Subgroups — #35557394

Order 21248.

706 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128118	C₂	1	1	yes	maximal, characteristic
2	#131×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#24909223	C₂	1	3	yes	maximal, complemented
4	#24909221	C₂	1	3	yes	maximal, complemented
5	#24909218	C₂	1	3	yes	maximal, complemented
6	C₂×Dic₈₃⋊C₁₆	C₂	1	3	yes	maximal, complemented
7	C₂×#504×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14581799	C₄	1	1	yes	characteristic
9	#41×Dic₈₃	C₄	1	1	yes	characteristic
10	#19128118	—	2	4	no	—
11	C₂×#14581732	C₂²	1	3	yes	—
12	C₂²×C₈₃⋊C₁₆	C₂²	1	3	yes	—
13	#19127659	C₄	1	3	yes	—
14	#19127656	C₂²	1	3	yes	—
15	M₁₆×Dic₈₃	C₄	1	3	yes	—
16	C₂×C₈×Dic₈₃	C₂²	1	3	yes	—
17	Dic₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	Dic₈₃⋊C₁₆	C₂²	1	3	yes	—
19	Dic₈₃⋊C₁₆	C₂²	1	3	yes	—
20	Dic₈₃⋊C₁₆	C₂²	1	3	yes	—
21	Dic₈₃⋊C₁₆	C₂²	1	6	yes	—
22	C₂×#131×C₈₃	C₂²	1	1	yes	characteristic
23	#504×C₈₃	—	2	4	no	—
24	C₂×M₃₂×C₈₃	C₂²	1	3	yes	—
25	C₂²×C₁₆×C₈₃	C₂²	1	3	yes	—
26	#14581799	—	2	4	no	—
27	C₂×#13050574	C₂×C₄	1	3	yes	—
28	C₂²×C₈₃⋊C₈	C₂×C₄	1	3	yes	—
29	Q₈×Dic₈₃	C₈	1	1	yes	characteristic
30	Q₈×Dic₈₃	C₈	1	1	yes	characteristic
31	D₄×Dic₈₃	C₈	1	3	yes	—
32	D₄×Dic₈₃	C₈	1	3	yes	—
33	#14581751	C₂×C₄	1	3	yes	—
34	C₂×C₄×Dic₈₃	C₂×C₄	1	3	yes	—
35	#14581732	—	2	6	no	—
36	C₂×C₈₃⋊C₁₆	—	2	6	no	—
37	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₈₃⋊C₁₆	C₂³	1	3	yes	—
39	#14581702	C₂×C₄	1	6	yes	—
40	C₈×Dic₈₃	C₂×C₄	1	3	yes	—
41	C₈×Dic₈₃	C₂×C₄	1	3	yes	—
42	C₈×Dic₈₃	C₂³	1	1	yes	characteristic
43	C₈×Dic₈₃	C₂³	1	3	yes	—
44	C₂×#41×C₈₃	C₂×C₄	1	1	yes	characteristic
45	#131×C₈₃	D₄	1	4	yes	—
46	#131×C₈₃	Q₈	1	4	yes	—
47	C₂×M₁₆×C₈₃	C₂×C₄	1	3	yes	—
48	C₂²×C₈×C₈₃	C₂³	1	3	yes	—
49	M₃₂×C₈₃	—	2	6	no	—
50	C₂×C₁₆×C₈₃	—	2	6	no	—
51	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₈₃	C₂³	1	3	yes	—
53	C₂²×Dic₈₃	C₂×C₈	1	3	yes	—
54	C₂²×Dic₈₃	C₂×C₈	1	3	yes	—
55	#13050574	—	2	6	no	—
56	C₂×C₈₃⋊C₈	—	2	6	no	—
57	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈₃⋊C₈	C₂²×C₄	1	3	yes	—
59	C₄⋊Dic₈₃	C₂×C₈	1	3	yes	—
60	C₄⋊Dic₈₃	C₂×C₈	1	3	yes	—
61	C₂²⋊Dic₈₃	C₂×C₈	1	6	yes	—
62	C₄×Dic₈₃	C₂×C₈	1	3	yes	—
63	C₄×Dic₈₃	C₂×C₈	1	3	yes	—
64	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₈₃	C₂²×C₄	1	3	yes	—
66	C₈₃⋊C₁₆	—	2	1	no	—
67	C₈₃⋊C₁₆	—	2	3	no	—
68	#41×C₈₃	C₄⋊C₄	1	4	yes	—
69	#41×C₈₃	M₁₆	1	4	yes	—
70	C₂×Q₈×C₈₃	C₂×C₈	1	1	yes	characteristic
71	C₂×D₄×C₈₃	C₂×C₈	1	3	yes	—
72	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
73	M₁₆×C₈₃	C₄⋊C₄	1	12	yes	—
