Subgroups — #35423413

Order 21248.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₆₆	C₂	1	2	yes	maximal
2	C₂×#19127831	C₂	1	1	yes	maximal, characteristic
3	#24908238	C₂	1	2	yes	maximal
4	#24908235	C₂	1	4	yes	maximal
5	#24907602	C₂	1	2	yes	maximal
6	#24907601	C₂	1	1	yes	maximal, characteristic
7	#24907368	C₂	1	2	yes	maximal
8	C₄×#108×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
10	#19127831	—	2	2	no	—
11	#19127831	C₄	1	4	yes	—
12	#19127831	C₄	1	4	yes	—
13	#19127829	—	2	4	no	—
14	C₈×Dic₁₆₆	—	2	2	no	—
15	C₂×#14581727	C₂²	1	4	yes	—
16	#19127725	C₂²	1	2	yes	—
17	#19127724	C₂²	1	4	yes	—
18	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×Dic₈₃⋊C₈	C₂²	1	2	yes	—
20	M₁₆×Dic₈₃	C₂²	1	4	yes	—
21	C₂×#14581702	C₂²	1	2	yes	—
22	#19127466	C₂²	1	2	yes	—
23	#19127466	C₂²	1	2	yes	—
24	#19127465	C₂²	1	4	yes	—
25	C₂×#108×C₈₃	C₂²	1	2	yes	—
26	C₄×M₁₆×C₈₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
28	#341×C₈₃	C₂²	1	1	yes	characteristic
29	#341×C₈₃	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₆₆	Q₈	1	1	yes	characteristic
31	C₂×#13050574	C₂³	1	4	yes	—
32	C₂²×C₈₃⋊C₈	C₂³	1	2	yes	—
33	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₆₆	D₄	1	2	yes	—
36	C₄×Dic₁₆₆	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₈₃	Q₈	1	1	yes	characteristic
38	C₂×Dic₈₃⋊C₄	Q₈	1	2	yes	—
39	C₂×C₄×Dic₈₃	C₂³	1	2	yes	—
40	#14581727	—	2	8	no	—
41	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
42	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
43	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
44	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
45	Dic₈₃⋊C₈	—	2	4	no	—
46	Dic₈₃⋊C₈	—	2	4	no	—
47	Dic₈₃⋊C₈	C₂×C₄	1	8	yes	—
48	#14581702	—	2	4	no	—
49	#14581702	C₂×C₄	1	8	yes	—
50	C₈×Dic₈₃	—	2	4	no	—
51	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
52	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
56	#108×C₈₃	—	2	4	no	—
57	#97×C₈₃	—	2	2	no	—
58	C₄×C₈×C₈₃	—	2	2	no	—
59	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
61	C₂×Dic₁₆₆	—	4	2	no	—
62	C₂×Dic₁₆₆	C₄⋊C₄	1	2	yes	—
63	C₂×Dic₁₆₆	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₈₃	C₂×Q₈	1	2	yes	—
65	C₂²×Dic₈₃	C₂×Q₈	1	2	yes	—
66	#13050574	—	2	4	no	—
67	#13050574	—	2	4	no	—
68	C₂×C₈₃⋊C₈	—	2	4	no	—
69	C₂×C₈₃⋊C₈	—	2	4	no	—
70	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
71	C₄⋊Dic₈₃	C₄⋊C₄	1	2	yes	—
72	C₄⋊Dic₈₃	C₄⋊C₄	1	2	yes	—
73	Dic₈₃⋊C₄	C₄⋊C₄	1	4	yes	—
74	Dic₈₃⋊C₄	C₄⋊C₄	1	4	yes	—
75	C₄×Dic₈₃	C₂²×C₄	1	4	yes	—
76	C₄×Dic₈₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₈₃	—	2	2	no	—
81	M₁₆×C₈₃	—	2	2	no	—
82	M₁₆×C₈₃	—	2	2	no	—
83	M₁₆×C₈₃	#41	1	4	yes	—
84	C₂×C₈×C₈₃	—	2	2	no	—
85	C₂×C₈×C₈₃	—	2	2	no	—
86	C₂×C₈×C₈₃	—	2	2	no	—
87	C₂×C₈×C₈₃	—	2	2	no	—
88	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₈₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₈₃	—	2	2	no	—
96	C₂×C₄×C₈₃	—	4	1	no	—
97	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
104	C₂×C₄×C₈₃	2₋¹⁺⁴	1	2	yes	—
105	C₂×Dic₈₃	—	4	2	no	—
106	C₂×Dic₈₃	—	4	2	no	—
107	C₂×Dic₈₃	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×Dic₈₃	C₂×C₄⋊C₄	1	4	yes	—
109	Dic₁₆₆	—	2	4	no	—
110	Dic₁₆₆	—	4	2	no	—
111	C₈×C₈₃	—	2	2	no	—
112	C₈×C₈₃	—	2	2	no	—
113	C₈×C₈₃	—	2	4	no	—
114	C₈×C₈₃	—	4	2	no	—
115	C₈₃⋊C₈	—	2	4	no	—
116	C₈₃⋊C₈	—	4	4	no	—
117	C₂²×C₈₃	—	2	1	no	—
118	C₂²×C₈₃	—	2	1	no	—
119	C₂²×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₈₃	#557	1	2	yes	—
121	C₄×C₈₃	—	2	1	no	—
122	C₄×C₈₃	—	2	1	no	—
123	C₄×C₈₃	—	2	1	no	—
124	C₄×C₈₃	#430	1	2	yes	—
125	C₄×C₈₃	C₄×#41	1	1	yes	characteristic
126	C₄×C₈₃	#519	1	1	yes	characteristic
127	C₄×C₈₃	#548	1	2	yes	—
128	Dic₈₃	—	2	4	no	—
129	Dic₈₃	—	2	4	no	—
130	#22747	—	83	1	no	maximal, Sylow
131	C₂×C₈₃	—	2	2	no	—
132	C₂×C₈₃	#2351	1	1	yes	characteristic
133	C₂×C₈₃	C₂×#430	1	1	yes	characteristic
134	C₂×C₈₃	#2916	1	1	yes	characteristic
135	Q₈×M₁₆	—	83	2	no	—
