Subgroups — #35322919

Order 21248.

462 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128081	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127522	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127516	C₂	1	2	yes	maximal
4	#24909182	C₂	1	2	yes	maximal
5	C₈₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	—	2	1	no	—
7	C₂²×Dic₃₃₂	C₂²	1	2	yes	—
8	C₂×#14581792	—	2	1	no	—
9	C₂×#14581732	C₂²	1	2	yes	—
10	#19128081	—	2	2	no	—
11	#19128081	C₄	1	4	yes	complemented
12	#19127522	—	2	2	no	—
13	#19127516	—	2	4	no	—
14	#19127504	—	2	4	no	—
15	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
16	#427×C₈₃	C₂²	1	1	yes	characteristic
17	#405×C₈₃	—	2	1	no	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	#339×C₈₃	—	2	1	no	—
20	C₂²×Dic₁₆₆	—	2	2	no	—
21	C₂×Dic₃₃₂	—	2	4	no	—
22	C₂×Dic₃₃₂	—	2	2	no	—
23	C₂×Dic₃₃₂	—	2	2	no	—
24	C₂×Dic₃₃₂	—	2	2	no	—
25	C₂×Dic₃₃₂	—	2	2	no	—
26	C₂×Dic₃₃₂	—	2	2	no	—
27	C₂×Dic₃₃₂	—	4	1	no	—
28	C₂×Dic₃₃₂	—	4	1	no	—
29	C₂×Dic₃₃₂	—	4	2	no	—
30	C₂×Dic₃₃₂	C₂×C₄	1	4	yes	—
31	#14581792	—	2	4	no	—
32	#14581792	—	4	2	no	—
33	C₂×C₄⋊Dic₈₃	—	4	2	no	—
34	#14581732	—	2	2	no	—
35	#14581732	—	2	2	no	—
36	C₂×C₈₃⋊C₁₆	—	2	2	no	—
37	C₂×M₁₆×C₈₃	—	2	1	no	—
38	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
40	#117×C₈₃	—	2	1	no	—
41	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
42	#108×C₈₃	—	2	1	no	—
43	#108×C₈₃	D₄	1	2	yes	—
44	#107×C₈₃	D₄	1	2	yes	—
45	#106×C₈₃	—	2	1	no	—
46	#97×C₈₃	—	4	1	no	—
47	C₄×C₈×C₈₃	—	4	1	no	—
48	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₈₃	D₄	1	2	yes	—
50	C₂×Dic₁₆₆	—	2	4	no	—
51	C₂×Dic₁₆₆	—	4	2	no	—
52	C₂×Dic₁₆₆	—	4	2	no	—
53	C₂×Dic₁₆₆	—	4	2	no	—
54	C₂×Dic₁₆₆	—	4	2	no	—
55	C₂×Dic₁₆₆	—	4	2	no	—
56	C₂²×Dic₈₃	—	4	2	no	—
57	Dic₃₃₂	—	2	4	no	—
58	Dic₃₃₂	—	2	4	no	—
59	Dic₃₃₂	—	4	4	no	—
60	Dic₃₃₂	—	4	4	no	—
61	Dic₃₃₂	—	4	2	no	—
62	Dic₃₃₂	—	4	2	no	—
63	C₄⋊Dic₈₃	—	4	8	no	—
64	C₈₃⋊C₁₆	—	4	2	no	—
65	C₂²×C₄×C₈₃	—	2	1	no	—
66	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
67	M₁₆×C₈₃	—	2	2	no	—
68	M₁₆×C₈₃	—	4	1	no	—
69	C₂×C₈×C₈₃	—	4	1	no	—
70	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
72	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
73	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
74	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
80	C₄⋊C₄×C₈₃	—	2	2	no	—
81	C₂²⋊C₄×C₈₃	—	4	1	no	—
82	C₄²×C₈₃	—	2	1	no	—
83	C₄²×C₈₃	—	4	1	no	—
84	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
85	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄×C₈₃	—	2	4	no	—
87	C₂×C₄×C₈₃	—	2	2	no	—
88	C₂×C₄×C₈₃	—	4	2	no	—
89	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×Dic₈₃	—	4	4	no	—
96	C₂×Dic₈₃	—	8	2	no	—
97	C₂×Dic₈₃	—	8	2	no	—
98	Dic₁₆₆	—	4	4	no	—
99	Dic₁₆₆	—	4	4	no	—
100	Dic₁₆₆	—	8	2	no	—
101	Dic₁₆₆	—	8	2	no	—
102	C₈×C₈₃	—	4	2	no	—
103	C₈×C₈₃	#102	1	2	yes	—
104	C₈×C₈₃	#102	1	2	yes	—
105	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
106	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
107	C₈×C₈₃	#124	1	1	yes	characteristic
108	C₈×C₈₃	#127	1	1	yes	characteristic
109	C₂²×C₈₃	—	2	2	no	—
110	C₂²×C₈₃	#418	1	1	yes	characteristic
111	C₂²×C₈₃	#459	1	1	yes	characteristic
112	C₂²×C₈₃	#460	1	1	yes	characteristic
113	C₄×C₈₃	—	2	4	no	—
114	C₄×C₈₃	—	8	2	no	—
115	C₄×C₈₃	#390	1	1	yes	characteristic
116	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
117	C₄×C₈₃	#459	1	1	yes	characteristic
118	C₄×C₈₃	#460	1	1	yes	characteristic
119	Dic₈₃	—	8	4	no	—
120	#13262	—	83	1	no	maximal, Sylow
121	C₂×C₈₃	—	2	1	no	—
122	C₂×C₈₃	—	2	1	no	—
123	C₂×C₈₃	#1982	1	1	yes	characteristic
124	C₂×C₈₃	#2214	1	2	yes	—
125	C₂×#494	—	83	1	no	—
126	C₂×#364	—	83	1	no	—
127	C₂×#362	—	83	2	no	—
128	#2141	—	83	2	no	—
129	#1772	D₈₃	1	1	yes	characteristic
130	C₈₃	#13262	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₁₆	—	83	2	no	—
132	C₂²×Q₁₆	—	166	1	no	—
133	C₂×C₄⋊Q₈	—	166	1	no	—
134	C₂×M₃₂	—	83	2	no	—
