Subgroups — #35295653

Order 21248.

430 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	C₄×#14581724	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₈₃	C₂	1	2	yes	maximal
3	C₂×#19127454	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×#14581724	C₄	1	2	yes	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₈×Dic₈₃	—	2	2	no	—
9	C₄×C₄⋊Dic₈₃	C₂²	1	2	yes	—
10	#19127464	C₄	1	2	yes	—
11	#19127455	—	2	2	no	—
12	#19127454	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#322×C₈₃	C₄	1	4	yes	complemented
17	C₂²×C₈₃⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₈₃	—	2	2	no	—
20	#14581724	C₈	1	8	yes	complemented
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	4	yes	—
23	C₈×Dic₈₃	—	4	4	no	—
24	C₈₃⋊#95	—	2	2	no	—
25	C₂²×C₈×C₈₃	—	2	1	no	—
26	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	D₄	1	2	yes	—
29	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₈₃	—	2	1	no	—
31	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₃	Q₈	1	2	yes	—
36	C₂²×Dic₈₃	—	2	4	no	—
37	C₂×C₈₃⋊C₈	—	2	4	no	—
38	C₂×C₈₃⋊C₈	—	4	2	no	—
39	C₂×C₈₃⋊C₈	—	4	2	no	—
40	C₄⋊Dic₈₃	—	2	4	no	—
41	C₄⋊Dic₈₃	C₂×C₈	1	8	yes	—
42	C₄×Dic₈₃	—	4	2	no	—
43	C₄×Dic₈₃	—	4	2	no	—
44	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₈₃	—	2	2	no	—
48	C₂×C₈×C₈₃	—	2	2	no	—
49	C₂×C₈×C₈₃	—	2	2	no	—
50	C₂×C₈×C₈₃	—	2	2	no	—
51	C₂×C₈×C₈₃	—	4	2	no	—
52	C₂×C₈×C₈₃	C₄²	1	2	yes	—
53	C₂×C₈×C₈₃	C₄²	1	2	yes	—
54	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₈₃	C₂×C₈	1	4	yes	—
56	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₈₃	#41	1	1	yes	characteristic
60	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₃	—	2	2	no	—
62	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₈₃	#131	1	2	yes	—
70	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₈₃	#137	1	2	yes	—
72	C₂×Dic₈₃	—	2	8	no	—
73	C₂×Dic₈₃	—	4	4	no	—
74	C₂×Dic₈₃	—	4	4	no	—
75	C₈×C₈₃	—	2	2	no	—
76	C₈×C₈₃	—	2	4	no	—
77	C₈×C₈₃	—	4	2	no	—
78	C₈×C₈₃	C₄×C₈	1	4	yes	complemented
79	C₈₃⋊C₈	—	4	4	no	—
80	C₂²×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₈₃	C₈×D₄	1	2	yes	—
82	C₂²×C₈₃	#440	1	2	yes	—
83	C₂²×C₈₃	#444	1	2	yes	—
84	C₄×C₈₃	—	2	1	no	—
85	C₄×C₈₃	—	2	1	no	—
86	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄×C₈₃	#405	1	1	yes	characteristic
88	C₄×C₈₃	#428	1	2	yes	—
89	C₄×C₈₃	#441	1	1	yes	characteristic
90	C₄×C₈₃	#442	1	1	yes	characteristic
91	C₄×C₈₃	C₈×Q₈	1	2	yes	—
92	Dic₈₃	—	4	8	no	—
93	#8285	—	83	1	no	maximal, Sylow
94	C₂×C₈₃	SD₁₆⋊C₈	1	4	yes	—
95	C₂×C₈₃	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₈₃	#1773	1	1	yes	characteristic
97	C₂×C₈₃	#1776	1	1	yes	characteristic
98	C₄×#106	—	83	1	no	—
99	C₈×C₄⋊C₄	—	83	2	no	—
100	C₂×#335	—	83	1	no	—
101	C₂×#322	D₈₃	1	1	yes	characteristic
102	#1279	—	83	2	no	—
103	C₈₃	#8285	1	1	yes	characteristic, complemented, Sylow
104	C₂×#106	—	83	2	no	—
105	C₂×C₄⋊C₈	—	83	2	no	—
106	C₂×C₄×C₈	—	166	2	no	—
107	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	83	2	no	—
111	#340	—	83	2	no	—
112	#336	—	166	2	no	—
113	#335	—	83	4	no	—
114	#322	Dic₈₃	1	4	yes	complemented
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	166	2	no	—
117	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
119	C₂²×C₈	D₃₃₂	1	2	yes	—
120	C₂×C₄⋊C₄	—	83	4	no	—
121	C₂×C₄²	—	166	2	no	—
122	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
123	#106	—	83	8	no	—
124	C₄⋊C₈	—	83	4	no	—
125	C₄⋊C₈	—	166	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	332	4	no	—
128	C₄×C₈	Dic₁₆₆	1	2	yes	—
129	C₄×C₈	C₄×D₈₃	1	2	yes	—
130	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
131	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
132	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
133	#95	—	166	2	no	—
134	C₂²×C₄	—	166	4	no	—
135	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
137	C₂²×C₄	#13050590	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	332	2	no	—
144	C₂×C₈	—	332	2	no	—
145	C₂×C₈	—	166	4	no	—
146	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
147	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
148	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
149	C₂×C₈	C₈×D₈₃	1	4	yes	—
150	C₄⋊C₄	—	83	8	no	—
151	C₄⋊C₄	—	166	4	no	—
152	C₄²	—	332	2	no	—
153	C₄²	—	332	2	no	—
154	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
155	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
156	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
157	C₄²	#13050590	1	1	yes	characteristic
158	C₂³	C₄×D₃₃₂	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	332	4	no	—
161	C₂×C₄	—	332	4	no	—
162	C₂×C₄	—	166	8	no	—
163	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
165	C₂×C₄	#14581751	1	1	yes	characteristic
166	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
167	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
169	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
170	C₂×C₄	#14581798	1	2	yes	—
171	C₂×C₄	#14581820	1	2	yes	—
172	C₂×C₄	#14581825	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	2	4	no	—
175	C₈	—	4	2	no	—
176	C₈	—	332	4	no	—
177	C₈	C₈×Dic₈₃	1	4	yes	complemented
178	C₂²	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
179	C₂²	C₈×D₃₃₂	1	2	yes	—
180	C₂²	#19127803	1	2	yes	—
181	C₂²	#19127824	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	332	8	no	—
185	C₄	C₂×C₈×Dic₈₃	1	1	yes	characteristic
186	C₄	#19127656	1	1	yes	characteristic
187	C₄	#19127743	1	2	yes	—
188	C₄	#19127809	1	1	yes	characteristic
189	C₄	#19127814	1	1	yes	characteristic
190	C₄	C₈×Dic₁₆₆	1	2	yes	—
191	C₂	#24906277	1	4	yes	—
192	C₂	C₈×C₄⋊Dic₈₃	1	1	yes	characteristic
193	C₂	#24907347	1	1	yes	characteristic
194	C₂	#24907360	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₃₂
commutator	C₄×C₈₃	C₂×C₄×C₈
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	#8285
LCS	G ▷ 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
12. #19127454	C₄	semidirect
16. #322×C₈₃	C₄	semidirect
20. #14581724	C₈	semidirect
78. C₈×C₈₃	C₄×C₈	semidirect
103. C₈₃	#8285	semidirect
114. #322	Dic₈₃	semidirect
177. C₈	C₈×Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581724	2
C₈×C₄⋊Dic₈₃	2
C₂×#19127454	2
#24905303	2
C₂×#322×C₈₃	2
#8285	83