—	2	2	no	—
92	C₂²×C₄×C₇₃	#41	1	2	yes	—
93	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
94	C₂×C₈×C₇₃	—	2	2	no	—
95	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₇₃	#41	1	4	yes	—
102	C₄²×C₇₃	—	2	2	no	—
103	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
104	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
108	C₂²×D₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	2	4	no	—
110	C₂×C₄×C₇₃	—	4	2	no	—
111	C₂×C₄×C₇₃	—	4	4	no	—
112	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
118	C₂²⋊D₇₃	—	2	8	no	—
119	C₂×Dic₇₃	—	2	2	no	—
120	C₂×Dic₇₃	—	2	4	no	—
121	D₂₉₂	—	2	2	no	—
122	D₂₉₂	C₂×C₄²	1	4	yes	—
123	C₄×D₇₃	—	2	8	no	—
124	C₄×D₇₃	—	8	4	no	—
125	Dic₁₄₆	—	2	2	no	—
126	Dic₁₄₆	C₂×C₄²	1	4	yes	—
127	C₈×C₇₃	—	2	2	no	—
128	C₈×C₇₃	C₂×C₄²	1	2	yes	—
129	C₈×C₇₃	C₂×C₄²	1	2	yes	—
130	C₈×C₇₃	C₄×D₄	1	8	yes	—
131	#10596831	—	2	8	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₇₃	C₄×#41	1	2	yes	—
135	C₂×D₇₃	—	2	4	no	—
136	C₂×D₇₃	—	4	4	no	—
137	C₄×C₇₃	—	2	2	no	—
138	C₄×C₇₃	—	2	2	no	—
139	C₄×C₇₃	—	4	4	no	—
140	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
141	C₄×C₇₃	C₂×C₄×D₄	1	2	yes	—
142	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
143	Dic₇₃	—	2	4	no	—
144	Dic₇₃	—	4	4	no	—
145	#20513	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	—	2	2	no	—
147	C₂×C₇₃	C₄²×D₄	1	1	yes	characteristic
148	C₂×C₇₃	#2957	1	2	yes	—
149	D₇₃	—	4	4	no	—
150	C₂×#444	—	73	2	no	—
151	C₄×#135	—	73	1	no	—
152	#2875	—	73	2	no	—
153	C₄×#108	—	73	1	no	—
154	#1773	—	73	2	no	—
155	#1764	—	73	4	no	—
156	#1759	—	73	2	no	—
157	#1726	D₇₃	1	1	yes	characteristic, complemented
158	C₇₃	#20513	1	1	yes	characteristic, complemented, Sylow
159	C₂×#135	—	73	1	no	—
160	C₂×#131	—	73	4	no	—
161	C₄×#41	—	73	2	no	—
162	#444	—	73	16	no	—
163	C₄×Q₁₆	—	73	2	no	—
164	C₄×SD₁₆	—	73	4	no	—
165	C₄×D₈	—	73	2	no	—
166	C₂×#108	—	73	2	no	—
167	#427	—	73	1	no	—
168	C₂×#104	—	73	4	no	—
169	#416	—	73	2	no	—
170	#405	—	73	4	no	—
171	#405	—	146	4	no	—
172	C₄×M₁₆	—	73	2	no	—
173	C₄×M₁₆	—	146	2	no	—
174	C₂×#97	—	146	2	no	—
175	C₂×#97	C₂×D₇₃	1	2	yes	—
176	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
178	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
179	#376	C₂×D₇₃	1	1	yes	characteristic
180	#341	—	73	2	no	—
181	#339	—	73	4	no	—
182	C₂×#41	—	73	2	no	—
183	#135	—	73	8	no	—
184	C₂×Q₁₆	—	73	1	no	—
185	C₂×SD₁₆	—	73	2	no	—
186	C₂×D₈	—	73	1	no	—
187	#131	—	73	16	no	—
188	#131	—	146	8	no	—
189	C₂×M₁₆	—	73	4	no	—
190	C₂×M₁₆	—	146	4	no	—
191	C₂²×C₈	—	146	4	no	—
192	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
195	C₄×Q₈	—	73	4	no	—
196	C₄×D₄	—	73	4	no	—
197	C₄×D₄	—	146	4	no	—
198	#117	—	73	2	no	—
199	#117	—	146	2	no	—
200	C₂×C₄²	—	146	2	no	—
201	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
202	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
203	#108	—	73	8	no	—
204	#107	—	73	2	no	—
205	#106	—	73	2	no	—
206	#104	—	73	16	no	—
207	#103	—	73	4	no	—
208	#102	—	73	4	no	—
209	#97	—	146	4	no	—
210	#97	—	146	2	no	—
211	#97	C₄×D₇₃	1	8	yes	complemented
212	#97	D₂₉₂	1	2	yes	—
213	#97	D₂₉₂	1	2	yes	—
214	C₄×C₈	—	2	4	no	—
215	C₄×C₈	—	146	4	no	—
