Subgroups — #33239397

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988704	C₂	1	2	yes	maximal, complemented
2	#22611817	C₂	1	4	yes	maximal, complemented
3	#22611812	C₂	1	4	yes	maximal, complemented
4	#22611726	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611659	C₂	1	2	yes	maximal, complemented
6	M₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#3414	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556598	—	2	2	no	—
9	C₂×#14556531	C₂²	1	1	yes	characteristic
10	#17988704	—	2	2	no	—
11	#17988704	—	2	2	no	—
12	#17988704	C₂²	1	4	yes	complemented
13	#17988704	C₂²	1	4	yes	complemented
14	C₂×#14556527	C₂²	1	2	yes	—
15	C₂×#14556518	—	2	2	no	—
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	#17988417	C₂²	1	4	yes	complemented
18	#17988394	C₂²	1	8	yes	complemented
19	#17988362	C₂²	1	2	yes	complemented
20	C₁₆×Dic₇₃	C₂²	1	2	yes	complemented
21	C₂×#135×C₇₃	C₂²	1	2	yes	—
22	#510×C₇₃	C₄	1	4	yes	complemented
23	#509×C₇₃	C₄	1	4	yes	complemented
24	#508×C₇₃	C₂²	1	4	yes	—
25	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
26	#14556598	—	2	4	no	—
27	#14556598	—	2	4	no	—
28	#14556598	—	4	2	no	—
29	#14556598	—	4	2	no	—
30	C₂×#11924012	—	2	2	no	—
31	C₂×#11924012	—	4	2	no	—
32	C₂²×#10596831	—	4	2	no	—
33	C₂²×#10596831	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
35	#14556531	—	2	2	no	—
36	C₂×#11924003	C₂³	1	2	yes	—
37	#14556527	—	2	2	no	—
38	#14556527	C₂³	1	4	yes	—
39	#14556518	—	2	4	no	—
40	#14556518	—	2	4	no	—
41	C₈×Dic₇₃	D₄	1	1	yes	characteristic
42	C₈×Dic₇₃	D₄	1	1	yes	characteristic
43	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
44	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
45	C₂×#41×C₇₃	—	2	2	no	—
46	#135×C₇₃	—	2	2	no	—
47	#135×C₇₃	—	2	2	no	—
48	#135×C₇₃	C₂×C₄	1	4	yes	—
49	#135×C₇₃	C₂³	1	4	yes	—
50	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
51	C₂×SD₁₆×C₇₃	—	2	2	no	—
52	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	Q₃₂×C₇₃	C₂×C₄	1	4	yes	complemented
55	SD₃₂×C₇₃	C₂×C₄	1	8	yes	complemented
56	D₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
57	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
58	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
59	C₂²×Dic₇₃	—	2	1	no	—
60	#11924012	—	2	4	no	—
61	#11924012	—	4	2	no	—
62	#11924012	—	4	4	no	—
63	#11924012	—	4	4	no	—
64	C₂×#10596831	—	4	2	no	—
65	C₂×#10596831	—	4	4	no	—
66	C₂×#10596831	—	4	4	no	—
67	C₂×#10596831	D₈	1	2	yes	—
68	C₂×#10596831	D₈	1	2	yes	—
69	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
70	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
71	C₄×Dic₇₃	—	2	1	no	—
72	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
73	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
74	#11924003	—	2	2	no	—
75	#41×C₇₃	—	2	4	no	—
76	#41×C₇₃	—	2	4	no	—
77	#41×C₇₃	—	4	2	no	—
78	#41×C₇₃	—	4	2	no	—
79	C₂×Q₈×C₇₃	—	2	2	no	—
80	C₂×D₄×C₇₃	—	2	2	no	—
81	C₂×D₄×C₇₃	—	4	2	no	—
82	C₂²×C₄×C₇₃	—	4	2	no	—
83	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₇₃	—	2	2	no	—
