Subgroups — #33239021

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556530	C₂	1	1	yes	maximal, characteristic
2	#427×D₇₃	C₂	1	2	yes	maximal
3	#22611718	C₂	1	4	yes	maximal, complemented
4	#22611716	C₂	1	4	yes	maximal, complemented
5	#22611715	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611713	C₂	1	2	yes	maximal, complemented
7	C₇₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	#117×D₇₃	—	2	2	no	—
10	C₂×#14556530	C₄	1	8	yes	complemented
11	C₂×#14556530	C₂²	1	2	yes	—
12	C₂×#14556530	C₂²	1	2	yes	—
13	C₂²×#11924003	C₂²	1	1	yes	characteristic
14	#17988625	C₂²	1	2	yes	—
15	#17988624	C₂²	1	4	yes	—
16	#17988623	—	2	2	no	—
17	#17988622	—	2	2	no	—
18	#107×D₇₃	C₂²	1	4	yes	—
19	#106×D₇₃	—	2	2	no	—
20	#17988415	C₂²	1	2	yes	complemented
21	#17988414	C₂²	1	4	yes	complemented
22	#17988413	C₂²	1	2	yes	complemented
23	#17988412	C₂²	1	4	yes	complemented
24	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
25	#427×C₇₃	C₂²	1	2	yes	—
26	#367×C₇₃	C₂²	1	2	yes	complemented
27	#366×C₇₃	C₂²	1	2	yes	complemented
28	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
29	C₂²×#10596831	D₄	1	1	yes	characteristic
30	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₇₃	D₄	1	2	yes	—
32	C₂×C₈×D₇₃	Q₈	1	4	yes	—
33	C₂×C₈×D₇₃	C₂³	1	2	yes	—
34	#14556550	—	2	2	no	—
35	#14556549	—	2	4	no	—
36	#14556548	—	4	2	no	—
37	#14556547	—	4	2	no	—
38	C₄⋊C₄×D₇₃	—	2	4	no	—
39	C₂²⋊C₄×D₇₃	—	4	2	no	—
40	C₄²×D₇₃	—	4	2	no	—
41	#14556530	C₂×C₄	1	8	yes	complemented
42	#14556530	C₂×C₄	1	8	yes	complemented
43	C₂×#11924003	C₂×C₄	1	4	yes	complemented
44	C₂×#11924003	C₂³	1	2	yes	—
45	#14556525	C₂³	1	4	yes	—
46	#14556524	—	2	4	no	—
47	#14556523	—	2	2	no	—
48	#14556522	—	2	4	no	—
49	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	#117×C₇₃	—	2	2	no	—
51	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
54	#107×C₇₃	C₂³	1	4	yes	—
55	#106×C₇₃	—	2	2	no	—
56	C₂³×D₇₃	D₈	1	1	yes	characteristic
57	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
58	C₂×C₄×D₇₃	—	4	4	no	—
59	C₂×C₄×D₇₃	C₄⋊C₄	1	4	yes	—
60	C₂×C₄×D₇₃	D₈	1	2	yes	—
61	C₂×C₄×D₇₃	Q₁₆	1	4	yes	—
62	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
63	C₂×#10596831	C₄⋊C₄	1	4	yes	—
64	C₂×#10596831	C₂×D₄	1	2	yes	—
65	C₈×D₇₃	C₄⋊C₄	1	8	yes	—
66	C₈×D₇₃	C₂²×C₄	1	4	yes	—
67	C₈×D₇₃	C₂×Q₈	1	4	yes	—
68	C₄⋊Dic₇₃	—	2	4	no	—
69	#11924007	—	4	4	no	—
70	C₂²⋊Dic₇₃	—	4	2	no	—
71	#11924005	—	4	4	no	—
72	C₄×Dic₇₃	—	4	2	no	—
73	C₄×Dic₇₃	—	4	4	no	—
74	#11924003	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₇₃	—	2	4	no	—
81	C₂²⋊C₄×C₇₃	—	4	2	no	—
82	C₄²×C₇₃	—	4	2	no	—
83	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
84	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
85	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
86	C₂²×D₇₃	—	4	2	no	—
87	C₂²×D₇₃	#107	1	4	yes	—
88	C₂²×D₇₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₇₃	—	4	4	no	—
90	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₇₃	—	4	4	no	—
95	C₂×Dic₇₃	#107	1	4	yes	—
96	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
97	C₄×D₇₃	—	8	8	no	—
98	C₄×D₇₃	#107	1	8	yes	—
99	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₄×D₇₃	C₂×Q₁₆	1	4	yes	—
101	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
104	#10596831	C₂×C₄⋊C₄	1	4	yes	—
105	C₂²×C₇₃	—	2	2	no	—
106	C₂²×C₇₃	C₂×#107	1	2	yes	—
107	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
108	C₂×D₇₃	—	2	4	no	—
109	C₂×D₇₃	—	4	4	no	—
110	C₂×D₇₃	C₂×#107	1	4	yes	—
111	C₄×C₇₃	—	8	4	no	—
112	C₄×C₇₃	C₂×#107	1	2	yes	—
113	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
115	Dic₇₃	—	8	4	no	—
116	Dic₇₃	C₂×#107	1	4	yes	—
117	#33409	—	73	1	no	maximal, Sylow
118	C₂×C₇₃	—	2	2	no	—
119	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
120	C₂×C₇₃	#3414	1	2	yes	—
121	D₇₃	—	2	4	no	—
122	C₂²×M₃₂	—	73	1	no	—
123	C₂×#427	—	73	2	no	—
124	#2159	—	73	4	no	—
125	#2159	—	73	4	no	—
126	#2158	—	73	1	no	—
127	#2158	—	73	2	no	—
128	#2158	D₇₃	1	1	yes	characteristic, complemented
129	C₇₃	#33409	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₈	—	73	1	no	—
131	C₂×#117	—	146	2	no	—
132	C₂×M₃₂	—	73	8	no	—
133	C₂×M₃₂	—	73	2	no	—
134	C₂×M₃₂	—	73	2	no	—
135	C₂²×C₁₆	—	73	1	no	—
