Subgroups — #35319749

Order 21248.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₈₃	C₂	1	2	yes	maximal
3	C₈×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	#24905540	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₈₃	C₂²	1	2	yes	—
10	C₄×C₄⋊Dic₈₃	C₄	1	1	yes	characteristic
11	C₄×C₈₃⋊C₁₆	C₄	1	4	yes	complemented
12	C₁₆×Dic₈₃	—	2	4	no	—
13	#19127455	C₄	1	2	yes	—
14	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₈₃	C₄	1	4	yes	complemented
17	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄⋊Dic₈₃	C₈	1	2	yes	—
19	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
20	C₂×C₈₃⋊C₁₆	—	2	1	no	—
21	C₂×C₈₃⋊C₁₆	—	2	1	no	—
22	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₈₃⋊C₁₆	Q₈	1	2	yes	—
26	C₄⋊(C₈₃⋊C₈)	C₈	1	4	yes	—
27	C₈×Dic₈₃	—	2	2	no	—
28	C₈×Dic₈₃	—	2	2	no	—
29	C₈₃⋊#95	C₈	1	2	yes	—
30	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₈₃	—	2	2	no	—
34	C₂×C₁₆×C₈₃	—	2	2	no	—
35	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
38	C₂²×Dic₈₃	C₂×C₈	1	4	yes	—
39	C₂×C₈₃⋊C₈	—	2	2	no	—
40	C₂×C₈₃⋊C₈	—	2	2	no	—
41	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
42	C₄⋊Dic₈₃	C₁₆	1	8	yes	complemented
43	C₄×Dic₈₃	—	2	2	no	—
44	C₄×Dic₈₃	—	2	2	no	—
45	C₈₃⋊C₁₆	—	2	2	no	—
46	C₈₃⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
47	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₃	—	2	1	no	—
51	C₂×C₈×C₈₃	—	2	1	no	—
52	C₂×C₈×C₈₃	C₄²	1	2	yes	—
53	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
59	C₄²×C₈₃	C₄²	1	2	yes	—
60	C₄²×C₈₃	C₂×C₈	1	2	yes	—
61	C₁₆×C₈₃	—	2	4	no	—
62	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₈₃	—	2	2	no	—
64	C₂×C₄×C₈₃	C₄×C₈	1	2	yes	—
65	C₂×C₄×C₈₃	C₄×C₈	1	2	yes	—
66	C₂×C₄×C₈₃	C₂×C₁₆	1	4	yes	—
67	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
73	C₂×Dic₈₃	—	2	4	no	—
74	C₂×Dic₈₃	—	2	4	no	—
75	C₂×Dic₈₃	C₂×C₁₆	1	8	yes	—
76	C₈×C₈₃	—	2	2	no	—
77	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₈₃	#117	1	1	yes	characteristic
79	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
80	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
81	C₈₃⋊C₈	—	2	4	no	—
82	C₂²×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
84	C₂²×C₈₃	C₈×Q₈	1	1	yes	characteristic
85	C₂²×C₈₃	C₂²×C₁₆	1	2	yes	—
86	C₂²×C₈₃	#504	1	2	yes	—
87	C₄×C₈₃	—	2	2	no	—
88	C₄×C₈₃	C₄×C₁₆	1	4	yes	complemented
89	C₄×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₈₃	#405	1	1	yes	characteristic
91	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
92	C₄×C₈₃	C₈×Q₈	1	1	yes	characteristic
93	Dic₈₃	—	2	8	no	—
94	C₁₆×C₄⋊C₄	—	83	1	no	maximal, Sylow
95	C₂×C₈₃	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₈₃	C₂×C₄×C₁₆	1	1	yes	characteristic
97	C₂×C₈₃	#2108	1	1	yes	characteristic
98	C₂×C₈₃	D₄×C₁₆	1	2	yes	—
99	C₂×C₈₃	Q₈×C₁₆	1	2	yes	—
100	C₂×C₄⋊C₁₆	D₈₃	1	1	yes	characteristic
101	C₂×C₄×C₁₆	—	83	1	no	—
102	C₂×C₄×C₁₆	—	83	2	no	—
103	C₈×C₄⋊C₄	—	83	1	no	—
104	#1374	—	83	2	no	—
105	C₈₃	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₁₆	—	83	1	no	—
107	C₂²×C₁₆	—	83	1	no	—
108	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
109	C₂×C₄⋊C₈	—	83	1	no	—
110	C₂×C₄×C₈	—	83	2	no	—
111	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
112	C₄×C₄⋊C₄	—	83	1	no	—
