Subgroups — #24912833

Order 10624.

484 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×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	#19127634	C₂	1	4	yes	maximal
3	#19127608	C₂	1	2	yes	maximal
4	#19127607	C₂	1	2	yes	maximal
5	#19127599	C₂	1	2	yes	maximal
6	#19127597	C₂	1	2	yes	maximal
7	#19127571	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₆₆	—	2	4	no	—
11	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
15	C₂×Dic₈₃⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₈₃⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
18	C₈₃⋊#95	C₂²	1	4	yes	—
19	C₈₃⋊#95	C₂²	1	4	yes	—
20	C₈₃⋊#95	C₂²	1	4	yes	—
21	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₈₃	—	2	4	no	—
23	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₈₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×Dic₁₆₆	—	2	2	no	—
28	C₂×Dic₁₆₆	—	2	2	no	—
29	C₂×Dic₁₆₆	Q₈	1	4	yes	complemented
30	C₂²×Dic₈₃	C₂³	1	4	yes	—
31	C₂²×Dic₈₃	C₂³	1	2	yes	—
32	C₂²×Dic₈₃	C₂³	1	2	yes	—
33	C₄⋊Dic₈₃	—	2	2	no	—
34	C₄⋊Dic₈₃	—	2	4	no	—
35	C₄⋊Dic₈₃	—	2	4	no	—
36	C₄⋊Dic₈₃	—	4	2	no	—
37	Dic₈₃⋊C₄	—	2	4	no	—
38	Dic₈₃⋊C₄	—	4	4	no	—
39	C₄×Dic₈₃	—	2	4	no	—
40	C₂×Q₈×C₈₃	—	2	2	no	—
41	C₂×Q₈×C₈₃	—	2	2	no	—
42	C₂×Q₈×C₈₃	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₈₃	—	2	4	no	—
49	C₄⋊C₄×C₈₃	—	2	2	no	—
50	C₄²×C₈₃	—	2	4	no	—
51	C₄²×C₈₃	D₄	1	4	yes	—
52	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
53	Q₈×C₈₃	—	2	8	no	—
54	C₂×C₄×C₈₃	—	2	2	no	—
55	C₂×C₄×C₈₃	—	2	2	no	—
56	C₂×C₄×C₈₃	—	2	1	no	—
57	C₂×C₄×C₈₃	—	2	1	no	—
58	C₂×C₄×C₈₃	—	2	2	no	—
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₈₃	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₈₃	#41	1	2	yes	—
68	C₂×C₄×C₈₃	#41	1	2	yes	—
69	C₂×Dic₈₃	—	2	2	no	—
70	C₂×Dic₈₃	—	2	2	no	—
71	C₂×Dic₈₃	—	2	2	no	—
72	C₂×Dic₈₃	—	2	4	no	—
73	C₂×Dic₈₃	—	2	4	no	—
74	C₂×Dic₈₃	—	2	2	no	—
75	C₂×Dic₈₃	—	2	4	no	—
76	C₂×Dic₈₃	—	2	2	no	—
77	C₂×Dic₈₃	C₂×Q₈	1	4	yes	—
78	Dic₁₆₆	—	2	8	no	—
79	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₈₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₈₃	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₈₃	C₂×#41	1	1	yes	characteristic
83	C₂²×C₈₃	C₂×#41	1	1	yes	characteristic
84	C₂²×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
85	C₂²×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
86	C₄×C₈₃	—	2	4	no	—
87	C₄×C₈₃	—	2	2	no	—
88	C₄×C₈₃	—	4	2	no	—
89	C₄×C₈₃	C₂²⋊Q₈	1	4	yes	—
90	C₄×C₈₃	C₂²⋊Q₈	1	4	yes	—
91	Dic₈₃	—	2	4	no	—
92	Dic₈₃	—	4	2	no	—
93	Dic₈₃	—	4	4	no	—
94	C₂×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₈₃	#537	1	1	yes	characteristic
97	C₂×C₈₃	#557	1	1	yes	characteristic
98	C₂×C₈₃	#557	1	1	yes	characteristic
99	C₂×C₈₃	Q₈²	1	1	yes	characteristic
100	C₂×C₈₃	#559	1	1	yes	characteristic
101	#2582	—	83	1	no	maximal, Sylow
102	C₈₃	#2582	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄×Q₈	—	83	1	no	—
104	C₂×C₄×Q₈	D₈₃	1	1	yes	characteristic
105	#398	—	83	4	no	—
106	#391	—	83	2	no	—
107	#391	—	83	2	no	—
108	#389	—	83	2	no	—
