Subgroups — #34013303

Order 18688.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624970	C₂	1	2	yes	maximal
2	#22624970	C₂	1	2	yes	maximal
3	#22624969	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924031	C₂²	1	4	yes	—
6	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
13	C₂×#11924031	—	2	4	no	—
14	C₂×#11924031	—	2	4	no	—
15	C₂×#11924031	—	2	4	no	—
16	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
17	C₂²×#10596831	C₂³	1	1	yes	characteristic
18	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596831	—	2	2	no	—
25	C₈×Dic₇₃	—	2	2	no	—
26	C₈×Dic₇₃	—	2	2	no	—
27	C₈×Dic₇₃	—	2	2	no	—
28	C₈×Dic₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	—	2	2	no	—
33	#11924031	—	4	4	no	—
34	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
38	C₂×#10596831	—	2	1	no	—
39	C₂×#10596831	—	2	1	no	—
40	C₂×#10596831	M₁₆	1	2	yes	—
41	C₂×#10596831	#41	1	2	yes	—
42	C₂×#10596831	#41	1	2	yes	—
43	C₂×#10596831	#41	1	2	yes	—
44	C₄×Dic₇₃	—	2	2	no	—
45	C₄×Dic₇₃	—	2	1	no	—
46	C₄×Dic₇₃	—	2	1	no	—
47	C₄×Dic₇₃	—	2	1	no	—
48	C₄×Dic₇₃	—	2	1	no	—
49	C₄×Dic₇₃	—	2	2	no	—
50	C₄×Dic₇₃	—	2	2	no	—
51	C₄×Dic₇₃	C₁₆	1	4	yes	complemented
52	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	2	1	no	—
59	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
60	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
61	C₄²×C₇₃	—	2	1	no	—
62	C₄²×C₇₃	—	2	1	no	—
63	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	1	no	—
65	C₂×C₄×C₇₃	—	2	1	no	—
66	C₂×C₄×C₇₃	—	2	1	no	—
67	C₂×C₄×C₇₃	—	2	1	no	—
68	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
69	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
70	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×Dic₇₃	—	2	1	no	—
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	—	2	1	no	—
79	C₂×Dic₇₃	—	2	1	no	—
80	C₂×Dic₇₃	—	2	1	no	—
81	C₂×Dic₇₃	—	2	1	no	—
82	C₂×Dic₇₃	—	2	1	no	—
83	C₂×Dic₇₃	—	2	1	no	—
84	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
85	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
86	C₂×Dic₇₃	M₃₂	1	2	yes	—
87	C₂×Dic₇₃	M₃₂	1	2	yes	—
88	C₈×C₇₃	—	2	2	no	—
89	C₈×C₇₃	—	2	2	no	—
90	#10596831	—	2	2	no	—
91	#10596831	#126	1	4	yes	—
92	C₂²×C₇₃	#431	1	1	yes	characteristic
93	C₂²×C₇₃	#431	1	1	yes	characteristic
94	C₂²×C₇₃	#431	1	1	yes	characteristic
95	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
96	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
97	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
98	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
99	C₄×C₇₃	—	2	2	no	—
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	#383	1	1	yes	characteristic
102	C₄×C₇₃	#432	1	1	yes	characteristic
103	C₄×C₇₃	#432	1	1	yes	characteristic
104	C₄×C₇₃	#432	1	1	yes	characteristic
105	Dic₇₃	—	2	2	no	—
106	Dic₇₃	—	2	2	no	—
107	Dic₇₃	—	2	2	no	—
108	Dic₇₃	—	2	2	no	—
109	#12894	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	#1839	1	1	yes	characteristic
111	C₂×C₇₃	#2162	1	1	yes	characteristic
112	C₂×C₇₃	#2162	1	1	yes	characteristic
113	C₂×C₇₃	#2162	1	1	yes	characteristic
114	C₂×C₇₃	#2163	1	1	yes	characteristic
115	C₂×C₇₃	#2163	1	1	yes	characteristic
116	C₂×C₇₃	#2163	1	1	yes	characteristic
117	C₄²×C₈	—	73	1	no	—
118	#1374	—	73	2	no	—
119	#1374	—	73	2	no	—
120	#1374	—	73	2	no	—
