Subgroups — #35316022

Order 21248.

238 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909266	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#24905533	C₂	1	2	yes	maximal, complemented
4	#24905527	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
10	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₈₃⋊C₁₆)	—	2	1	no	—
12	#19127477	—	2	1	no	—
13	#431×C₈₃	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₈₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#322×C₈₃	C₄	1	2	yes	—
17	C₈²×C₈₃	C₂²	1	2	yes	—
18	C₂×C₈₃⋊C₁₆	—	2	2	no	—
19	C₂×C₈₃⋊C₁₆	—	2	1	no	—
20	C₂×C₈₃⋊C₁₆	—	2	1	no	—
21	C₂×C₈₃⋊C₁₆	—	4	2	no	—
22	C₂×M₁₆×C₈₃	—	2	1	no	—
23	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₈×C₈₃	C₈	1	2	yes	—
27	C₂²⋊C₈×C₈₃	C₈	1	2	yes	—
28	#97×C₈₃	—	2	1	no	—
29	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
35	C₈₃⋊C₁₆	—	4	1	no	—
36	C₈₃⋊C₁₆	—	4	2	no	—
37	C₈₃⋊C₁₆	—	4	1	no	—
38	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
41	M₁₆×C₈₃	—	2	4	no	—
42	C₂×C₈×C₈₃	—	2	2	no	—
43	C₂×C₈×C₈₃	—	2	1	no	—
44	C₂×C₈×C₈₃	—	2	1	no	—
45	C₂×C₈×C₈₃	—	4	1	no	—
46	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₈₃	M₁₆	1	1	yes	characteristic
54	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂³×C₈₃	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₈₃	—	2	2	no	—
57	C₂×C₄×C₈₃	—	2	1	no	—
58	C₂×C₄×C₈₃	—	2	1	no	—
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₈₃	—	2	4	no	—
67	C₈×C₈₃	—	2	1	no	—
68	C₈×C₈₃	—	2	1	no	—
69	C₈×C₈₃	—	2	1	no	—
70	C₈×C₈₃	—	2	4	no	—
71	C₈×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₈×C₈₃	#137	1	1	yes	characteristic
73	C₂²×C₈₃	—	2	1	no	—
74	C₂²×C₈₃	—	2	1	no	—
75	C₂²×C₈₃	#364	1	2	yes	—
76	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₄×C₈₃	—	2	1	no	—
78	C₄×C₈₃	—	2	1	no	—
79	C₄×C₈₃	—	2	1	no	—
80	C₄×C₈₃	—	2	1	no	—
81	C₄×C₈₃	#428	1	1	yes	characteristic
82	C₄×C₈₃	#430	1	1	yes	characteristic
83	C₄×C₈₃	#504	1	2	yes	—
84	#12016	—	83	1	no	maximal, Sylow
85	C₂×C₈₃	—	2	2	no	—
86	C₂×C₈₃	#1565	1	1	yes	characteristic
87	C₂×C₈₃	#2151	1	1	yes	characteristic
88	C₂×C₈₃	C₂×#364	1	1	yes	characteristic
89	#2163	—	83	1	no	—
90	C₂×C₄⋊C₁₆	—	83	1	no	—
91	C₈×M₁₆	D₈₃	1	1	yes	characteristic
92	#1370	—	83	2	no	—
93	#1367	—	83	2	no	—
94	C₈₃	#12016	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	166	1	no	—
96	#431	Dic₈₃	1	1	yes	characteristic
97	C₄×M₁₆	Dic₈₃	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
99	C₄⋊C₁₆	—	83	1	no	—
100	C₄⋊C₁₆	—	83	2	no	—
101	C₄⋊C₁₆	—	83	1	no	—
102	C₄⋊C₁₆	—	166	1	no	—
103	C₂²⋊C₁₆	—	166	1	no	—
104	#346	—	166	1	no	—
105	#322	Dic₈₃	1	2	yes	—
106	C₈²	C₂×D₈₃	1	2	yes	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
109	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
111	C₂×C₁₆	—	332	2	no	—
112	C₂×C₁₆	—	166	1	no	—
113	C₂×C₁₆	—	166	1	no	—
114	C₂×C₁₆	—	166	2	no	—
115	C₄⋊C₈	C₈₃⋊C₈	1	2	yes	—
116	C₂²⋊C₈	C₈₃⋊C₈	1	2	yes	—
117	#97	—	2	1	no	—
118	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
119	C₄×C₈	D₃₃₂	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
121	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
122	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
123	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
126	C₂²×C₄	#13050574	1	1	yes	characteristic
127	M₁₆	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
134	C₂×C₈	C₂×C₈₃⋊C₈	1	4	yes	—
135	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
137	C₄²	C₄⋊Dic₈₃	1	1	yes	characteristic
138	C₄²	C₂×C₈₃⋊C₈	1	1	yes	characteristic
139	C₄²	#13050574	1	1	yes	characteristic
140	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
141	C₁₆	—	332	2	no	—
142	C₁₆	—	332	1	no	—
143	C₁₆	—	332	1	no	—
144	C₂³	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
150	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
151	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
153	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
154	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	2	4	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	#14581820	1	1	yes	characteristic
161	C₈	#14581825	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#19127522	1	2	yes	—
165	C₂²	C₂×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#19127743	1	1	yes	characteristic
171	C₄	#19127750	1	1	yes	characteristic
172	C₄	#19128118	1	2	yes	—
173	C₂	—	2	2	no	—
174	C₂	#24906215	1	1	yes	characteristic
175	C₂	#24909222	1	1	yes	characteristic
176	C₂	C₂×#19127522	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈×M₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₈
prime radical	C₂³×C₈₃	C₄⋊C₈
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₈×M₁₆	D₈₃
hypocenter	C₈₃	#12016
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #24905533	C₂	semidirect
4. #24905527	C₂	semidirect
94. C₈₃	#12016	semidirect

Maximal subgroups

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

subgroup	index
#24909266	2
C₂×C₄⋊(C₈₃⋊C₁₆)	2
#24905533	2
#24905527	2
C₈×M₁₆×C₈₃	2
#12016	83