Subgroups — #35323025

Order 21248.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 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	#24907620	C₂	1	1	yes	maximal, characteristic
3	#24905569	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
5	#19128258	—	2	1	no	—
6	#19128106	C₄	1	2	yes	complemented
7	#19128103	—	2	1	no	—
8	#19128103	C₄	1	2	yes	complemented
9	#19128099	—	2	1	no	—
10	#19127739	—	2	1	no	—
11	#19127739	C₂²	1	2	yes	—
12	C₂×#14581724	—	2	1	no	—
13	C₂×#14581724	C₂²	1	2	yes	—
14	#19127571	—	2	1	no	—
15	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#346×C₈₃	—	2	1	no	—
18	C₄×C₁₆×C₈₃	—	2	1	no	—
19	#14581792	—	2	2	no	—
20	#14581784	—	2	2	no	—
21	#14581771	—	4	2	no	—
22	C₄×Dic₁₆₆	—	4	2	no	—
23	#14581751	—	2	2	no	—
24	C₂×C₄⋊Dic₈₃	—	2	2	no	—
25	C₂×C₄⋊Dic₈₃	—	4	2	no	—
26	#14581726	—	2	2	no	—
27	#14581726	C₂×C₄	1	4	yes	—
28	#14581724	—	2	2	no	—
29	#14581724	—	2	2	no	—
30	#14581724	—	2	4	no	—
31	#14581724	—	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	M₃₂×C₈₃	—	2	2	no	—
36	M₃₂×C₈₃	Q₈	1	4	yes	complemented
37	C₂×C₁₆×C₈₃	—	2	2	no	—
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	C₂²×Dic₈₃	—	4	2	no	—
44	C₄⋊Dic₈₃	—	2	4	no	—
45	C₄⋊Dic₈₃	—	4	2	no	—
46	C₄⋊Dic₈₃	—	4	2	no	—
47	C₄⋊Dic₈₃	—	4	2	no	—
48	C₄⋊Dic₈₃	—	8	4	no	—
49	Dic₈₃⋊C₄	—	4	4	no	—
50	C₂²⋊Dic₈₃	—	4	2	no	—
51	C₄×Dic₈₃	—	4	2	no	—
52	C₂²×C₄×C₈₃	D₈	1	2	yes	—
53	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
57	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
58	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₈₃	#41	1	2	yes	—
62	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₈₃	SD₁₆	1	2	yes	—
65	C₁₆×C₈₃	—	2	4	no	—
66	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	4	no	—
68	C₂×C₄×C₈₃	#102	1	2	yes	—
69	C₂×C₄×C₈₃	#102	1	2	yes	—
70	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×Dic₈₃	—	4	4	no	—
77	C₂×Dic₈₃	—	8	4	no	—
78	C₂×Dic₈₃	—	8	2	no	—
79	Dic₁₆₆	—	8	4	no	—
80	C₈×C₈₃	#103	1	2	yes	—
81	C₈×C₈₃	#103	1	2	yes	—
82	C₈×C₈₃	C₄×Q₈	1	2	yes	—
83	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
84	C₂²×C₈₃	—	2	2	no	—
85	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
86	C₂²×C₈₃	#486	1	1	yes	characteristic
87	C₂²×C₈₃	#500	1	1	yes	characteristic
88	C₄×C₈₃	—	2	2	no	—
89	C₄×C₈₃	#360	1	4	yes	—
90	C₄×C₈₃	#391	1	1	yes	characteristic
91	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
92	C₄×C₈₃	#487	1	1	yes	characteristic
93	C₄×C₈₃	#498	1	1	yes	characteristic
94	Dic₈₃	—	8	4	no	—
95	Dic₈₃	—	16	2	no	—
96	#13287	—	83	1	no	maximal, Sylow
97	C₂×C₈₃	—	2	1	no	—
98	C₂×C₈₃	—	2	1	no	—
99	C₂×C₈₃	#1990	1	1	yes	characteristic
100	C₂×C₈₃	C₂×#360	1	2	yes	—
101	#3160	—	83	1	no	—
102	C₄×M₃₂	D₈₃	1	1	yes	characteristic
103	#1844	—	83	1	no	—
104	#1386	—	83	4	no	—
105	C₈₃	#13287	1	1	yes	characteristic, complemented, Sylow
106	#533	—	166	1	no	—
107	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
108	#500	—	83	2	no	—
109	#499	—	83	2	no	—
110	#499	—	166	1	no	—
111	#498	—	166	1	no	—
112	#427	—	83	2	no	—
113	#427	—	166	1	no	—
114	C₂×#106	—	83	2	no	—