121	C₇₃	#12894	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₁₆	—	73	4	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₂×C₄×C₈	#64408	1	1	yes	characteristic
129	C₄³	—	73	1	no	—
130	C₂²×C₈	—	73	1	no	—
131	C₂²×C₈	—	73	1	no	—
132	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
133	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
134	C₂×C₄²	—	73	1	no	—
135	C₂×C₄²	—	73	1	no	—
136	C₂×C₄²	—	73	1	no	—
137	C₂×C₄²	—	73	1	no	—
138	C₂×C₄²	—	73	1	no	—
139	C₂×C₄²	—	73	1	no	—
140	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
141	C₂×C₁₆	—	146	4	no	—
142	C₂×C₁₆	—	146	4	no	—
143	C₂×C₁₆	—	146	4	no	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	146	2	no	—
146	C₄×C₈	—	146	2	no	—
147	C₄×C₈	—	146	2	no	—
148	C₄×C₈	—	146	2	no	—
149	C₄×C₈	—	146	2	no	—
150	C₂²×C₄	—	73	1	no	—
151	C₂²×C₄	—	73	1	no	—
152	C₂²×C₄	—	73	1	no	—
153	C₂²×C₄	—	73	1	no	—
154	C₂²×C₄	#11924036	1	1	yes	characteristic
155	C₂²×C₄	#11924036	1	1	yes	characteristic
156	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	73	2	no	—
162	C₂×C₈	—	73	2	no	—
163	C₂×C₈	—	73	2	no	—
164	C₂×C₈	—	73	2	no	—
165	C₂×C₈	—	146	1	no	—
166	C₂×C₈	—	146	1	no	—
167	C₂×C₈	#11924037	1	2	yes	—
168	C₂×C₈	#11924038	1	2	yes	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	73	4	no	—
172	C₄²	—	146	2	no	—
173	C₄²	—	146	2	no	—
174	C₄²	—	146	1	no	—
175	C₄²	—	146	1	no	—
176	C₄²	—	146	2	no	—
177	C₄²	—	146	1	no	—
178	C₄²	—	146	1	no	—
179	C₁₆	—	292	4	no	—
180	C₂³	C₂×#11924036	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	73	2	no	—
186	C₂×C₄	—	73	2	no	—
187	C₂×C₄	—	73	2	no	—
188	C₂×C₄	—	73	2	no	—
189	C₂×C₄	—	146	1	no	—
190	C₂×C₄	—	146	1	no	—
191	C₂×C₄	—	146	1	no	—
192	C₂×C₄	—	146	1	no	—
193	C₂×C₄	—	146	1	no	—
194	C₂×C₄	—	146	1	no	—
195	C₂×C₄	—	146	1	no	—
196	C₂×C₄	—	146	1	no	—
197	C₂×C₄	#14556664	1	2	yes	—
198	C₂×C₄	#14556666	1	2	yes	—
199	C₂×C₄	#14556670	1	1	yes	characteristic
200	C₂×C₄	#14556670	1	1	yes	characteristic
201	C₂×C₄	#14556670	1	1	yes	characteristic
202	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
203	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
204	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	73	4	no	—
208	C₈	—	146	2	no	—
209	C₂²	#17989604	1	1	yes	characteristic
210	C₂²	#17989605	1	1	yes	characteristic
211	C₂²	#17989606	1	1	yes	characteristic
212	C₂²	C₂×#14556664	1	1	yes	characteristic
213	C₂²	#17989643	1	1	yes	characteristic
214	C₂²	#17989643	1	1	yes	characteristic
215	C₂²	C₂×#14556666	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	146	2	no	—
219	C₄	—	146	2	no	—
220	C₄	—	146	2	no	—
221	C₄	—	146	2	no	—
222	C₄	#17989532	1	1	yes	characteristic
223	C₄	#17989607	1	1	yes	characteristic
224	C₄	#17989608	1	1	yes	characteristic
225	C₄	#17989609	1	1	yes	characteristic
226	C₂	#22625859	1	1	yes	characteristic
227	C₂	#22626289	1	1	yes	characteristic
228	C₂	#22626290	1	1	yes	characteristic
229	C₂	#22626291	1	1	yes	characteristic
230	C₂	#22626293	1	1	yes	characteristic
231	C₂	#22626294	1	1	yes	characteristic
232	C₂	#22626295	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12894
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
51. C₄×Dic₇₃	C₁₆	semidirect
121. C₇₃	#12894	semidirect

Maximal subgroups

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

subgroup	index
#22624970	2
#22624969	2
C₄×C₈×Dic₇₃	2
#12894	73