Subgroups — #35557904

Order 21248.

490 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	#24916397	C₂	1	1	yes	maximal, characteristic
2	#24909369	C₂	1	8	yes	maximal, complemented
3	#24909221	C₂	1	2	yes	maximal
4	#24909219	C₂	1	1	yes	maximal, characteristic
5	#24909059	C₂	1	2	yes	maximal
6	C₂×#346×C₈₃	C₂	1	1	yes	maximal, characteristic
7	#19128258	C₄	1	1	yes	characteristic
8	C₂×#14581732	C₂²	1	2	yes	—
9	#19127831	C₄	1	4	yes	—
10	C₈×Dic₁₆₆	C₂²	1	4	yes	—
11	#19127727	C₄	1	2	yes	—
12	#19127725	C₄	1	1	yes	characteristic
13	#19127656	C₂²	1	2	yes	—
14	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	4	yes	—
15	Dic₈₃⋊C₁₆	C₂²	1	8	yes	—
16	#19127476	C₂²	1	4	yes	—
17	C₁₆×Dic₈₃	C₂²	1	4	yes	—
18	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	#346×C₈₃	C₂²	1	4	yes	—
22	C₂×#13050574	C₂×C₄	1	2	yes	—
23	#14581792	C₈	1	2	yes	—
24	#14581784	C₈	1	2	yes	—
25	#14581771	C₈	1	4	yes	—
26	C₄×Dic₁₆₆	C₂×C₄	1	4	yes	—
27	#14581751	C₂×C₄	1	2	yes	—
28	#14581732	—	2	4	no	—
29	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
30	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	4	yes	—
31	Dic₈₃⋊C₈	C₂×C₄	1	8	yes	—
32	#14581702	C₂×C₄	1	4	yes	—
33	C₈×Dic₈₃	C₂³	1	4	yes	—
34	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₈₃	—	2	2	no	—
38	C₂×C₁₆×C₈₃	Q₈	1	4	yes	complemented
39	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
40	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
41	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₈₃	C₂³	1	2	yes	—
43	C₂×Dic₁₆₆	C₂×C₈	1	4	yes	—
44	#13050574	—	2	4	no	—
45	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
46	C₄⋊Dic₈₃	C₂×C₈	1	4	yes	—
47	C₄⋊Dic₈₃	C₂×C₈	1	4	yes	—
48	Dic₈₃⋊C₄	C₂×C₈	1	8	yes	—
49	C₂²⋊Dic₈₃	C₂×C₈	1	4	yes	—
50	C₄×Dic₈₃	C₂²×C₄	1	4	yes	—
51	C₈₃⋊C₁₆	—	2	4	no	—
52	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₃	—	2	2	no	—
55	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₈₃	#41	1	2	yes	—
59	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
60	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
63	C₁₆×C₈₃	—	2	2	no	—
64	C₁₆×C₈₃	C₂×Q₈	1	4	yes	—
65	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₈₃	—	4	2	no	—
67	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#131	1	2	yes	—
72	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₈₃	C₂²×C₈	1	8	yes	—
74	Dic₁₆₆	—	4	4	no	—
75	C₈×C₈₃	—	2	2	no	—
76	C₈×C₈₃	C₄×Q₈	1	2	yes	—
77	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
79	C₈₃⋊C₈	—	2	4	no	—
80	C₂²×C₈₃	—	2	2	no	—
81	C₂²×C₈₃	C₈×Q₈	1	2	yes	—
82	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₈₃	—	2	2	no	—
84	C₄×C₈₃	—	2	2	no	—
85	C₄×C₈₃	C₈×Q₈	1	2	yes	—
86	C₄×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
87	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
88	Dic₈₃	—	2	8	no	—
89	#33475	—	83	1	no	maximal, Sylow
90	C₂×C₈₃	—	2	2	no	—
91	C₂×C₈₃	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂×C₈₃	#3407	1	2	yes	—
93	#2960	—	83	1	no	—
94	#2183	—	83	8	no	—
95	#2151	—	83	2	no	—
96	#2150	—	83	1	no	—
97	#2108	—	83	2	no	—
98	C₂×#346	D₈₃	1	1	yes	characteristic
99	C₈₃	#33475	1	1	yes	characteristic, complemented, Sylow
