Subgroups — #35298097

Order 21248.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 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	#24905322	C₂	1	4	yes	maximal
4	C₈₃×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581728	—	2	1	no	—
6	C₂×#14581728	—	2	1	no	—
7	C₂×#14581726	—	2	1	no	—
8	C₂×#14581724	—	2	1	no	—
9	#19127725	C₂²	1	2	yes	—
10	#19127567	C₂²	1	2	yes	—
11	#431×C₈₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	#322×C₈₃	C₄	1	2	yes	complemented
14	C₈²×C₈₃	C₄	1	2	yes	complemented
15	C₂×#13050574	—	2	2	no	—
16	C₂×C₄⋊Dic₈₃	—	2	2	no	—
17	#14581728	—	4	2	no	—
18	#14581728	—	4	2	no	—
19	#14581726	—	4	2	no	—
20	#14581724	—	4	2	no	—
21	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
22	C₈₃⋊#95	—	2	4	no	—
23	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₈₃	—	2	2	no	—
27	C₂²⋊C₈×C₈₃	—	2	2	no	—
28	C₄×C₈×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	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	#41	1	2	yes	—
41	C₂×C₈×C₈₃	—	2	4	no	—
42	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	D₈	1	2	yes	—
47	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
49	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
50	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₈₃	#41	1	2	yes	—
53	C₂³×C₈₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₈₃	—	4	2	no	—
55	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	#117	1	2	yes	—
57	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₈₃	#125	1	2	yes	—
59	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₈₃	C₂×SD₁₆	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	—	4	4	no	—
66	C₈×C₈₃	#106	1	2	yes	—
67	C₈×C₈₃	#106	1	2	yes	—
68	C₈×C₈₃	#107	1	2	yes	—
69	C₈×C₈₃	#107	1	2	yes	—
70	C₈₃⋊C₈	—	8	2	no	—
71	C₂²×C₈₃	—	2	2	no	—
72	C₂²×C₈₃	#382	1	1	yes	characteristic
73	C₂²×C₈₃	#486	1	1	yes	characteristic
74	C₂²×C₈₃	#487	1	1	yes	characteristic
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	—	2	2	no	—
77	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
78	C₄×C₈₃	C₂×#107	1	1	yes	characteristic
79	C₄×C₈₃	#486	1	1	yes	characteristic
80	C₄×C₈₃	#487	1	1	yes	characteristic
81	Dic₈₃	—	16	2	no	—
82	#8793	—	83	1	no	maximal, Sylow
83	C₂×C₈₃	—	2	1	no	—
84	C₂×C₈₃	—	2	1	no	—
85	C₂×C₈₃	#1833	1	2	yes	—
86	C₂×C₈₃	#1834	1	1	yes	characteristic
87	#1848	—	83	1	no	—
88	#1846	—	83	1	no	—
89	#1450	D₈₃	1	1	yes	characteristic
90	#1286	—	83	4	no	—
91	C₈₃	#8793	1	1	yes	characteristic, complemented, Sylow
92	#431	C₂×D₈₃	1	2	yes	—
93	C₂×#108	—	166	1	no	—
94	C₂×#108	—	166	1	no	—
95	C₂×#107	—	166	1	no	—
96	C₂×#106	—	166	1	no	—
97	#423	—	83	2	no	—
98	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
99	#382	—	83	2	no	—
100	#322	Dic₈₃	1	2	yes	complemented
101	C₈²	Dic₈₃	1	2	yes	complemented
102	C₂×M₁₆	—	166	2	no	—
103	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
104	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	166	2	no	—
106	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
107	#108	—	332	2	no	—
108	#108	—	332	2	no	—
109	#107	—	332	2	no	—
110	#106	—	332	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	166	4	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
115	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
117	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
119	#95	—	166	4	no	—
120	C₂²×C₄	—	332	2	no	—
121	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
122	C₂²×C₄	#13050590	1	2	yes	—
123	M₁₆	—	332	2	no	—
124	M₁₆	—	332	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	332	2	no	—
127	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
129	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
130	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
131	C₂×C₈	D₆₆₄	1	2	yes	—
132	C₂×C₈	#13050579	1	2	yes	—
133	C₂×C₈	#13050579	1	2	yes	—
134	C₂×C₈	Dic₃₃₂	1	2	yes	—
135	C₄⋊C₄	—	332	2	no	—
136	C₄⋊C₄	—	332	2	no	—
137	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
138	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
139	C₄²	#13050590	1	2	yes	—
140	C₂³	#14581780	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	664	4	no	—
143	C₂×C₄	—	664	2	no	—
144	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
145	C₂×C₄	#14581751	1	2	yes	—
146	C₂×C₄	#14581780	1	1	yes	characteristic
147	C₂×C₄	#14581784	1	2	yes	—
148	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
149	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
150	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
152	C₈	—	4	4	no	—
153	C₈	—	664	2	no	—
154	C₈	#14581724	1	2	yes	—
155	C₈	#14581724	1	2	yes	—
156	C₈	#14581726	1	2	yes	—
157	C₈	#14581726	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#19127567	1	1	yes	characteristic
160	C₂²	#19128048	1	1	yes	characteristic
161	C₂²	#19128052	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	1328	2	no	—
165	C₄	C₂×#14581724	1	1	yes	characteristic
166	C₄	C₂×#14581726	1	1	yes	characteristic
167	C₄	#19128048	1	1	yes	characteristic
168	C₄	#19128052	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#24907587	1	2	yes	—
172	C₂	#24907591	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19127567
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1450	C₂
socle	C₂²×C₈₃	#382
prime radical	C₂³×C₈₃	#124
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1450	D₈₃
hypocenter	C₈₃	#8793
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
13. #322×C₈₃	C₄	semidirect
14. C₈²×C₈₃	C₄	semidirect
91. C₈₃	#8793	semidirect
100. #322	Dic₈₃	semidirect
101. C₈²	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
#24905322	2
C₈₃×#1450	2
#8793	83