Subgroups — #35303646

Order 21248.

518 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	#24908416	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
4	C₂×#19127447	C₂	1	2	yes	maximal
5	C₂×#494×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₈₃	—	2	1	no	—
9	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
10	#19127463	—	2	4	no	—
11	#19127455	—	2	1	no	—
12	#19127454	—	2	2	no	—
13	#19127447	—	2	4	no	—
14	C₂²×Q₁₆×C₈₃	—	2	1	no	—
15	C₂²×Q₁₆×C₈₃	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
17	#494×C₈₃	—	2	2	no	—
18	#494×C₈₃	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂²×C₈₃⋊C₈	—	2	1	no	—
21	C₂²×C₈₃⋊C₈	—	2	2	no	—
22	C₂×C₄⋊Dic₈₃	D₄	1	2	yes	—
23	C₂×C₄×Dic₈₃	—	2	1	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
26	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
27	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
28	C₈×Dic₈₃	—	4	2	no	—
29	C₈₃⋊#95	—	2	1	no	—
30	C₂²×Q₈×C₈₃	—	2	2	no	—
31	C₂×Q₁₆×C₈₃	—	2	4	no	—
32	C₂×Q₁₆×C₈₃	—	2	4	no	—
33	C₂×Q₁₆×C₈₃	—	4	2	no	—
34	C₂×Q₁₆×C₈₃	—	4	2	no	—
35	C₂×Q₁₆×C₈₃	—	4	2	no	—
36	C₂×Q₁₆×C₈₃	—	4	2	no	—
37	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₄⋊Q₈×C₈₃	—	2	2	no	—
40	C₄⋊Q₈×C₈₃	—	2	2	no	—
41	C₄⋊Q₈×C₈₃	—	2	2	no	—
42	C₄⋊Q₈×C₈₃	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
45	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₈₃	D₄	1	2	yes	—
48	C₂²×Dic₈₃	—	2	2	no	—
49	C₂×C₈₃⋊C₈	—	2	2	no	—
50	C₂×C₈₃⋊C₈	—	4	1	no	—
51	C₂×C₈₃⋊C₈	—	4	1	no	—
52	C₂×C₈₃⋊C₈	—	4	2	no	—
53	C₂×C₈₃⋊C₈	—	4	4	no	—
54	C₄⋊Dic₈₃	—	2	2	no	—
55	C₄⋊Dic₈₃	Q₁₆	1	4	yes	complemented
56	C₄×Dic₈₃	—	4	1	no	—
57	C₄×Dic₈₃	—	4	1	no	—
58	C₂×Q₈×C₈₃	—	2	4	no	—
59	C₂×Q₈×C₈₃	—	4	2	no	—
60	C₂×Q₈×C₈₃	—	4	2	no	—
61	C₂×Q₈×C₈₃	—	4	2	no	—
62	C₂×Q₈×C₈₃	—	4	2	no	—
63	C₂×Q₈×C₈₃	—	4	2	no	—
64	C₂²×C₄×C₈₃	—	4	2	no	—
65	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
68	Q₁₆×C₈₃	—	4	8	no	—
69	Q₁₆×C₈₃	—	4	8	no	—
70	C₂×C₈×C₈₃	—	2	1	no	—
71	C₂×C₈×C₈₃	—	2	1	no	—
72	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₈₃	D₈	1	2	yes	—
75	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
76	C₄⋊C₄×C₈₃	—	2	4	no	—
77	C₄⋊C₄×C₈₃	—	2	4	no	—
78	C₄⋊C₄×C₈₃	—	4	2	no	—
79	C₄⋊C₄×C₈₃	—	4	2	no	—
80	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₃	#41	1	1	yes	characteristic
84	C₂³×C₈₃	#121	1	1	yes	characteristic
85	Q₈×C₈₃	—	4	4	no	—
86	Q₈×C₈₃	—	4	4	no	—
87	Q₈×C₈₃	—	8	2	no	—
88	Q₈×C₈₃	—	8	2	no	—
89	C₂×C₄×C₈₃	—	2	2	no	—
90	C₂×C₄×C₈₃	—	4	4	no	—
91	C₂×C₄×C₈₃	—	8	2	no	—
92	C₂×C₄×C₈₃	—	8	2	no	—
93	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₈₃	#135	1	2	yes	—
103	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×Dic₈₃	—	2	4	no	—
106	C₂×Dic₈₃	—	4	2	no	—
107	C₂×Dic₈₃	—	4	2	no	—
108	C₈×C₈₃	—	2	2	no	—
109	C₈×C₈₃	#102	1	4	yes	—
110	C₈₃⋊C₈	—	4	2	no	—
111	C₈₃⋊C₈	—	8	2	no	—
112	C₂²×C₈₃	#390	1	1	yes	characteristic
113	C₂²×C₈₃	#459	1	1	yes	characteristic
114	C₂²×C₈₃	#460	1	1	yes	characteristic
115	C₂²×C₈₃	#467	1	2	yes	—
116	C₂²×C₈₃	#473	1	2	yes	—
117	C₄×C₈₃	—	2	1	no	—
118	C₄×C₈₃	—	2	1	no	—
119	C₄×C₈₃	—	8	4	no	—
120	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
121	C₄×C₈₃	#418	1	1	yes	characteristic
122	C₄×C₈₃	C₄×Q₁₆	1	2	yes	—
123	C₄×C₈₃	#459	1	1	yes	characteristic
124	C₄×C₈₃	#460	1	1	yes	characteristic
125	C₄×C₈₃	#494	1	1	yes	characteristic
126	C₄×C₈₃	#495	1	1	yes	characteristic
127	Dic₈₃	—	4	4	no	—
128	#9733	—	83	1	no	maximal, Sylow
129	C₂×C₈₃	#1669	1	2	yes	—
130	C₂×C₈₃	#1670	1	2	yes	—
131	C₂×C₈₃	#1969	1	1	yes	characteristic
132	C₂×C₈₃	#1979	1	1	yes	characteristic
133	C₂×C₈₃	#1982	1	1	yes	characteristic
134	C₂×#494	D₈₃	1	1	yes	characteristic
135	#1988	—	83	2	no	—
136	C₈×C₄⋊C₄	—	83	1	no	—
137	C₂×#335	—	83	1	no	—
138	C₂×#332	—	83	2	no	—
139	C₈₃	#9733	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×Q₁₆	—	2	1	no	—
142	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
143	#494	—	2	2	no	—
