Subgroups — #35322922

Order 21248.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#24905540	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₆₆	C₄	1	1	yes	characteristic
6	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
7	C₈×Dic₁₆₆	—	2	2	no	—
8	C₈×Dic₁₆₆	—	2	2	no	—
9	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×Dic₈₃⋊C₈	C₄	1	2	yes	—
11	C₂×C₈×Dic₈₃	C₂²	1	2	yes	—
12	C₄×C₈₃⋊C₁₆	—	2	2	no	—
13	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₈₃	—	2	2	no	—
16	C₂²×Dic₁₆₆	C₈	1	1	yes	characteristic
17	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₆₆	—	2	2	no	—
19	C₄×Dic₁₆₆	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	C₈	1	1	yes	characteristic
21	C₂×Dic₈₃⋊C₄	C₈	1	2	yes	—
22	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
23	C₂×C₈₃⋊C₁₆	—	2	2	no	—
24	C₂×C₈₃⋊C₁₆	—	2	2	no	—
25	C₂×C₈₃⋊C₁₆	Q₈	1	4	yes	complemented
26	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
27	Dic₈₃⋊C₈	—	2	4	no	—
28	C₈×Dic₈₃	—	2	2	no	—
29	C₈×Dic₈₃	—	2	2	no	—
30	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₈₃	—	2	2	no	—
34	C₂×C₁₆×C₈₃	—	2	4	no	—
35	C₄×C₈×C₈₃	D₄	1	2	yes	—
36	C₄×C₈×C₈₃	D₄	1	2	yes	—
37	C₂×Dic₁₆₆	—	2	2	no	—
38	C₂×Dic₁₆₆	—	2	2	no	—
39	C₂×Dic₁₆₆	C₁₆	1	4	yes	complemented
40	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
41	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
42	C₂×C₈₃⋊C₈	—	2	2	no	—
43	C₂×C₈₃⋊C₈	—	2	4	no	—
44	C₄⋊Dic₈₃	—	2	2	no	—
45	Dic₈₃⋊C₄	—	2	4	no	—
46	C₄×Dic₈₃	—	2	2	no	—
47	C₄×Dic₈₃	—	2	2	no	—
48	C₈₃⋊C₁₆	—	2	4	no	—
49	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₃	—	2	2	no	—
53	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₈₃	#41	1	2	yes	—
58	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₈₃	M₁₆	1	2	yes	—
60	C₁₆×C₈₃	—	4	2	no	—
61	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₈₃	—	2	2	no	—
63	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₈₃	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₈₃	M₃₂	1	2	yes	—
67	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₈₃	#131	1	2	yes	—
71	C₂×Dic₈₃	—	2	2	no	—
72	C₂×Dic₈₃	—	2	2	no	—
73	C₂×Dic₈₃	—	2	2	no	—
74	C₂×Dic₈₃	—	2	4	no	—
75	C₂×Dic₈₃	C₂×C₁₆	1	4	yes	—
76	Dic₁₆₆	—	2	8	no	—
77	C₈×C₈₃	—	2	2	no	—
78	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
79	C₈×C₈₃	#124	1	1	yes	characteristic
80	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₈₃⋊C₈	—	4	2	no	—
82	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂²×C₈₃	C₈×Q₈	1	2	yes	—
84	C₂²×C₈₃	C₂²×C₁₆	1	1	yes	characteristic
85	C₂²×C₈₃	C₂×M₃₂	1	1	yes	characteristic
86	C₂²×C₈₃	#504	1	2	yes	—
87	C₄×C₈₃	—	2	2	no	—
88	C₄×C₈₃	C₂²⋊C₁₆	1	4	yes	—
89	C₄×C₈₃	#391	1	1	yes	characteristic
90	C₄×C₈₃	#408	1	1	yes	characteristic
91	C₄×C₈₃	#446	1	2	yes	—
92	Dic₈₃	—	2	4	no	—
93	Dic₈₃	—	4	2	no	—
94	#13263	—	83	1	no	maximal, Sylow
95	C₂×C₈₃	#1984	1	1	yes	characteristic
96	C₂×C₈₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂×C₈₃	#2113	1	1	yes	characteristic
98	C₂×C₈₃	Q₈×C₁₆	1	2	yes	—
99	C₂×C₈₃	#2183	1	2	yes	—
100	C₂×C₈×Q₈	—	83	1	no	—
101	C₂×C₄⋊C₁₆	D₈₃	1	1	yes	characteristic
102	C₂×C₄×C₁₆	—	83	1	no	—
103	#1374	—	83	4	no	—
104	C₈₃	#13263	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	83	1	no	—
