Subgroups — #35319713

Order 21248.

414 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	1	2	yes	maximal
2	C₂×#19127477	C₂	1	2	yes	maximal
3	#24905541	C₂	1	2	yes	maximal
4	C₄²×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₁₆	C₂²	1	4	yes	—
6	C₄⋊(C₈₃⋊C₁₆)	C₄	1	8	yes	complemented
7	#19127477	—	2	4	no	—
8	C₂×C₄×C₈×C₈₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₈₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₈₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₄³×C₈₃	C₄	1	1	yes	characteristic
15	C₂×C₈₃⋊C₁₆	—	2	4	no	—
16	C₂×C₈₃⋊C₁₆	—	2	4	no	—
17	C₂×C₈₃⋊C₁₆	C₂×C₄	1	8	yes	—
18	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₈₃	C₈	1	2	yes	—
23	C₂×C₄²×C₈₃	C₈	1	2	yes	—
24	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₃	—	2	2	no	—
28	C₄×C₈×C₈₃	—	2	2	no	—
29	C₄×C₈×C₈₃	—	2	2	no	—
30	C₄×C₈×C₈₃	C₈	1	4	yes	—
31	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₈₃	D₄	1	2	yes	—
33	C₄×C₈×C₈₃	Q₈	1	2	yes	—
34	C₈₃⋊C₁₆	—	2	8	no	—
35	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₃	—	2	1	no	—
41	C₂×C₈×C₈₃	—	2	1	no	—
42	C₂×C₈×C₈₃	—	2	1	no	—
43	C₂×C₈×C₈₃	—	2	1	no	—
44	C₂×C₈×C₈₃	C₄²	1	2	yes	—
45	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
49	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₈₃	#41	1	2	yes	—
53	C₄²×C₈₃	—	2	4	no	—
54	C₄²×C₈₃	—	2	4	no	—
55	C₄²×C₈₃	—	2	1	no	—
56	C₄²×C₈₃	—	2	1	no	—
57	C₄²×C₈₃	C₄²	1	2	yes	—
58	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
59	C₄²×C₈₃	C₂×C₈	1	2	yes	—
60	C₄²×C₈₃	M₁₆	1	2	yes	—
61	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₈₃	—	2	2	no	—
63	C₂×C₄×C₈₃	—	2	2	no	—
64	C₂×C₄×C₈₃	—	2	2	no	—
65	C₂×C₄×C₈₃	—	2	2	no	—
66	C₂×C₄×C₈₃	—	2	2	no	—
67	C₂×C₄×C₈₃	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₈₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₈₃	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₈₃	M₃₂	1	4	yes	—
71	C₂×C₄×C₈₃	M₃₂	1	4	yes	—
72	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
78	C₈×C₈₃	—	2	2	no	—
79	C₈×C₈₃	—	2	2	no	—
80	C₈×C₈₃	C₄⋊C₈	1	4	yes	—
81	C₈×C₈₃	C₄×D₄	1	2	yes	—
82	C₈×C₈₃	C₄×Q₈	1	2	yes	—
83	C₂²×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₈₃	#431	1	1	yes	characteristic
86	C₂²×C₈₃	C₂×M₃₂	1	2	yes	—
87	C₂²×C₈₃	C₂×M₃₂	1	2	yes	—
88	C₄×C₈₃	—	2	2	no	—
89	C₄×C₈₃	—	2	4	no	—
90	C₄×C₈₃	—	2	4	no	—
91	C₄×C₈₃	#346	1	4	yes	complemented
92	C₄×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₈₃	C₄×M₁₆	1	1	yes	characteristic
94	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₈₃	#431	1	1	yes	characteristic
96	#12653	—	83	1	no	maximal, Sylow
97	C₂×C₈₃	C₄×C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₈₃	C₂×#346	1	1	yes	characteristic
99	C₂×C₈₃	C₄×M₃₂	1	1	yes	characteristic
100	C₂×C₈₃	#2150	1	2	yes	—
101	C₂×C₈₃	#2163	1	2	yes	—
102	C₂×C₄⋊C₁₆	—	83	2	no	—
103	C₂×#346	—	83	2	no	—
104	C₄²×C₈	D₈₃	1	1	yes	characteristic
105	#1374	—	83	2	no	—
106	C₈₃	#12653	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₁₆	—	83	4	no	—
