Subgroups — #32985263

Order 18688.

442 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988992	C₂	1	1	yes	maximal, characteristic
2	#22610879	C₂	1	2	yes	maximal
3	C₂×#17988338	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988335	C₂	1	2	yes	maximal
5	C₇₃×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	2	yes	—
7	#17988992	—	2	2	no	—
8	#17988992	C₄	1	4	yes	complemented
9	C₂×#14556525	—	2	1	no	—
10	C₂×#14556525	—	2	1	no	—
11	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
12	#17988349	—	2	4	no	—
13	#17988338	—	2	2	no	—
14	#17988335	—	2	4	no	—
15	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₇₃	—	2	1	no	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
19	#336×C₇₃	—	2	1	no	—
20	C₂²×Dic₁₄₆	—	2	2	no	—
21	C₂²×#10596831	—	2	2	no	—
22	#14556591	—	2	2	no	—
23	#14556591	—	2	2	no	—
24	#14556591	—	2	2	no	—
25	#14556591	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₇₃	—	2	2	no	—
27	C₂×C₄⋊Dic₇₃	—	2	2	no	—
28	#14556525	—	2	4	no	—
29	#14556525	—	2	4	no	—
30	C₄⋊#10596831	—	2	2	no	—
31	C₄⋊#10596831	—	2	2	no	—
32	C₂²×C₈×C₇₃	—	2	1	no	—
33	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
36	C₂×C₄²×C₇₃	—	2	1	no	—
37	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₇₃	—	2	1	no	—
39	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
40	#97×C₇₃	—	4	2	no	—
41	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₇₃	D₄	1	2	yes	—
43	#95×C₇₃	—	2	1	no	—
44	C₂×Dic₁₄₆	—	2	4	no	—
45	C₂×Dic₁₄₆	—	4	2	no	—
46	C₂×Dic₁₄₆	—	8	4	no	—
47	C₂²×Dic₇₃	—	4	2	no	—
48	C₂×#10596831	—	4	2	no	—
49	C₂×#10596831	—	4	4	no	—
50	C₄⋊Dic₇₃	—	2	4	no	—
51	C₄⋊Dic₇₃	—	2	4	no	—
52	C₄⋊Dic₇₃	—	4	2	no	—
53	C₄⋊Dic₇₃	—	4	2	no	—
54	C₂²×C₄×C₇₃	—	2	2	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	2	no	—
61	C₂×C₈×C₇₃	—	4	1	no	—
62	C₂×C₈×C₇₃	—	4	1	no	—
63	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
66	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
67	C₄⋊C₄×C₇₃	—	2	4	no	—
68	C₄²×C₇₃	—	4	1	no	—
69	C₄²×C₇₃	—	4	1	no	—
70	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇₃	#41	1	1	yes	characteristic
74	C₂³×C₇₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	—	4	2	no	—
77	C₂×C₄×C₇₃	—	4	2	no	—
78	C₂×C₄×C₇₃	—	4	2	no	—
79	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#137	1	4	yes	—
90	C₂×Dic₇₃	—	4	4	no	—
91	C₂×Dic₇₃	—	8	4	no	—
92	Dic₁₄₆	—	8	4	no	—
93	Dic₁₄₆	—	8	4	no	—
94	C₈×C₇₃	—	2	2	no	—
95	C₈×C₇₃	—	4	2	no	—
96	C₈×C₇₃	#103	1	4	yes	—
97	#10596831	—	8	2	no	—
98	C₂²×C₇₃	#390	1	1	yes	characteristic
99	C₂²×C₇₃	#461	1	1	yes	characteristic
100	C₂²×C₇₃	#462	1	1	yes	characteristic
101	C₂²×C₇₃	#470	1	2	yes	—
102	C₂²×C₇₃	#472	1	2	yes	—
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	—	2	1	no	—
105	C₄×C₇₃	—	8	2	no	—
106	C₄×C₇₃	C₂×#103	1	1	yes	characteristic
107	C₄×C₇₃	#419	1	1	yes	characteristic
108	C₄×C₇₃	#441	1	2	yes	—
109	C₄×C₇₃	#461	1	1	yes	characteristic
110	C₄×C₇₃	#462	1	1	yes	characteristic
111	C₄×C₇₃	#497	1	2	yes	—
112	Dic₇₃	—	8	4	no	—
113	#9739	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	#1695	1	2	yes	—
115	C₂×C₇₃	#1696	1	2	yes	—
116	C₂×C₇₃	#1971	1	1	yes	characteristic
117	C₂×C₇₃	#1978	1	1	yes	characteristic
118	C₂×C₇₃	#1981	1	1	yes	characteristic
119	C₂×#500	—	73	1	no	—
120	#1987	—	73	2	no	—
121	#1770	D₇₃	1	1	yes	characteristic
122	C₂×#334	—	73	1	no	—
123	C₂×#333	—	73	2	no	—
124	C₇₃	#9739	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄⋊Q₈	—	73	2	no	—
126	#500	—	73	4	no	—
127	#500	—	146	2	no	—
128	C₂×#107	—	146	1	no	—
129	C₂×#107	—	146	1	no	—
130	C₂×C₄⋊C₈	—	73	2	no	—
131	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
