Subgroups — #35315993

Order 21248.

242 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24906211	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
3	#24905537	C₂	1	2	yes	maximal, complemented
4	#24905535	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#19127754	—	2	1	no	—
7	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
8	#19127454	—	2	1	no	—
9	#19127454	C₂²	1	1	yes	characteristic
10	#19127454	C₂²	1	2	yes	—
11	#19127454	C₂²	1	1	yes	characteristic
12	#19127452	—	2	1	no	—
13	C₂×M₃₂×C₈₃	C₄	1	1	yes	characteristic
14	C₂×M₃₂×C₈₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#346×C₈₃	C₄	1	2	yes	—
17	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
18	C₂²×C₈₃⋊C₈	—	4	1	no	—
19	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
20	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
23	C₂²⋊(C₈₃⋊C₈)	—	4	1	no	—
24	C₄×C₈₃⋊C₈	—	4	1	no	—
25	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
28	M₃₂×C₈₃	—	2	2	no	—
29	M₃₂×C₈₃	C₈	1	4	yes	complemented
30	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
36	C₂×C₈₃⋊C₈	—	4	2	no	—
37	C₂×C₈₃⋊C₈	—	4	1	no	—
38	C₂×C₈₃⋊C₈	—	4	1	no	—
39	C₂×C₈₃⋊C₈	—	8	2	no	—
40	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₈₃	Q₁₆	1	1	yes	characteristic
43	C₂×C₈×C₈₃	—	2	2	no	—
44	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
48	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₈₃	D₈	1	1	yes	characteristic
52	C₄²×C₈₃	Q₁₆	1	1	yes	characteristic
53	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₁₆×C₈₃	—	2	2	no	—
55	C₁₆×C₈₃	C₂×C₈	1	4	yes	complemented
56	C₂³×C₈₃	#107	1	1	yes	characteristic
57	C₂×C₄×C₈₃	—	2	1	no	—
58	C₂×C₄×C₈₃	—	2	1	no	—
59	C₂×C₄×C₈₃	—	4	1	no	—
60	C₂×C₄×C₈₃	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
63	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#108	1	2	yes	—
65	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₈₃	—	2	1	no	—
69	C₈×C₈₃	—	2	1	no	—
70	C₈×C₈₃	C₄⋊C₈	1	2	yes	—
71	C₈×C₈₃	C₂²×C₈	1	1	yes	characteristic
72	C₈×C₈₃	C₂×M₁₆	1	1	yes	characteristic
73	C₈₃⋊C₈	—	8	1	no	—
74	C₈₃⋊C₈	—	8	2	no	—
75	C₈₃⋊C₈	—	8	1	no	—
76	C₂²×C₈₃	—	2	1	no	—
77	C₂²×C₈₃	—	2	1	no	—
78	C₂²×C₈₃	#335	1	2	yes	—
79	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
80	C₄×C₈₃	—	2	1	no	—
81	C₄×C₈₃	—	2	1	no	—
82	C₄×C₈₃	—	2	1	no	—
83	C₄×C₈₃	#335	1	2	yes	—
84	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
86	C₄×C₈₃	#509	1	1	yes	characteristic
87	C₄×C₈₃	#510	1	1	yes	characteristic
88	#12009	—	83	1	no	maximal, Sylow
89	C₂×C₈₃	—	2	2	no	—
90	C₂×C₈₃	C₂×#335	1	1	yes	characteristic
91	C₂×C₈₃	#2159	1	1	yes	characteristic
92	C₂×C₈₃	#2161	1	1	yes	characteristic
93	C₄×M₃₂	D₈₃	1	1	yes	characteristic
94	#1564	—	83	1	no	—
95	C₂×#335	—	83	1	no	—
96	#1372	—	83	2	no	—
97	#1371	—	83	2	no	—
98	C₈₃	#12009	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₃₂	Dic₈₃	1	1	yes	characteristic
100	C₂×M₃₂	Dic₈₃	1	1	yes	characteristic
101	#431	—	166	1	no	—
102	C₂×C₄⋊C₈	—	166	1	no	—
103	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
104	#346	Dic₈₃	1	2	yes	—
105	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
106	#335	—	83	2	no	—
107	#335	—	83	1	no	—
108	#335	—	83	1	no	—
109	#335	—	166	1	no	—
110	#334	—	166	1	no	—
111	C₂²×C₈	—	332	1	no	—
112	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
114	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	C₈₃⋊C₈	1	4	yes	complemented
117	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
118	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
119	C₄⋊C₈	—	332	1	no	—
120	C₄⋊C₈	—	166	1	no	—
121	C₄⋊C₈	—	166	1	no	—
122	C₄⋊C₈	—	166	2	no	—
123	C₂²⋊C₈	—	332	1	no	—
124	C₄×C₈	—	332	1	no	—
125	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
126	C₄×C₈	D₃₃₂	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
128	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
129	C₂²×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
130	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
131	C₂²×C₄	Dic₃₃₂	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	332	1	no	—
134	C₂×C₈	—	332	1	no	—
135	C₂×C₈	—	332	2	no	—
136	C₂×C₈	—	664	2	no	—
137	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
138	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
139	C₂×C₈	C₂×C₈₃⋊C₈	1	2	yes	—
140	C₂×C₈	#13050574	1	2	yes	—
141	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
143	C₄²	C₄⋊Dic₈₃	1	1	yes	characteristic
144	C₄²	D₆₆₄	1	1	yes	characteristic
145	C₄²	Dic₃₃₂	1	1	yes	characteristic
146	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
147	C₁₆	—	2	2	no	—
148	C₁₆	C₂×C₈₃⋊C₈	1	4	yes	complemented
149	C₂³	#14581726	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	2	yes	—
154	C₂×C₄	#14581726	1	1	yes	characteristic
155	C₂×C₄	#14581726	1	1	yes	characteristic
156	C₂×C₄	#14581726	1	1	yes	characteristic
157	C₂×C₄	#14581728	1	2	yes	—
158	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
159	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	664	2	no	—
164	C₈	—	664	1	no	—
165	C₈	—	664	1	no	—
166	C₈	C₄⋊(C₈₃⋊C₈)	1	2	yes	—
167	C₈	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
168	C₈	C₂×#13050574	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#19127454	1	2	yes	—
172	C₂²	C₂×#14581726	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#19127454	1	2	yes	—
177	C₄	C₂×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
178	C₄	C₂×#14581728	1	1	yes	characteristic
179	C₄	#19128139	1	1	yes	characteristic
180	C₄	#19128144	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×#19127454	1	1	yes	characteristic
183	C₂	#24909251	1	1	yes	characteristic
184	C₂	#24909258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₄
commutator	C₈×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₄×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#107
prime radical	C₂³×C₈₃	#107
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#12009
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #24905537	C₂	semidirect
4. #24905535	C₂	semidirect
29. M₃₂×C₈₃	C₈	semidirect
55. C₁₆×C₈₃	C₂×C₈	semidirect
98. C₈₃	#12009	semidirect
116. M₃₂	C₈₃⋊C₈	semidirect
148. C₁₆	C₂×C₈₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#24906211	2
C₂×#19127454	2
#24905537	2
#24905535	2
C₄×M₃₂×C₈₃	2
#12009	83