Subgroups — #24916550

Order 10624.

364 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#19127831	C₂	1	2	yes	maximal, complemented
3	#19127830	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₆₆	C₂	1	2	yes	maximal, complemented
5	#19127758	C₂	1	1	yes	maximal, characteristic
6	#19127755	C₂	1	2	yes	maximal
7	#19127751	C₂	1	2	yes	maximal
8	#431×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₆₆	C₄	1	1	yes	characteristic
10	C₄×Dic₁₆₆	C₄	1	2	yes	—
11	C₄×Dic₁₆₆	C₄	1	1	yes	characteristic
12	C₄×Dic₁₆₆	C₄	1	1	yes	characteristic
13	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₆₆	C₂²	1	2	yes	—
15	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₈₃	C₄	1	1	yes	characteristic
17	C₂×Dic₈₃⋊C₄	C₄	1	2	yes	—
18	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
19	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
20	Dic₈₃⋊C₈	C₂²	1	4	yes	—
21	Dic₈₃⋊C₈	C₂²	1	4	yes	—
22	C₂²⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
23	#14581703	C₂²	1	2	yes	—
24	#14581702	C₂²	1	4	yes	—
25	C₈×Dic₈₃	C₂²	1	4	yes	—
26	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
28	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₈₃	C₂²	1	2	yes	—
31	C₂×Dic₁₆₆	—	2	4	no	—
32	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₁₆₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
36	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
37	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
38	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
41	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
42	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
43	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
44	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
45	C₄×Dic₈₃	C₂×C₄	1	2	yes	—
46	C₄×Dic₈₃	C₂×C₄	1	2	yes	—
47	C₄×Dic₈₃	C₂³	1	2	yes	—
48	C₄×Dic₈₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₈₃	C₂³	1	2	yes	—
53	C₂×C₈×C₈₃	C₂³	1	2	yes	—
54	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₈₃	C₂³	1	1	yes	characteristic
57	C₄²×C₈₃	C₂³	1	1	yes	characteristic
58	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	—	2	2	no	—
62	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₈₃	—	2	4	no	—
69	C₂×Dic₈₃	—	2	4	no	—
70	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
71	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
72	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
74	Dic₁₆₆	—	2	2	no	—
75	Dic₁₆₆	—	2	4	no	—
76	Dic₁₆₆	M₁₆	1	4	yes	—
77	C₈×C₈₃	—	2	2	no	—
78	C₈×C₈₃	—	2	2	no	—
79	C₈₃⋊C₈	—	2	4	no	—
80	C₂²×C₈₃	—	2	1	no	—
81	C₂²×C₈₃	—	2	1	no	—
82	C₂²×C₈₃	#131	1	2	yes	—
83	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₈₃	—	2	1	no	—
85	C₄×C₈₃	—	2	1	no	—
86	C₄×C₈₃	—	2	1	no	—
87	C₄×C₈₃	—	2	1	no	—
88	C₄×C₈₃	C₂×M₁₆	1	2	yes	—
89	C₄×C₈₃	2₋¹⁺⁴	1	2	yes	—
90	Dic₈₃	—	2	2	no	—
91	Dic₈₃	—	2	2	no	—
92	Dic₈₃	—	2	2	no	—
93	Dic₈₃	C₂×M₁₆	1	4	yes	—
94	C₂×C₈₃	—	2	2	no	—
95	C₂×C₈₃	#519	1	1	yes	characteristic
96	C₂×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
97	C₂×C₈₃	C₂×#131	1	1	yes	characteristic
98	#2984	—	83	1	no	maximal, Sylow
99	C₈₃	#2984	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄×Q₈	—	83	1	no	—
101	#446	—	83	4	no	—
102	#446	—	83	2	no	—
103	C₈×Q₈	—	83	2	no	—
104	#432	—	83	1	no	—
105	#432	—	83	2	no	—
106	#431	—	83	2	no	—
107	#431	D₈₃	1	1	yes	characteristic
108	C₂²×Q₈	—	83	1	no	—
109	C₄×Q₈	—	83	1	no	—
110	C₄×Q₈	—	83	2	no	—
111	C₄×Q₈	—	83	1	no	—
112	C₄×Q₈	—	83	1	no	—
113	C₄×Q₈	—	83	2	no	—
114	C₄×Q₈	—	83	1	no	—
115	C₂×C₄⋊C₄	—	83	2	no	—
116	C₂×C₄⋊C₄	—	83	1	no	—
117	C₂×C₄²	—	83	2	no	—
118	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
119	C₄⋊C₈	—	83	4	no	—
120	C₄⋊C₈	—	83	4	no	—
121	C₄⋊C₈	—	83	2	no	—
122	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
123	C₂²⋊C₈	—	83	2	no	—
124	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
125	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
126	#97	—	83	4	no	—
127	#97	—	83	2	no	—
128	C₄×C₈	—	83	4	no	—
129	C₄×C₈	C₂×D₈₃	1	2	yes	—
130	C₂×Q₈	—	83	1	no	—
131	C₂×Q₈	—	83	2	no	—
132	C₂×Q₈	—	83	1	no	—
133	C₂×Q₈	—	166	4	no	—
134	C₂²×C₄	—	83	2	no	—
135	C₂²×C₄	—	83	2	no	—
136	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
139	C₂×C₈	—	83	4	no	—
140	C₂×C₈	C₂²×D₈₃	1	2	yes	—
141	C₂×C₈	C₂²×D₈₃	1	2	yes	—
142	C₄⋊C₄	—	83	2	no	—
143	C₄⋊C₄	—	83	2	no	—
144	C₄⋊C₄	—	83	2	no	—
145	C₄⋊C₄	—	83	2	no	—
146	C₄⋊C₄	—	83	2	no	—
147	C₄⋊C₄	—	83	1	no	—
148	C₄⋊C₄	—	83	1	no	—
149	C₄²	—	83	2	no	—
150	C₄²	—	83	2	no	—
151	C₄²	—	83	2	no	—
152	C₄²	—	83	2	no	—
153	C₄²	C₄×D₈₃	1	1	yes	characteristic
154	C₄²	C₄×D₈₃	1	1	yes	characteristic
155	C₄²	C₂²×D₈₃	1	1	yes	characteristic
156	C₄²	C₂²×D₈₃	1	1	yes	characteristic
157	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
158	Q₈	—	83	4	no	—
159	Q₈	—	166	4	no	—
160	Q₈	—	166	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	83	2	no	—
165	C₂×C₄	—	83	2	no	—
166	C₂×C₄	—	83	2	no	—
167	C₂×C₄	—	83	2	no	—
168	C₂×C₄	—	166	4	no	—
169	C₂×C₄	—	166	4	no	—
170	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	166	4	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	#14581798	1	2	yes	—
182	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	83	4	no	—
188	C₄	—	166	2	no	—
189	C₄	—	166	2	no	—
190	C₄	—	166	2	no	—
191	C₄	M₁₆×D₈₃	1	2	yes	—
192	C₄	#14581839	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#19128178	1	1	yes	characteristic
195	C₂	C₂×M₁₆×D₈₃	1	1	yes	characteristic
196	C₂	C₂×#14581798	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	#431×C₈₃	C₂
prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	#431	D₈₃
hypocenter	C₈₃	#2984
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #19127831	C₂	semidirect
3. #19127830	C₂	semidirect
4. C₈×Dic₁₆₆	C₂	semidirect
99. C₈₃	#2984	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
#19127831	2
#19127830	2
C₈×Dic₁₆₆	2
#19127758	2
#19127755	2
#19127751	2
#431×C₈₃	2
#2984	83