113	C₄⋊C₁₆	Dic₈₃	1	4	yes	complemented
114	C₄×C₁₆	—	83	4	no	—
115	C₄×C₁₆	—	166	4	no	—
116	#336	—	83	2	no	—
117	C₂²×C₈	—	83	2	no	—
118	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	83	2	no	—
121	C₂×C₄²	—	83	2	no	—
122	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	83	2	no	—
126	C₂×C₁₆	—	83	2	no	—
127	C₂×C₁₆	—	83	2	no	—
128	C₂×C₁₆	—	83	2	no	—
129	C₂×C₁₆	—	166	1	no	—
130	C₂×C₁₆	—	166	1	no	—
131	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	—
132	C₄⋊C₈	—	83	4	no	—
133	C₄×C₈	—	166	2	no	—
134	C₄×C₈	—	166	2	no	—
135	C₄×C₈	C₄×D₈₃	1	2	yes	—
136	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
137	#95	—	83	2	no	—
138	C₂²×C₄	—	83	4	no	—
139	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
140	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	83	4	no	—
145	C₂×C₈	—	166	2	no	—
146	C₂×C₈	—	166	2	no	—
147	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
148	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
149	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
150	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
152	C₂×C₈	#13050589	1	1	yes	characteristic
153	C₂×C₈	#13050591	1	1	yes	characteristic
154	C₄⋊C₄	—	83	8	no	—
155	C₄²	—	166	2	no	—
156	C₄²	—	166	2	no	—
157	C₄²	C₄×Dic₈₃	1	2	yes	—
158	C₄²	C₈×D₈₃	1	2	yes	—
159	C₁₆	—	2	4	no	—
160	C₁₆	—	83	4	no	—
161	C₁₆	—	166	2	no	—
162	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	83	8	no	—
165	C₂×C₄	—	166	4	no	—
166	C₂×C₄	—	166	4	no	—
167	C₂×C₄	C₈×Dic₈₃	1	2	yes	—
168	C₂×C₄	C₈×Dic₈₃	1	2	yes	—
169	C₂×C₄	C₁₆×D₈₃	1	4	yes	—
170	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
171	C₂×C₄	#14581754	1	1	yes	characteristic
172	C₂×C₄	#14581757	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
174	C₂×C₄	#14581800	1	1	yes	characteristic
175	C₂×C₄	#14581800	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	166	4	no	—
178	C₈	C₄⋊C₄×D₈₃	1	1	yes	characteristic
179	C₈	#14581750	1	1	yes	characteristic
180	C₈	D₄×Dic₈₃	1	1	yes	characteristic
181	C₈	Q₈×Dic₈₃	1	1	yes	characteristic
182	C₂²	C₂×C₈×Dic₈₃	1	1	yes	characteristic
183	C₂²	#19127766	1	1	yes	characteristic
184	C₂²	#19127825	1	1	yes	characteristic
185	C₂²	C₂×C₁₆×D₈₃	1	2	yes	—
186	C₂²	#19128119	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	166	8	no	—
189	C₄	C₁₆×Dic₈₃	1	4	yes	complemented
190	C₄	C₄⋊C₄×Dic₈₃	1	1	yes	characteristic
191	C₄	#19127659	1	1	yes	characteristic
192	C₄	#19127766	1	1	yes	characteristic
193	C₄	#19127825	1	1	yes	characteristic
194	C₂	#24907328	1	1	yes	characteristic
195	C₂	C₂×C₁₆×Dic₈₃	1	1	yes	characteristic
196	C₂	#24909062	1	1	yes	characteristic
197	C₂	#24909282	1	2	yes	—
198	C₂	#24909365	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator	C₂×C₈₃	C₂×C₄×C₁₆
Fitting	C₂×C₄⋊C₁₆×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₃
hypocenter	C₈₃	C₁₆×C₄⋊C₄
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
11. C₄×C₈₃⋊C₁₆	C₄	semidirect
16. C₄⋊C₁₆×C₈₃	C₄	semidirect
42. C₄⋊Dic₈₃	C₁₆	semidirect
46. C₈₃⋊C₁₆	C₄⋊C₄	semidirect
88. C₄×C₈₃	C₄×C₁₆	semidirect
105. C₈₃	C₁₆×C₄⋊C₄	semidirect
113. C₄⋊C₁₆	Dic₈₃	semidirect
189. 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
C₂×C₄×C₈₃⋊C₁₆	2
C₂×C₁₆×Dic₈₃	2
C₈×C₄⋊Dic₈₃	2
#24905540	2
C₂×C₄⋊C₁₆×C₈₃	2
C₁₆×C₄⋊C₄	83