Subgroups — #35304404

Order 21248.

512 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907316	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#24905304	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14581703	—	2	1	no	—
8	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
9	#19127456	—	2	1	no	—
10	#19127456	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₈₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₈₃	—	2	1	no	—
13	#391×C₈₃	—	2	1	no	—
14	#389×C₈₃	—	2	1	no	—
15	#389×C₈₃	C₄	1	2	yes	—
16	#384×C₈₃	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₈₃	—	2	1	no	—
18	C₄×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
19	C₄³×C₈₃	C₂²	1	1	yes	characteristic
20	C₂²×C₈₃⋊C₈	—	2	2	no	—
21	C₂²×C₈₃⋊C₈	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	Q₈	1	4	yes	complemented
25	#14581703	—	4	2	no	—
26	C₄×C₈₃⋊C₈	—	2	4	no	—
27	C₂²×Q₈×C₈₃	—	2	1	no	—
28	C₄⋊Q₈×C₈₃	—	2	4	no	—
29	C₄×Q₈×C₈₃	—	4	4	no	—
30	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₈₃	—	2	2	no	—
37	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
44	#95×C₈₃	—	2	2	no	—
45	#95×C₈₃	—	2	2	no	—
46	C₂×C₈₃⋊C₈	—	2	4	no	—
47	C₂×C₈₃⋊C₈	—	2	4	no	—
48	C₂×C₈₃⋊C₈	—	4	2	no	—
49	C₂×C₈₃⋊C₈	—	4	2	no	—
50	C₂×C₈₃⋊C₈	—	4	2	no	—
51	C₂×C₈₃⋊C₈	—	4	2	no	—
52	C₂×Q₈×C₈₃	—	2	4	no	—
53	C₂×Q₈×C₈₃	—	4	2	no	—
54	C₂×Q₈×C₈₃	—	4	2	no	—
55	C₂²×C₄×C₈₃	—	2	2	no	—
56	C₂²×C₄×C₈₃	—	2	2	no	—
57	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₈₃	—	2	4	no	—
65	C₄⋊C₄×C₈₃	—	2	2	no	—
66	C₄⋊C₄×C₈₃	—	2	4	no	—
67	C₄⋊C₄×C₈₃	—	4	2	no	—
68	C₄⋊C₄×C₈₃	—	4	2	no	—
69	C₄⋊C₄×C₈₃	—	4	4	no	—
70	C₄⋊C₄×C₈₃	M₁₆	1	4	yes	—
71	C₄²×C₈₃	—	2	2	no	—
72	C₄²×C₈₃	—	2	2	no	—
73	C₄²×C₈₃	—	2	2	no	—
74	C₄²×C₈₃	—	2	2	no	—
75	C₄²×C₈₃	—	4	1	no	—
76	C₄²×C₈₃	—	4	4	no	—
77	C₄²×C₈₃	SD₁₆	1	2	yes	—
78	C₄²×C₈₃	Q₁₆	1	2	yes	—
79	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₈₃	#41	1	2	yes	—
81	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	Q₈×C₈₃	—	4	8	no	—
83	C₂×C₄×C₈₃	—	2	1	no	—
84	C₂×C₄×C₈₃	—	2	1	no	—
85	C₂×C₄×C₈₃	—	2	1	no	—
86	C₂×C₄×C₈₃	—	2	1	no	—
87	C₂×C₄×C₈₃	—	2	1	no	—
88	C₂×C₄×C₈₃	—	2	1	no	—
89	C₂×C₄×C₈₃	—	2	1	no	—
90	C₂×C₄×C₈₃	—	2	1	no	—
91	C₂×C₄×C₈₃	—	2	4	no	—
92	C₂×C₄×C₈₃	—	2	1	no	—
93	C₂×C₄×C₈₃	—	2	1	no	—
94	C₂×C₄×C₈₃	—	2	1	no	—
95	C₂×C₄×C₈₃	—	4	2	no	—
96	C₂×C₄×C₈₃	—	4	2	no	—
97	C₂×C₄×C₈₃	—	4	2	no	—
98	C₂×C₄×C₈₃	—	4	2	no	—
99	C₂×C₄×C₈₃	—	4	2	no	—
100	C₂×C₄×C₈₃	#103	1	2	yes	—
101	C₂×C₄×C₈₃	#103	1	2	yes	—
102	C₂×C₄×C₈₃	#104	1	2	yes	—
103	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
106	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₈₃	#131	1	2	yes	—
109	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
112	C₈₃⋊C₈	—	4	4	no	—
113	C₈₃⋊C₈	—	4	4	no	—
114	C₂²×C₈₃	#391	1	1	yes	characteristic
115	C₂²×C₈₃	#407	1	1	yes	characteristic
116	C₂²×C₈₃	#408	1	1	yes	characteristic
117	C₂²×C₈₃	C₂×#103	1	1	yes	characteristic
118	C₂²×C₈₃	#418	1	1	yes	characteristic
119	C₂²×C₈₃	C₂×#104	1	1	yes	characteristic
120	C₂²×C₈₃	#421	1	1	yes	characteristic
121	C₄×C₈₃	—	2	1	no	—
122	C₄×C₈₃	—	2	1	no	—
123	C₄×C₈₃	—	2	2	no	—
124	C₄×C₈₃	—	2	2	no	—
125	C₄×C₈₃	—	2	2	no	—
126	C₄×C₈₃	—	4	4	no	—
127	C₄×C₈₃	—	4	1	no	—
128	C₄×C₈₃	—	8	2	no	—
129	C₄×C₈₃	#446	1	4	yes	—
130	C₄×C₈₃	#476	1	2	yes	—
131	C₄×C₈₃	#477	1	2	yes	—
132	#9868	—	83	1	no	maximal, Sylow
133	C₂×C₈₃	#1487	1	1	yes	characteristic
134	C₂×C₈₃	#1490	1	1	yes	characteristic
135	C₂×C₈₃	#1493	1	1	yes	characteristic