74	C₂×C₈×C₈₃	C₂²×C₄	1	3	yes	—
75	C₂×C₈×C₈₃	C₂×D₄	1	6	yes	—
76	C₂×C₈×C₈₃	C₂×Q₈	1	6	yes	—
77	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
78	C₁₆×C₈₃	—	2	1	no	—
79	C₁₆×C₈₃	—	2	3	no	—
80	C₂³×C₈₃	C₂²×C₈	1	3	yes	—
81	Q₈×C₈₃	C₄⋊C₈	1	4	yes	—
82	D₄×C₈₃	C₄⋊C₈	1	12	yes	—
83	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	6	yes	—
84	C₂×C₄×C₈₃	C₂²×C₈	1	3	yes	—
85	C₂×C₄×C₈₃	C₂×M₁₆	1	6	yes	—
86	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₈₃	—	2	6	no	—
88	C₂×Dic₈₃	—	4	3	no	—
89	C₂×Dic₈₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×Dic₈₃	C₂²×C₈	1	3	yes	—
91	C₂×Dic₈₃	C₂²×C₈	1	3	yes	—
92	C₂×Dic₈₃	C₂²×C₈	1	1	yes	characteristic
93	C₈×C₈₃	C₂×C₄⋊C₄	1	6	yes	—
94	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
95	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
96	C₈₃⋊C₈	—	2	1	no	—
97	C₈₃⋊C₈	—	2	3	no	—
98	C₂²×C₈₃	—	2	3	no	—
99	C₂²×C₈₃	—	2	3	no	—
100	C₂²×C₈₃	C₂×C₄⋊C₈	1	6	yes	—
101	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
102	C₄×C₈₃	C₂×C₄⋊C₈	1	6	yes	—
103	C₄×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
105	Dic₈₃	—	2	3	no	—
106	Dic₈₃	—	2	1	no	—
107	Dic₈₃	—	2	3	no	—
108	Dic₈₃	#504	1	2	yes	complemented
109	#33398	—	83	1	no	maximal, Sylow
110	C₂×C₈₃	—	2	6	no	—
111	C₂×C₈₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₈₃	C₂×#504	1	1	yes	characteristic
113	C₂×C₈₃	#3407	1	1	yes	characteristic
114	C₂×#504	—	83	1	no	—
115	C₂×#504	D₈₃	1	1	yes	characteristic
116	C₄×#131	—	83	1	no	—
117	#2151	—	83	3	no	—
118	#2151	—	83	3	no	—
119	#2150	—	83	3	no	—
120	C₂×C₄⋊C₁₆	—	83	3	no	—
121	C₈₃	#33398	1	1	yes	characteristic, complemented, Sylow
122	C₂×#131	—	83	1	no	—
123	C₂×#131	C₂×D₈₃	1	1	yes	characteristic
124	C₄×#41	—	83	1	no	—
125	#504	—	2	4	no	—
126	#504	—	166	4	no	—
127	C₂×M₃₂	—	83	3	no	—
128	C₂×M₃₂	C₂×D₈₃	1	3	yes	—
129	C₂²×C₁₆	—	83	3	no	—
130	C₂²×C₁₆	C₂×D₈₃	1	3	yes	—
131	#405	—	83	3	no	—
132	#405	—	83	3	no	—
133	C₄×M₁₆	—	83	3	no	—
134	C₂×C₄×C₈	—	83	3	no	—
135	C₄⋊C₁₆	—	83	1	no	—
136	C₄⋊C₁₆	—	83	3	no	—
137	C₄⋊C₁₆	—	83	3	no	—
138	C₄⋊C₁₆	—	83	3	no	—
139	C₄⋊C₁₆	—	83	6	no	—
140	C₂×#41	C₄×D₈₃	1	1	yes	characteristic
141	#131	—	166	4	no	—
142	#131	Dic₁₆₆	1	4	yes	—
143	#131	C₂²⋊D₈₃	1	4	yes	—
144	C₂×M₁₆	—	83	3	no	—
145	C₂×M₁₆	C₄×D₈₃	1	3	yes	—
146	C₂²×C₈	—	83	3	no	—
147	C₂²×C₈	C₂²×D₈₃	1	3	yes	—
148	C₄×Q₈	—	83	1	no	—
149	C₄×Q₈	—	83	1	no	—
150	C₄×D₄	—	83	3	no	—
151	C₄×D₄	—	83	3	no	—
152	#117	—	83	3	no	—
153	C₂×C₄²	—	83	3	no	—
154	M₃₂	—	2	6	no	—
155	M₃₂	—	166	6	no	—
156	C₂×C₁₆	—	2	6	no	—
157	C₂×C₁₆	—	83	1	no	—
