Subgroups — #35291313

Order 21248.

266 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909219	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581732	—	2	1	no	—
5	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₈₃	—	2	1	no	—
7	C₄×C₄⋊Dic₈₃	C₄	1	1	yes	characteristic
8	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₈₃⋊C₁₆)	C₄	1	2	yes	complemented
10	#19127455	—	2	1	no	—
11	C₂×M₃₂×C₈₃	—	2	1	no	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	#346×C₈₃	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₈₃	—	2	1	no	—
15	C₂²×C₈₃⋊C₈	—	2	1	no	—
16	C₂×C₄⋊Dic₈₃	—	2	1	no	—
17	C₂×C₄×Dic₈₃	—	2	1	no	—
18	#14581732	—	4	1	no	—
19	#14581732	—	4	1	no	—
20	C₂×C₈₃⋊C₁₆	—	2	2	no	—
21	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
22	C₄⋊(C₈₃⋊C₈)	C₈	1	2	yes	—
23	C₈×Dic₈₃	—	4	1	no	—
24	C₈×Dic₈₃	—	4	1	no	—
25	C₈₃⋊#95	C₈	1	2	yes	—
26	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₈₃	—	2	2	no	—
30	M₃₂×C₈₃	—	2	2	no	—
31	C₂×C₁₆×C₈₃	—	2	2	no	—
32	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
36	C₂²×Dic₈₃	—	2	2	no	—
37	C₂×C₈₃⋊C₈	—	2	2	no	—
38	C₂×C₈₃⋊C₈	—	4	1	no	—
39	C₂×C₈₃⋊C₈	—	4	1	no	—
40	C₄⋊Dic₈₃	—	2	4	no	—
41	C₄×Dic₈₃	—	4	1	no	—
42	C₄×Dic₈₃	—	4	1	no	—
43	C₈₃⋊C₁₆	—	4	2	no	—
44	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
47	C₂×C₈×C₈₃	—	2	1	no	—
48	C₂×C₈×C₈₃	—	2	1	no	—
49	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
54	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₈₃	Q₁₆	1	1	yes	characteristic
57	C₄²×C₈₃	C₄²	1	1	yes	characteristic
58	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₈₃	M₁₆	1	1	yes	characteristic
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	—	4	1	no	—
65	C₂×C₄×C₈₃	#95	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
70	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#104	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#104	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₄⋊C₈	1	1	yes	characteristic
75	C₂×Dic₈₃	—	2	4	no	—
76	C₂×Dic₈₃	—	4	2	no	—
77	C₂×Dic₈₃	—	4	2	no	—
78	C₈×C₈₃	—	2	1	no	—
79	C₈×C₈₃	—	2	1	no	—
80	C₈×C₈₃	#102	1	1	yes	characteristic
81	C₈×C₈₃	#103	1	1	yes	characteristic
82	C₈×C₈₃	#106	1	1	yes	characteristic
83	C₈×C₈₃	#107	1	1	yes	characteristic
84	C₈₃⋊C₈	—	4	2	no	—
85	C₂²×C₈₃	—	2	1	no	—
86	C₂²×C₈₃	—	2	1	no	—
87	C₂²×C₈₃	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₈₃	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₈₃	#336	1	1	yes	characteristic
90	C₄×C₈₃	—	2	1	no	—
91	C₄×C₈₃	—	2	1	no	—
92	C₄×C₈₃	—	2	1	no	—
93	C₄×C₈₃	D₄⋊C₈	1	1	yes	characteristic
94	C₄×C₈₃	Q₈⋊C₈	1	1	yes	characteristic
95	C₄×C₈₃	#339	1	1	yes	characteristic
96	C₄×C₈₃	#340	1	1	yes	characteristic
97	C₄×C₈₃	#364	1	2	yes	—
98	Dic₈₃	—	4	4	no	—
99	#7169	—	83	1	no	maximal, Sylow
100	C₂×C₈₃	—	2	1	no	—
101	C₂×C₈₃	—	2	1	no	—
102	C₂×C₈₃	#1279	1	1	yes	characteristic
103	C₂×C₈₃	#1376	1	1	yes	characteristic
104	C₂×C₈₃	#1378	1	1	yes	characteristic
105	#2150	—	83	1	no	—
106	C₄×M₃₂	D₈₃	1	1	yes	characteristic
107	C₈×C₄⋊C₄	—	83	1	no	—
108	C₈₃	#7169	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₃₂	—	2	1	no	—
110	C₂×M₃₂	—	166	1	no	—
111	C₂×C₄⋊C₈	—	83	1	no	—
112	C₂×C₄×C₈	—	166	1	no	—
113	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
114	C₄×C₄⋊C₄	—	83	1	no	—
115	C₄⋊C₁₆	—	83	2	no	—
116	C₄⋊C₁₆	—	166	1	no	—
117	#346	Dic₈₃	1	2	yes	complemented