100	#533	—	83	1	no	—
101	C₂×M₃₂	—	83	2	no	—
102	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
103	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
104	#446	—	83	4	no	—
105	C₈×Q₈	—	83	4	no	—
106	#424	—	83	2	no	—
107	#423	—	83	1	no	—
108	#405	—	83	2	no	—
109	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
110	C₄⋊C₁₆	—	83	8	no	—
111	C₄⋊C₁₆	—	83	4	no	—
112	#346	—	83	4	no	—
113	#346	C₂×D₈₃	1	4	yes	—
114	C₄×C₁₆	—	83	4	no	—
115	C₂×M₁₆	—	83	2	no	—
116	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
118	C₄⋊Q₈	—	83	2	no	—
119	#125	—	83	2	no	—
120	C₂²⋊Q₈	—	83	4	no	—
121	C₄×Q₈	—	83	4	no	—
122	#117	—	83	2	no	—
123	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
124	M₃₂	—	166	4	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	83	4	no	—
127	C₂×C₁₆	Dic₁₆₆	1	4	yes	complemented
128	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
129	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
130	C₄⋊C₈	—	83	8	no	—
131	C₄⋊C₈	—	83	4	no	—
132	#97	—	83	4	no	—
133	C₄×C₈	—	83	4	no	—
134	C₄×C₈	C₄×D₈₃	1	2	yes	—
135	C₄×C₈	C₂²×D₈₃	1	2	yes	—
136	C₂×Q₈	—	83	4	no	—
137	C₂²×C₄	C₈×D₈₃	1	2	yes	—
138	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
139	M₁₆	—	166	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	83	4	no	—
142	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
143	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
144	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
145	C₂×C₈	#13050590	1	2	yes	—
146	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
147	C₄⋊C₄	—	83	4	no	—
148	C₄⋊C₄	—	83	8	no	—
149	C₄⋊C₄	—	83	4	no	—
150	C₂²⋊C₄	—	83	4	no	—
151	C₄²	—	83	4	no	—
152	C₄²	C₈×D₈₃	1	1	yes	characteristic
153	C₄²	C₈×D₈₃	1	1	yes	characteristic
154	C₄²	C₂×C₄×D₈₃	1	2	yes	—
155	C₁₆	—	2	2	no	—
156	C₁₆	—	166	4	no	—
157	C₁₆	C₂×Dic₁₆₆	1	4	yes	—
158	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
159	Q₈	—	332	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	83	8	no	—
162	C₂×C₄	C₄×Dic₁₆₆	1	2	yes	—
163	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
165	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
166	C₂×C₄	#14581798	1	2	yes	—
167	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	166	4	no	—
170	C₈	C₄×Dic₁₆₆	1	2	yes	—
171	C₈	C₂²×Dic₁₆₆	1	1	yes	characteristic
172	C₈	C₂×#13050590	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	C₈×Dic₁₆₆	1	2	yes	—
175	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	166	8	no	—
179	C₄	C₈×Dic₁₆₆	1	2	yes	—
180	C₄	C₂×C₄×Dic₁₆₆	1	1	yes	characteristic
181	C₄	C₂×#14581798	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×C₈×Dic₁₆₆	1	1	yes	characteristic
184	C₂	#24920565	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₈
Fitting	C₂×#346×C₈₃	C₂
prime radical	C₂³×C₈₃	C₂²×C₈
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×#346	D₈₃
hypocenter	C₈₃	#33475
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #24909369	C₂	semidirect
38. C₂×C₁₆×C₈₃	Q₈	semidirect
99. C₈₃	#33475	semidirect
127. C₂×C₁₆	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
#24916397	2
#24909369	2
#24909221	2
#24909219	2
#24909059	2
C₂×#346×C₈₃	2
#33475	83