132	C₂×#97	—	2	1	no	—
133	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
134	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
135	#340	—	146	4	no	—
136	#336	—	2	1	no	—
137	#334	—	146	2	no	—
138	#333	—	146	4	no	—
139	C₂²×Q₈	—	146	2	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	146	2	no	—
142	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
144	C₄⋊Q₈	—	73	4	no	—
145	C₄⋊Q₈	—	146	2	no	—
146	C₄⋊Q₈	—	146	2	no	—
147	C₄⋊Q₈	—	146	2	no	—
148	C₂×C₄⋊C₄	—	146	2	no	—
149	C₂×C₄⋊C₄	—	146	2	no	—
150	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
153	#107	—	146	4	no	—
154	#107	—	146	4	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	146	2	no	—
157	C₄⋊C₈	—	146	2	no	—
158	C₄⋊C₈	D₂₉₂	1	2	yes	—
159	#97	—	4	2	no	—
160	C₄×C₈	C₄×D₇₃	1	2	yes	—
161	C₄×C₈	D₂₉₂	1	2	yes	—
162	#95	—	2	1	no	—
163	C₂×Q₈	—	292	2	no	—
164	C₂×Q₈	—	584	4	no	—
165	C₂×Q₈	—	146	4	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	292	2	no	—
168	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
170	C₂²×C₄	#11924028	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	292	2	no	—
177	C₂×C₈	—	292	4	no	—
178	C₂×C₈	#11924005	1	2	yes	—
179	C₂×C₈	#11924005	1	2	yes	—
180	C₂×C₈	#11924016	1	2	yes	—
181	C₂×C₈	#11924018	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	292	2	no	—
184	C₄⋊C₄	—	292	2	no	—
185	C₄⋊C₄	—	146	4	no	—
186	C₄⋊C₄	—	146	4	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	4	1	no	—
189	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
190	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
191	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
192	C₄²	#11924028	1	1	yes	characteristic
193	C₂³	#14556565	1	1	yes	characteristic
194	Q₈	—	584	4	no	—
195	Q₈	—	584	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	292	4	no	—
201	C₂×C₄	—	584	4	no	—
202	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
203	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
204	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
205	C₂×C₄	#14556565	1	1	yes	characteristic
206	C₂×C₄	#14556579	1	1	yes	characteristic
207	C₂×C₄	#14556588	1	1	yes	characteristic
208	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
209	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
210	C₂×C₄	#14556621	1	1	yes	characteristic
211	C₂×C₄	#14556621	1	1	yes	characteristic
212	C₂×C₄	#14556624	1	4	yes	—
213	C₈	—	2	2	no	—
214	C₈	—	4	2	no	—
215	C₈	—	584	2	no	—
216	C₈	#14556514	1	4	yes	—
217	C₂²	#17988489	1	1	yes	characteristic
218	C₂²	#17988792	1	1	yes	characteristic
219	C₂²	#17988797	1	1	yes	characteristic
220	C₂²	#17988845	1	2	yes	—
221	C₂²	#17988857	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	2	no	—
225	C₄	—	584	4	no	—
226	C₄	C₂×#14556514	1	1	yes	characteristic
227	C₄	#17988595	1	1	yes	characteristic
228	C₄	#17988695	1	2	yes	—
229	C₄	#17988792	1	1	yes	characteristic
230	C₄	#17988797	1	1	yes	characteristic
231	C₄	#17988981	1	2	yes	—
232	C₂	#22609454	1	2	yes	—
233	C₂	#22609459	1	2	yes	—
234	C₂	#22610802	1	1	yes	characteristic
235	C₂	#22610841	1	1	yes	characteristic
236	C₂	#22610854	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556565
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1770	C₂
socle, prime radical	C₂³×C₇₃	#121
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1770	D₇₃
hypocenter	C₇₃	#9739
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
8. #17988992	C₄	semidirect
124. C₇₃	#9739	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988992	2
#22610879	2
C₂×#17988338	2
C₂×#17988335	2
C₇₃×#1770	2
#9739	73