Subgroups — #35320705

Order 21248.

404 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	C₂×C₂²⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#24907501	C₂	1	1	yes	maximal, characteristic
3	#24905558	C₂	1	4	yes	maximal, complemented
4	C₂×#427×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14581728	C₂²	1	2	yes	—
7	#19127665	—	2	1	no	—
8	C₂²⋊(C₈₃⋊C₁₆)	—	2	1	no	—
9	C₂²⋊(C₈₃⋊C₁₆)	—	2	1	no	—
10	#19127466	—	2	1	no	—
11	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₈₃	—	2	1	no	—
13	#427×C₈₃	—	2	1	no	—
14	#427×C₈₃	—	2	1	no	—
15	#427×C₈₃	C₄	1	2	yes	—
16	#427×C₈₃	C₂²	1	2	yes	—
17	C₂×#107×C₈₃	C₄	1	1	yes	characteristic
18	C₂×#107×C₈₃	C₄	1	1	yes	characteristic
19	C₂×#106×C₈₃	—	2	1	no	—
20	C₂×#13050574	—	2	2	no	—
21	C₂×C₈₃⋊C₁₆	—	2	2	no	—
22	C₂×C₈₃⋊C₁₆	—	2	4	no	—
23	#14581728	—	2	4	no	—
24	C₂²⋊(C₈₃⋊C₈)	—	4	1	no	—
25	C₂²⋊(C₈₃⋊C₈)	—	4	1	no	—
26	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₃	—	2	4	no	—
28	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
34	#117×C₈₃	—	2	2	no	—
35	#117×C₈₃	—	2	2	no	—
36	#117×C₈₃	—	4	1	no	—
37	#117×C₈₃	—	4	1	no	—
38	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₈₃	—	4	1	no	—
41	C₂×C₄²×C₈₃	—	4	1	no	—
42	#107×C₈₃	—	2	1	no	—
43	#107×C₈₃	—	2	1	no	—
44	#107×C₈₃	C₂×C₄	1	2	yes	—
45	#107×C₈₃	C₂×C₄	1	2	yes	—
46	#106×C₈₃	—	2	2	no	—
47	#106×C₈₃	—	2	2	no	—
48	#13050574	—	4	2	no	—
49	#13050574	—	4	2	no	—
50	C₂×C₈₃⋊C₈	—	4	2	no	—
51	C₈₃⋊C₁₆	—	4	2	no	—
52	C₂⁴×C₈₃	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	—	2	2	no	—
54	C₂²×C₄×C₈₃	—	2	2	no	—
55	C₂²×C₄×C₈₃	—	4	1	no	—
56	C₂²×C₄×C₈₃	—	4	1	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	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₈₃	—	2	2	no	—
64	C₂×C₈×C₈₃	—	2	2	no	—
65	C₂×C₈×C₈₃	—	2	2	no	—
66	C₂×C₈×C₈₃	—	2	2	no	—
67	C₂×C₈×C₈₃	—	2	2	no	—
68	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₃	#41	1	4	yes	—
73	C₄⋊C₄×C₈₃	—	2	2	no	—
74	C₄⋊C₄×C₈₃	—	2	2	no	—
75	C₄⋊C₄×C₈₃	—	4	1	no	—
76	C₄⋊C₄×C₈₃	—	4	1	no	—
77	C₂²⋊C₄×C₈₃	—	4	2	no	—
78	C₂²⋊C₄×C₈₃	—	4	2	no	—
79	C₄²×C₈₃	—	4	2	no	—
80	C₄²×C₈₃	—	4	2	no	—
81	C₂³×C₈₃	—	2	2	no	—
82	C₂³×C₈₃	—	2	2	no	—
83	C₂³×C₈₃	—	4	1	no	—
84	C₂³×C₈₃	#102	1	1	yes	characteristic
85	C₂³×C₈₃	#102	1	1	yes	characteristic
86	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₃	—	2	2	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	1	no	—
92	C₂×C₄×C₈₃	—	2	2	no	—
93	C₂×C₄×C₈₃	—	2	2	no	—
94	C₂×C₄×C₈₃	—	4	2	no	—
95	C₂×C₄×C₈₃	—	4	2	no	—
96	C₂×C₄×C₈₃	—	8	2	no	—
97	C₂×C₄×C₈₃	—	8	2	no	—
98	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₃	#123	1	2	yes	—
102	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈₃	#137	1	2	yes	—
104	C₈×C₈₃	—	2	1	no	—
105	C₈×C₈₃	—	2	1	no	—
106	C₈×C₈₃	#117	1	2	yes	—
107	C₈×C₈₃	#123	1	2	yes	—
108	C₈₃⋊C₈	—	8	2	no	—
109	C₂²×C₈₃	—	2	1	no	—
110	C₂²×C₈₃	—	2	1	no	—
111	C₂²×C₈₃	—	2	1	no	—
112	C₂²×C₈₃	—	2	1	no	—
113	C₂²×C₈₃	—	2	1	no	—
114	C₂²×C₈₃	—	2	1	no	—
115	C₂²×C₈₃	—	2	1	no	—
116	C₂²×C₈₃	—	4	4	no	—
117	C₂²×C₈₃	#359	1	2	yes	—
118	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
119	C₂²×C₈₃	#483	1	2	yes	—
120	C₄×C₈₃	—	2	1	no	—
121	C₄×C₈₃	—	2	1	no	—
122	C₄×C₈₃	—	8	2	no	—
123	C₄×C₈₃	—	8	2	no	—
124	C₄×C₈₃	#381	1	1	yes	characteristic
125	C₄×C₈₃	#419	1	1	yes	characteristic
126	C₄×C₈₃	#480	1	2	yes	—
127	#12869	—	83	1	no	maximal, Sylow
128	C₂×C₈₃	—	2	2	no	—
129	C₂×C₈₃	—	2	2	no	—
130	C₂×C₈₃	—	4	1	no	—
