Subgroups — #32977184

Order 18688.

446 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₈×Dic₇₃	C₂	1	2	yes	maximal
2	C₄×#14556501	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
4	#22607765	C₂	1	2	yes	maximal
5	C₂×#334×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
7	C₂×#14556501	C₄	1	2	yes	—
8	C₂×#14556501	C₄	1	2	yes	—
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	—	2	2	no	—
11	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
12	#17988341	—	2	2	no	—
13	#17988340	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#334×C₇₃	C₄	1	4	yes	complemented
17	C₂²×#10596831	—	2	2	no	—
18	C₂²×#10596831	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
21	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	C₂×C₄	1	4	yes	—
24	#14556501	—	2	4	no	—
25	#14556501	C₈	1	8	yes	complemented
26	C₄×#10596831	—	2	1	no	—
27	C₄×#10596831	—	2	1	no	—
28	C₈×Dic₇₃	—	4	4	no	—
29	C₂²×C₈×C₇₃	—	2	2	no	—
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₇₃	—	2	2	no	—
34	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
37	C₂²×Dic₇₃	C₄⋊C₄	1	4	yes	—
38	C₂×#10596831	—	2	2	no	—
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	—	2	4	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₂×#10596831	C₂×C₈	1	4	yes	—
44	C₄×Dic₇₃	—	2	2	no	—
45	C₄×Dic₇₃	—	2	2	no	—
46	C₄×Dic₇₃	—	4	2	no	—
47	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
48	C₄×Dic₇₃	M₁₆	1	4	yes	—
49	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	—	2	4	no	—
54	C₂×C₈×C₇₃	—	4	2	no	—
55	C₂×C₈×C₇₃	—	4	2	no	—
56	C₂×C₈×C₇₃	C₄²	1	2	yes	—
57	C₂×C₈×C₇₃	C₄²	1	2	yes	—
58	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
59	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₇₃	#41	1	1	yes	characteristic
62	C₄²×C₇₃	#41	1	1	yes	characteristic
63	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₇₃	#137	1	2	yes	—
75	C₂×Dic₇₃	—	2	4	no	—
76	C₂×Dic₇₃	—	2	4	no	—
77	C₂×Dic₇₃	C₄⋊C₈	1	8	yes	—
78	C₈×C₇₃	—	2	2	no	—
79	C₈×C₇₃	—	4	4	no	—
80	C₈×C₇₃	C₄×C₈	1	4	yes	complemented
81	#10596831	—	2	4	no	—
82	#10596831	—	4	4	no	—
83	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
85	C₂²×C₇₃	#428	1	2	yes	—
86	C₂²×C₇₃	#430	1	2	yes	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
91	C₄×C₇₃	#431	1	2	yes	—
92	C₄×C₇₃	#440	1	2	yes	—
93	C₄×C₇₃	#441	1	1	yes	characteristic
94	C₄×C₇₃	#442	1	1	yes	characteristic
95	Dic₇₃	—	2	8	no	—
96	#8275	—	73	1	no	maximal, Sylow
97	C₂×C₇₃	#1565	1	4	yes	—
98	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₇₃	#1776	1	1	yes	characteristic
100	C₂×C₇₃	#1778	1	1	yes	characteristic
101	C₄×C₄⋊C₈	—	73	2	no	—
102	C₄×#97	—	73	1	no	—
103	C₂×#335	—	73	1	no	—
104	C₂×#334	D₇₃	1	1	yes	characteristic
105	#1277	—	73	2	no	—
106	C₇₃	#8275	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊C₈	—	73	2	no	—
108	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
109	C₂×#97	—	73	2	no	—
110	C₂×#97	—	73	2	no	—
111	C₂×C₄×C₈	—	73	1	no	—
112	C₂×C₄×C₈	—	146	2	no	—
113	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
114	C₄³	—	73	1	no	—
115	#336	—	146	2	no	—
116	#335	—	73	4	no	—
117	#334	Dic₇₃	1	4	yes	complemented
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	73	2	no	—
120	C₂²×C₈	—	146	2	no	—
121	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
122	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
123	C₂×C₄²	—	73	2	no	—
124	C₂×C₄²	—	73	2	no	—
125	C₂×C₄²	—	73	2	no	—
126	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	73	4	no	—
129	C₄⋊C₈	—	146	2	no	—
130	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
131	#97	—	73	8	no	—
132	#97	—	146	4	no	—
133	C₄×C₈	—	292	4	no	—
134	C₄×C₈	—	146	1	no	—
135	C₄×C₈	—	146	1	no	—
136	C₄×C₈	C₄×D₇₃	1	2	yes	—
137	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
138	C₂²×C₄	—	73	4	no	—
139	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
141	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	292	2	no	—
147	C₂×C₈	—	292	2	no	—
148	C₂×C₈	—	73	4	no	—
149	C₂×C₈	—	146	4	no	—
150	C₂×C₈	—	146	2	no	—
151	C₂×C₈	—	146	2	no	—
152	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
153	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
154	C₂×C₈	C₈×D₇₃	1	4	yes	—
155	C₄²	—	292	2	no	—
156	C₄²	—	73	4	no	—
157	C₄²	—	73	4	no	—
158	C₄²	—	146	2	no	—
159	C₄²	—	146	2	no	—
160	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
161	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
162	C₄²	#11924027	1	1	yes	characteristic
163	C₄²	#11924029	1	1	yes	characteristic
164	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	73	8	no	—
167	C₂×C₄	—	146	4	no	—
168	C₂×C₄	—	146	4	no	—
169	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
170	C₂×C₄	#14556549	1	1	yes	characteristic
171	C₂×C₄	#14556553	1	1	yes	characteristic
172	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
173	C₂×C₄	#14556556	1	1	yes	characteristic
174	C₂×C₄	Q₈×Dic₇₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
176	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
177	C₂×C₄	#14556618	1	2	yes	—
178	C₂×C₄	#14556620	1	2	yes	—
179	C₈	—	2	2	no	—
180	C₈	—	4	4	no	—
181	C₈	—	292	4	no	—
182	C₈	—	146	4	no	—
183	C₈	C₈×Dic₇₃	1	4	yes	complemented
184	C₂²	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
185	C₂²	C₄⋊C₈×D₇₃	1	2	yes	—
186	C₂²	#17988626	1	2	yes	—
187	C₂²	#17988634	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	146	8	no	—
191	C₄	C₂×C₈×Dic₇₃	1	1	yes	characteristic
192	C₄	M₁₆×Dic₇₃	1	1	yes	characteristic
193	C₄	#17988639	1	2	yes	—
194	C₄	#17988688	1	2	yes	—
195	C₄	#17988693	1	1	yes	characteristic
196	C₄	#17988698	1	1	yes	characteristic
197	C₂	#22608680	1	4	yes	—
198	C₂	C₄⋊C₈×Dic₇₃	1	1	yes	characteristic
199	C₂	#22609831	1	1	yes	characteristic
200	C₂	#22609839	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#334×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#334	D₇₃
hypocenter	C₇₃	#8275
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
13. #17988340	C₄	semidirect
16. #334×C₇₃	C₄	semidirect
25. #14556501	C₈	semidirect
80. C₈×C₇₃	C₄×C₈	semidirect
106. C₇₃	#8275	semidirect
117. #334	Dic₇₃	semidirect
183. C₈	C₈×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×Dic₇₃	2
C₄×#14556501	2
C₂×#17988340	2
#22607765	2
C₂×#334×C₇₃	2
#8275	73