Subgroups — #33239412

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989019	C₂	1	1	yes	maximal, characteristic
2	#22618849	C₂	1	2	yes	maximal, complemented
3	#22611818	C₂	1	4	yes	maximal, complemented
4	#22611813	C₂	1	4	yes	maximal, complemented
5	C₂×#17988417	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611660	C₂	1	2	yes	maximal, complemented
7	C₇₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556613	C₂²	1	2	yes	—
9	#17989353	—	2	2	no	—
10	#17989019	C₄	1	2	yes	complemented
11	#17989019	C₄	1	2	yes	complemented
12	#17989019	C₂²	1	2	yes	—
13	#17989019	C₂²	1	2	yes	—
14	C₂×Dic₅₈₄	C₄	1	1	yes	characteristic, complemented
15	C₂×#14556536	C₄	1	2	yes	complemented
16	C₂×D₁₁₆₈	C₄	1	1	yes	characteristic, complemented
17	#17988710	—	2	4	no	—
18	#17988705	C₂²	1	8	yes	complemented
19	C₂×#14556527	C₂²	1	2	yes	—
20	#17988605	—	2	2	no	—
21	#17988417	C₂²	1	2	yes	complemented
22	#17988417	C₂²	1	2	yes	complemented
23	#17988395	C₂²	1	8	yes	complemented
24	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
26	#405×C₇₃	C₂²	1	1	yes	characteristic
27	#346×C₇₃	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₇₃	C₂²	1	2	yes	complemented
29	C₂×#11924028	—	2	2	no	—
30	#14556613	—	2	2	no	—
31	#14556613	—	2	2	no	—
32	#14556613	C₂×C₄	1	4	yes	—
33	#14556613	C₂³	1	4	yes	—
34	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
35	C₂×#11924017	—	2	2	no	—
36	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
37	#14556599	—	4	4	no	—
38	#14556597	—	2	8	no	—
39	C₂×#11924012	—	2	2	no	—
40	M₁₆×D₇₃	—	4	4	no	—
41	Dic₅₈₄	C₂×C₄	1	2	yes	—
42	Dic₅₈₄	C₂×C₄	1	2	yes	—
43	#14556536	C₂×C₄	1	4	yes	—
44	#14556536	C₂×C₄	1	4	yes	—
45	D₁₁₆₈	C₂×C₄	1	2	yes	—
46	D₁₁₆₈	C₂×C₄	1	2	yes	—
47	#14556527	—	2	2	no	—
48	#14556527	C₂³	1	4	yes	—
49	#14556519	—	2	8	no	—
50	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
52	#117×C₇₃	D₄	1	1	yes	characteristic
53	M₃₂×C₇₃	—	2	1	no	—
54	M₃₂×C₇₃	—	2	1	no	—
55	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
60	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
62	#97×C₇₃	D₄	1	2	yes	—
63	C₄×C₈×C₇₃	C₂³	1	2	yes	—
64	#11924028	—	2	4	no	—
65	#11924028	—	2	4	no	—
66	#11924028	—	4	2	no	—
67	#11924028	—	4	2	no	—
68	C₂×Dic₁₄₆	—	2	2	no	—
69	C₂×D₂₉₂	—	2	2	no	—
70	C₂×C₂²⋊D₇₃	—	4	2	no	—
71	C₂×C₄×D₇₃	—	4	2	no	—
72	Dic₂₉₂	—	2	2	no	—
73	Dic₂₉₂	C₂²×C₄	1	4	yes	—
74	#11924017	—	2	4	no	—
75	#11924017	—	2	4	no	—
76	D₅₈₄	—	2	2	no	—
77	D₅₈₄	C₂²×C₄	1	4	yes	—
78	#11924012	—	2	4	no	—
79	#11924012	—	4	2	no	—
80	#11924011	—	4	8	no	—
81	C₂×#10596831	—	4	2	no	—
82	C₈×D₇₃	—	4	8	no	—
83	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
84	M₁₆×C₇₃	D₈	1	2	yes	—
85	M₁₆×C₇₃	D₈	1	2	yes	—
86	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₇₃	—	2	1	no	—
94	C₂²⋊C₄×C₇₃	—	2	1	no	—
95	C₄²×C₇₃	C₂×D₄	1	2	yes	—
96	C₁₆×C₇₃	—	2	2	no	—
97	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₇₃	#135	1	1	yes	characteristic
102	C₂²×D₇₃	—	4	2	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
110	C₂²⋊D₇₃	—	4	4	no	—
111	C₂²⋊D₇₃	—	4	4	no	—
112	C₂×Dic₇₃	—	4	2	no	—
113	D₂₉₂	—	2	4	no	—
114	D₂₉₂	—	4	2	no	—
115	C₄×D₇₃	—	4	4	no	—
116	C₄×D₇₃	—	4	4	no	—
117	Dic₁₄₆	—	2	4	no	—
118	Dic₁₄₆	—	4	2	no	—