144	#494	Dic₈₃	1	4	yes	complemented
145	C₂×C₄⋊C₈	—	83	1	no	—
146	C₂×C₄⋊C₈	—	83	2	no	—
147	C₂×C₄×C₈	—	166	1	no	—
148	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
149	C₄×C₄⋊C₄	—	83	1	no	—
150	#340	—	166	4	no	—
151	#336	—	166	1	no	—
152	#335	—	166	2	no	—
153	#332	—	166	4	no	—
154	C₂²×Q₈	—	2	2	no	—
155	C₂×Q₁₆	—	2	4	no	—
156	C₂×Q₁₆	—	2	4	no	—
157	C₂×Q₁₆	—	4	2	no	—
158	C₂×Q₁₆	—	4	2	no	—
159	C₂×Q₁₆	—	4	2	no	—
160	C₂×Q₁₆	—	4	2	no	—
161	C₂²×C₈	—	166	2	no	—
162	C₂²×C₈	—	166	1	no	—
163	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	C₂×Dic₈₃	1	4	yes	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	83	2	no	—
172	C₂×C₄²	—	166	1	no	—
173	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
174	C₄⋊C₈	—	83	2	no	—
175	C₄⋊C₈	—	166	2	no	—
176	C₄⋊C₈	—	166	2	no	—
177	C₄⋊C₈	—	166	1	no	—
178	C₄×C₈	—	332	2	no	—
179	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
180	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
181	#95	—	166	1	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	166	2	no	—
190	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
192	C₂²×C₄	#13050589	1	1	yes	characteristic
193	Q₁₆	—	4	8	no	—
194	Q₁₆	—	4	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	332	1	no	—
198	C₂×C₈	—	332	1	no	—
199	C₂×C₈	—	332	2	no	—
200	C₂×C₈	—	332	4	no	—
201	C₂×C₈	—	166	2	no	—
202	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
204	C₂×C₈	#13050575	1	2	yes	—
205	C₂×C₈	#13050578	1	2	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	83	4	no	—
211	C₄⋊C₄	—	166	2	no	—
212	C₄²	—	332	1	no	—
213	C₄²	—	332	1	no	—
214	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
215	C₄²	D₄×D₈₃	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
217	C₄²	#13050589	1	1	yes	characteristic
218	C₂³	#14581763	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	8	2	no	—
222	Q₈	—	8	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	332	2	no	—
228	C₂×C₄	—	332	2	no	—
229	C₂×C₄	—	166	4	no	—
230	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
231	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
232	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
233	C₂×C₄	#14581763	1	1	yes	characteristic
234	C₂×C₄	#14581779	1	1	yes	characteristic
235	C₂×C₄	#14581788	1	1	yes	characteristic
236	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
237	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
238	C₂×C₄	Q₁₆×D₈₃	1	2	yes	—
239	C₂×C₄	#14581815	1	2	yes	—
240	C₂×C₄	#14581818	1	1	yes	characteristic
241	C₂×C₄	#14581818	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	332	2	no	—
244	C₈	—	664	2	no	—
245	C₈	#14581713	1	4	yes	—
246	C₂²	#19127604	1	1	yes	characteristic
247	C₂²	#19127893	1	1	yes	characteristic
248	C₂²	#19127898	1	1	yes	characteristic
249	C₂²	#19127939	1	2	yes	—
250	C₂²	#19127973	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	4	no	—
254	C₄	—	332	4	no	—
255	C₄	C₂×#14581713	1	1	yes	characteristic
256	C₄	#19127706	1	1	yes	characteristic
257	C₄	Q₁₆×Dic₈₃	1	2	yes	—
258	C₄	#19127893	1	1	yes	characteristic
259	C₄	#19127898	1	1	yes	characteristic
260	C₄	#19128079	1	1	yes	characteristic
261	C₄	#19128086	1	1	yes	characteristic
262	C₂	#24906825	1	2	yes	—
263	C₂	#24906830	1	2	yes	—
264	C₂	#24908325	1	1	yes	characteristic
265	C₂	#24908379	1	1	yes	characteristic
266	C₂	#24908391	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581763
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#494×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#121
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#494	D₈₃
hypocenter	C₈₃	#9733
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
18. #494×C₈₃	C₄	semidirect
55. C₄⋊Dic₈₃	Q₁₆	semidirect
139. C₈₃	#9733	semidirect
144. #494	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24908416	2
C₈×C₄⋊Dic₈₃	2
C₂×#19127454	2
C₂×#19127447	2
C₂×#494×C₈₃	2
#9733	83