115	C₂×#106	—	166	1	no	—
116	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
117	#384	—	166	1	no	—
118	#346	—	2	1	no	—
119	C₄×C₁₆	—	2	1	no	—
120	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
122	C₄⋊Q₈	—	166	2	no	—
123	#125	—	166	2	no	—
124	C₂²⋊Q₈	—	332	2	no	—
125	C₄×Q₈	—	332	2	no	—
126	#117	—	166	2	no	—
127	C₂×C₄⋊C₄	—	332	2	no	—
128	C₂×C₄⋊C₄	—	166	2	no	—
129	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	Dic₁₆₆	1	4	yes	complemented
132	C₂×C₁₆	—	2	2	no	—
133	#107	—	83	4	no	—
134	#107	—	166	2	no	—
135	#106	—	332	2	no	—
136	#106	—	166	4	no	—
137	#106	—	166	2	no	—
138	#106	—	166	2	no	—
139	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
140	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
143	C₂×Q₈	—	332	2	no	—
144	C₂²×C₄	—	332	2	no	—
145	C₂²×C₄	D₆₆₄	1	2	yes	—
146	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
149	C₂×C₈	#13050579	1	2	yes	—
150	C₂×C₈	Dic₃₃₂	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
153	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
154	C₂×C₈	#13050590	1	2	yes	—
155	C₄⋊C₄	—	332	4	no	—
156	C₄⋊C₄	—	332	2	no	—
157	C₄⋊C₄	—	332	2	no	—
158	C₄⋊C₄	—	332	2	no	—
159	C₄⋊C₄	—	664	4	no	—
160	C₄⋊C₄	—	166	4	no	—
161	C₂²⋊C₄	—	332	2	no	—
162	C₄²	—	332	2	no	—
163	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
164	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
165	C₄²	#13050579	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
168	Q₈	—	664	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	332	4	no	—
171	C₂×C₄	—	664	4	no	—
172	C₂×C₄	—	664	2	no	—
173	C₂×C₄	#14581714	1	2	yes	—
174	C₂×C₄	#14581714	1	2	yes	—
175	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#14581780	1	1	yes	characteristic
177	C₂×C₄	#14581792	1	1	yes	characteristic
178	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
179	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
180	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
181	C₈	#14581717	1	2	yes	—
182	C₈	#14581717	1	2	yes	—
183	C₈	C₄×Dic₁₆₆	1	2	yes	—
184	C₈	#14581771	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#14581714	1	1	yes	characteristic
187	C₂²	#19128048	1	1	yes	characteristic
188	C₂²	#19128106	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	1328	2	no	—
191	C₄	—	664	4	no	—
192	C₄	#19127512	1	4	yes	—
193	C₄	#19127607	1	1	yes	characteristic
194	C₄	C₂×#14581717	1	1	yes	characteristic
195	C₄	#19128052	1	1	yes	characteristic
196	C₄	#19128099	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#24908424	1	1	yes	characteristic
200	C₂	C₂×#19127512	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14581714
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₄×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#102
prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#13287
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
6. #19128106	C₄	semidirect
8. #19128103	C₄	semidirect
36. M₃₂×C₈₃	Q₈	semidirect
105. C₈₃	#13287	semidirect
131. M₃₂	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
#24907620	2
#24905569	2
C₄×M₃₂×C₈₃	2
#13287	83