Subgroups — #35310325

Order 21248.

714 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14581728	C₂	1	2	yes	maximal
2	#24907202	C₂	1	2	yes	maximal
3	C₂×C₈₃⋊#338	C₂	1	2	yes	maximal
4	C₂×#405×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13050574	C₂²	1	4	yes	—
6	C₂×#14581728	—	2	4	no	—
7	C₂×#14581728	—	2	4	no	—
8	C₂×#14581728	C₄	1	8	yes	complemented
9	#19127470	—	2	4	no	—
10	C₈₃⋊#338	—	2	4	no	—
11	C₂²×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₈₃	C₂²	1	1	yes	characteristic
14	#405×C₈₃	—	2	4	no	—
15	C₂×#97×C₈₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₄	1	2	yes	—
17	C₂×#13050574	—	2	4	no	—
18	C₂×#13050574	—	2	4	no	—
19	C₂×#13050574	—	2	4	no	—
20	C₂×#13050574	—	2	4	no	—
21	C₂×#13050574	—	2	4	no	—
22	C₂×#13050574	C₂×C₄	1	8	yes	—
23	C₂²×C₈₃⋊C₈	—	2	4	no	—
24	#14581728	—	2	16	no	—
25	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₈₃	—	2	2	no	—
27	C₂×M₁₆×C₈₃	—	2	2	no	—
28	C₂×M₁₆×C₈₃	D₄	1	2	yes	—
29	C₂×M₁₆×C₈₃	Q₈	1	2	yes	—
30	C₂²×C₈×C₈₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₃	D₄	1	2	yes	—
35	C₂²×C₈×C₈₃	D₄	1	4	yes	—
36	C₂²×C₈×C₈₃	Q₈	1	2	yes	—
37	#117×C₈₃	—	2	2	no	—
38	#117×C₈₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₈₃	D₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₈₃	Q₈	1	2	yes	—
41	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
42	#97×C₈₃	—	2	4	no	—
43	C₄×C₈×C₈₃	—	2	4	no	—
44	#13050574	—	2	8	no	—
45	#13050574	—	2	8	no	—
46	#13050574	—	4	4	no	—
47	#13050574	—	4	4	no	—
48	C₂×C₈₃⋊C₈	—	2	8	no	—
49	C₂×C₈₃⋊C₈	—	4	4	no	—
50	C₂×C₈₃⋊C₈	—	4	4	no	—
51	C₂⁴×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	#41	1	2	yes	—
62	C₂²×C₄×C₈₃	#41	1	2	yes	—
63	M₁₆×C₈₃	—	2	4	no	—
64	M₁₆×C₈₃	—	2	4	no	—
65	C₂×C₈×C₈₃	—	2	2	no	—
66	C₂×C₈×C₈₃	—	2	2	no	—
67	C₂×C₈×C₈₃	C₄²	1	2	yes	—
68	C₂×C₈×C₈₃	C₄²	1	2	yes	—
69	C₂×C₈×C₈₃	C₂²⋊C₄	1	8	yes	—
70	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₈₃	C₄⋊C₄	1	8	yes	—
73	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
76	C₄⋊C₄×C₈₃	—	2	4	no	—
77	C₂²⋊C₄×C₈₃	—	2	4	no	—
78	C₄²×C₈₃	—	2	2	no	—
79	C₄²×C₈₃	—	2	2	no	—
80	C₂³×C₈₃	—	4	2	no	—
81	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₈₃	C₄×Q₈	1	2	yes	—
83	C₂³×C₈₃	C₂²⋊Q₈	1	2	yes	—
84	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₃	—	2	4	no	—
87	C₂×C₄×C₈₃	—	4	4	no	—
88	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₈₃	#121	1	2	yes	—
100	C₂×C₄×C₈₃	#121	1	2	yes	—
101	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₈₃	#124	1	2	yes	—
103	C₂×C₄×C₈₃	#125	1	2	yes	—
104	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
106	C₈×C₈₃	—	2	4	no	—
107	C₈×C₈₃	#95	1	4	yes	—
108	C₈×C₈₃	#95	1	4	yes	—
109	C₈₃⋊C₈	—	4	8	no	—
110	C₂²×C₈₃	—	2	2	no	—
111	C₂²×C₈₃	—	2	4	no	—
112	C₂²×C₈₃	—	4	2	no	—
113	C₂²×C₈₃	—	4	1	no	—
114	C₂²×C₈₃	—	4	1	no	—
115	C₂²×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₈₃	#380	1	1	yes	characteristic
117	C₂²×C₈₃	#384	1	1	yes	characteristic
118	C₂²×C₈₃	#389	1	2	yes	—
119	C₂²×C₈₃	#391	1	2	yes	—
120	C₄×C₈₃	—	4	4	no	—
121	C₄×C₈₃	C₂×#95	1	1	yes	characteristic
122	C₄×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₈₃	#380	1	1	yes	characteristic
124	C₄×C₈₃	#382	1	1	yes	characteristic
125	C₄×C₈₃	#389	1	2	yes	—
126	C₄×C₈₃	#390	1	2	yes	—
127	#10867	—	83	1	no	maximal, Sylow
128	C₂×C₈₃	—	2	1	no	—
129	C₂×C₈₃	—	2	1	no	—
130	C₂×C₈₃	—	2	2	no	—
131	C₂×C₈₃	—	2	2	no	—
132	C₂×C₈₃	C₄⋊#95	1	1	yes	characteristic
133	C₂×C₈₃	#1813	1	2	yes	—
134	C₂²×#108	—	83	2	no	—
135	C₂×#405	D₈₃	1	1	yes	characteristic
136	#1742	—	83	2	no	—
137	C₂×#338	—	83	2	no	—
138	C₈₃	#10867	1	1	yes	characteristic, complemented, Sylow
139	C₂²×M₁₆	—	83	4	no	—
140	C₂²×M₁₆	C₂×D₈₃	1	1	yes	characteristic
141	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
142	C₂×#117	C₂×D₈₃	1	1	yes	characteristic
143	C₂×#108	—	83	8	no	—
144	C₂×#108	—	166	4	no	—
145	C₂×#108	—	166	4	no	—
146	#405	—	2	4	no	—
147	C₂×#97	Dic₈₃	1	2	yes	—
148	C₂×C₄×C₈	Dic₈₃	1	2	yes	—
149	#343	—	166	4	no	—