108	C₂×C₄×C₈	Dic₈₃	1	1	yes	characteristic
109	C₂×C₄×C₈	Dic₈₃	1	1	yes	characteristic
110	C₂×C₄×C₈	Dic₈₃	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
114	C₄³	Dic₈₃	1	1	yes	characteristic
115	C₄⋊C₁₆	—	83	8	no	—
116	#346	—	166	4	no	—
117	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
121	C₂×C₄²	C₈₃⋊C₈	1	2	yes	—
122	C₂×C₄²	C₈₃⋊C₈	1	2	yes	—
123	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
126	C₂×C₁₆	—	83	8	no	—
127	C₂×C₁₆	—	166	4	no	—
128	C₂×C₁₆	—	166	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	C₈₃⋊C₈	1	4	yes	—
133	C₄×C₈	Dic₁₆₆	1	2	yes	—
134	C₄×C₈	C₄×D₈₃	1	4	yes	—
135	C₄×C₈	D₃₃₂	1	2	yes	—
136	C₂²×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
139	C₂²×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
146	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
147	C₂×C₈	C₂×C₈₃⋊C₈	1	2	yes	—
148	C₂×C₈	C₂×C₈₃⋊C₈	1	2	yes	—
149	C₂×C₈	#13050574	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
151	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
153	C₂×C₈	#13050590	1	2	yes	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	1	no	—
158	C₄²	C₄×Dic₈₃	1	2	yes	—
159	C₄²	C₄⋊Dic₈₃	1	2	yes	—
160	C₄²	C₂×C₈₃⋊C₈	1	2	yes	—
161	C₄²	#13050574	1	2	yes	—
162	C₁₆	—	166	8	no	—
163	C₂³	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₄×C₈₃⋊C₈	1	2	yes	—
170	C₂×C₄	C₄×C₈₃⋊C₈	1	2	yes	—
171	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	4	yes	—
172	C₂×C₄	#14581732	1	4	yes	—
173	C₂×C₄	#14581732	1	4	yes	—
174	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
176	C₂×C₄	#14581751	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
178	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
179	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	C₄⋊(C₈₃⋊C₈)	1	4	yes	—
183	C₈	C₄×D₃₃₂	1	2	yes	—
184	C₈	C₄×Dic₁₆₆	1	2	yes	—
185	C₂²	C₂×C₄×C₈₃⋊C₈	1	1	yes	characteristic
186	C₂²	C₂×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
187	C₂²	#19127754	1	1	yes	characteristic
188	C₂²	C₂×#14581732	1	2	yes	—
189	C₂²	C₂×#14581732	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	#19127477	1	4	yes	complemented
194	C₄	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
195	C₄	C₄×#13050574	1	1	yes	characteristic
196	C₄	C₂×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
197	C₄	#19127754	1	1	yes	characteristic
198	C₂	C₄×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
199	C₂	C₂×#19127477	1	1	yes	characteristic
200	C₂	C₄×#14581732	1	1	yes	characteristic
201	C₂	#24909219	1	2	yes	—
202	C₂	#24909266	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂×C₄×C₈
Fitting	C₄²×C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₄²×C₈	D₈₃
hypocenter	C₈₃	#12653
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. C₄⋊(C₈₃⋊C₁₆)	C₄	semidirect
91. C₄×C₈₃	#346	semidirect
106. C₈₃	#12653	semidirect
193. C₄	#19127477	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₈₃⋊C₁₆)	2
C₂×#19127477	2
#24905541	2
C₄²×C₈×C₈₃	2
#12653	83