Subgroups — #35298212

Order 21248.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907601	C₂	1	2	yes	maximal
2	#24906222	C₂	1	2	yes	maximal
3	#24906219	C₂	1	2	yes	maximal
4	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic
5	#19127725	C₂²	1	4	yes	—
6	#19127466	—	2	4	no	—
7	#19127454	—	2	4	no	—
8	#19127452	—	2	4	no	—
9	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
12	C₈²×C₈₃	C₄	1	1	yes	characteristic
13	C₈²×C₈₃	C₄	1	1	yes	characteristic
14	C₈²×C₈₃	C₄	1	1	yes	characteristic
15	C₈²×C₈₃	C₄	1	1	yes	characteristic
16	C₂×#13050574	—	2	4	no	—
17	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
18	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
19	C₂²×C₈×C₈₃	D₄	1	2	yes	—
20	C₂²×C₈×C₈₃	Q₈	1	2	yes	—
21	C₂²×C₈×C₈₃	Q₈	1	2	yes	—
22	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₃	D₄	1	2	yes	—
30	C₄×C₈×C₈₃	D₄	1	2	yes	—
31	C₄×C₈×C₈₃	Q₈	1	2	yes	—
32	#13050574	—	4	8	no	—
33	C₂×C₈₃⋊C₈	—	4	4	no	—
34	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₈₃	—	2	4	no	—
38	C₂×C₈×C₈₃	—	2	4	no	—
39	C₂×C₈×C₈₃	—	2	4	no	—
40	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
46	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈₃	—	2	2	no	—
52	C₂×C₄×C₈₃	—	2	2	no	—
53	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₈₃	#135	1	4	yes	—
60	C₈×C₈₃	—	2	2	no	—
61	C₈×C₈₃	—	2	2	no	—
62	C₈×C₈₃	—	2	2	no	—
63	C₈×C₈₃	—	2	2	no	—
64	C₈×C₈₃	#108	1	4	yes	—
65	C₈×C₈₃	#108	1	4	yes	—
66	C₈₃⋊C₈	—	8	4	no	—
67	C₂²×C₈₃	—	2	1	no	—
68	C₂²×C₈₃	—	2	1	no	—
69	C₂²×C₈₃	#384	1	1	yes	characteristic
70	C₂²×C₈₃	#499	1	2	yes	—
71	C₄×C₈₃	—	2	1	no	—
72	C₄×C₈₃	—	2	1	no	—
73	C₄×C₈₃	#427	1	2	yes	—
74	C₄×C₈₃	C₂×#108	1	2	yes	—
75	C₄×C₈₃	C₂×#108	1	2	yes	—
76	C₄×C₈₃	#495	1	2	yes	—
77	#8824	—	83	1	no	maximal, Sylow
78	C₂×C₈₃	—	2	2	no	—
79	C₂×C₈₃	#1846	1	1	yes	characteristic
80	C₂×C₈₃	#1848	1	1	yes	characteristic
81	C₂×C₈₃	#1848	1	1	yes	characteristic
82	#1837	—	83	2	no	—
83	#1567	—	83	2	no	—
84	#1566	—	83	2	no	—
85	C₂×C₈²	D₈₃	1	1	yes	characteristic
86	C₈₃	#8824	1	1	yes	characteristic, complemented, Sylow
87	#423	—	83	4	no	—
88	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
91	#341	—	166	4	no	—
92	#335	—	166	4	no	—
93	#334	—	166	4	no	—
94	C₈²	Dic₈₃	1	1	yes	characteristic
95	C₈²	Dic₈₃	1	1	yes	characteristic
96	C₈²	Dic₈₃	1	1	yes	characteristic
97	C₈²	Dic₈₃	1	1	yes	characteristic
98	C₂×M₁₆	—	166	4	no	—
99	C₂²×C₈	Dic₁₆₆	1	2	yes	—
100	C₂²×C₈	Dic₁₆₆	1	2	yes	—
101	C₂²×C₈	D₃₃₂	1	2	yes	—
102	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
103	C₄⋊C₈	—	166	4	no	—
104	C₄⋊C₈	—	166	4	no	—
105	C₄×C₈	Dic₁₆₆	1	2	yes	—
106	C₄×C₈	D₃₃₂	1	2	yes	—
107	C₄×C₈	D₃₃₂	1	2	yes	—
108	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
114	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
117	M₁₆	—	332	8	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	332	4	no	—
122	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
123	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
124	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
125	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
126	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
128	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
129	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
130	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
131	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
132	C₂³	#14581792	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
138	C₂×C₄	#14581789	1	1	yes	characteristic
139	C₂×C₄	#14581792	1	1	yes	characteristic
140	C₂×C₄	#14581792	1	1	yes	characteristic
141	C₂×C₄	#14581814	1	4	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	664	4	no	—
147	C₈	#14581728	1	4	yes	—
148	C₈	#14581728	1	4	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#19127571	1	1	yes	characteristic
152	C₂²	#19128103	1	2	yes	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#19127739	1	2	yes	—
156	C₄	C₂×#14581728	1	2	yes	—
157	C₄	C₂×#14581728	1	2	yes	—
158	C₄	#19128085	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#24907627	1	1	yes	characteristic
161	C₂	#24907635	1	1	yes	characteristic
162	C₂	#24907635	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581789
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle	C₂²×C₈₃	#384
prime radical	C₂³×C₈₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	#8824
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
86. C₈₃	#8824	semidirect

Maximal subgroups

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

subgroup	index
#24907601	2
#24906222	2
#24906219	2
C₂×C₈²×C₈₃	2
#8824	83