136	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
137	#427	—	73	2	no	—
138	#427	—	73	4	no	—
139	#427	—	146	2	no	—
140	#427	—	146	2	no	—
141	#427	C₂×D₇₃	1	2	yes	—
142	C₂×#107	—	73	4	no	—
143	C₂×#106	—	146	2	no	—
144	#367	—	73	2	no	—
145	#367	—	73	4	no	—
146	#367	C₂×D₇₃	1	2	yes	complemented
147	#366	—	73	2	no	—
148	#366	—	73	4	no	—
149	#366	C₂×D₇₃	1	2	yes	complemented
150	C₂³×C₄	—	73	1	no	—
151	C₂²×C₈	—	73	1	no	—
152	C₂²×C₈	—	73	4	no	—
153	C₂²×C₈	—	73	2	no	—
154	C₂²×C₈	—	73	4	no	—
155	C₂²×C₈	—	73	2	no	—
156	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
157	#117	—	2	2	no	—
158	#117	—	292	2	no	—
159	#117	—	292	2	no	—
160	#117	—	146	4	no	—
161	#117	—	146	2	no	—
162	C₂×C₄⋊C₄	—	146	4	no	—
163	C₂×C₂²⋊C₄	—	292	2	no	—
164	C₂×C₄²	—	292	2	no	—
165	M₃₂	—	73	8	no	—
166	M₃₂	—	73	8	no	—
167	C₂×C₁₆	—	73	4	no	—
168	C₂×C₁₆	—	73	2	no	—
169	C₂×C₁₆	C₄×D₇₃	1	4	yes	complemented
170	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
171	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
172	#107	—	73	4	no	—
173	#107	—	146	4	no	—
174	#107	C₂²×D₇₃	1	4	yes	—
175	#106	—	2	2	no	—
176	#106	—	146	4	no	—
177	#106	—	146	2	no	—
178	C₂⁴	—	73	1	no	—
179	C₂²×C₄	—	292	4	no	—
180	C₂²×C₄	—	73	1	no	—
181	C₂²×C₄	—	73	4	no	—
182	C₂²×C₄	—	73	2	no	—
183	C₂²×C₄	—	73	4	no	—
184	C₂²×C₄	—	73	2	no	—
185	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
186	C₂×C₈	—	73	4	no	—
187	C₂×C₈	—	73	2	no	—
188	C₂×C₈	—	73	8	no	—
189	C₂×C₈	—	73	4	no	—
190	C₂×C₈	—	73	4	no	—
191	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
192	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
193	C₂×C₈	Q₈×D₇₃	1	2	yes	—
194	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	292	4	no	—
197	C₄⋊C₄	—	146	4	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	292	4	no	—
200	C₂²⋊C₄	—	292	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	292	4	no	—
203	C₄²	—	292	2	no	—
204	C₁₆	—	73	4	no	—
205	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
206	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
207	C₂³	—	292	2	no	—
208	C₂³	—	73	4	no	—
209	C₂³	—	73	2	no	—
210	C₂³	D₈×D₇₃	1	1	yes	characteristic
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	292	4	no	—
213	C₂×C₄	—	584	8	no	—
214	C₂×C₄	—	73	4	no	—
215	C₂×C₄	—	73	2	no	—
216	C₂×C₄	—	73	8	no	—
217	C₂×C₄	—	73	4	no	—
218	C₂×C₄	—	73	4	no	—
219	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
220	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
221	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
223	C₈	—	73	4	no	—
224	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
225	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
226	C₈	C₂×Q₈×D₇₃	1	1	yes	characteristic
227	C₂²	—	2	2	no	—
228	C₂²	—	292	4	no	—
229	C₂²	—	73	4	no	—
230	C₂²	—	146	4	no	—
231	C₂²	#107×D₇₃	1	2	yes	—
232	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
233	C₄	—	8	4	no	—
234	C₄	—	584	4	no	—
235	C₄	—	73	4	no	—
236	C₄	#107×D₇₃	1	2	yes	—
237	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
238	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	—	146	4	no	—
241	C₂	C₂×#107×D₇₃	1	1	yes	characteristic
242	C₂	#22623064	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2158	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂³×D₇₃	D₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2158	D₇₃
hypocenter	C₇₃	#33409
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #22611718	C₂	semidirect
4. #22611716	C₂	semidirect
5. #22611715	C₂	semidirect
6. #22611713	C₂	semidirect
7. C₇₃×#2158	C₂	semidirect
10. C₂×#14556530	C₄	semidirect
20. #17988415	C₂²	semidirect
21. #17988414	C₂²	semidirect
22. #17988413	C₂²	semidirect
23. #17988412	C₂²	semidirect
26. #367×C₇₃	C₂²	semidirect
27. #366×C₇₃	C₂²	semidirect
41. #14556530	C₂×C₄	semidirect
42. #14556530	C₂×C₄	semidirect
43. C₂×#11924003	C₂×C₄	semidirect
51. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
128. #2158	D₇₃	semidirect
129. C₇₃	#33409	semidirect
146. #367	C₂×D₇₃	semidirect
149. #366	C₂×D₇₃	semidirect
169. C₂×C₁₆	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556530	2
#427×D₇₃	2
#22611718	2
#22611716	2
#22611715	2
#22611713	2
C₇₃×#2158	2
#33409	73