216	C₄×C₈	—	146	2	no	—
217	C₄×C₈	C₄×D₇₃	1	2	yes	—
218	C₄×C₈	C₄×D₇₃	1	2	yes	—
219	#95	—	2	2	no	—
220	#41	—	73	8	no	—
221	#41	—	146	4	no	—
222	C₂×Q₈	—	73	2	no	—
223	C₂×D₄	—	73	2	no	—
224	C₂×D₄	—	146	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	146	4	no	—
227	C₂²×C₄	—	146	2	no	—
228	C₂²×C₄	#11924028	1	2	yes	—
229	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
230	Q₁₆	—	73	4	no	—
231	SD₁₆	—	73	8	no	—
232	D₈	—	73	4	no	—
233	M₁₆	—	73	8	no	—
234	M₁₆	—	146	4	no	—
235	M₁₆	—	146	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	146	4	no	—
238	C₂×C₈	—	146	16	no	—
239	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
243	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
244	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
245	C₂×C₈	#11924028	1	4	yes	—
246	C₄⋊C₄	—	73	4	no	—
247	C₄⋊C₄	—	146	4	no	—
248	C₂²⋊C₄	—	146	2	no	—
249	C₂²⋊C₄	—	146	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	146	4	no	—
252	C₄²	—	146	2	no	—
253	C₄²	C₂×C₄×D₇₃	1	4	yes	—
254	C₄²	C₂×C₄×D₇₃	1	2	yes	—
255	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
256	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
257	C₂³	—	146	2	no	—
258	C₂³	C₂×#11924028	1	1	yes	characteristic
259	Q₈	—	73	4	no	—
260	Q₈	—	146	2	no	—
261	D₄	—	73	4	no	—
262	D₄	—	146	8	no	—
263	D₄	—	146	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	584	4	no	—
268	C₂×C₄	—	146	4	no	—
269	C₂×C₄	—	146	2	no	—
270	C₂×C₄	—	146	8	no	—
271	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
272	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
274	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
276	C₂×C₄	C₂×#11924028	1	2	yes	—
277	C₈	—	2	2	no	—
278	C₈	—	146	8	no	—
279	C₈	C₄²×D₇₃	1	2	yes	—
280	C₈	C₄²×D₇₃	1	2	yes	—
281	C₈	C₄×D₂₉₂	1	8	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	292	4	no	—
284	C₂²	—	146	4	no	—
285	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
286	C₂²	C₄×#11924028	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	4	no	—
290	C₄	—	292	4	no	—
291	C₄	—	146	4	no	—
292	C₄	C₂×C₄²×D₇₃	1	1	yes	characteristic
293	C₄	C₂×C₄×D₂₉₂	1	2	yes	—
294	C₄	C₄×#11924028	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	292	4	no	—
297	C₂	C₄²×D₂₉₂	1	1	yes	characteristic
298	C₂	#22618849	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1726	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#14556613	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1726	D₇₃
hypocenter	C₇₃	#20513
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
3. #22618356	C₂	semidirect
4. C₄×#14556527	C₂	semidirect
5. #22609815	C₂	semidirect
6. #22609768	C₂	semidirect
7. #22609742	C₂	semidirect
8. C₇₃×#1726	C₂	semidirect
12. #17988710	C₄	semidirect
13. C₄×Dic₂₉₂	C₄	semidirect
14. C₄×#11924017	C₄	semidirect
15. C₄×D₅₈₄	C₄	semidirect
38. #14556597	C₂×C₄	semidirect
48. #14556527	C₂×C₄	semidirect
51. #14556519	C₂×C₄	semidirect
63. #97×C₇₃	C₂×C₄	semidirect
157. #1726	D₇₃	semidirect
158. C₇₃	#20513	semidirect
211. #97	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×#17988710	2
C₄×#14556613	2
#22618356	2
C₄×#14556527	2
#22609815	2
#22609768	2
#22609742	2
C₇₃×#1726	2
#20513	73