119	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
120	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
121	C₈×C₇₃	C₂×D₈	1	4	yes	—
122	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
123	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
124	#10596831	—	4	4	no	—
125	C₂²×C₇₃	—	4	1	no	—
126	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
127	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
128	C₂²×C₇₃	C₂×#135	1	1	yes	characteristic
129	C₂×D₇₃	—	4	4	no	—
130	C₂×D₇₃	—	16	2	no	—
131	C₄×C₇₃	—	4	2	no	—
132	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
133	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
134	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
135	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
136	Dic₇₃	—	4	4	no	—
137	#33465	—	73	1	no	maximal, Sylow
138	C₂×C₇₃	—	2	1	no	—
139	C₂×C₇₃	—	2	1	no	—
140	C₂×C₇₃	—	2	1	no	—
141	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
142	D₇₃	—	8	4	no	—
143	C₂×#508	—	73	1	no	—
144	#2957	—	73	2	no	—
145	#2179	—	73	4	no	—
146	#2178	—	73	4	no	—
147	C₂×#368	—	73	1	no	—
148	#2143	—	73	2	no	—
149	#2108	D₇₃	1	1	yes	characteristic, complemented
150	C₇₃	#33465	1	1	yes	characteristic, complemented, Sylow
151	C₂×#135	—	73	2	no	—
152	#568	—	146	2	no	—
153	#508	—	73	2	no	—
154	#508	—	73	2	no	—
155	#508	—	73	2	no	—
156	#508	—	73	2	no	—
157	C₂×Q₃₂	—	73	1	no	—
158	C₂×SD₃₂	—	73	2	no	—
159	C₂×D₁₆	—	73	1	no	—
160	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
161	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
162	#444	—	146	4	no	—
163	#443	—	73	8	no	—
164	C₂×#108	—	73	2	no	—
165	#421	—	146	2	no	—
166	#405	C₂×D₇₃	1	1	yes	characteristic
167	#368	—	73	2	no	—
168	#368	—	73	2	no	—
169	#359	—	73	8	no	—
170	#346	C₂×D₇₃	1	2	yes	complemented
171	C₄×C₁₆	C₂×D₇₃	1	2	yes	complemented
172	C₂×#41	—	146	2	no	—
173	#135	—	73	4	no	—
174	#135	—	73	4	no	—
175	#135	—	146	2	no	—
176	#135	—	146	2	no	—
177	C₂×Q₁₆	—	73	2	no	—
178	C₂×SD₁₆	—	146	2	no	—
179	C₂×D₈	—	73	2	no	—
180	#131	—	292	4	no	—
181	#131	—	146	8	no	—
182	C₂×M₁₆	—	292	4	no	—
183	C₂×M₁₆	—	146	2	no	—
184	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
186	#117	D₂₉₂	1	1	yes	characteristic
187	Q₃₂	—	73	2	no	—
188	Q₃₂	—	73	2	no	—
189	SD₃₂	—	73	4	no	—
190	SD₃₂	—	73	4	no	—
191	D₁₆	—	73	2	no	—
192	D₁₆	—	73	2	no	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	2	1	no	—
195	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
196	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
197	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
198	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
199	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
200	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
201	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
202	#108	—	73	4	no	—
203	#108	—	146	2	no	—
204	#104	—	146	8	no	—
205	#97	D₂₉₂	1	2	yes	—
206	C₄×C₈	C₂²×D₇₃	1	2	yes	—
207	#41	—	292	2	no	—
208	#41	—	292	2	no	—
209	#41	—	146	4	no	—
210	#41	—	146	4	no	—
211	C₂×Q₈	—	146	2	no	—
212	C₂×D₄	—	292	2	no	—
213	C₂×D₄	—	146	2	no	—
214	C₂²×C₄	—	292	2	no	—
215	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
216	Q₁₆	—	73	4	no	—
217	Q₁₆	—	146	2	no	—
