Subgroups — #35320780

Order 21248.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907635	C₂	1	1	yes	maximal, characteristic
2	#24907627	C₂	1	1	yes	maximal, characteristic
3	#24905562	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581728	—	2	1	no	—
6	C₂×#14581728	C₂²	1	2	yes	—
7	C₂×#14581726	—	2	1	no	—
8	C₂×#14581724	C₂²	1	2	yes	—
9	#19127725	—	2	1	no	—
10	#19127567	—	2	1	no	—
11	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	#346×C₈₃	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₈₃	C₄	1	2	yes	complemented
15	C₂×#13050574	—	2	2	no	—
16	C₂×C₄⋊Dic₈₃	—	2	2	no	—
17	#14581728	—	2	4	no	—
18	#14581728	—	4	2	no	—
19	#14581726	—	4	2	no	—
20	#14581724	—	2	4	no	—
21	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
22	C₈₃⋊#95	—	4	2	no	—
23	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
26	M₃₂×C₈₃	—	2	2	no	—
27	M₃₂×C₈₃	—	2	2	no	—
28	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
33	C₂²×Dic₈₃	—	4	2	no	—
34	#13050574	—	4	2	no	—
35	#13050574	—	4	2	no	—
36	C₂×C₈₃⋊C₈	—	4	2	no	—
37	C₄⋊Dic₈₃	—	4	2	no	—
38	C₄⋊Dic₈₃	—	4	2	no	—
39	C₂²×C₄×C₈₃	SD₁₆	1	2	yes	—
40	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₃	D₈	1	2	yes	—
44	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
45	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₈₃	#41	1	2	yes	—
48	C₂×C₈×C₈₃	#41	1	2	yes	—
49	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₈₃	SD₁₆	1	2	yes	—
52	C₁₆×C₈₃	—	2	4	no	—
53	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₄×C₈₃	—	4	2	no	—
55	C₂×C₄×C₈₃	#106	1	2	yes	—
56	C₂×C₄×C₈₃	#106	1	2	yes	—
57	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₈₃	—	8	4	no	—
64	C₂×Dic₈₃	—	8	2	no	—
65	C₈×C₈₃	#107	1	2	yes	—
66	C₈×C₈₃	#107	1	2	yes	—
67	C₈×C₈₃	#117	1	2	yes	—
68	C₈×C₈₃	#125	1	2	yes	—
69	C₈₃⋊C₈	—	8	2	no	—
70	C₂²×C₈₃	—	2	2	no	—
71	C₂²×C₈₃	C₂×#106	1	1	yes	characteristic
72	C₂²×C₈₃	#486	1	1	yes	characteristic
73	C₂²×C₈₃	#487	1	1	yes	characteristic
74	C₄×C₈₃	—	2	2	no	—
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	#382	1	1	yes	characteristic
77	C₄×C₈₃	C₂×#107	1	1	yes	characteristic
78	C₄×C₈₃	#486	1	1	yes	characteristic
79	C₄×C₈₃	#487	1	1	yes	characteristic
80	Dic₈₃	—	16	2	no	—
81	#12887	—	83	1	no	maximal, Sylow
82	C₂×C₈₃	—	2	1	no	—
83	C₂×C₈₃	—	2	1	no	—
84	C₂×C₈₃	#1834	1	1	yes	characteristic
85	C₂×C₈₃	#2155	1	2	yes	—
86	C₄×M₃₂	D₈₃	1	1	yes	characteristic
87	#1848	—	83	1	no	—
88	#1846	—	83	1	no	—
89	#1383	—	83	4	no	—
90	C₈₃	#12887	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
92	C₂×#108	—	83	2	no	—
93	C₂×#108	—	166	1	no	—
94	C₂×#107	—	166	1	no	—
95	C₂×#106	—	83	2	no	—
96	#423	—	166	1	no	—
97	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
98	#382	—	166	1	no	—
99	#346	Dic₈₃	1	2	yes	complemented
100	C₄×C₁₆	Dic₈₃	1	2	yes	complemented
101	C₂×M₁₆	—	166	2	no	—
102	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	166	2	no	—
105	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	—
109	#108	—	332	2	no	—
110	#108	—	166	4	no	—
111	#107	—	332	2	no	—
112	#106	—	166	4	no	—
113	C₄⋊C₈	—	332	2	no	—
114	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
115	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
118	#95	—	332	2	no	—
119	C₂²×C₄	—	332	2	no	—
120	C₂²×C₄	#13050579	1	2	yes	—
121	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
122	M₁₆	—	332	2	no	—
123	M₁₆	—	332	2	no	—
124	C₂×C₈	—	332	2	no	—
125	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
126	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
127	C₂×C₈	D₆₆₄	1	2	yes	—
128	C₂×C₈	Dic₃₃₂	1	2	yes	—
129	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
131	C₂×C₈	#13050590	1	2	yes	—
132	C₂×C₈	#13050590	1	2	yes	—
133	C₄⋊C₄	—	332	2	no	—
134	C₄⋊C₄	—	332	2	no	—
135	C₄²	C₄⋊Dic₈₃	1	1	yes	characteristic
136	C₄²	C₄⋊Dic₈₃	1	1	yes	characteristic
137	C₄²	#13050579	1	2	yes	—
138	C₁₆	—	2	4	no	—
139	C₂³	C₂×#13050579	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	664	4	no	—
142	C₂×C₄	—	664	2	no	—
143	C₂×C₄	#14581724	1	2	yes	—
144	C₂×C₄	#14581724	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
146	C₂×C₄	#14581780	1	1	yes	characteristic
147	C₂×C₄	#14581780	1	1	yes	characteristic
148	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
149	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
150	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
151	C₈	—	664	2	no	—
152	C₈	#14581726	1	2	yes	—
153	C₈	#14581726	1	2	yes	—
154	C₈	#14581751	1	2	yes	—
155	C₈	#14581784	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#14581724	1	1	yes	characteristic
158	C₂²	#19128048	1	1	yes	characteristic
159	C₂²	#19128052	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	1328	2	no	—
163	C₄	#19127567	1	1	yes	characteristic
164	C₄	C₂×#14581726	1	1	yes	characteristic
165	C₄	#19128048	1	1	yes	characteristic
166	C₄	#19128052	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#24907591	1	1	yes	characteristic
170	C₂	#24909236	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14581724
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₄×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#106
prime radical	C₂³×C₈₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#12887
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. #346×C₈₃	C₄	semidirect
14. C₄×C₁₆×C₈₃	C₄	semidirect
90. C₈₃	#12887	semidirect
99. #346	Dic₈₃	semidirect
100. C₄×C₁₆	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24907635	2
#24907627	2
#24905562	2
C₄×M₃₂×C₈₃	2
#12887	83