Subgroups — #32984667

Order 18688.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610690	C₂	1	2	yes	maximal
2	#22610091	C₂	1	1	yes	maximal, characteristic
3	#22609798	C₂	1	1	yes	maximal, characteristic
4	#22607815	C₂	1	2	yes	maximal
5	C₂×#334×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556525	—	2	2	no	—
7	C₂×#14556525	—	2	1	no	—
8	C₂×#14556525	—	2	1	no	—
9	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
10	C₂×#14556501	—	2	1	no	—
11	#17988457	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
13	#17988350	—	2	2	no	—
14	#17988341	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#334×C₇₃	C₄	1	4	yes	complemented
18	C₂²×#10596831	—	2	1	no	—
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×C₄⋊Dic₇₃	—	2	2	no	—
21	C₂×C₄⋊Dic₇₃	—	2	2	no	—
22	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₃	—	2	1	no	—
25	#14556525	—	4	2	no	—
26	#14556525	—	4	4	no	—
27	#14556525	—	4	2	no	—
28	C₄⋊#10596831	—	2	1	no	—
29	C₄⋊#10596831	Q₈	1	2	yes	—
30	#14556501	—	4	2	no	—
31	#14556498	—	2	1	no	—
32	C₂²×C₈×C₇₃	—	2	2	no	—
33	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₇₃	—	2	2	no	—
37	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₃	Q₈	1	2	yes	—
40	C₂²×Dic₇₃	—	2	1	no	—
41	C₂²×Dic₇₃	—	2	1	no	—
42	C₂²×Dic₇₃	—	4	2	no	—
43	C₂×#10596831	—	2	2	no	—
44	C₂×#10596831	—	4	1	no	—
45	C₂×#10596831	—	4	1	no	—
46	C₄⋊Dic₇₃	—	2	2	no	—
47	C₄⋊Dic₇₃	—	2	2	no	—
48	C₄⋊Dic₇₃	—	4	2	no	—
49	C₄⋊Dic₇₃	—	4	2	no	—
50	C₄⋊Dic₇₃	—	4	2	no	—
51	C₄⋊Dic₇₃	—	4	2	no	—
52	C₄⋊Dic₇₃	—	8	2	no	—
53	C₄×Dic₇₃	—	4	1	no	—
54	C₄×Dic₇₃	—	4	1	no	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₇₃	—	2	1	no	—
59	C₂×C₈×C₇₃	—	2	1	no	—
60	C₂×C₈×C₇₃	—	2	4	no	—
61	C₂×C₈×C₇₃	—	4	2	no	—
62	C₂×C₈×C₇₃	—	4	2	no	—
63	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₇₃	D₈	1	2	yes	—
66	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
67	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₇₃	#41	1	1	yes	characteristic
71	C₂³×C₇₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	1	no	—
74	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#125	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#137	1	2	yes	—
85	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
86	C₂×Dic₇₃	—	4	1	no	—
87	C₂×Dic₇₃	—	4	1	no	—
88	C₂×Dic₇₃	—	4	1	no	—
89	C₂×Dic₇₃	—	4	1	no	—
90	C₂×Dic₇₃	—	4	1	no	—
91	C₂×Dic₇₃	—	4	1	no	—
92	C₂×Dic₇₃	—	8	2	no	—
93	C₂×Dic₇₃	—	8	2	no	—
94	C₂×Dic₇₃	—	8	2	no	—
95	C₈×C₇₃	—	2	2	no	—
96	C₈×C₇₃	—	4	4	no	—
97	C₈×C₇₃	#107	1	4	yes	complemented
98	#10596831	—	4	2	no	—
99	C₂²×C₇₃	#389	1	1	yes	characteristic
100	C₂²×C₇₃	#459	1	1	yes	characteristic
101	C₂²×C₇₃	#462	1	1	yes	characteristic
102	C₂²×C₇₃	#469	1	1	yes	characteristic
103	C₂²×C₇₃	#470	1	1	yes	characteristic
104	C₂²×C₇₃	#472	1	1	yes	characteristic
105	C₂²×C₇₃	#473	1	1	yes	characteristic
106	C₄×C₇₃	—	2	1	no	—
107	C₄×C₇₃	—	2	1	no	—
108	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
109	C₄×C₇₃	#428	1	1	yes	characteristic
110	C₄×C₇₃	#440	1	2	yes	—
111	C₄×C₇₃	#474	1	1	yes	characteristic
112	C₄×C₇₃	#477	1	1	yes	characteristic
113	C₄×C₇₃	#501	1	2	yes	—
114	Dic₇₃	—	8	1	no	—
115	Dic₇₃	—	8	1	no	—
116	Dic₇₃	—	16	2	no	—
117	#9620	—	73	1	no	maximal, Sylow
118	C₂×C₇₃	#1687	1	1	yes	characteristic
119	C₂×C₇₃	#1689	1	1	yes	characteristic
120	C₂×C₇₃	#1694	1	1	yes	characteristic
121	C₂×C₇₃	#1696	1	1	yes	characteristic
122	C₂×C₇₃	#1944	1	1	yes	characteristic
123	C₂×C₇₃	#1947	1	1	yes	characteristic
124	C₂×C₇₃	#1952	1	1	yes	characteristic
125	#1947	—	73	2	no	—
126	#1845	—	73	1	no	—
127	#1770	—	73	1	no	—
128	C₂×#334	D₇₃	1	1	yes	characteristic
129	#1296	—	73	2	no	—
130	C₇₃	#9620	1	1	yes	characteristic, complemented, Sylow
131	C₂×#107	—	146	2	no	—
132	C₂×#107	—	146	1	no	—
133	C₂×#107	—	146	1	no	—
134	C₂×C₄⋊C₈	—	73	1	no	—
135	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
136	C₂×#97	—	146	1	no	—
137	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
138	#384	—	73	2	no	—
139	C₄×C₄⋊C₄	—	73	1	no	—
140	#340	—	146	2	no	—
141	#336	—	146	1	no	—
