Subgroups — #32977162

Order 18688.

342 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22609676	C₂	1	1	yes	maximal, characteristic
2	#22608677	C₂	1	1	yes	maximal, characteristic
3	#22608677	C₂	1	1	yes	maximal, characteristic
4	#22608214	C₂	1	1	yes	maximal, characteristic
5	#22607763	C₂	1	1	yes	maximal, characteristic
6	#22607763	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#1452	C₂	1	1	yes	maximal, characteristic
8	#17988640	C₂²	1	1	yes	characteristic
9	#17988640	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
11	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
12	#17988341	C₄	1	2	yes	—
13	#17988341	C₄	1	2	yes	—
14	#17988339	C₄	1	2	yes	complemented
15	#17988339	C₄	1	2	yes	complemented
16	#17988337	C₄	1	2	yes	complemented
17	#17988337	C₄	1	2	yes	complemented
18	#17988308	—	2	1	no	—
19	C₈×#10596831	—	2	1	no	—
20	#431×C₇₃	C₂²	1	1	yes	characteristic
21	#431×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	#322×C₇₃	—	2	1	no	—
24	#322×C₇₃	—	2	1	no	—
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₇₃	—	2	1	no	—
28	C₂×C₄×Dic₇₃	—	2	1	no	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₄⋊#10596831	C₂×C₄	1	2	yes	—
31	C₄⋊#10596831	C₂×C₄	1	2	yes	—
32	C₂²⋊#10596831	—	2	1	no	—
33	C₂²⋊#10596831	—	2	1	no	—
34	C₄×#10596831	—	2	1	no	—
35	C₄×#10596831	—	2	1	no	—
36	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
37	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
38	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
39	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
44	C₂²⋊C₈×C₇₃	—	2	1	no	—
45	C₂²⋊C₈×C₇₃	—	2	1	no	—
46	C₄×C₈×C₇₃	—	2	1	no	—
47	C₄×C₈×C₇₃	—	2	1	no	—
48	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₃	Q₈	1	2	yes	—
51	C₂²×Dic₇₃	—	2	2	no	—
52	C₂×#10596831	—	2	2	no	—
53	C₂×#10596831	—	2	2	no	—
54	C₂×#10596831	—	2	2	no	—
55	C₂×#10596831	—	2	2	no	—
56	C₂×#10596831	C₄²	1	1	yes	characteristic
57	C₂×#10596831	C₄²	1	1	yes	characteristic
58	C₂×#10596831	C₄²	1	1	yes	characteristic
59	C₂×#10596831	C₄²	1	1	yes	characteristic
60	C₄×Dic₇₃	—	2	4	no	—
61	C₄×Dic₇₃	—	2	2	no	—
62	C₄×Dic₇₃	—	2	2	no	—
63	C₄×Dic₇₃	—	4	2	no	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
67	C₂×C₈×C₇₃	—	2	2	no	—
68	C₂×C₈×C₇₃	—	2	2	no	—
69	C₂×C₈×C₇₃	—	2	2	no	—
70	C₂×C₈×C₇₃	—	2	2	no	—
71	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
77	C₂³×C₇₃	#117	1	1	yes	characteristic
78	C₂×C₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	—	2	2	no	—
80	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#131	1	2	yes	—
87	C₂×C₄×C₇₃	#131	1	2	yes	—
88	C₂×Dic₇₃	—	2	4	no	—
89	C₂×Dic₇₃	—	4	2	no	—
90	C₂×Dic₇₃	—	4	2	no	—
91	C₈×C₇₃	—	2	2	no	—
92	C₈×C₇₃	—	2	2	no	—
93	C₈×C₇₃	—	4	2	no	—
94	C₈×C₇₃	—	4	2	no	—
95	#10596831	—	2	1	no	—
96	#10596831	—	2	1	no	—
97	#10596831	—	2	1	no	—
98	#10596831	—	2	1	no	—
99	#10596831	—	4	2	no	—
100	#10596831	—	4	2	no	—
101	C₂²×C₇₃	—	2	1	no	—
102	C₂²×C₇₃	—	2	1	no	—
103	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₇₃	#433	1	2	yes	—
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	—	2	1	no	—
107	C₄×C₇₃	#405	1	1	yes	characteristic
108	C₄×C₇₃	#405	1	1	yes	characteristic
109	C₄×C₇₃	#424	1	2	yes	—
110	C₄×C₇₃	#443	1	2	yes	—
111	C₄×C₇₃	#444	1	2	yes	—
112	Dic₇₃	—	4	4	no	—
113	#8270	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	—	2	2	no	—
115	C₂×C₇₃	#1768	1	1	yes	characteristic
116	C₂×C₇₃	#1772	1	1	yes	characteristic
117	C₂×C₇₃	#1773	1	1	yes	characteristic
118	#1744	—	73	1	no	—
119	#1564	—	73	1	no	—
120	#1564	—	73	1	no	—
121	#1452	D₇₃	1	1	yes	characteristic
122	#1450	—	73	1	no	—
123	#1276	—	73	1	no	—
124	#1276	—	73	1	no	—
125	C₇₃	#8270	1	1	yes	characteristic, complemented, Sylow
126	#431	—	73	1	no	—
127	#431	—	73	1	no	—