131	C₂×C₈₃	#1823	1	1	yes	characteristic
132	C₂×C₈₃	C₂×#359	1	1	yes	characteristic
133	C₂×C₈₃	#2143	1	1	yes	characteristic
134	C₂×#427	D₈₃	1	1	yes	characteristic
135	C₂×C₂²⋊C₁₆	—	83	1	no	—
136	#1810	—	83	1	no	—
137	#1382	—	83	4	no	—
138	C₈₃	#12869	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
140	C₂×#117	—	2	1	no	—
141	C₂²×C₁₆	—	83	2	no	—
142	C₂×#108	—	83	2	no	—
143	#427	—	2	1	no	—
144	#427	—	2	1	no	—
145	#427	Dic₈₃	1	2	yes	—
146	#427	C₂×D₈₃	1	2	yes	—
147	C₂×#107	Dic₈₃	1	1	yes	characteristic
148	C₂×#107	Dic₈₃	1	1	yes	characteristic
149	C₂×#106	—	2	1	no	—
150	#407	—	166	1	no	—
151	C₂²⋊C₁₆	—	166	1	no	—
152	C₂²⋊C₁₆	—	166	1	no	—
153	#341	—	166	1	no	—
154	C₂³×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
155	C₂×M₁₆	—	166	2	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
158	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
163	#117	—	2	2	no	—
164	#117	—	2	2	no	—
165	#117	—	4	1	no	—
166	#117	—	4	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	—	4	1	no	—
171	C₂×C₁₆	—	166	2	no	—
172	C₂×C₁₆	—	166	4	no	—
173	#108	—	166	4	no	—
174	#107	—	2	1	no	—
175	#107	—	2	1	no	—
176	#107	C₂×Dic₈₃	1	2	yes	—
177	#107	C₂×Dic₈₃	1	2	yes	—
178	#106	—	2	2	no	—
179	#106	—	2	2	no	—
180	C₂²⋊C₈	—	332	1	no	—
181	C₂²⋊C₈	—	332	1	no	—
182	C₂⁴	#13050578	1	1	yes	characteristic
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
189	C₂²×C₄	#13050575	1	1	yes	characteristic
190	C₂²×C₄	#13050575	1	1	yes	characteristic
191	C₂²×C₄	#13050578	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
193	M₁₆	—	332	2	no	—
194	M₁₆	—	332	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	332	2	no	—
201	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
203	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
204	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
205	C₂×C₈	#13050590	1	4	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₁₆	—	332	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	1	no	—
218	C₂³	#14581713	1	1	yes	characteristic
219	C₂³	#14581713	1	1	yes	characteristic
220	C₂³	C₂×#13050578	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	#14581713	1	1	yes	characteristic
233	C₂×C₄	#14581713	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
235	C₂×C₄	#14581775	1	2	yes	—
236	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
237	C₂×C₄	#14581823	1	2	yes	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	664	2	no	—
241	C₈	#14581751	1	2	yes	—
242	C₈	#14581775	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	#19127508	1	2	yes	—
252	C₂²	C₂×#14581713	1	1	yes	characteristic
253	C₂²	#19128031	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	#19127564	1	1	yes	characteristic
259	C₄	#19127710	1	1	yes	characteristic
260	C₄	#19128014	1	2	yes	—
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	4	1	no	—
264	C₂	#24907549	1	1	yes	characteristic
265	C₂	C₂×#19127508	1	1	yes	characteristic
266	C₂	#24909191	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#427×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#102
prime radical	C₂⁴×C₈₃	SD₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#427	D₈₃
hypocenter	C₈₃	#12869
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #24905558	C₂	semidirect
138. C₈₃	#12869	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₈₃⋊C₁₆)	2
#24907501	2
#24905558	2
C₂×#427×C₈₃	2
#12869	83