Subgroups — #35298141

Order 21248.

462 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907615	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127451	C₂	1	4	yes	maximal
4	C₈₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈₃⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
11	#19127451	—	2	8	no	—
12	C₂×C₄⋊C₈×C₈₃	—	2	2	no	—
13	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
14	C₄³×C₈₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈₃⋊C₈	—	2	2	no	—
16	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
19	C₄⋊(C₈₃⋊C₈)	—	4	4	no	—
20	C₄×C₈₃⋊C₈	—	2	4	no	—
21	C₄×C₈₃⋊C₈	—	2	2	no	—
22	C₄×C₈₃⋊C₈	C₈	1	4	yes	complemented
23	C₄×C₈₃⋊C₈	D₄	1	4	yes	—
24	C₄×C₈₃⋊C₈	Q₈	1	4	yes	—
25	C₂²×C₈×C₈₃	—	2	2	no	—
26	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₃	D₄	1	2	yes	—
29	C₂×C₄²×C₈₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₈₃	—	2	4	no	—
32	C₄⋊C₈×C₈₃	—	4	4	no	—
33	C₂×C₈₃⋊C₈	—	2	2	no	—
34	C₂×C₈₃⋊C₈	—	2	2	no	—
35	C₂×C₈₃⋊C₈	—	4	2	no	—
36	C₂×C₈₃⋊C₈	—	4	4	no	—
37	C₂×C₈₃⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂×C₈₃⋊C₈	C₄⋊C₄	1	4	yes	—
39	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
40	C₂×C₈₃⋊C₈	M₁₆	1	4	yes	—
41	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₈₃	—	4	2	no	—
47	C₂×C₈×C₈₃	—	4	4	no	—
48	C₄²×C₈₃	—	2	4	no	—
49	C₄²×C₈₃	—	2	2	no	—
50	C₄²×C₈₃	C₂×C₈	1	2	yes	—
51	C₄²×C₈₃	M₁₆	1	2	yes	—
52	C₄²×C₈₃	SD₁₆	1	4	yes	—
53	C₄²×C₈₃	SD₁₆	1	4	yes	—
54	C₄²×C₈₃	C₂×D₄	1	2	yes	—
55	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
56	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₈₃	—	2	2	no	—
58	C₂×C₄×C₈₃	—	2	2	no	—
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	C₄⋊C₈	1	4	yes	—
61	C₂×C₄×C₈₃	#106	1	4	yes	—
62	C₂×C₄×C₈₃	#106	1	4	yes	—
63	C₂×C₄×C₈₃	#108	1	4	yes	—
64	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
68	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
73	C₈×C₈₃	—	8	2	no	—
74	C₈₃⋊C₈	—	2	4	no	—
75	C₈₃⋊C₈	—	8	2	no	—
76	C₈₃⋊C₈	C₄⋊C₈	1	8	yes	complemented
77	C₂²×C₈₃	#384	1	1	yes	characteristic
78	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₈₃	#423	1	1	yes	characteristic
80	C₂²×C₈₃	C₂×#106	1	2	yes	—
81	C₂²×C₈₃	C₂×#108	1	2	yes	—
82	C₄×C₈₃	—	2	4	no	—
83	C₄×C₈₃	—	2	2	no	—
84	C₄×C₈₃	#334	1	8	yes	complemented
85	C₄×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
86	C₄×C₈₃	#423	1	2	yes	—
87	C₄×C₈₃	#492	1	1	yes	characteristic
88	C₄×C₈₃	#492	1	1	yes	characteristic
89	C₄×C₈₃	#498	1	2	yes	—
90	#8805	—	83	1	no	maximal, Sylow
91	C₂×C₈₃	C₂×#334	1	2	yes	—
92	C₂×C₈₃	#1566	1	2	yes	—
93	C₂×C₈₃	#1842	1	1	yes	characteristic
94	C₂×C₈₃	#1844	1	1	yes	characteristic
95	C₂×C₈₃	#1848	1	1	yes	characteristic
96	#1842	—	83	1	no	—
97	#1842	D₈₃	1	1	yes	characteristic
98	C₄²×C₈	—	83	1	no	—
99	C₂×#334	—	83	4	no	—
100	C₈₃	#8805	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊C₈	—	2	2	no	—
102	C₂×C₄⋊C₈	—	83	2	no	—
103	C₂×C₄⋊C₈	—	166	2	no	—
104	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