109	#389	—	83	2	no	—
110	#384	—	83	1	no	—
111	C₂²×Q₈	—	83	1	no	—
112	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
113	C₄×Q₈	—	2	4	no	—
114	C₄×Q₈	—	166	4	no	—
115	C₂×C₄⋊C₄	—	83	2	no	—
116	C₂×C₄⋊C₄	—	83	4	no	—
117	C₂×C₄⋊C₄	—	83	1	no	—
118	C₂×C₄⋊C₄	—	83	2	no	—
119	C₂×C₄⋊C₄	—	83	2	no	—
120	C₂×C₄⋊C₄	—	83	2	no	—
121	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
123	C₂×C₄²	—	83	2	no	—
124	C₂×C₄²	C₂×D₈₃	1	2	yes	—
125	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
126	#95	—	83	4	no	—
127	#95	—	83	4	no	—
128	#95	—	83	4	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	83	4	no	—
132	C₂×Q₈	—	166	2	no	—
133	C₂×Q₈	—	166	2	no	—
134	C₂×Q₈	Dic₁₆₆	1	4	yes	complemented
135	C₂²×C₄	—	83	2	no	—
136	C₂²×C₄	—	83	4	no	—
137	C₂²×C₄	—	83	2	no	—
138	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
139	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
141	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	332	4	no	—
146	C₄⋊C₄	—	332	2	no	—
147	C₄⋊C₄	—	166	4	no	—
148	C₄⋊C₄	—	166	2	no	—
149	C₄⋊C₄	—	166	4	no	—
150	C₄⋊C₄	—	166	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	166	4	no	—
153	C₄²	C₂²⋊D₈₃	1	4	yes	—
154	C₂³	C₂³×D₈₃	1	1	yes	characteristic
155	Q₈	—	2	8	no	—
156	Q₈	—	166	8	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	83	4	no	—
165	C₂×C₄	—	166	2	no	—
166	C₂×C₄	—	166	4	no	—
167	C₂×C₄	—	166	4	no	—
168	C₂×C₄	—	166	4	no	—
169	C₂×C₄	—	166	2	no	—
170	C₂×C₄	—	166	2	no	—
171	C₂×C₄	—	166	2	no	—
172	C₂×C₄	—	166	2	no	—
173	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
175	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
176	C₂×C₄	Q₈×D₈₃	1	2	yes	—
177	C₂×C₄	C₂×Dic₁₆₆	1	4	yes	—
178	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
179	C₂×C₄	#13050590	1	2	yes	—
180	C₂×C₄	#13050591	1	2	yes	—
181	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
182	C₂²	C₂×Q₈×D₈₃	1	1	yes	characteristic
183	C₂²	C₂²×Dic₁₆₆	1	1	yes	characteristic
184	C₂²	C₂×#13050590	1	1	yes	characteristic
185	C₂²	C₂×#13050591	1	1	yes	characteristic
186	C₂²	#14581838	1	1	yes	characteristic
187	C₂²	#14581839	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	332	4	no	—
192	C₄	—	332	2	no	—
193	C₄	—	166	4	no	—
194	C₄	#14581768	1	4	yes	—
195	C₄	#14581771	1	4	yes	—
196	C₂	C₂×#14581768	1	1	yes	characteristic
197	C₂	C₂×#14581771	1	1	yes	characteristic
198	C₂	#19128279	1	1	yes	characteristic
199	C₂	#19128429	1	1	yes	characteristic
200	C₂	#19128430	1	1	yes	characteristic
201	C₂	Q₈×Dic₁₆₆	1	1	yes	characteristic
202	C₂	#19128436	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₃
commutator, socle, prime radical, cosocle	C₂³×C₈₃	C₂⁴
Fitting	C₂×C₄×Q₈×C₈₃	C₂
hypercenter	C₂×C₄×Q₈	D₈₃
hypocenter	C₈₃	#2582
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
29. C₂×Dic₁₆₆	Q₈	semidirect
42. C₂×Q₈×C₈₃	Q₈	semidirect
102. C₈₃	#2582	semidirect
134. C₂×Q₈	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₆₆	2
#19127634	2
#19127608	2
#19127607	2
#19127599	2
#19127597	2
#19127571	2
C₂×C₄×Q₈×C₈₃	2
#2582	83