Subgroups — #34003952

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625478	C₂	1	2	yes	maximal
2	#22625134	C₂	1	2	yes	maximal
3	#22624759	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	#17989603	C₂²	1	4	yes	—
6	#17989450	C₄	1	4	yes	complemented
7	#17989448	C₄	1	4	yes	complemented
8	#17989428	—	2	2	no	—
9	C₈×#10596833	—	2	2	no	—
10	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
17	C₄⋊#10596833	C₂×C₄	1	8	yes	—
18	C₂²⋊#10596833	—	2	4	no	—
19	C₄×#10596833	—	2	4	no	—
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	—	2	2	no	—
23	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
25	C₄×#10596831	—	2	2	no	—
26	C₈×Dic₇₃	—	2	2	no	—
27	C₈×Dic₇₃	C₈	1	4	yes	complemented
28	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
29	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₈×Dic₇₃	D₄	1	1	yes	characteristic
31	C₈×Dic₇₃	D₄	1	1	yes	characteristic
32	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
33	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₃	C₈	1	4	yes	complemented
38	C₂×#10596833	—	2	8	no	—
39	C₂²×Dic₇₃	—	2	1	no	—
40	C₂²×Dic₇₃	#41	1	2	yes	—
41	C₂×#10596831	—	2	1	no	—
42	C₂×#10596831	—	2	1	no	—
43	C₂×#10596831	C₄²	1	2	yes	—
44	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
46	C₂×#10596831	C₂×C₈	1	2	yes	—
47	C₂×#10596831	M₁₆	1	2	yes	—
48	C₄×Dic₇₃	—	2	8	no	—
49	C₄×Dic₇₃	—	2	1	no	—
50	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
51	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
52	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
53	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	—	2	1	no	—
55	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
63	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
64	C₄²×C₇₃	—	2	1	no	—
65	C₄²×C₇₃	C₂×C₈	1	2	yes	—
66	C₂³×C₇₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	2	4	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂²×C₈	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×Dic₇₃	—	2	4	no	—
76	C₂×Dic₇₃	—	2	2	no	—
77	C₂×Dic₇₃	—	2	4	no	—
78	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
79	C₂×Dic₇₃	C₄×Q₈	1	2	yes	—
80	#10596833	—	4	8	no	—
81	C₈×C₇₃	C₄×C₈	1	2	yes	—
82	C₈×C₇₃	C₄×C₈	1	2	yes	—
83	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
84	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
85	#10596831	—	2	2	no	—
86	#10596831	C₄⋊C₈	1	2	yes	—
87	#10596831	C₄⋊C₈	1	2	yes	—
88	C₂²×C₇₃	—	2	2	no	—
89	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₇₃	#431	1	1	yes	characteristic
91	C₂²×C₇₃	#431	1	1	yes	characteristic
92	C₄×C₇₃	—	2	4	no	—
93	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
94	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
95	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	Dic₇₃	—	2	4	no	—
98	Dic₇₃	—	2	2	no	—
99	Dic₇₃	#443	1	4	yes	—
100	#8266	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	—	2	1	no	—
102	C₂×C₇₃	—	2	1	no	—
103	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₇₃	#1772	1	2	yes	—
105	C₄²×C₈	—	73	1	no	—
106	#1564	—	73	2	no	—
107	#1450	—	73	2	no	—
108	#1276	—	73	2	no	—
109	C₇₃	#8266	1	1	yes	characteristic, complemented, Sylow
110	#431	—	73	4	no	—
111	C₂×C₄×C₈	—	73	1	no	—
112	C₂×C₄×C₈	—	73	1	no	—
113	C₂×C₄×C₈	—	73	1	no	—
114	C₂×C₄×C₈	—	73	1	no	—
