Subgroups — #35298161

Order 21248.

362 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	#24907579	C₂	1	2	yes	maximal
2	#24907208	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127453	C₂	1	2	yes	maximal
4	C₂×#19127452	C₂	1	1	yes	maximal, characteristic
5	C₂×#334×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₄²×Dic₈₃	C₂²	1	1	yes	characteristic
9	#19127455	—	2	4	no	—
10	#19127455	C₄	1	2	yes	—
11	#19127455	C₄	1	2	yes	—
12	#19127453	—	2	4	no	—
13	#19127452	—	2	2	no	—
14	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#334×C₈₃	—	2	2	no	—
17	C₂²×C₈₃⋊C₈	—	2	2	no	—
18	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₈₃	D₄	1	2	yes	—
21	C₂×C₄×Dic₈₃	Q₈	1	2	yes	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
25	C₄×C₈₃⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₈×C₈₃	—	2	2	no	—
27	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₈₃	—	2	2	no	—
31	C₄⋊C₈×C₈₃	—	2	2	no	—
32	C₄×C₈×C₈₃	D₄	1	2	yes	—
33	C₄×C₈×C₈₃	Q₈	1	2	yes	—
34	C₂²×Dic₈₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈₃⋊C₈	—	2	2	no	—
36	C₂×C₈₃⋊C₈	—	2	2	no	—
37	C₂×C₈₃⋊C₈	—	4	2	no	—
38	C₂×C₈₃⋊C₈	—	4	4	no	—
39	C₂×C₈₃⋊C₈	C₄⋊C₄	1	4	yes	—
40	C₄×Dic₈₃	—	2	2	no	—
41	C₄×Dic₈₃	—	2	2	no	—
42	C₄×Dic₈₃	—	2	4	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	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₈₃	—	2	2	no	—
48	C₂×C₈×C₈₃	—	2	2	no	—
49	C₂×C₈×C₈₃	—	4	2	no	—
50	C₂×C₈×C₈₃	—	4	4	no	—
51	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
53	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
57	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	—	2	2	no	—
59	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈₃	#131	1	2	yes	—
66	C₂×C₄×C₈₃	#131	1	2	yes	—
67	C₂×C₄×C₈₃	#135	1	4	yes	—
68	C₂×Dic₈₃	—	2	4	no	—
69	C₂×Dic₈₃	—	2	4	no	—
70	C₂×Dic₈₃	—	2	4	no	—
71	C₈×C₈₃	—	2	2	no	—
72	C₈×C₈₃	—	2	2	no	—
73	C₈×C₈₃	—	8	2	no	—
74	C₈₃⋊C₈	—	2	4	no	—
75	C₈₃⋊C₈	—	8	2	no	—
76	C₂²×C₈₃	#384	1	1	yes	characteristic
77	C₂²×C₈₃	#424	1	2	yes	—
78	C₂²×C₈₃	#427	1	2	yes	—
79	C₂²×C₈₃	#430	1	2	yes	—
80	C₄×C₈₃	—	2	1	no	—
81	C₄×C₈₃	—	2	1	no	—
82	C₄×C₈₃	#424	1	2	yes	—
83	C₄×C₈₃	#424	1	1	yes	characteristic
84	C₄×C₈₃	#424	1	1	yes	characteristic
85	C₄×C₈₃	#495	1	2	yes	—
86	C₄×C₈₃	#499	1	2	yes	—
87	Dic₈₃	—	4	4	no	—
88	#8811	—	83	1	no	maximal, Sylow
89	C₂×C₈₃	#1572	1	4	yes	—
90	C₂×C₈₃	#1843	1	1	yes	characteristic
91	C₂×C₈₃	#1846	1	1	yes	characteristic
92	C₂×C₈₃	#1849	1	1	yes	characteristic
93	#1831	—	83	2	no	—
94	#1744	—	83	1	no	—
95	C₂×#335	—	83	2	no	—
96	C₂×#334	—	83	1	no	—
97	C₂×#334	D₈₃	1	1	yes	characteristic
98	C₈₃	#8811	1	1	yes	characteristic, complemented, Sylow
99	C₂×C₄⋊C₈	—	83	2	no	—
100	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
101	C₂×C₄×C₈	—	83	1	no	—
102	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
103	C₄³	—	83	1	no	—
104	#336	—	83	2	no	—
105	#336	—	83	2	no	—
106	#336	—	166	4	no	—
107	#335	—	166	4	no	—
108	#334	—	2	2	no	—
109	#334	—	166	2	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	83	2	no	—
112	C₂²×C₈	—	166	2	no	—
113	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
115	C₂×C₄²	—	83	2	no	—
116	C₂×C₄²	—	83	2	no	—
117	C₂×C₄²	—	83	2	no	—
118	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	166	2	no	—
122	C₄⋊C₈	—	166	2	no	—
123	C₄×C₈	—	83	2	no	—
124	C₄×C₈	—	83	2	no	—
125	C₄×C₈	Dic₁₆₆	1	2	yes	—
126	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
127	C₂²×C₄	—	83	4	no	—
128	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
129	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
130	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	332	2	no	—
136	C₂×C₈	—	332	4	no	—
137	C₂×C₈	—	83	4	no	—
138	C₂×C₈	—	166	2	no	—
139	C₂×C₈	—	166	2	no	—
140	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
141	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
142	C₄²	—	332	2	no	—
143	C₄²	—	166	2	no	—
144	C₄²	—	166	2	no	—
145	C₄²	—	166	4	no	—
146	C₄²	D₄×D₈₃	1	1	yes	characteristic
147	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
148	C₄²	Q₈×D₈₃	1	1	yes	characteristic
149	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
150	C₂³	C₄⋊C₄×D₈₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	166	4	no	—
153	C₂×C₄	—	166	4	no	—
154	C₂×C₄	—	166	4	no	—
155	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#14581788	1	1	yes	characteristic
158	C₂×C₄	#14581790	1	1	yes	characteristic
159	C₂×C₄	#14581791	1	1	yes	characteristic
160	C₂×C₄	#14581791	1	1	yes	characteristic
161	C₂×C₄	#14581798	1	2	yes	—
162	C₂×C₄	#14581800	1	2	yes	—
163	C₂×C₄	#14581813	1	4	yes	—
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	8	2	no	—
167	C₈	—	664	2	no	—
168	C₈	—	166	4	no	—
169	C₂²	#19127570	1	1	yes	characteristic
170	C₂²	#19127726	1	2	yes	—
171	C₂²	#19127736	1	2	yes	—
172	C₂²	#19127748	1	2	yes	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	332	4	no	—
176	C₄	#19127726	1	2	yes	—
177	C₄	#19127727	1	1	yes	characteristic
178	C₄	#19127729	1	1	yes	characteristic
179	C₄	#19128082	1	2	yes	—
180	C₄	#19128101	1	2	yes	—
181	C₂	#24906240	1	4	yes	—
182	C₂	#24907616	1	1	yes	characteristic
183	C₂	#24907624	1	1	yes	characteristic
184	C₂	#24907636	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₃
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#334×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#334	D₈₃
hypocenter	C₈₃	#8811
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
98. C₈₃	#8811	semidirect

Maximal subgroups

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

subgroup	index
#24907579	2
#24907208	2
C₂×#19127453	2
C₂×#19127452	2
C₂×#334×C₈₃	2
#8811	83