Subgroups — #32984655

Order 18688.

370 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610695	C₂	1	2	yes	maximal
2	#22610088	C₂	1	1	yes	maximal, characteristic
3	#22609798	C₂	1	1	yes	maximal, characteristic
4	#22607815	C₂	1	2	yes	maximal
5	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556523	—	2	2	no	—
7	C₂×#14556523	—	2	1	no	—
8	C₂×#14556523	—	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	#335×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	2	yes	—
23	C₂×C₄×Dic₇₃	—	2	1	no	—
24	#14556523	—	4	2	no	—
25	#14556523	—	4	4	no	—
26	#14556523	—	4	2	no	—
27	C₄⋊#10596831	—	2	1	no	—
28	C₄⋊#10596831	Q₈	1	2	yes	—
29	#14556501	—	4	2	no	—
30	#14556498	—	2	1	no	—
31	C₂²×C₈×C₇₃	—	2	2	no	—
32	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₃	—	2	2	no	—
36	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₃	Q₈	1	2	yes	—
39	C₂²×Dic₇₃	—	2	2	no	—
40	C₂²×Dic₇₃	—	4	2	no	—
41	C₂×#10596831	—	2	2	no	—
42	C₂×#10596831	—	4	1	no	—
43	C₂×#10596831	—	4	1	no	—
44	C₄⋊Dic₇₃	—	2	2	no	—
45	C₄⋊Dic₇₃	—	2	2	no	—
46	C₄⋊Dic₇₃	—	4	2	no	—
47	C₄⋊Dic₇₃	—	4	2	no	—
48	C₄⋊Dic₇₃	—	4	2	no	—
49	C₄⋊Dic₇₃	—	4	2	no	—
50	C₄⋊Dic₇₃	—	8	2	no	—
51	C₄×Dic₇₃	—	4	1	no	—
52	C₄×Dic₇₃	—	4	1	no	—
53	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	2	4	no	—
59	C₂×C₈×C₇₃	—	4	2	no	—
60	C₂×C₈×C₇₃	—	4	2	no	—
61	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
64	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
65	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₇₃	#41	1	1	yes	characteristic
69	C₂³×C₇₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#125	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#137	1	2	yes	—
82	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
83	C₂×Dic₇₃	—	4	2	no	—
84	C₂×Dic₇₃	—	4	2	no	—
85	C₂×Dic₇₃	—	4	2	no	—
86	C₂×Dic₇₃	—	8	2	no	—
87	C₂×Dic₇₃	—	8	2	no	—
88	C₂×Dic₇₃	—	8	2	no	—
89	C₈×C₇₃	—	2	2	no	—
90	C₈×C₇₃	—	4	4	no	—
91	C₈×C₇₃	#106	1	4	yes	complemented
92	#10596831	—	4	2	no	—
93	C₂²×C₇₃	#389	1	1	yes	characteristic
94	C₂²×C₇₃	#460	1	1	yes	characteristic
95	C₂²×C₇₃	#461	1	1	yes	characteristic
96	C₂²×C₇₃	#468	1	2	yes	—
97	C₂²×C₇₃	#471	1	2	yes	—
98	C₄×C₇₃	—	2	1	no	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
101	C₄×C₇₃	#428	1	1	yes	characteristic
102	C₄×C₇₃	#441	1	1	yes	characteristic
103	C₄×C₇₃	#442	1	1	yes	characteristic
104	C₄×C₇₃	#475	1	1	yes	characteristic
105	C₄×C₇₃	#476	1	1	yes	characteristic
106	C₄×C₇₃	#501	1	2	yes	—
107	Dic₇₃	—	8	2	no	—
108	Dic₇₃	—	16	2	no	—
109	#9618	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	#1686	1	2	yes	—
111	C₂×C₇₃	#1691	1	2	yes	—
112	C₂×C₇₃	#1943	1	1	yes	characteristic
113	C₂×C₇₃	#1948	1	1	yes	characteristic
114	C₂×C₇₃	#1951	1	1	yes	characteristic
115	#1948	—	73	2	no	—
116	#1844	—	73	1	no	—
117	#1770	—	73	1	no	—
118	C₂×#335	D₇₃	1	1	yes	characteristic
119	#1296	—	73	2	no	—
120	C₇₃	#9618	1	1	yes	characteristic, complemented, Sylow
121	C₂×#106	—	146	2	no	—
122	C₂×#106	—	146	1	no	—
123	C₂×#106	—	146	1	no	—
124	C₂×C₄⋊C₈	—	73	1	no	—
125	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
126	C₂×#97	—	146	1	no	—
127	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
128	#384	—	73	2	no	—
129	C₄×C₄⋊C₄	—	73	1	no	—
130	#340	—	146	2	no	—
131	#336	—	146	1	no	—
132	#335	Dic₇₃	1	4	yes	complemented
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	146	1	no	—
135	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
136	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	73	2	no	—