218	SD₁₆	—	146	4	no	—
219	SD₁₆	—	146	4	no	—
220	D₈	—	73	4	no	—
221	D₈	—	146	2	no	—
222	M₁₆	—	292	2	no	—
223	M₁₆	—	292	8	no	—
224	M₁₆	—	146	4	no	—
225	M₁₆	D₅₈₄	1	2	yes	—
226	M₁₆	D₅₈₄	1	2	yes	—
227	C₂×C₈	—	292	2	no	—
228	C₂×C₈	—	292	8	no	—
229	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
231	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
232	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
233	C₂×C₈	#11924028	1	1	yes	characteristic
234	C₂×C₈	#11924028	1	1	yes	characteristic
235	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	C₂×D₂₉₂	1	2	yes	—
239	C₁₆	—	2	2	no	—
240	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
241	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
242	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
243	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
244	C₂³	—	292	2	no	—
245	C₂³	#14556613	1	1	yes	characteristic
246	Q₈	—	292	2	no	—
247	Q₈	—	146	4	no	—
248	D₄	—	292	4	no	—
249	D₄	—	292	4	no	—
250	D₄	—	292	2	no	—
251	D₄	—	146	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	292	2	no	—
254	C₂×C₄	—	292	4	no	—
255	C₂×C₄	—	292	4	no	—
256	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
257	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
258	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
259	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
260	C₂×C₄	#14556613	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
262	C₈	—	292	4	no	—
263	C₈	C₄×D₂₉₂	1	1	yes	characteristic
264	C₈	C₄×D₂₉₂	1	1	yes	characteristic
265	C₈	C₂×D₅₈₄	1	4	yes	—
266	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
267	C₈	C₂×#11924028	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	292	4	no	—
270	C₂²	—	1168	2	no	—
271	C₂²	C₄×D₅₈₄	1	1	yes	characteristic
272	C₂²	C₄×D₅₈₄	1	1	yes	characteristic
273	C₂²	C₂×#14556613	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	292	4	no	—
276	C₄	C₄×D₅₈₄	1	1	yes	characteristic
277	C₄	C₄×D₅₈₄	1	1	yes	characteristic
278	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
279	C₄	C₂²×D₅₈₄	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	584	4	no	—
284	C₂	C₂×C₄×D₅₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₅₈₄
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2108	C₂
socle	C₂×C₇₃	C₂×C₄×D₈
prime radical	C₂×#17989019	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2108	D₇₃
hypocenter	C₇₃	#33465
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #22618849	C₂	semidirect
3. #22611818	C₂	semidirect
4. #22611813	C₂	semidirect
5. C₂×#17988417	C₂	semidirect
6. #22611660	C₂	semidirect
7. C₇₃×#2108	C₂	semidirect
10. #17989019	C₄	semidirect
11. #17989019	C₄	semidirect
14. C₂×Dic₅₈₄	C₄	semidirect
15. C₂×#14556536	C₄	semidirect
16. C₂×D₁₁₆₈	C₄	semidirect
18. #17988705	C₂²	semidirect
21. #17988417	C₂²	semidirect
22. #17988417	C₂²	semidirect
23. #17988395	C₂²	semidirect
27. #346×C₇₃	C₂²	semidirect
28. C₄×C₁₆×C₇₃	C₂²	semidirect
149. #2108	D₇₃	semidirect
150. C₇₃	#33465	semidirect
170. #346	C₂×D₇₃	semidirect
171. C₄×C₁₆	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989019	2
#22618849	2
#22611818	2
#22611813	2
C₂×#17988417	2
#22611660	2
C₇₃×#2108	2
#33465	73