Subgroups — #35302971

Order 21248.

378 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	#24908181	C₂	1	2	yes	maximal
2	C₄×#14581726	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127452	C₂	1	1	yes	maximal, characteristic
4	#24905303	C₂	1	2	yes	maximal
5	C₂×#322×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581726	C₄	1	2	yes	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×Dic₈₃⋊C₈	—	2	2	no	—
9	C₄×C₄⋊Dic₈₃	C₂²	1	2	yes	—
10	#19127464	C₄	1	1	yes	characteristic
11	#19127464	C₄	1	1	yes	characteristic
12	#19127455	—	2	2	no	—
13	#19127452	—	2	2	no	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#322×C₈₃	—	2	2	no	—
18	C₂²×C₈₃⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₈₃	—	2	2	no	—
22	#14581726	—	2	4	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	Dic₈₃⋊C₈	—	4	4	no	—
26	C₈₃⋊#95	—	2	2	no	—
27	C₂²×C₈×C₈₃	—	2	1	no	—
28	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	D₄	1	2	yes	—
31	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₈₃	—	2	1	no	—
33	C₄×C₈×C₈₃	—	2	1	no	—
34	C₄×C₈×C₈₃	—	2	1	no	—
35	C₄×C₈×C₈₃	D₄	1	2	yes	—
36	C₄×C₈×C₈₃	Q₈	1	2	yes	—
37	C₄×C₈×C₈₃	Q₈	1	2	yes	—
38	C₂²×Dic₈₃	—	2	2	no	—
39	C₂²×Dic₈₃	—	2	2	no	—
40	C₂×C₈₃⋊C₈	—	4	2	no	—
41	C₂×C₈₃⋊C₈	—	4	2	no	—
42	C₂×C₈₃⋊C₈	—	4	2	no	—
43	C₄⋊Dic₈₃	—	2	2	no	—
44	C₄⋊Dic₈₃	—	2	2	no	—
45	C₄⋊Dic₈₃	—	2	2	no	—
46	C₄⋊Dic₈₃	M₁₆	1	4	yes	—
47	C₄×Dic₈₃	—	4	2	no	—
48	C₄×Dic₈₃	—	4	2	no	—
49	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₈₃	—	2	4	no	—
53	C₂×C₈×C₈₃	—	2	2	no	—
54	C₂×C₈×C₈₃	—	2	2	no	—
55	C₂×C₈×C₈₃	—	2	2	no	—
56	C₂×C₈×C₈₃	—	2	2	no	—
57	C₂×C₈×C₈₃	—	2	2	no	—
58	C₂×C₈×C₈₃	—	2	2	no	—
59	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
61	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₈₃	#41	1	1	yes	characteristic
65	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	—	2	2	no	—
67	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₈₃	#131	1	2	yes	—
75	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₈₃	#137	1	2	yes	—
77	C₂×Dic₈₃	—	2	4	no	—
78	C₂×Dic₈₃	—	4	2	no	—
79	C₂×Dic₈₃	—	4	2	no	—
80	C₂×Dic₈₃	—	4	2	no	—
81	C₂×Dic₈₃	—	4	2	no	—
82	C₂×Dic₈₃	—	4	2	no	—
83	C₈×C₈₃	—	2	4	no	—
84	C₈×C₈₃	—	2	2	no	—
85	C₈×C₈₃	—	2	2	no	—
86	C₈×C₈₃	—	4	2	no	—
87	C₈₃⋊C₈	—	8	2	no	—
88	C₂²×C₈₃	#389	1	1	yes	characteristic
89	C₂²×C₈₃	#435	1	2	yes	—
90	C₂²×C₈₃	#441	1	1	yes	characteristic
91	C₂²×C₈₃	#442	1	1	yes	characteristic
92	C₂²×C₈₃	#443	1	2	yes	—
93	C₄×C₈₃	—	2	1	no	—
94	C₄×C₈₃	—	2	1	no	—
95	C₄×C₈₃	#423	1	1	yes	characteristic
96	C₄×C₈₃	#424	1	1	yes	characteristic
97	C₄×C₈₃	#446	1	2	yes	—
98	C₄×C₈₃	#468	1	2	yes	—
99	C₄×C₈₃	#501	1	2	yes	—
100	Dic₈₃	—	4	4	no	—
101	Dic₈₃	—	8	2	no	—
102	#9599	—	83	1	no	maximal, Sylow
103	C₂×C₈₃	#1586	1	2	yes	—
104	C₂×C₈₃	#1587	1	2	yes	—
105	C₂×C₈₃	#1940	1	1	yes	characteristic
106	C₂×C₈₃	#1943	1	1	yes	characteristic
107	C₂×C₈₃	#1945	1	1	yes	characteristic
108	#1939	—	83	2	no	—
109	C₄×#107	—	83	1	no	—
110	C₂×#334	—	83	1	no	—
111	C₂×#322	D₈₃	1	1	yes	characteristic
112	#1279	—	83	2	no	—
113	C₈₃	#9599	1	1	yes	characteristic, complemented, Sylow
114	C₂×#107	—	83	2	no	—
115	C₂×C₄⋊C₈	—	83	2	no	—
116	C₂×C₄⋊C₈	—	166	2	no	—
117	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	83	2	no	—