85	Q₁₆×C₇₃	C₂²×C₄	1	4	yes	—
86	SD₁₆×C₇₃	—	2	4	no	—
87	SD₁₆×C₇₃	—	2	4	no	—
88	D₈×C₇₃	—	2	2	no	—
89	D₈×C₇₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₃	#41	1	2	yes	—
93	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
94	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
95	C₂³×C₇₃	—	4	2	no	—
96	C₂³×C₇₃	#135	1	1	yes	characteristic
97	Q₈×C₇₃	—	2	4	no	—
98	Q₈×C₇₃	—	4	2	no	—
99	D₄×C₇₃	—	2	4	no	—
100	D₄×C₇₃	—	4	4	no	—
101	D₄×C₇₃	—	4	4	no	—
102	D₄×C₇₃	—	4	2	no	—
103	C₂×C₄×C₇₃	—	4	2	no	—
104	C₂×C₄×C₇₃	—	4	4	no	—
105	C₂×C₄×C₇₃	—	4	4	no	—
106	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₇₃	—	2	1	no	—
111	C₂×Dic₇₃	—	2	1	no	—
112	C₂×Dic₇₃	—	4	2	no	—
113	C₈×C₇₃	C₄×D₄	1	2	yes	—
114	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
115	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
116	#10596831	—	4	4	no	—
117	#10596831	C₂×D₈	1	1	yes	characteristic
118	#10596831	C₂×D₈	1	2	yes	—
119	#10596831	C₂×D₈	1	1	yes	characteristic
120	C₂²×C₇₃	—	4	1	no	—
121	C₂²×C₇₃	—	4	4	no	—
122	C₂²×C₇₃	—	8	4	no	—
123	C₂²×C₇₃	C₄×D₈	1	2	yes	—
124	C₂²×C₇₃	C₂×#135	1	1	yes	characteristic
125	C₄×C₇₃	—	4	4	no	—
126	C₄×C₇₃	C₄×D₈	1	2	yes	—
127	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
129	Dic₇₃	—	2	2	no	—
130	Dic₇₃	—	4	1	no	—
131	#33463	—	73	1	no	maximal, Sylow
132	C₂×C₇₃	—	2	1	no	—
133	C₂×C₇₃	—	2	2	no	—
134	C₂×C₇₃	—	8	4	no	—
135	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
136	#3414	D₇₃	1	1	yes	characteristic
137	C₂×#443	—	73	2	no	—
138	#2179	—	73	4	no	—
139	#2178	—	73	4	no	—
140	#2161	—	73	1	no	—
141	#2143	—	73	2	no	—
142	C₄×M₃₂	—	73	1	no	—
143	C₇₃	#33463	1	1	yes	characteristic, complemented, Sylow
144	C₂×#135	C₂×D₇₃	1	2	yes	—
145	C₂×#131	—	146	2	no	—
146	#510	Dic₇₃	1	4	yes	complemented
147	#509	Dic₇₃	1	4	yes	complemented
148	#508	C₂×D₇₃	1	4	yes	—
149	C₂×M₃₂	—	73	1	no	—
150	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
151	#443	—	73	4	no	—
152	#443	—	73	4	no	—
153	#443	—	146	2	no	—
154	#443	—	146	2	no	—
155	C₂×#108	—	73	2	no	—
156	C₂×#104	—	146	2	no	—
157	C₂×C₄×C₈	—	73	1	no	—
158	#368	—	73	4	no	—
159	#359	—	73	8	no	—
160	#346	—	73	2	no	—
161	C₄×C₁₆	—	73	2	no	—
162	C₂×#41	—	2	2	no	—
163	#135	—	2	2	no	—
164	#135	—	2	2	no	—
165	#135	C₂×Dic₇₃	1	4	yes	—
166	#135	C₂²×D₇₃	1	4	yes	—
167	C₂×Q₁₆	C₂×Dic₇₃	1	2	yes	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×D₈	C₂×Dic₇₃	1	2	yes	—
170	#131	—	292	2	no	—
171	#131	—	292	2	no	—
172	#131	—	146	4	no	—
173	#131	—	146	4	no	—
174	C₂×M₁₆	—	292	2	no	—
175	C₂×M₁₆	—	146	2	no	—
176	C₂²×C₈	—	292	2	no	—
177	C₂²×C₈	—	73	1	no	—
178	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
179	C₂×C₄²	—	73	1	no	—
180	Q₃₂	C₂×Dic₇₃	1	4	yes	complemented
181	SD₃₂	C₂×Dic₇₃	1	8	yes	complemented
182	D₁₆	C₂×Dic₇₃	1	4	yes	complemented
183	M₃₂	—	146	2	no	—
184	M₃₂	C₂×Dic₇₃	1	4	yes	—
185	C₂×C₁₆	—	73	2	no	—
186	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
187	#108	—	73	4	no	—
188	#108	—	146	2	no	—
189	#104	—	146	4	no	—
190	#104	—	146	4	no	—