128	#431	C₂×D₇₃	1	1	yes	characteristic
129	#431	C₂×D₇₃	1	1	yes	characteristic
130	C₂×C₄×C₈	—	73	1	no	—
131	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
132	C₄³	—	73	1	no	—
133	#336	—	73	2	no	—
134	#336	—	73	2	no	—
135	#335	—	73	2	no	—
136	#335	—	73	2	no	—
137	#334	—	73	2	no	—
138	#334	—	73	2	no	—
139	#322	—	2	1	no	—
140	#322	—	2	1	no	—
141	#322	—	146	1	no	—
142	C₈²	—	146	1	no	—
143	C₂²×C₈	—	73	1	no	—
144	C₂²×C₈	—	73	1	no	—
145	C₂²×C₈	C₄×D₇₃	1	2	yes	—
146	C₂×C₄²	—	73	2	no	—
147	C₂×C₄²	—	146	1	no	—
148	C₂×C₄²	—	146	1	no	—
149	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
150	C₄⋊C₈	—	73	2	no	—
151	C₄⋊C₈	—	73	2	no	—
152	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
153	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
154	C₂²⋊C₈	—	2	1	no	—
155	C₂²⋊C₈	—	2	1	no	—
156	C₂²⋊C₈	—	146	1	no	—
157	C₂²⋊C₈	—	146	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	73	1	no	—
161	C₄×C₈	—	73	1	no	—
162	C₄×C₈	—	73	1	no	—
163	C₄×C₈	—	73	1	no	—
164	C₄×C₈	—	146	1	no	—
165	C₄×C₈	—	146	1	no	—
166	C₄×C₈	Dic₁₄₆	1	2	yes	—
167	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
169	C₂²×C₄	—	146	2	no	—
170	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
171	C₂²×C₄	#11924028	1	1	yes	characteristic
172	C₂²×C₄	#11924028	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	73	1	no	—
178	C₂×C₈	—	73	1	no	—
179	C₂×C₈	—	73	1	no	—
180	C₂×C₈	—	73	1	no	—
181	C₂×C₈	—	146	2	no	—
182	C₂×C₈	—	146	2	no	—
183	C₂×C₈	—	146	2	no	—
184	C₂×C₈	—	146	2	no	—
185	C₂×C₈	#11924007	1	2	yes	—
186	C₂×C₈	#11924007	1	2	yes	—
187	C₄²	—	292	2	no	—
188	C₄²	—	146	4	no	—
189	C₄²	—	146	2	no	—
190	C₄²	—	146	2	no	—
191	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
192	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
193	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
194	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
195	C₂³	#14556548	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	292	2	no	—
199	C₂×C₄	—	292	2	no	—
200	C₂×C₄	—	146	4	no	—
201	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
202	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
203	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
204	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
205	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
206	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
207	C₂×C₄	#14556599	1	2	yes	—
208	C₂×C₄	#14556599	1	2	yes	—
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	2	no	—
212	C₈	—	4	2	no	—
213	C₈	—	292	2	no	—
214	C₈	—	292	2	no	—
215	C₈	—	146	1	no	—
216	C₈	—	146	1	no	—
217	C₈	—	146	1	no	—
218	C₈	—	146	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₄×#11924007	1	1	yes	characteristic
222	C₂²	#17988649	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	292	4	no	—
226	C₄	#17988544	1	1	yes	characteristic
227	C₄	#17988544	1	1	yes	characteristic
228	C₄	#17988615	1	2	yes	—
229	C₄	#17988702	1	2	yes	—
230	C₄	#17988707	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	#22609791	1	1	yes	characteristic
233	C₂	#22609811	1	1	yes	characteristic
234	C₂	#22609816	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1452	C₂
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1452	D₇₃
hypocenter	C₇₃	#8270
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

Complements

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

N	H ≅ G/N	type
14. #17988339	C₄	semidirect
15. #17988339	C₄	semidirect
16. #17988337	C₄	semidirect
17. #17988337	C₄	semidirect
125. C₇₃	#8270	semidirect

Maximal subgroups

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

subgroup	index
#22609676	2
#22608677	2
#22608214	2
#22607763	2
C₇₃×#1452	2
#8270	73