136	C₂×#446	—	83	1	no	—
137	#1950	—	83	4	no	—
138	#1950	—	83	2	no	—
139	#1838	—	83	2	no	—
140	#1837	—	83	1	no	—
141	#1778	—	83	2	no	—
142	C₄×#108	D₈₃	1	1	yes	characteristic
143	C₈₃	#22747	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄×Q₈	—	83	1	no	—
145	#446	—	83	4	no	—
146	#446	—	83	4	no	—
147	#446	—	166	2	no	—
148	#446	—	166	4	no	—
149	C₈×Q₈	—	166	2	no	—
150	C₂×#108	—	83	4	no	—
151	C₂×#108	C₂×D₈₃	1	2	yes	—
152	#423	—	83	4	no	—
153	#423	—	83	2	no	—
154	C₂×C₄⋊C₈	—	83	2	no	—
155	C₂×C₄⋊C₈	—	83	1	no	—
156	C₄×M₁₆	—	83	4	no	—
157	C₄×M₁₆	C₂×D₈₃	1	2	yes	—
158	C₂×#97	—	83	2	no	—
159	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
160	#341	—	83	4	no	—
161	#341	—	83	2	no	—
162	#341	—	83	2	no	—
163	#341	C₂×D₈₃	1	1	yes	characteristic
164	#341	C₂×D₈₃	1	1	yes	characteristic
165	C₂²×Q₈	—	83	1	no	—
166	C₂×M₁₆	—	83	4	no	—
167	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
168	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
169	C₂²×C₈	—	83	2	no	—
170	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
172	C₄×Q₈	—	83	2	no	—
173	C₄×Q₈	—	83	2	no	—
174	C₄×Q₈	—	83	2	no	—
175	C₄×Q₈	—	83	2	no	—
176	C₂×C₄⋊C₄	—	83	2	no	—
177	C₂×C₄⋊C₄	—	83	1	no	—
178	C₂×C₄²	—	83	2	no	—
179	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
180	#108	—	2	4	no	—
181	#108	—	166	8	no	—
182	C₄⋊C₈	—	83	8	no	—
183	C₄⋊C₈	—	83	2	no	—
184	C₄⋊C₈	—	83	2	no	—
185	C₄⋊C₈	—	166	4	no	—
186	C₄⋊C₈	—	166	4	no	—
187	C₄⋊C₈	—	166	2	no	—
188	C₄⋊C₈	—	166	2	no	—
189	#97	—	2	2	no	—
190	#97	—	83	8	no	—
191	#97	—	166	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	166	4	no	—
194	C₄×C₈	C₄×D₈₃	1	2	yes	—
195	C₄×C₈	C₄×D₈₃	1	2	yes	—
196	C₂×Q₈	—	332	2	no	—
197	C₂×Q₈	—	83	2	no	—
198	C₂×Q₈	—	83	2	no	—
199	C₂²×C₄	—	83	2	no	—
200	C₂²×C₄	—	83	2	no	—
201	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
202	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	166	4	no	—
208	M₁₆	—	166	4	no	—
209	M₁₆	#13050590	1	4	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	83	4	no	—
215	C₂×C₈	—	166	4	no	—
216	C₂×C₈	—	166	4	no	—
217	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
219	C₄⋊C₄	—	83	4	no	—
220	C₄⋊C₄	—	83	4	no	—
221	C₄⋊C₄	—	83	2	no	—
222	C₄⋊C₄	—	83	2	no	—
223	C₄²	—	83	4	no	—
224	C₄²	—	83	4	no	—
225	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
226	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
227	C₄²	D₄×D₈₃	1	1	yes	characteristic
228	C₄²	D₄×D₈₃	1	1	yes	characteristic
229	C₂³	C₂×Q₈×D₈₃	1	1	yes	characteristic
230	Q₈	—	332	2	no	—
231	Q₈	—	166	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	332	2	no	—
235	C₂×C₄	—	332	2	no	—
236	C₂×C₄	—	83	4	no	—
237	C₂×C₄	—	83	4	no	—
238	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
240	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
241	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#13050590	1	2	yes	—
245	C₂×C₄	#14581839	1	2	yes	—
246	C₈	—	2	4	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	332	4	no	—
251	C₈	—	166	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂×C₄⋊C₄×D₈₃	1	1	yes	characteristic
255	C₂²	#19128429	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	166	4	no	—
260	C₄	—	166	4	no	—
261	C₄	#19127748	1	2	yes	—
262	C₄	C₄×#13050590	1	1	yes	characteristic
263	C₄	#19128178	1	1	yes	characteristic
264	C₄	#19128360	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#24910383	1	1	yes	characteristic
267	C₂	C₂×#19127748	1	1	yes	characteristic
268	C₂	#24916071	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₄×#108×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₈₃	C₂²×Q₈
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	C₄×#108	D₈₃
hypocenter	C₈₃	#22747
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
143. C₈₃	#22747	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₆₆	2
C₂×#19127831	2
#24908238	2
#24908235	2
#24907602	2
#24907601	2
#24907368	2
C₄×#108×C₈₃	2
#22747	83