138	C₂×C₄⋊C₄	—	146	2	no	—
139	C₂×C₄⋊C₄	—	146	2	no	—
140	C₂×C₄⋊C₄	—	146	2	no	—
141	C₂×C₄²	—	146	1	no	—
142	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
143	#106	—	292	2	no	—
144	#106	—	292	2	no	—
145	#106	—	292	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	73	2	no	—
148	C₄⋊C₈	—	146	1	no	—
149	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
150	#97	—	292	2	no	—
151	C₄×C₈	Dic₁₄₆	1	2	yes	—
152	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
153	#95	—	146	1	no	—
154	C₂²×C₄	—	292	2	no	—
155	C₂²×C₄	—	146	2	no	—
156	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
158	C₂²×C₄	#11924029	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	292	1	no	—
165	C₂×C₈	—	292	1	no	—
166	C₂×C₈	—	146	2	no	—
167	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
168	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
169	C₂×C₈	#11924017	1	2	yes	—
170	C₂×C₈	#11924017	1	2	yes	—
171	C₄⋊C₄	—	292	2	no	—
172	C₄⋊C₄	—	292	2	no	—
173	C₄⋊C₄	—	292	2	no	—
174	C₄⋊C₄	—	292	2	no	—
175	C₄⋊C₄	—	584	2	no	—
176	C₄⋊C₄	—	146	2	no	—
177	C₄⋊C₄	—	146	2	no	—
178	C₄²	—	292	1	no	—
179	C₄²	—	292	1	no	—
180	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
181	C₄²	Q₈×D₇₃	1	1	yes	characteristic
182	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
183	C₄²	#11924027	1	1	yes	characteristic
184	C₂³	#14556566	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	292	2	no	—
187	C₂×C₄	—	292	2	no	—
188	C₂×C₄	—	292	2	no	—
189	C₂×C₄	—	584	2	no	—
190	C₂×C₄	—	584	2	no	—
191	C₂×C₄	—	584	2	no	—
192	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
193	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
194	C₂×C₄	Q₈×Dic₇₃	1	1	yes	characteristic
195	C₂×C₄	#14556571	1	1	yes	characteristic
196	C₂×C₄	#14556582	1	1	yes	characteristic
197	C₂×C₄	#14556590	1	1	yes	characteristic
198	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
199	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
200	C₂×C₄	#14556615	1	2	yes	—
201	C₂×C₄	#14556619	1	1	yes	characteristic
202	C₂×C₄	#14556623	1	2	yes	—
203	C₂×C₄	#14556624	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	4	no	—
206	C₈	—	292	2	no	—
207	C₈	#14556523	1	4	yes	complemented
208	C₂²	#17988486	1	1	yes	characteristic
209	C₂²	#17988788	1	1	yes	characteristic
210	C₂²	#17988793	1	1	yes	characteristic
211	C₂²	#17988834	1	2	yes	—
212	C₂²	#17988851	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	584	2	no	—
216	C₄	—	1168	2	no	—
217	C₄	C₂×#14556523	1	1	yes	characteristic
218	C₄	#17988629	1	1	yes	characteristic
219	C₄	#17988694	1	1	yes	characteristic
220	C₄	#17988699	1	1	yes	characteristic
221	C₄	#17988873	1	1	yes	characteristic
222	C₄	#17988878	1	1	yes	characteristic
223	C₄	#17988995	1	2	yes	—
224	C₂	#22609404	1	2	yes	—
225	C₂	#22609431	1	2	yes	—
226	C₂	#22610670	1	1	yes	characteristic
227	C₂	#22610695	1	1	yes	characteristic
228	C₂	#22610712	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556566
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#335×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#121
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	#9618
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. #335×C₇₃	C₄	semidirect
91. C₈×C₇₃	#106	semidirect
120. C₇₃	#9618	semidirect
132. #335	Dic₇₃	semidirect
207. C₈	#14556523	semidirect

Maximal subgroups

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

subgroup	index
#22610695	2
#22610088	2
#22609798	2
#22607815	2
C₂×#335×C₇₃	2
#9618	73