105	C₂×C₄×C₈	—	83	1	no	—
106	C₂×C₄×C₈	—	83	2	no	—
107	C₂×C₄×C₈	—	83	2	no	—
108	C₂×C₄×C₈	—	83	1	no	—
109	C₄³	C₂×D₈₃	1	1	yes	characteristic
110	#334	—	166	8	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	83	2	no	—
113	C₂²×C₈	—	83	2	no	—
114	C₂²×C₈	—	166	2	no	—
115	C₂×C₄²	Dic₁₆₆	1	2	yes	—
116	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
117	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊D₈₃	1	2	yes	—
119	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
120	C₄⋊C₈	—	2	4	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄⋊C₈	—	332	4	no	—
123	C₄⋊C₈	—	166	4	no	—
124	C₄×C₈	—	83	4	no	—
125	C₄×C₈	—	83	4	no	—
126	C₄×C₈	—	83	4	no	—
127	C₄×C₈	—	166	4	no	—
128	C₄×C₈	—	166	2	no	—
129	C₂²×C₄	Dic₈₃⋊C₄	1	2	yes	—
130	C₂²×C₄	Dic₈₃⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	332	2	no	—
137	C₂×C₈	—	332	4	no	—
138	C₂×C₈	—	83	4	no	—
139	C₂×C₈	—	83	4	no	—
140	C₂×C₈	—	83	4	no	—
141	C₂×C₈	—	83	4	no	—
142	C₂×C₈	—	166	2	no	—
143	C₂×C₈	—	166	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	4	no	—
146	C₄²	C₈×D₈₃	1	2	yes	—
147	C₄²	#13050573	1	2	yes	—
148	C₄²	#13050577	1	4	yes	—
149	C₄²	#13050578	1	4	yes	—
150	C₄²	D₄×D₈₃	1	2	yes	—
151	C₄²	Q₈×D₈₃	1	2	yes	—
152	C₂³	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	Dic₈₃⋊C₈	1	4	yes	—
157	C₂×C₄	#14581723	1	4	yes	—
158	C₂×C₄	#14581723	1	4	yes	—
159	C₂×C₄	#14581727	1	4	yes	—
160	C₂×C₄	C₄⋊C₄×D₈₃	1	2	yes	—
161	C₂×C₄	#14581788	1	1	yes	characteristic
162	C₂×C₄	#14581790	1	1	yes	characteristic
163	C₂×C₄	#14581791	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
165	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
166	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
167	C₂×C₄	C₂×#13050577	1	2	yes	—
168	C₂×C₄	C₂×#13050578	1	2	yes	—
169	C₈	—	8	2	no	—
170	C₈	—	83	8	no	—
171	C₈	—	664	2	no	—
172	C₈	—	166	4	no	—
173	C₂²	#19127570	1	1	yes	characteristic
174	C₂²	C₂×Dic₈₃⋊C₈	1	1	yes	characteristic
175	C₂²	#19127724	1	1	yes	characteristic
176	C₂²	C₂×#14581723	1	2	yes	—
177	C₂²	C₂×#14581727	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	2	4	no	—
180	C₄	#19127451	1	8	yes	complemented
181	C₄	C₄⋊C₈×D₈₃	1	2	yes	—
182	C₄	#19127723	1	2	yes	—
183	C₄	#19128073	1	1	yes	characteristic
184	C₄	#19128075	1	1	yes	characteristic
185	C₄	#19128097	1	2	yes	—
186	C₂	C₂×#19127451	1	2	yes	—
187	C₂	#24906218	1	2	yes	—
188	C₂	#24907614	1	1	yes	characteristic
189	C₂	#24907619	1	1	yes	characteristic
190	C₂	#24907634	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1842	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1842	D₈₃
hypocenter	C₈₃	#8805
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
22. C₄×C₈₃⋊C₈	C₈	semidirect
76. C₈₃⋊C₈	C₄⋊C₈	semidirect
84. C₄×C₈₃	#334	semidirect
100. C₈₃	#8805	semidirect
180. C₄	#19127451	semidirect

Maximal subgroups

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

subgroup	index
#24907615	2
C₄²×C₈₃⋊C₈	2
C₂×#19127451	2
C₈₃×#1842	2
#8805	83