150	#338	—	166	4	no	—
151	C₂³×C₄	C₂²×D₈₃	1	1	yes	characteristic
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	83	8	no	—
155	C₂×M₁₆	—	166	4	no	—
156	C₂×M₁₆	—	166	4	no	—
157	C₂×M₁₆	—	166	4	no	—
158	C₂×M₁₆	—	166	4	no	—
159	C₂×M₁₆	—	166	4	no	—
160	C₂×M₁₆	Dic₁₆₆	1	2	yes	—
161	C₂×M₁₆	D₃₃₂	1	2	yes	—
162	C₂²×C₈	—	166	4	no	—
163	C₂²×C₈	Dic₁₆₆	1	2	yes	—
164	C₂²×C₈	C₄×D₈₃	1	4	yes	—
165	C₂²×C₈	D₃₃₂	1	2	yes	—
166	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
167	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
168	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
169	C₂²×C₈	C₂²⋊D₈₃	1	4	yes	—
170	#117	—	2	2	no	—
171	#117	—	2	2	no	—
172	C₂×C₄⋊C₄	D₃₃₂	1	2	yes	—
173	C₂×C₂²⋊C₄	Dic₁₆₆	1	2	yes	—
174	C₂×C₄²	C₂×Dic₈₃	1	2	yes	—
175	#108	—	166	16	no	—
176	#97	—	2	4	no	—
177	C₄×C₈	—	2	4	no	—
178	C₂⁴	C₂×Dic₁₆₆	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₈₃	1	4	yes	—
180	C₂²×C₄	C₂×C₄×D₈₃	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
183	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
186	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
187	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
188	C₂²×C₄	#13050590	1	2	yes	—
189	C₂²×C₄	#13050590	1	2	yes	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	332	4	no	—
193	M₁₆	—	332	4	no	—
194	M₁₆	—	166	8	no	—
195	M₁₆	—	166	8	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	332	4	no	—
199	C₂×C₈	—	332	4	no	—
200	C₂×C₈	—	166	8	no	—
201	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
202	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
203	C₂×C₈	(C₂×D₈₃)⋊C₄	1	8	yes	—
204	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
205	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
206	C₂×C₈	Dic₈₃⋊C₄	1	8	yes	—
207	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
208	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
209	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
216	C₂³	C₄×Dic₁₆₆	1	2	yes	—
217	C₂³	#14581771	1	2	yes	—
218	C₂³	#14581792	1	1	yes	characteristic
219	C₂³	#14581792	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
223	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	2	yes	—
224	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
225	C₂×C₄	C₂×Dic₈₃⋊C₄	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
228	C₂×C₄	#14581751	1	1	yes	characteristic
229	C₂×C₄	#14581751	1	1	yes	characteristic
230	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
231	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
232	C₂×C₄	C₄×Dic₁₆₆	1	2	yes	—
233	C₂×C₄	#14581766	1	2	yes	—
234	C₂×C₄	#14581766	1	2	yes	—
235	C₂×C₄	#14581771	1	2	yes	—
236	C₂×C₄	#14581780	1	2	yes	—
237	C₂×C₄	#14581784	1	2	yes	—
238	C₂×C₄	#14581789	1	1	yes	characteristic
239	C₂×C₄	#14581792	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	332	8	no	—
242	C₈	C₈₃⋊#95	1	4	yes	—
243	C₈	C₈₃⋊#95	1	4	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
250	C₂²	#19127561	1	1	yes	characteristic
251	C₂²	#19127571	1	1	yes	characteristic
252	C₂²	#19127599	1	2	yes	—
253	C₂²	#19127607	1	2	yes	—
254	C₄	—	4	4	no	—
255	C₄	C₂×C₈₃⋊#95	1	1	yes	characteristic
256	C₄	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
257	C₄	#19127561	1	1	yes	characteristic
258	C₄	#19127567	1	1	yes	characteristic
259	C₄	#19127599	1	2	yes	—
260	C₄	#19127603	1	2	yes	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	2	no	—
264	C₂	—	2	2	no	—
265	C₂	#24905693	1	1	yes	characteristic
266	C₂	#24907513	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14581751
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₃
Fitting	C₂×#405×C₈₃	C₂
socle	C₂²×C₈₃	#380
prime radical	C₂⁴×C₈₃	C₂×Q₈
cosocle	C₂³×C₄×C₈₃	C₂³
hypercenter	C₂×#405	D₈₃
hypocenter	C₈₃	#10867
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. C₂×#14581728	C₄	semidirect
138. C₈₃	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14581728	2
#24907202	2
C₂×C₈₃⋊#338	2
C₂×#405×C₈₃	2
#10867	83