142	#334	Dic₇₃	1	4	yes	complemented
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	146	1	no	—
145	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
146	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	73	1	no	—
148	C₂×C₄⋊C₄	—	73	1	no	—
149	C₂×C₄⋊C₄	—	146	2	no	—
150	C₂×C₄⋊C₄	—	146	2	no	—
151	C₂×C₄⋊C₄	—	146	2	no	—
152	C₂×C₄²	—	146	1	no	—
153	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
154	#107	—	292	2	no	—
155	#107	—	292	2	no	—
156	#107	—	292	4	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	73	2	no	—
159	C₄⋊C₈	—	146	1	no	—
160	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
161	#97	—	292	2	no	—
162	C₄×C₈	Dic₁₄₆	1	2	yes	—
163	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
164	#95	—	146	1	no	—
165	C₂²×C₄	—	292	2	no	—
166	C₂²×C₄	—	146	1	no	—
167	C₂²×C₄	—	146	1	no	—
168	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
170	C₂²×C₄	#11924029	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	292	1	no	—
177	C₂×C₈	—	292	1	no	—
178	C₂×C₈	—	146	2	no	—
179	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
180	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
181	C₂×C₈	D₅₈₄	1	2	yes	—
182	C₂×C₈	Dic₂₉₂	1	2	yes	—
183	C₄⋊C₄	—	292	2	no	—
184	C₄⋊C₄	—	292	2	no	—
185	C₄⋊C₄	—	292	2	no	—
186	C₄⋊C₄	—	292	2	no	—
187	C₄⋊C₄	—	584	2	no	—
188	C₄⋊C₄	—	146	2	no	—
189	C₄⋊C₄	—	146	2	no	—
190	C₄²	—	292	1	no	—
191	C₄²	—	292	1	no	—
192	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
193	C₄²	Q₈×D₇₃	1	1	yes	characteristic
194	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
195	C₄²	#11924027	1	1	yes	characteristic
196	C₂³	#14556566	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	292	1	no	—
200	C₂×C₄	—	292	1	no	—
201	C₂×C₄	—	292	1	no	—
202	C₂×C₄	—	292	1	no	—
203	C₂×C₄	—	292	1	no	—
204	C₂×C₄	—	292	1	no	—
205	C₂×C₄	—	584	2	no	—
206	C₂×C₄	—	584	2	no	—
207	C₂×C₄	—	584	2	no	—
208	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
209	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
210	C₂×C₄	Q₈×Dic₇₃	1	1	yes	characteristic
211	C₂×C₄	#14556571	1	1	yes	characteristic
212	C₂×C₄	#14556582	1	1	yes	characteristic
213	C₂×C₄	#14556590	1	1	yes	characteristic
214	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
215	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
216	C₂×C₄	#14556618	1	2	yes	—
217	C₂×C₄	#14556619	1	1	yes	characteristic
218	C₂×C₄	#14556620	1	2	yes	—
219	C₂×C₄	#14556624	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	292	2	no	—
223	C₈	#14556525	1	4	yes	complemented
224	C₂²	#17988486	1	1	yes	characteristic
225	C₂²	#17988783	1	1	yes	characteristic
226	C₂²	#17988798	1	1	yes	characteristic
227	C₂²	#17988841	1	1	yes	characteristic
228	C₂²	#17988846	1	1	yes	characteristic
229	C₂²	#17988858	1	1	yes	characteristic
230	C₂²	#17988863	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	584	1	no	—
234	C₄	—	584	1	no	—
235	C₄	—	1168	2	no	—
236	C₄	C₂×#14556525	1	1	yes	characteristic
237	C₄	#17988629	1	1	yes	characteristic
238	C₄	#17988688	1	2	yes	—
239	C₄	#17988868	1	1	yes	characteristic
240	C₄	#17988883	1	1	yes	characteristic
241	C₄	#17988994	1	2	yes	—
242	C₂	#22609411	1	1	yes	characteristic
243	C₂	#22609421	1	1	yes	characteristic
244	C₂	#22609450	1	1	yes	characteristic
245	C₂	#22609460	1	1	yes	characteristic
246	C₂	#22610675	1	1	yes	characteristic
247	C₂	#22610690	1	1	yes	characteristic
248	C₂	#22610717	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556566
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#334×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#121
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#334	D₇₃
hypocenter	C₇₃	#9620
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
17. #334×C₇₃	C₄	semidirect
97. C₈×C₇₃	#107	semidirect
130. C₇₃	#9620	semidirect
142. #334	Dic₇₃	semidirect
223. C₈	#14556525	semidirect

Maximal subgroups

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

subgroup	index
#22610690	2
#22610091	2
#22609798	2
#22607815	2
C₂×#334×C₇₃	2
#9620	73