191	C₄×C₈	—	73	1	no	—
192	C₄×C₈	—	73	1	no	—
193	C₄×C₈	—	73	1	no	—
194	C₄×C₈	—	73	1	no	—
195	#41	—	2	4	no	—
196	#41	—	2	4	no	—
197	#41	—	4	2	no	—
198	#41	—	4	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	146	1	no	—
204	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
205	Q₁₆	—	2	2	no	—
206	Q₁₆	C₂²×Dic₇₃	1	4	yes	—
207	SD₁₆	—	2	4	no	—
208	SD₁₆	—	2	4	no	—
209	D₈	—	2	2	no	—
210	D₈	C₂²×Dic₇₃	1	4	yes	—
211	M₁₆	—	292	2	no	—
212	M₁₆	—	292	4	no	—
213	M₁₆	—	292	4	no	—
214	M₁₆	—	146	4	no	—
215	C₂×C₈	—	292	2	no	—
216	C₂×C₈	—	292	4	no	—
217	C₂×C₈	—	292	4	no	—
218	C₂×C₈	—	73	2	no	—
219	C₂×C₈	—	73	2	no	—
220	C₂×C₈	—	73	1	no	—
221	C₂×C₈	—	73	1	no	—
222	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
223	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
224	C₂×C₈	#11924027	1	2	yes	—
225	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
226	C₄²	—	73	1	no	—
227	C₄²	—	73	1	no	—
228	C₄²	—	146	1	no	—
229	C₁₆	—	146	2	no	—
230	C₁₆	C₂²×Dic₇₃	1	4	yes	—
231	C₂³	—	4	2	no	—
232	C₂³	#14556610	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	4	2	no	—
235	D₄	—	2	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	292	2	no	—
243	C₂×C₄	—	146	1	no	—
244	C₂×C₄	—	146	1	no	—
245	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
246	C₂×C₄	D₈×D₇₃	1	2	yes	—
247	C₂×C₄	#14556610	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
249	C₈	—	292	4	no	—
250	C₈	—	73	1	no	—
251	C₈	—	73	2	no	—
252	C₈	—	73	1	no	—
253	C₈	D₄×Dic₇₃	1	2	yes	—
254	C₈	C₂³×Dic₇₃	1	1	yes	characteristic
255	C₈	C₂×#11924027	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	8	4	no	—
259	C₂²	D₈×Dic₇₃	1	2	yes	—
260	C₂²	C₂×#14556610	1	1	yes	characteristic
261	C₄	—	4	4	no	—
262	C₄	—	292	1	no	—
263	C₄	—	146	2	no	—
264	C₄	D₈×Dic₇₃	1	2	yes	—
265	C₄	C₂×D₄×Dic₇₃	1	1	yes	characteristic
266	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₂×D₈×Dic₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting, prime radical	C₇₃×#3414	C₂
socle	C₂×C₇₃	C₂×C₄×D₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#3414	D₇₃
hypocenter	C₇₃	#33463
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #3414

Complements

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

N	H ≅ G/N	type
1. C₂×#17988704	C₂	semidirect
2. #22611817	C₂	semidirect
3. #22611812	C₂	semidirect
4. #22611726	C₂	semidirect
5. #22611659	C₂	semidirect
6. M₃₂×Dic₇₃	C₂	semidirect
12. #17988704	C₂²	semidirect
13. #17988704	C₂²	semidirect
17. #17988417	C₂²	semidirect
18. #17988394	C₂²	semidirect
19. #17988362	C₂²	semidirect
20. C₁₆×Dic₇₃	C₂²	semidirect
22. #510×C₇₃	C₄	semidirect
23. #509×C₇₃	C₄	semidirect
54. Q₃₂×C₇₃	C₂×C₄	semidirect
55. SD₃₂×C₇₃	C₂×C₄	semidirect
56. D₁₆×C₇₃	C₂×C₄	semidirect
143. C₇₃	#33463	semidirect
146. #510	Dic₇₃	semidirect
147. #509	Dic₇₃	semidirect
180. Q₃₂	C₂×Dic₇₃	semidirect
181. SD₃₂	C₂×Dic₇₃	semidirect
182. D₁₆	C₂×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988704	2
#22611817	2
#22611812	2
#22611726	2
#22611659	2
M₃₂×Dic₇₃	2
C₇₃×#3414	2
#33463	73