136	C₂×C₈₃	#1496	1	1	yes	characteristic
137	C₂×C₈₃	#1985	1	1	yes	characteristic
138	C₂×C₈₃	#1986	1	1	yes	characteristic
139	C₂×C₈₃	#1987	1	1	yes	characteristic
140	C₄×C₄⋊Q₈	D₈₃	1	1	yes	characteristic
141	#1767	—	83	1	no	—
142	C₄×C₄⋊C₈	—	83	1	no	—
143	#1279	—	83	4	no	—
144	C₈₃	#9868	1	1	yes	characteristic, complemented, Sylow
145	C₂×C₄⋊Q₈	Dic₈₃	1	1	yes	characteristic
146	C₂×C₄×Q₈	—	2	1	no	—
147	C₂×C₄⋊C₈	—	83	2	no	—
148	C₂×C₄⋊C₈	—	166	1	no	—
149	C₂×#97	—	166	1	no	—
150	C₂×C₄×C₈	—	83	2	no	—
151	#391	—	2	1	no	—
152	#389	—	2	1	no	—
153	#389	Dic₈₃	1	2	yes	—
154	#384	Dic₈₃	1	1	yes	characteristic
155	C₄×C₄⋊C₄	—	2	1	no	—
156	C₄×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
157	C₄³	C₂×D₈₃	1	1	yes	characteristic
158	#336	—	166	1	no	—
159	#336	—	166	1	no	—
160	C₂²×Q₈	—	2	1	no	—
161	C₂²×C₈	—	166	2	no	—
162	C₂²×C₈	—	166	2	no	—
163	C₄⋊Q₈	—	2	4	no	—
164	C₄×Q₈	—	4	4	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
170	C₂×C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
179	C₄⋊C₈	—	83	4	no	—
180	C₄⋊C₈	—	166	4	no	—
181	C₄⋊C₈	—	166	2	no	—
182	#97	—	332	2	no	—
183	C₄×C₈	—	166	4	no	—
184	#95	—	2	2	no	—
185	#95	—	2	2	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
198	C₂×C₈	—	332	2	no	—
199	C₂×C₈	—	332	2	no	—
200	C₂×C₈	—	332	2	no	—
201	C₂×C₈	—	332	2	no	—
202	C₂×C₈	—	166	4	no	—
203	C₂×C₈	—	166	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	#13050574	1	4	yes	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	4	no	—
217	C₄²	#13050577	1	2	yes	—
218	C₄²	#13050580	1	2	yes	—
219	C₄²	Q₈×D₈₃	1	2	yes	—
220	C₄²	#13050591	1	2	yes	—
221	C₂³	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
222	Q₈	—	4	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	#14581716	1	2	yes	—
241	C₂×C₄	#14581716	1	2	yes	—
242	C₂×C₄	#14581719	1	2	yes	—
243	C₂×C₄	Q₈×Dic₈₃	1	2	yes	—
244	C₂×C₄	#14581768	1	2	yes	—
245	C₂×C₄	#14581781	1	1	yes	characteristic
246	C₂×C₄	#14581790	1	1	yes	characteristic
247	C₂×C₄	C₂×#13050574	1	2	yes	—
248	C₂×C₄	#14581799	1	2	yes	—
249	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
250	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
251	C₂×C₄	#14581818	1	2	yes	—
252	C₈	—	332	4	no	—
253	C₈	—	332	4	no	—
254	C₂²	#19127608	1	1	yes	characteristic
255	C₂²	#19127665	1	1	yes	characteristic
256	C₂²	#19127668	1	1	yes	characteristic
257	C₂²	C₂×#14581716	1	1	yes	characteristic
258	C₂²	#19127706	1	1	yes	characteristic
259	C₂²	C₂×#14581719	1	1	yes	characteristic
260	C₂²	#19127718	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	4	no	—
267	C₄	—	4	1	no	—
268	C₄	—	8	2	no	—
269	C₄	#19127832	1	4	yes	—
270	C₄	#19127988	1	2	yes	—
271	C₄	#19127993	1	2	yes	—
272	C₂	#24905892	1	1	yes	characteristic
273	C₂	#24905906	1	1	yes	characteristic
274	C₂	#24905918	1	1	yes	characteristic
275	C₂	#24905932	1	1	yes	characteristic
276	C₂	#24908404	1	1	yes	characteristic
277	C₂	#24908408	1	1	yes	characteristic
278	C₂	#24908412	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₃
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₄×C₄⋊Q₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₈₃
hypocenter	C₈₃	#9868
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₄⋊(C₈₃⋊C₈)	Q₈	semidirect
144. C₈₃	#9868	semidirect

Maximal subgroups

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

subgroup	index
#24907316	2
C₄×C₄⋊(C₈₃⋊C₈)	2
#24905304	2
C₄×C₄⋊Q₈×C₈₃	2
#9868	83