118	C₄×C₁₆	—	2	1	no	—
119	#336	—	166	1	no	—
120	C₂²×C₈	—	166	1	no	—
121	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	166	1	no	—
124	C₂×C₄²	—	166	1	no	—
125	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
126	M₃₂	—	2	2	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	—	332	1	no	—
129	M₃₂	—	332	1	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	166	2	no	—
132	C₄⋊C₈	—	83	2	no	—
133	C₄⋊C₈	—	166	1	no	—
134	C₄×C₈	—	332	1	no	—
135	C₄×C₈	—	332	1	no	—
136	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
137	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
138	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
140	#95	—	83	2	no	—
141	C₂²×C₄	—	166	2	no	—
142	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
144	C₂²×C₄	#13050573	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	332	1	no	—
148	C₂×C₈	—	332	1	no	—
149	C₂×C₈	—	166	2	no	—
150	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
152	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
153	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
154	C₂×C₈	D₆₆₄	1	1	yes	characteristic
155	C₂×C₈	#13050579	1	1	yes	characteristic
156	C₂×C₈	#13050579	1	1	yes	characteristic
157	C₂×C₈	Dic₃₃₂	1	1	yes	characteristic
158	C₄⋊C₄	—	166	4	no	—
159	C₄²	—	332	1	no	—
160	C₄²	—	332	1	no	—
161	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
162	C₄²	Dic₈₃⋊C₄	1	1	yes	characteristic
163	C₄²	C₈×D₈₃	1	1	yes	characteristic
164	C₄²	#13050573	1	1	yes	characteristic
165	C₁₆	—	2	4	no	—
166	C₁₆	—	332	2	no	—
167	C₂³	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	332	2	no	—
171	C₂×C₄	—	332	2	no	—
172	C₂×C₄	—	166	4	no	—
173	C₂×C₄	C₈₃⋊#95	1	1	yes	characteristic
174	C₂×C₄	C₈×Dic₈₃	1	1	yes	characteristic
175	C₂×C₄	#14581702	1	1	yes	characteristic
176	C₂×C₄	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
177	C₂×C₄	#14581714	1	1	yes	characteristic
178	C₂×C₄	#14581717	1	1	yes	characteristic
179	C₂×C₄	#14581720	1	1	yes	characteristic
180	C₂×C₄	#14581720	1	1	yes	characteristic
181	C₂×C₄	Dic₈₃⋊C₈	1	1	yes	characteristic
182	C₂×C₄	Dic₈₃⋊C₈	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	332	2	no	—
186	C₈	#14581714	1	1	yes	characteristic
187	C₈	#14581717	1	1	yes	characteristic
188	C₈	#14581724	1	1	yes	characteristic
189	C₈	#14581726	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#19127430	1	1	yes	characteristic
193	C₂²	#19127433	1	1	yes	characteristic
194	C₂²	#19127455	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	332	4	no	—
199	C₄	#19127430	1	1	yes	characteristic
200	C₄	#19127433	1	1	yes	characteristic
201	C₄	#19127461	1	1	yes	characteristic
202	C₄	#19127464	1	1	yes	characteristic
203	C₄	#19127521	1	2	yes	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#24905303	1	1	yes	characteristic
207	C₂	#24905544	1	1	yes	characteristic
208	C₂	#24905550	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₈₃
Fitting	C₄×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	#336
prime radical	C₂³×C₈₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₈₃	C₂²
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#7169
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₈₃⋊C₁₆)	C₄	semidirect
13. #346×C₈₃	C₄	semidirect
108. C₈₃	#7169	semidirect
117. #346	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24909219	2
C₈×C₄⋊Dic₈₃	2
C₄×M₃₂×C₈₃	2
#7169	83