135	#494	—	83	4	no	—
136	#494	—	166	2	no	—
137	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
138	#427	C₂×D₈₃	1	1	yes	characteristic
139	#405	—	2	1	no	—
140	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
141	#364	—	166	2	no	—
142	#362	—	166	4	no	—
143	#358	—	166	4	no	—
144	#339	—	2	1	no	—
145	C₂²×Q₈	—	166	2	no	—
146	C₂×Q₁₆	—	332	1	no	—
147	C₂×Q₁₆	—	332	1	no	—
148	C₂×Q₁₆	—	332	2	no	—
149	C₂×Q₁₆	—	83	4	no	—
150	C₂×Q₁₆	—	166	4	no	—
151	C₂×Q₁₆	—	166	2	no	—
152	C₂×Q₁₆	—	166	2	no	—
153	C₂×Q₁₆	—	166	2	no	—
154	C₂×Q₁₆	—	166	2	no	—
155	C₂×Q₁₆	—	166	2	no	—
156	C₂×M₁₆	—	2	1	no	—
157	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
159	C₄⋊Q₈	—	332	2	no	—
160	C₄⋊Q₈	—	166	4	no	—
161	#117	—	2	1	no	—
162	C₂×C₄⋊C₄	—	332	2	no	—
163	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
164	M₃₂	—	166	2	no	—
165	M₃₂	—	166	2	no	—
166	C₂×C₁₆	—	166	2	no	—
167	#108	—	2	1	no	—
168	#108	D₃₃₂	1	2	yes	—
169	#107	D₃₃₂	1	2	yes	—
170	#106	—	2	1	no	—
171	#97	—	4	1	no	—
172	C₄×C₈	—	4	1	no	—
173	C₄×C₈	C₄×D₈₃	1	2	yes	—
174	C₄×C₈	D₃₃₂	1	2	yes	—
175	C₂×Q₈	—	332	2	no	—
176	C₂×Q₈	—	332	2	no	—
177	C₂×Q₈	—	332	2	no	—
178	C₂×Q₈	—	332	2	no	—
179	C₂×Q₈	—	332	2	no	—
180	C₂×Q₈	—	166	4	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	332	2	no	—
183	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
184	Q₁₆	—	332	2	no	—
185	Q₁₆	—	332	2	no	—
186	Q₁₆	—	332	4	no	—
187	Q₁₆	—	332	4	no	—
188	Q₁₆	—	166	4	no	—
189	Q₁₆	—	166	4	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	4	1	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
194	C₂×C₈	#13050575	1	1	yes	characteristic
195	C₂×C₈	#13050575	1	1	yes	characteristic
196	C₂×C₈	#13050577	1	2	yes	—
197	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
198	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
199	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
200	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
201	C₂×C₈	#13050590	1	1	yes	characteristic
202	C₂×C₈	#13050590	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	332	8	no	—
205	C₂²⋊C₄	—	4	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	4	1	no	—
208	C₄²	(C₂×D₈₃)⋊C₄	1	2	yes	—
209	C₁₆	—	332	2	no	—
210	C₂³	#14581817	1	1	yes	characteristic
211	Q₈	—	332	4	no	—
212	Q₈	—	332	4	no	—
213	Q₈	—	664	2	no	—
214	Q₈	—	664	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	332	4	no	—
219	C₂×C₄	—	664	2	no	—
220	C₂×C₄	—	664	2	no	—
221	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	#14581766	1	1	yes	characteristic
223	C₂×C₄	#14581766	1	1	yes	characteristic
224	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
225	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
226	C₂×C₄	#14581817	1	1	yes	characteristic
227	C₈	—	4	2	no	—
228	C₈	#14581712	1	2	yes	—
229	C₈	#14581712	1	2	yes	—
230	C₈	C₄×D₃₃₂	1	1	yes	characteristic
231	C₈	C₄×D₃₃₂	1	1	yes	characteristic
232	C₈	#14581780	1	1	yes	characteristic
233	C₈	#14581789	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	#19127705	1	1	yes	characteristic
236	C₂²	#19127896	1	1	yes	characteristic
237	C₂²	#19127901	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	8	2	no	—
240	C₄	—	664	4	no	—
241	C₄	#19127603	1	1	yes	characteristic
242	C₄	C₂×#14581712	1	1	yes	characteristic
243	C₄	#19127896	1	1	yes	characteristic
244	C₄	#19127901	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	2	1	no	—
247	C₂	#24908390	1	1	yes	characteristic
248	C₂	#24909544	1	2	yes	—

	subgroup	G/H
center	C₂²	#19127896
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1772	C₂
socle	C₂²×C₈₃	#459
prime radical	C₂³×C₈₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1772	D₈₃
hypocenter	C₈₃	#13262
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
11. #19128081	C₄	semidirect
130. C₈₃	#13262	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128081	2
C₂×#19127522	2
C₂×#19127516	2
#24909182	2
C₈₃×#1772	2
#13262	83