121	#340	—	83	1	no	—
122	#340	—	83	1	no	—
123	#336	—	166	2	no	—
124	#334	—	166	2	no	—
125	#322	—	2	2	no	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	166	2	no	—
128	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
129	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
130	C₂²×C₈	D₃₃₂	1	2	yes	—
131	C₂×C₄⋊C₄	—	83	2	no	—
132	C₂×C₄⋊C₄	—	83	2	no	—
133	C₂×C₄²	—	166	2	no	—
134	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
135	#107	—	166	4	no	—
136	C₄⋊C₈	—	332	4	no	—
137	C₄⋊C₈	—	166	2	no	—
138	C₄⋊C₈	—	166	2	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	—	2	1	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	Dic₁₆₆	1	2	yes	—
143	C₄×C₈	Dic₁₆₆	1	2	yes	—
144	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
145	#95	—	166	2	no	—
146	C₂²×C₄	—	166	2	no	—
147	C₂²×C₄	—	166	2	no	—
148	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
150	C₂²×C₄	#13050590	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	332	2	no	—
159	C₂×C₈	—	332	2	no	—
160	C₂×C₈	—	332	2	no	—
161	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
162	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
163	C₄⋊C₄	—	83	4	no	—
164	C₄⋊C₄	—	166	2	no	—
165	C₄⋊C₄	—	166	2	no	—
166	C₄⋊C₄	—	166	2	no	—
167	C₄²	—	332	2	no	—
168	C₄²	—	332	2	no	—
169	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
170	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
171	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
172	C₄²	#13050590	1	1	yes	characteristic
173	C₂³	C₄×D₃₃₂	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	332	2	no	—
176	C₂×C₄	—	332	2	no	—
177	C₂×C₄	—	332	2	no	—
178	C₂×C₄	—	332	2	no	—
179	C₂×C₄	—	332	2	no	—
180	C₂×C₄	—	166	4	no	—
181	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
182	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
183	C₂×C₄	#14581766	1	1	yes	characteristic
184	C₂×C₄	#14581771	1	1	yes	characteristic
185	C₂×C₄	#14581784	1	1	yes	characteristic
186	C₂×C₄	#14581792	1	1	yes	characteristic
187	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
188	C₂×C₄	#14581800	1	2	yes	—
189	C₂×C₄	#14581820	1	2	yes	—
190	C₂×C₄	#14581825	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	664	2	no	—
196	C₂²	#19127599	1	1	yes	characteristic
197	C₂²	#19127774	1	2	yes	—
198	C₂²	#19127809	1	1	yes	characteristic
199	C₂²	#19127814	1	1	yes	characteristic
200	C₂²	#19127819	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	332	4	no	—
204	C₄	—	664	2	no	—
205	C₄	#19127724	1	1	yes	characteristic
206	C₄	#19127729	1	1	yes	characteristic
207	C₄	#19127829	1	2	yes	—
208	C₄	#19127947	1	2	yes	—
209	C₄	#19128110	1	2	yes	—
210	C₂	#24906329	1	2	yes	—
211	C₂	#24906336	1	2	yes	—
212	C₂	#24908186	1	1	yes	characteristic
213	C₂	#24908201	1	1	yes	characteristic
214	C₂	#24908211	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₃₂
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#322×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#322	D₈₃
hypocenter	C₈₃	#9599
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
113. C₈₃	#9599	semidirect

Maximal subgroups

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

subgroup	index
#24908181	2
C₄×#14581726	2
C₂×#19127452	2
#24905303	2
C₂×#322×C₈₃	2
#9599	83