115	C₂×C₄×C₈	—	73	1	no	—
116	C₂×C₄×C₈	#64408	1	1	yes	characteristic
117	C₄³	—	73	1	no	—
118	#335	—	73	4	no	—
119	#334	—	73	4	no	—
120	#322	—	146	2	no	—
121	C₈²	—	146	2	no	—
122	C₂²×C₈	—	73	1	no	—
123	C₂²×C₈	—	73	1	no	—
124	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
125	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
126	C₂×C₄²	—	73	1	no	—
127	C₂×C₄²	—	73	1	no	—
128	C₂×C₄²	—	146	2	no	—
129	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
130	C₄⋊C₈	—	73	8	no	—
131	C₂²⋊C₈	—	146	4	no	—
132	C₄×C₈	—	73	2	no	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	73	1	no	—
135	C₄×C₈	—	73	1	no	—
136	C₄×C₈	—	73	1	no	—
137	C₄×C₈	—	73	1	no	—
138	C₄×C₈	—	73	4	no	—
139	C₄×C₈	—	146	2	no	—
140	C₄×C₈	—	146	4	no	—
141	C₄×C₈	—	146	2	no	—
142	C₄×C₈	#10596833	1	4	yes	complemented
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	73	2	no	—
145	C₂²×C₄	—	146	1	no	—
146	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
147	C₂×C₈	—	73	2	no	—
148	C₂×C₈	—	73	2	no	—
149	C₂×C₈	—	73	1	no	—
150	C₂×C₈	—	73	1	no	—
151	C₂×C₈	—	73	2	no	—
152	C₂×C₈	—	146	8	no	—
153	C₂×C₈	—	146	1	no	—
154	C₂×C₈	—	146	1	no	—
155	C₂×C₈	C₄×#64408	1	1	yes	characteristic
156	C₂×C₈	C₄×#64408	1	1	yes	characteristic
157	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
158	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
159	C₂×C₈	C₂×#10596833	1	2	yes	—
160	C₂×C₈	C₂×#10596833	1	2	yes	—
161	C₂×C₈	#11924036	1	2	yes	—
162	C₂×C₈	#11924038	1	2	yes	—
163	C₄²	—	2	1	no	—
164	C₄²	—	73	2	no	—
165	C₄²	—	73	1	no	—
166	C₄²	—	73	1	no	—
167	C₄²	—	73	2	no	—
168	C₄²	—	146	8	no	—
169	C₄²	—	146	1	no	—
170	C₄²	C₂×#10596833	1	2	yes	—
171	C₂³	#14556670	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	73	2	no	—
175	C₂×C₄	—	73	2	no	—
176	C₂×C₄	—	146	4	no	—
177	C₂×C₄	—	146	4	no	—
178	C₂×C₄	—	146	2	no	—
179	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
181	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
182	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
183	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
184	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
185	C₈	—	292	8	no	—
186	C₈	—	73	2	no	—
187	C₈	—	73	2	no	—
188	C₈	—	146	2	no	—
189	C₈	C₄×#10596833	1	2	yes	—
190	C₈	C₄×#10596833	1	2	yes	—
191	C₈	C₄⋊#10596833	1	2	yes	—
192	C₈	C₄⋊#10596833	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	C₄×C₄⋊#64408	1	1	yes	characteristic
195	C₂²	#17989603	1	1	yes	characteristic
196	C₂²	#17989606	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	73	4	no	—
199	C₄	—	146	4	no	—
200	C₄	—	146	2	no	—
201	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
202	C₄	C₄×#11924038	1	1	yes	characteristic
203	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
204	C₄	#17989588	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₄×C₄⋊#10596833	1	1	yes	characteristic
208	C₂	#22625733	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#8266
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17989450	C₄	semidirect
7. #17989448	C₄	semidirect
27. C₈×Dic₇₃	C₈	semidirect
37. C₄×C₈×C₇₃	C₈	semidirect
109. C₇₃	#8266	semidirect
142. C₄×C₈	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22625478	2
#22625134	2
#22624759	2
C₄×C₈×Dic₇₃	2
#8266	73