158	C₂×C₁₆	—	83	3	no	—
159	C₂×C₁₆	—	166	6	no	—
160	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
161	C₂×C₁₆	C₂²×D₈₃	1	3	yes	—
162	#97	—	83	6	no	—
163	C₄×C₈	—	83	3	no	—
164	C₄×C₈	—	83	3	no	—
165	C₄×C₈	—	83	1	no	—
166	C₄×C₈	—	83	3	no	—
167	#41	Dic₈₃⋊C₄	1	4	yes	—
168	#41	#13050573	1	4	yes	—
169	C₂×Q₈	C₈×D₈₃	1	1	yes	characteristic
170	C₂×D₄	C₈×D₈₃	1	3	yes	—
171	C₂²×C₄	—	83	3	no	—
172	C₂²×C₄	—	83	3	no	—
173	C₂²×C₄	C₂×C₄×D₈₃	1	3	yes	—
174	M₁₆	—	166	6	no	—
175	M₁₆	Dic₈₃⋊C₄	1	12	yes	—
176	C₂×C₈	—	83	1	no	—
177	C₂×C₈	—	83	3	no	—
178	C₂×C₈	—	166	6	no	—
179	C₂×C₈	C₂×C₄×D₈₃	1	3	yes	—
180	C₂×C₈	C₂×C₂²⋊D₈₃	1	6	yes	—
181	C₂×C₈	C₂×Dic₁₆₆	1	6	yes	—
182	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
183	C₄⋊C₄	—	83	3	no	—
184	C₄⋊C₄	—	83	3	no	—
185	C₂²⋊C₄	—	83	6	no	—
186	C₄²	—	83	3	no	—
187	C₄²	—	83	3	no	—
188	C₄²	—	83	1	no	—
189	C₄²	—	83	3	no	—
190	C₁₆	—	2	1	no	—
191	C₁₆	—	2	3	no	—
192	C₁₆	—	166	1	no	—
193	C₁₆	—	166	3	no	—
194	C₂³	C₂×C₈×D₈₃	1	3	yes	—
195	Q₈	Dic₈₃⋊C₈	1	4	yes	—
196	D₄	Dic₈₃⋊C₈	1	12	yes	—
197	C₂×C₄	—	332	3	no	—
198	C₂×C₄	—	83	1	no	—
199	C₂×C₄	—	83	1	no	—
200	C₂×C₄	—	83	3	no	—
201	C₂×C₄	—	83	3	no	—
202	C₂×C₄	—	166	6	no	—
203	C₂×C₄	C₂×Dic₈₃⋊C₄	1	6	yes	—
204	C₂×C₄	C₂×C₈×D₈₃	1	3	yes	—
205	C₂×C₄	C₂×#13050573	1	6	yes	—
206	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
207	C₈	—	166	1	no	—
208	C₈	—	166	3	no	—
209	C₈	C₂×Dic₈₃⋊C₄	1	6	yes	—
210	C₈	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
211	C₈	C₂²×Dic₁₆₆	1	1	yes	characteristic
212	C₂²	—	2	3	no	—
213	C₂²	—	2	3	no	—
214	C₂²	C₂×Dic₈₃⋊C₈	1	6	yes	—
215	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
216	C₄	—	83	2	no	—
217	C₄	—	166	3	no	—
218	C₄	—	166	1	no	—
219	C₄	—	166	3	no	—
220	C₄	C₂×Dic₈₃⋊C₈	1	6	yes	—
221	C₄	C₂²×Dic₈₃⋊C₄	1	1	yes	characteristic
222	C₄	C₂²×#13050573	1	1	yes	characteristic
223	C₂	—	2	6	no	—
224	C₂	C₂²×Dic₈₃⋊C₈	1	1	yes	characteristic
225	C₂	#504×D₈₃	1	1	yes	characteristic
226	C₂	#24920564	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₈
Fitting	C₂×#504×C₈₃	C₂
prime radical	C₂×#41×C₈₃	C₂×C₄
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×#504	D₈₃
hypocenter	C₈₃	#33398
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
3. #24909223	C₂	semidirect
4. #24909221	C₂	semidirect
5. #24909218	C₂	semidirect
6. C₂×Dic₈₃⋊C₁₆	C₂	semidirect
108. Dic₈₃	#504	semidirect
121. C₈₃	#33398	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128118	2
#131×Dic₈₃	2
#24909223	2
#24909221	2
#24909218	2
C₂×Dic₈₃⋊C₁₆	2
C₂×#504×C₈₃	2
#33398	83