106	C₂²×C₁₆	—	83	2	no	—
107	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
108	C₈×Q₈	—	166	2	no	—
109	C₈×Q₈	—	166	2	no	—
110	C₂×C₄⋊C₈	—	83	2	no	—
111	C₂×C₄⋊C₈	—	83	1	no	—
112	C₂×C₄×C₈	—	83	2	no	—
113	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
114	C₄⋊C₁₆	—	2	2	no	—
115	C₄×C₁₆	—	166	2	no	—
116	C₂²×Q₈	—	83	1	no	—
117	C₂²×C₈	—	83	2	no	—
118	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
120	C₄×Q₈	—	166	2	no	—
121	C₄×Q₈	—	166	2	no	—
122	C₂×C₄⋊C₄	—	83	2	no	—
123	C₂×C₄⋊C₄	—	83	1	no	—
124	C₂×C₄²	—	83	2	no	—
125	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	4	no	—
128	C₂×C₁₆	—	83	4	no	—
129	C₂×C₁₆	—	166	2	no	—
130	C₂×C₁₆	—	166	2	no	—
131	C₄⋊C₈	—	166	4	no	—
132	C₄⋊C₈	—	166	2	no	—
133	C₄×C₈	—	166	2	no	—
134	C₄×C₈	—	166	2	no	—
135	C₄×C₈	D₃₃₂	1	2	yes	—
136	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
137	C₂×Q₈	—	83	4	no	—
138	C₂×Q₈	—	166	2	no	—
139	C₂×Q₈	—	166	2	no	—
140	C₂²×C₄	—	83	2	no	—
141	C₂²×C₄	—	83	2	no	—
142	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
143	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	166	4	no	—
147	C₂×C₈	—	166	2	no	—
148	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
149	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
150	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₈₃	1	2	yes	—
152	C₂×C₈	#13050589	1	2	yes	—
153	C₄⋊C₄	—	166	4	no	—
154	C₄⋊C₄	—	166	2	no	—
155	C₄²	—	166	2	no	—
156	C₄²	—	166	2	no	—
157	C₄²	(C₂×D₈₃)⋊C₄	1	2	yes	—
158	C₄²	#13050573	1	2	yes	—
159	C₁₆	—	4	2	no	—
160	C₁₆	—	166	4	no	—
161	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
162	Q₈	—	166	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	83	4	no	—
165	C₂×C₄	—	166	4	no	—
166	C₂×C₄	—	166	2	no	—
167	C₂×C₄	—	166	2	no	—
168	C₂×C₄	—	166	2	no	—
169	C₂×C₄	(C₂×D₈₃)⋊C₈	1	2	yes	—
170	C₂×C₄	(C₂×D₈₃)⋊C₈	1	2	yes	—
171	C₂×C₄	C₁₆×D₈₃	1	2	yes	—
172	C₂×C₄	#14581731	1	2	yes	—
173	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	#14581757	1	2	yes	—
175	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
176	C₂×C₄	#14581800	1	2	yes	—
177	C₈	—	2	2	no	—
178	C₈	—	332	2	no	—
179	C₈	#14581772	1	2	yes	—
180	C₈	#14581779	1	1	yes	characteristic
181	C₈	#14581790	1	1	yes	characteristic
182	C₂²	C₂×(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
183	C₂²	#19127825	1	2	yes	—
184	C₂²	C₂×C₁₆×D₈₃	1	1	yes	characteristic
185	C₂²	C₂×#14581731	1	1	yes	characteristic
186	C₂²	#19128119	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	332	2	no	—
189	C₄	—	166	4	no	—
190	C₄	(C₂×D₈₃)⋊C₁₆	1	4	yes	—
191	C₄	#19127606	1	1	yes	characteristic
192	C₄	#19127669	1	1	yes	characteristic
193	C₄	#19127828	1	2	yes	—
194	C₂	#24908398	1	1	yes	characteristic
195	C₂	C₂×(C₂×D₈₃)⋊C₁₆	1	1	yes	characteristic
196	C₂	#24909078	1	1	yes	characteristic
197	C₂	#24909365	1	2	yes	—
198	C₂	#24909368	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂²×C₁₆
Fitting	C₂×C₄⋊C₁₆×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₃
hypocenter	C₈₃	#13263
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
25. C₂×C₈₃⋊C₁₆	Q₈	semidirect
39. C₂×Dic₁₆₆	C₁₆	semidirect
104. C₈₃	#13263	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₆₆	2
C₂×C₄×C₈₃⋊C₁₆	2
#24905540	2
C₂×C₄⋊C₁₆×C₈₃	2
#13263	83