Subgroups — #35303015

Order 21248.

306 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581728	C₂	1	1	yes	maximal, characteristic
2	#24906216	C₂	1	2	yes	maximal
3	C₂×#19127452	C₂	1	1	yes	maximal, characteristic
4	#24905312	C₂	1	2	yes	maximal
5	C₈₃×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581728	C₄	1	1	yes	characteristic
7	C₂×#14581728	C₄	1	1	yes	characteristic
8	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
9	C₄×#13050574	C₂²	1	2	yes	—
10	#19127466	C₄	1	2	yes	—
11	#19127453	—	2	2	no	—
12	#19127452	—	2	1	no	—
13	#19127452	—	2	1	no	—
14	#19127451	—	2	2	no	—
15	#431×C₈₃	C₂²	1	1	yes	characteristic
16	#431×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	#322×C₈₃	—	2	1	no	—
19	C₈²×C₈₃	—	2	1	no	—
20	C₂×#13050574	C₂×C₄	1	2	yes	—
21	C₂×#13050574	C₂×C₄	1	2	yes	—
22	C₂²×C₈₃⋊C₈	—	2	2	no	—
23	#14581728	—	2	2	no	—
24	#14581728	C₈	1	4	yes	complemented
25	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
26	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
27	#14581703	—	2	2	no	—
28	C₄×C₈₃⋊C₈	—	2	2	no	—
29	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₈₃	—	2	1	no	—
33	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
34	C₂²⋊C₈×C₈₃	—	2	1	no	—
35	C₂²⋊C₈×C₈₃	Q₈	1	2	yes	—
36	C₄×C₈×C₈₃	—	2	1	no	—
37	C₄×C₈×C₈₃	—	2	1	no	—
38	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
42	#13050574	—	2	2	no	—
43	#13050574	—	2	2	no	—
44	#13050574	—	2	2	no	—
45	#13050574	C₂×C₈	1	4	yes	—
46	C₂×C₈₃⋊C₈	—	2	2	no	—
47	C₂×C₈₃⋊C₈	—	2	2	no	—
48	C₂×C₈₃⋊C₈	—	4	2	no	—
49	C₂×C₈₃⋊C₈	—	8	2	no	—
50	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₈₃	—	2	1	no	—
54	C₂×C₈×C₈₃	—	2	1	no	—
55	C₂×C₈×C₈₃	—	2	2	no	—
56	C₂×C₈×C₈₃	—	2	2	no	—
57	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
63	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₈₃	#41	1	1	yes	characteristic
67	C₂³×C₈₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈₃	—	2	2	no	—
69	C₂×C₄×C₈₃	—	4	1	no	—
70	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
80	C₈×C₈₃	—	2	1	no	—
81	C₈×C₈₃	—	2	1	no	—
82	C₈×C₈₃	—	4	2	no	—
83	C₈×C₈₃	—	4	2	no	—
84	C₈×C₈₃	C₄⋊C₈	1	2	yes	—
85	C₈×C₈₃	C₄⋊C₈	1	2	yes	—
86	C₈₃⋊C₈	—	2	4	no	—
87	C₈₃⋊C₈	—	4	2	no	—
88	C₈₃⋊C₈	—	8	2	no	—
89	C₂²×C₈₃	—	2	1	no	—
90	C₂²×C₈₃	—	2	1	no	—
91	C₂²×C₈₃	#389	1	1	yes	characteristic
92	C₂²×C₈₃	C₈×Q₈	1	1	yes	characteristic
93	C₂²×C₈₃	#446	1	1	yes	characteristic
94	C₄×C₈₃	—	2	1	no	—
95	C₄×C₈₃	—	2	1	no	—
96	C₄×C₈₃	—	2	1	no	—
97	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₈₃	#424	1	1	yes	characteristic
99	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
100	C₄×C₈₃	#436	1	1	yes	characteristic
101	C₄×C₈₃	#471	1	2	yes	—
102	#9607	—	83	1	no	maximal, Sylow
103	C₂×C₈₃	—	2	1	no	—
104	C₂×C₈₃	—	2	1	no	—
105	C₂×C₈₃	#1939	1	1	yes	characteristic
106	C₂×C₈₃	#1951	1	1	yes	characteristic
107	C₂×C₈₃	#1953	1	1	yes	characteristic
108	C₄×#108	—	83	1	no	—
109	#1565	—	83	2	no	—
110	C₂×#334	—	83	1	no	—
111	#1450	D₈₃	1	1	yes	characteristic
112	#1283	—	83	2	no	—
113	C₈₃	#9607	1	1	yes	characteristic, complemented, Sylow
114	#431	C₂×D₈₃	1	1	yes	characteristic
115	#431	C₂×D₈₃	1	1	yes	characteristic
116	C₂×#108	—	83	1	no	—
117	C₂×#108	—	83	1	no	—
118	C₂×C₄⋊C₈	—	83	2	no	—
