Subgroups — #35323021

Order 21248.

342 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917977	C₂	1	1	yes	maximal, characteristic
2	#24907635	C₂	1	1	yes	maximal, characteristic
3	#24905559	C₂	1	4	yes	maximal
4	C₂×#346×C₈₃	C₂	1	1	yes	maximal, characteristic
5	#19128258	—	2	1	no	—
6	#19128106	C₄	1	2	yes	—
7	#19128103	—	2	1	no	—
8	#19128103	C₄	1	2	yes	—
9	#19128099	—	2	1	no	—
10	C₂×#14581728	—	2	1	no	—
11	C₂×#14581728	C₂²	1	2	yes	—
12	#19127739	—	2	1	no	—
13	#19127739	C₂²	1	2	yes	—
14	#19127725	—	2	1	no	—
15	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#346×C₈₃	—	2	1	no	—
18	#346×C₈₃	—	2	1	no	—
19	C₂×#13050574	—	2	2	no	—
20	#14581792	—	2	2	no	—
21	#14581784	—	2	2	no	—
22	#14581771	—	4	2	no	—
23	C₄×Dic₁₆₆	—	4	2	no	—
24	#14581751	—	2	2	no	—
25	#14581728	—	2	4	no	—
26	#14581728	—	4	2	no	—
27	#14581726	—	2	2	no	—
28	#14581726	C₂×C₄	1	4	yes	—
29	#14581724	—	2	2	no	—
30	#14581724	—	2	2	no	—
31	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
32	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₃	—	2	2	no	—
36	C₂×C₁₆×C₈₃	—	2	2	no	—
37	C₂×C₁₆×C₈₃	Q₈	1	4	yes	complemented
38	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
42	C₂×Dic₁₆₆	—	4	2	no	—
43	#13050574	—	4	2	no	—
44	#13050574	—	4	2	no	—
45	C₂×C₈₃⋊C₈	—	4	2	no	—
46	C₄⋊Dic₈₃	—	2	4	no	—
47	C₄⋊Dic₈₃	—	4	2	no	—
48	Dic₈₃⋊C₄	—	4	4	no	—
49	C₂²⋊Dic₈₃	—	4	2	no	—
50	C₄×Dic₈₃	—	4	2	no	—
51	C₂²×C₄×C₈₃	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
57	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₈₃	#41	1	2	yes	—
61	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₈₃	D₈	1	2	yes	—
64	C₁₆×C₈₃	—	2	4	no	—
65	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₃	—	4	2	no	—
67	C₂×C₄×C₈₃	#102	1	2	yes	—
68	C₂×C₄×C₈₃	#102	1	2	yes	—
69	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₈₃	—	4	4	no	—
76	Dic₁₆₆	—	8	4	no	—
77	C₈×C₈₃	#103	1	2	yes	—
78	C₈×C₈₃	#103	1	2	yes	—
79	C₈×C₈₃	C₄×Q₈	1	2	yes	—
80	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
81	C₈₃⋊C₈	—	8	2	no	—
82	C₂²×C₈₃	—	2	2	no	—
83	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
84	C₂²×C₈₃	#487	1	1	yes	characteristic
85	C₂²×C₈₃	#498	1	1	yes	characteristic
86	C₄×C₈₃	—	2	2	no	—
87	C₄×C₈₃	—	2	2	no	—
88	C₄×C₈₃	#391	1	1	yes	characteristic
89	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
90	C₄×C₈₃	#486	1	1	yes	characteristic
91	C₄×C₈₃	#500	1	1	yes	characteristic
92	Dic₈₃	—	8	4	no	—
93	#13286	—	83	1	no	maximal, Sylow
94	C₂×C₈₃	—	2	1	no	—
95	C₂×C₈₃	—	2	1	no	—
96	C₂×C₈₃	#1990	1	1	yes	characteristic
97	C₂×C₈₃	#2143	1	2	yes	—
98	#3160	—	83	1	no	—
99	C₂×#346	D₈₃	1	1	yes	characteristic
100	#1848	—	83	1	no	—
101	#1382	—	83	4	no	—
102	C₈₃	#13286	1	1	yes	characteristic, complemented, Sylow
103	#533	—	166	1	no	—
104	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
105	#500	—	83	2	no	—
106	#499	—	83	2	no	—
107	#499	—	166	1	no	—
108	#498	—	166	1	no	—
109	C₂×#108	—	83	2	no	—
110	C₂×#108	—	166	1	no	—
111	#427	—	83	2	no	—
112	#427	—	166	1	no	—
113	#423	—	166	1	no	—
114	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
115	#346	—	2	1	no	—
116	#346	—	2	1	no	—
117	C₂×M₁₆	—	166	2	no	—
118	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
120	C₄⋊Q₈	—	166	2	no	—
121	#125	—	166	2	no	—
122	C₂²⋊Q₈	—	332	2	no	—
123	C₄×Q₈	—	332	2	no	—
124	#117	—	166	2	no	—
125	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	Dic₁₆₆	1	4	yes	complemented
129	#108	—	332	2	no	—
130	#108	—	166	4	no	—
131	#107	—	83	4	no	—
132	#107	—	166	2	no	—
133	#106	—	166	2	no	—
134	#106	—	166	2	no	—
135	C₄⋊C₈	—	332	2	no	—
136	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
137	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
140	C₂×Q₈	—	332	2	no	—
141	C₂²×C₄	#13050579	1	2	yes	—
142	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
143	M₁₆	—	332	2	no	—
144	M₁₆	—	332	2	no	—
145	C₂×C₈	—	332	2	no	—
146	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	#13050579	1	2	yes	—
149	C₂×C₈	Dic₃₃₂	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
153	C₂×C₈	#13050590	1	2	yes	—
154	C₄⋊C₄	—	332	4	no	—
155	C₄⋊C₄	—	332	2	no	—
156	C₄⋊C₄	—	166	4	no	—
157	C₂²⋊C₄	—	332	2	no	—
158	C₄²	—	332	2	no	—
159	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
160	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
161	C₄²	D₆₆₄	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	C₂×#13050579	1	1	yes	characteristic
164	Q₈	—	664	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	332	4	no	—
167	C₂×C₄	#14581714	1	2	yes	—
168	C₂×C₄	#14581714	1	2	yes	—
169	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
170	C₂×C₄	#14581780	1	1	yes	characteristic
171	C₂×C₄	#14581792	1	1	yes	characteristic
172	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
173	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
175	C₈	—	664	2	no	—
176	C₈	#14581717	1	2	yes	—
177	C₈	#14581717	1	2	yes	—
178	C₈	C₄×Dic₁₆₆	1	2	yes	—
179	C₈	#14581771	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#14581714	1	1	yes	characteristic
182	C₂²	#19128052	1	1	yes	characteristic
183	C₂²	#19128099	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	664	4	no	—
187	C₄	#19127607	1	1	yes	characteristic
188	C₄	C₂×#14581717	1	1	yes	characteristic
189	C₄	#19128048	1	1	yes	characteristic
190	C₄	#19128106	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#24908424	1	1	yes	characteristic
194	C₂	#24909192	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₄
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#346×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#102
prime radical	C₂³×C₈₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#346	D₈₃
hypocenter	C₈₃	#13286
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
37. C₂×C₁₆×C₈₃	Q₈	semidirect
102. C₈₃	#13286	semidirect
128. C₂×C₁₆	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
#24917977	2
#24907635	2
#24905559	2
C₂×#346×C₈₃	2
#13286	83