Subgroups — #35319783

Order 21248.

308 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909236	C₂	1	2	yes	maximal
2	#24907342	C₂	1	2	yes	maximal
3	#24905567	C₂	1	1	yes	maximal, characteristic
4	#24905565	C₂	1	1	yes	maximal, characteristic
5	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581728	C₂²	1	2	yes	—
7	#19127739	C₂²	1	2	yes	—
8	#19127656	—	2	2	no	—
9	#19127524	C₄	1	8	yes	complemented
10	#19127461	—	2	2	no	—
11	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	#346×C₈₃	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₈₃	C₄	1	2	yes	complemented
16	C₂×#13050574	—	2	2	no	—
17	#14581751	—	2	2	no	—
18	#14581728	—	2	2	no	—
19	#14581728	C₂×C₄	1	4	yes	—
20	#14581726	—	2	2	no	—
21	#14581724	C₂×C₄	1	4	yes	—
22	#14581702	—	4	2	no	—
23	C₈×Dic₈₃	—	4	2	no	—
24	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
27	M₃₂×C₈₃	—	2	2	no	—
28	M₃₂×C₈₃	C₂×C₄	1	4	yes	—
29	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
35	#13050574	—	2	4	no	—
36	#13050574	—	4	2	no	—
37	C₂×C₈₃⋊C₈	—	4	2	no	—
38	C₄⋊Dic₈₃	—	2	4	no	—
39	C₂²⋊Dic₈₃	—	4	2	no	—
40	C₄×Dic₈₃	—	4	2	no	—
41	C₂²×C₄×C₈₃	—	2	1	no	—
42	C₂²×C₄×C₈₃	C₂×D₄	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	SD₁₆	1	2	yes	—
46	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
47	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₈₃	#41	1	2	yes	—
51	C₄²×C₈₃	—	2	1	no	—
52	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₈₃	—	2	2	no	—
55	C₁₆×C₈₃	C₄²	1	4	yes	—
56	C₂³×C₈₃	#135	1	1	yes	characteristic
57	C₂×C₄×C₈₃	—	2	1	no	—
58	C₂×C₄×C₈₃	—	2	1	no	—
59	C₂×C₄×C₈₃	—	4	2	no	—
60	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
63	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
64	C₂×Dic₈₃	—	4	4	no	—
65	C₈×C₈₃	#106	1	2	yes	—
66	C₈×C₈₃	#106	1	1	yes	characteristic
67	C₈×C₈₃	#106	1	1	yes	characteristic
68	C₈×C₈₃	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₈₃	#117	1	1	yes	characteristic
70	C₈×C₈₃	C₄×Q₈	1	2	yes	—
71	C₈₃⋊C₈	—	4	4	no	—
72	C₂²×C₈₃	—	4	1	no	—
73	C₂²×C₈₃	#427	1	1	yes	characteristic
74	C₂²×C₈₃	C₄×SD₁₆	1	2	yes	—
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	—	4	1	no	—
77	C₄×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
79	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
80	Dic₈₃	—	8	4	no	—
81	#12668	—	83	1	no	maximal, Sylow
82	C₂×C₈₃	—	2	2	no	—
83	C₂×C₈₃	—	2	1	no	—
84	C₂×C₈₃	C₄×#106	1	1	yes	characteristic
85	#2155	—	83	2	no	—
86	C₄×M₃₂	D₈₃	1	1	yes	characteristic
87	#1772	—	83	2	no	—
88	#1385	—	83	1	no	—
89	#1384	—	83	1	no	—
90	C₈₃	#12668	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
92	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
93	C₂×#108	—	83	2	no	—
94	#427	—	83	2	no	—
95	#405	—	166	2	no	—
96	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
97	#365	—	83	8	no	—
98	#346	Dic₈₃	1	2	yes	complemented
99	C₄×C₁₆	Dic₈₃	1	2	yes	complemented
100	#339	—	166	2	no	—
101	C₂×M₁₆	—	166	2	no	—
102	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
104	#117	—	166	2	no	—
105	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₄×D₈₃	1	4	yes	—
108	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
110	#108	—	83	4	no	—
111	#108	—	166	2	no	—
112	#107	—	166	2	no	—
113	#106	—	83	4	no	—
114	#97	—	332	2	no	—
115	C₄×C₈	—	332	2	no	—
116	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
117	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
122	M₁₆	—	332	2	no	—
123	M₁₆	—	166	4	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	#13050579	1	2	yes	—
128	C₂×C₈	#13050579	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
130	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
131	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
132	C₂×C₈	#13050590	1	2	yes	—
133	C₄⋊C₄	—	166	4	no	—
134	C₂²⋊C₄	—	332	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	332	2	no	—
137	C₄²	C₄⋊Dic₈₃	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₈₃	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	C₄×Dic₈₃	1	4	yes	—
141	C₂³	#14581814	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	332	4	no	—
146	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
147	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
148	C₂×C₄	C₂×#13050579	1	2	yes	—
149	C₂×C₄	#14581814	1	1	yes	characteristic
150	C₈	—	332	4	no	—
151	C₈	#14581724	1	2	yes	—
152	C₈	#14581724	1	1	yes	characteristic
153	C₈	#14581724	1	1	yes	characteristic
154	C₈	C₂×C₄×Dic₈₃	1	1	yes	characteristic
155	C₈	#14581751	1	1	yes	characteristic
156	C₈	C₄×Dic₁₆₆	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#19127739	1	1	yes	characteristic
159	C₂²	C₄×#13050579	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	664	4	no	—
163	C₄	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
164	C₄	C₂×#14581724	1	1	yes	characteristic
165	C₄	C₄×#13050579	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×#14581724	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14581724
commutator	C₈×C₈₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₄×M₃₂×C₈₃	C₂
socle	C₂×C₈₃	C₄×#106
prime radical	C₂³×C₈₃	#135
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#12668
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
9. #19127524	C₄	semidirect
14. #346×C₈₃	C₄	semidirect
15. C₄×C₁₆×C₈₃	C₄	semidirect
90. C₈₃	#12668	semidirect
98. #346	Dic₈₃	semidirect
99. 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
#24909236	2
#24907342	2
#24905567	2
#24905565	2
C₄×M₃₂×C₈₃	2
#12668	83