119	C₄×M₁₆	—	83	2	no	—
120	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
121	#341	—	83	2	no	—
122	#335	—	166	2	no	—
123	#334	—	166	2	no	—
124	#334	—	166	1	no	—
125	#334	—	166	1	no	—
126	#322	—	2	1	no	—
127	C₈²	—	2	1	no	—
128	C₂×M₁₆	—	83	2	no	—
129	C₂×M₁₆	—	83	2	no	—
130	C₂²×C₈	—	166	2	no	—
131	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
134	#108	—	83	4	no	—
135	#108	—	166	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	166	2	no	—
138	C₄⋊C₈	—	166	2	no	—
139	C₄⋊C₈	D₃₃₂	1	2	yes	—
140	C₂²⋊C₈	—	2	1	no	—
141	C₂²⋊C₈	Dic₁₆₆	1	2	yes	—
142	#97	—	166	2	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	—	166	2	no	—
146	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
147	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
152	C₂²×C₄	#13050590	1	1	yes	characteristic
153	M₁₆	—	83	4	no	—
154	M₁₆	—	166	2	no	—
155	M₁₆	—	166	2	no	—
156	M₁₆	—	166	2	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	332	2	no	—
162	C₂×C₈	—	664	2	no	—
163	C₂×C₈	—	166	2	no	—
164	C₂×C₈	—	166	2	no	—
165	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
166	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
167	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
168	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₈×D₈₃	1	2	yes	—
170	C₂×C₈	#13050573	1	2	yes	—
171	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
172	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
173	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
174	C₄²	#13050590	1	1	yes	characteristic
175	C₂³	C₄×Dic₁₆₆	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
179	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
180	C₂×C₄	#14581766	1	1	yes	characteristic
181	C₂×C₄	#14581771	1	1	yes	characteristic
182	C₂×C₄	#14581784	1	1	yes	characteristic
183	C₂×C₄	#14581792	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
185	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
186	C₂×C₄	#14581798	1	1	yes	characteristic
187	C₂×C₄	#14581798	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	2	no	—
191	C₈	—	4	2	no	—
192	C₈	—	332	2	no	—
193	C₈	—	664	2	no	—
194	C₈	—	166	4	no	—
195	C₈	Dic₈₃⋊C₈	1	2	yes	—
196	C₈	Dic₈₃⋊C₈	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#19127599	1	1	yes	characteristic
200	C₂²	C₈×Dic₁₆₆	1	1	yes	characteristic
201	C₂²	#19127831	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	C₂×Dic₈₃⋊C₈	1	1	yes	characteristic
206	C₄	#19127727	1	1	yes	characteristic
207	C₄	C₈×D₃₃₂	1	1	yes	characteristic
208	C₄	#19127780	1	1	yes	characteristic
209	C₄	#19127964	1	2	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#24908181	1	1	yes	characteristic
213	C₂	#24908243	1	1	yes	characteristic
214	C₂	#24908253	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581766
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1450	C₂
socle	C₂²×C₈₃	#389
prime radical	C₂³×C₈₃	C₄×Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1450	D₈₃
hypocenter	C₈₃	#9607
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
24. #14581728	C₈	semidirect
113. C₈₃	#9607	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581728	2
#24906216	2
C₂×#19127452	2
#24905312	2
C₈₃×#1450	2
#9607	83