Subgroups — #34001951

Order 18688.

582 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556646	C₂	1	2	yes	maximal
2	C₂×#17989429	C₂	1	2	yes	maximal
3	C₂×#17989428	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
5	#97×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556646	C₄	1	8	yes	—
7	C₂×C₄×#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	2	yes	—
9	#17989429	C₄	1	8	yes	complemented
10	#17989428	C₄	1	4	yes	complemented
11	C₈×#10596833	C₄	1	4	yes	complemented
12	C₂×#14556502	C₄	1	1	yes	characteristic
13	C₂×#14556501	C₄	1	1	yes	characteristic
14	C₂×#14556501	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
19	C₂²×#10596833	C₂×C₄	1	4	yes	—
20	C₂²×#10596833	C₂×C₄	1	4	yes	—
21	#14556646	—	2	16	no	—
22	C₄×#10596833	C₂×C₄	1	4	yes	—
23	C₄×#10596833	C₂×C₄	1	4	yes	—
24	C₄×#10596833	C₂×C₄	1	4	yes	—
25	C₄×#10596833	C₂×C₄	1	4	yes	—
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
31	#14556502	—	2	2	no	—
32	#14556501	—	2	2	no	—
33	#14556501	—	2	2	no	—
34	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
35	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
36	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	#97×C₇₃	—	2	2	no	—
41	C₂×#10596833	—	2	8	no	—
42	C₂×#10596833	C₄²	1	8	yes	—
43	C₂×#10596833	C₄²	1	8	yes	—
44	C₂×#10596833	M₁₆	1	16	yes	—
45	C₂²×Dic₇₃	C₄²	1	2	yes	—
46	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
47	C₂×#10596831	—	2	1	no	—
48	C₂×#10596831	—	2	1	no	—
49	C₂×#10596831	C₄²	1	2	yes	—
50	C₂×#10596831	C₄²	1	2	yes	—
51	C₂×#10596831	M₁₆	1	2	yes	—
52	C₂×#10596831	M₁₆	1	2	yes	—
53	C₄×Dic₇₃	—	2	1	no	—
54	C₄×Dic₇₃	—	2	1	no	—
55	C₄×Dic₇₃	—	2	8	no	—
56	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
58	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	—	2	1	no	—
62	C₂×C₈×C₇₃	—	2	1	no	—
63	C₂×C₈×C₇₃	C₄²	1	2	yes	—
64	C₂×C₈×C₇₃	C₄²	1	2	yes	—
65	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
66	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
67	C₄²×C₇₃	—	2	1	no	—
68	C₄²×C₇₃	—	2	1	no	—
69	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	4	no	—
72	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×C₄²	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×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₇₃	—	2	4	no	—
79	C₂×Dic₇₃	—	2	2	no	—
80	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
81	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
82	C₂×Dic₇₃	C₂×M₁₆	1	8	yes	—
83	#10596833	—	2	8	no	—
84	#10596833	#97	1	16	yes	complemented
85	C₈×C₇₃	—	2	2	no	—
86	C₈×C₇₃	#97	1	4	yes	complemented
87	#10596831	—	2	2	no	—
88	#10596831	#97	1	4	yes	complemented
89	C₂²×C₇₃	C₄³	1	1	yes	characteristic
90	C₂²×C₇₃	C₄×M₁₆	1	4	yes	—
91	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
92	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
93	C₄×C₇₃	—	4	2	no	—
94	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
95	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
96	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
98	Dic₇₃	—	4	2	no	—
99	Dic₇₃	C₂×#97	1	4	yes	—
100	Dic₇₃	C₄×M₁₆	1	4	yes	—
101	#7846	—	73	1	no	maximal, Sylow
102	C₂×C₇₃	#1455	1	2	yes	—
103	C₂×C₇₃	#1455	1	2	yes	—
104	C₂×C₇₃	C₄×#97	1	2	yes	—
105	C₂×C₇₃	C₄×#97	1	1	yes	characteristic
106	C₄×#97	—	73	2	no	—
107	C₄×#97	—	73	1	no	—
108	C₂×#322	—	73	1	no	—
109	C₂×#322	—	73	2	no	—
110	C₂×C₈²	—	73	1	no	—
111	C₇₃	#7846	1	1	yes	characteristic, complemented, Sylow
112	C₂×#97	—	73	8	no	—
113	C₂×#97	—	73	1	no	—
114	C₂×#97	—	73	1	no	—
115	C₂×#97	—	73	1	no	—
116	C₂×#97	#64408	1	1	yes	characteristic
117	C₂×C₄×C₈	—	73	2	no	—
118	C₂×C₄×C₈	—	73	2	no	—
119	C₂×C₄×C₈	—	73	1	no	—
120	C₂×C₄×C₈	—	73	1	no	—
121	C₄³	—	73	1	no	—
122	#322	—	73	4	no	—
123	#322	—	73	8	no	—
124	C₈²	—	73	4	no	—
125	C₂²×C₈	—	73	4	no	—
126	C₂²×C₈	—	73	1	no	—
127	C₂²×C₈	—	73	4	no	—
128	C₂²×C₈	—	73	1	no	—
129	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
130	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
131	C₂×C₄²	—	73	1	no	—
132	C₂×C₄²	—	73	4	no	—
133	C₂×C₄²	—	73	1	no	—
134	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
135	#97	—	2	2	no	—
136	#97	—	146	2	no	—
137	#97	—	146	2	no	—
138	#97	—	146	16	no	—
139	#97	—	146	2	no	—
140	C₄×C₈	—	73	4	no	—
141	C₄×C₈	—	73	4	no	—
142	C₄×C₈	—	73	4	no	—
143	C₄×C₈	—	73	4	no	—
144	C₄×C₈	—	73	2	no	—
145	C₄×C₈	—	73	2	no	—
146	C₄×C₈	—	73	4	no	—
147	C₂²×C₄	—	73	2	no	—
148	C₂²×C₄	—	73	2	no	—
149	C₂²×C₄	C₄×#64408	1	2	yes	—
150	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	73	8	no	—
154	C₂×C₈	—	73	16	no	—
155	C₂×C₈	—	73	2	no	—
156	C₂×C₈	—	73	2	no	—
157	C₂×C₈	—	73	2	no	—
158	C₂×C₈	—	73	8	no	—
159	C₂×C₈	—	73	2	no	—
160	C₂×C₈	—	146	8	no	—
161	C₂×C₈	—	146	1	no	—
162	C₂×C₈	—	146	1	no	—
163	C₂×C₈	C₄×#64408	1	2	yes	—
164	C₂×C₈	C₄×#64408	1	2	yes	—
165	C₂×C₈	#11924037	1	2	yes	—
166	C₂×C₈	#11924038	1	2	yes	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	73	1	no	—
170	C₄²	—	73	1	no	—
171	C₄²	—	73	2	no	—
172	C₄²	—	146	8	no	—
173	C₄²	—	146	1	no	—
174	C₄²	—	146	1	no	—
175	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	73	2	no	—
179	C₂×C₄	—	73	2	no	—
180	C₂×C₄	—	73	8	no	—
181	C₂×C₄	—	146	4	no	—
182	C₂×C₄	—	146	2	no	—
183	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
185	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
186	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
187	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
188	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	73	16	no	—
191	C₈	—	73	4	no	—
192	C₈	—	146	8	no	—
193	C₈	—	146	2	no	—
194	C₈	#14556646	1	4	yes	complemented
195	C₂²	C₄²×#64408	1	1	yes	characteristic
196	C₂²	M₁₆×#64408	1	4	yes	—
197	C₂²	C₄×#11924037	1	1	yes	characteristic
198	C₂²	C₄×#11924038	1	1	yes	characteristic
199	C₄	—	4	2	no	—
200	C₄	—	292	2	no	—
201	C₄	—	73	4	no	—
202	C₄	—	73	4	no	—
203	C₄	C₂×#14556646	1	1	yes	characteristic
204	C₄	#17989546	1	1	yes	characteristic
205	C₄	C₄×#11924037	1	1	yes	characteristic
206	C₄	C₄×#11924038	1	1	yes	characteristic
207	C₂	#22625154	1	2	yes	—
208	C₂	#22625155	1	2	yes	—
209	C₂	#97×#64408	1	2	yes	—
210	C₂	C₄×#14556646	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₄³
Fitting	C₂×#97×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#7846
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
9. #17989429	C₄	semidirect
10. #17989428	C₄	semidirect
11. C₈×#10596833	C₄	semidirect
84. #10596833	#97	semidirect
86. C₈×C₇₃	#97	semidirect
88. #10596831	#97	semidirect
111. C₇₃	#7846	semidirect
194. C₈	#14556646	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556646	2
C₂×#17989429	2
C₂×#17989428	2
C₂×C₈×#10596833	2
#97×Dic₇₃	2
#7846	73