Subgroups — #35314435

Order 21248.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909262	C₂	1	1	yes	maximal, characteristic
2	#24906222	C₂	1	1	yes	maximal, characteristic
3	#24905427	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Q₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
5	#19127725	—	2	1	no	—
6	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
7	C₂²⋊(C₈₃⋊C₁₆)	—	2	1	no	—
8	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
9	#19127454	—	2	1	no	—
10	#19127454	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₈₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₈₃	—	2	1	no	—
13	C₄×Q₁₆×C₈₃	—	2	1	no	—
14	C₄×Q₁₆×C₈₃	—	2	1	no	—
15	C₄×Q₁₆×C₈₃	C₄	1	2	yes	—
16	C₄×Q₁₆×C₈₃	C₂²	1	2	yes	—
17	C₂×#107×C₈₃	C₄	1	1	yes	characteristic
18	C₂×#103×C₈₃	—	2	1	no	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×#13050574	—	4	1	no	—
21	C₂×C₈₃⋊C₁₆	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
24	C₂²×Q₈×C₈₃	—	2	1	no	—
25	C₂×Q₁₆×C₈₃	—	2	4	no	—
26	C₂×Q₁₆×C₈₃	—	2	1	no	—
27	C₂×Q₁₆×C₈₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₈₃	—	2	2	no	—
31	C₄×Q₈×C₈₃	—	2	2	no	—
32	C₄×Q₈×C₈₃	—	4	1	no	—
33	C₄×Q₈×C₈₃	—	4	1	no	—
34	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
36	C₂×C₄²×C₈₃	—	4	1	no	—
37	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
38	#107×C₈₃	—	2	1	no	—
39	#107×C₈₃	C₂×C₄	1	2	yes	—
40	#103×C₈₃	—	2	2	no	—
41	#103×C₈₃	—	2	2	no	—
42	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
46	#13050574	—	8	2	no	—
47	C₂×C₈₃⋊C₈	—	4	2	no	—
48	C₈₃⋊C₁₆	—	4	2	no	—
49	C₂×Q₈×C₈₃	—	2	4	no	—
50	C₂×Q₈×C₈₃	—	2	2	no	—
51	C₂×Q₈×C₈₃	—	4	2	no	—
52	C₂×Q₈×C₈₃	—	4	1	no	—
53	C₂²×C₄×C₈₃	—	4	1	no	—
54	C₂²×C₄×C₈₃	—	4	1	no	—
55	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
58	Q₁₆×C₈₃	—	2	4	no	—
59	Q₁₆×C₈₃	—	2	2	no	—
60	Q₁₆×C₈₃	M₁₆	1	4	yes	complemented
61	C₂×C₈×C₈₃	—	2	2	no	—
62	C₂×C₈×C₈₃	—	4	1	no	—
63	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₈₃	—	2	2	no	—
68	C₄⋊C₄×C₈₃	—	4	1	no	—
69	C₄⋊C₄×C₈₃	—	4	2	no	—
70	C₄⋊C₄×C₈₃	—	4	2	no	—
71	C₄²×C₈₃	—	4	2	no	—
72	C₄²×C₈₃	—	4	2	no	—
73	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₈₃	D₈	1	1	yes	characteristic
75	C₄²×C₈₃	SD₁₆	1	1	yes	characteristic
76	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₈₃	#102	1	1	yes	characteristic
78	Q₈×C₈₃	—	2	4	no	—
79	Q₈×C₈₃	—	4	4	no	—
80	Q₈×C₈₃	—	4	2	no	—
81	C₂×C₄×C₈₃	—	2	1	no	—
82	C₂×C₄×C₈₃	—	2	1	no	—
83	C₂×C₄×C₈₃	—	4	1	no	—
84	C₂×C₄×C₈₃	—	4	4	no	—
85	C₂×C₄×C₈₃	—	4	2	no	—
86	C₂×C₄×C₈₃	—	4	2	no	—
87	C₂×C₄×C₈₃	—	8	2	no	—
88	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₈₃	SD₃₂	1	2	yes	—
92	C₂×C₄×C₈₃	Q₃₂	1	2	yes	—
93	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₈₃	—	2	1	no	—
97	C₈×C₈₃	—	2	1	no	—
98	C₈×C₈₃	—	2	1	no	—
99	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
100	C₈₃⋊C₈	—	8	2	no	—
101	C₂²×C₈₃	—	2	1	no	—
102	C₂²×C₈₃	—	2	1	no	—
103	C₂²×C₈₃	#358	1	2	yes	—
104	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
105	C₄×C₈₃	—	2	1	no	—
106	C₄×C₈₃	—	2	1	no	—
107	C₄×C₈₃	—	2	1	no	—
108	C₄×C₈₃	—	4	4	no	—
109	C₄×C₈₃	—	8	2	no	—
110	C₄×C₈₃	#358	1	2	yes	—
111	C₄×C₈₃	#407	1	1	yes	characteristic
112	C₄×C₈₃	#421	1	1	yes	characteristic
113	C₄×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
114	C₄×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
115	#11658	—	83	1	no	maximal, Sylow
116	C₂×C₈₃	—	2	2	no	—
117	C₂×C₈₃	#1486	1	1	yes	characteristic
118	C₂×C₈₃	C₂×#358	1	1	yes	characteristic
119	C₂×C₈₃	#2143	1	1	yes	characteristic
120	C₂×C₄×Q₁₆	D₈₃	1	1	yes	characteristic
121	#2162	—	83	1	no	—
122	#1567	—	83	1	no	—
123	#1332	—	83	4	no	—
124	C₈₃	#11658	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₁₆	Dic₈₃	1	1	yes	characteristic
126	C₂×C₄×Q₈	—	2	1	no	—
127	C₄×Q₁₆	—	2	1	no	—
128	C₄×Q₁₆	—	2	1	no	—
129	C₄×Q₁₆	Dic₈₃	1	2	yes	—
130	C₄×Q₁₆	C₂×D₈₃	1	2	yes	—
131	C₂×#107	Dic₈₃	1	1	yes	characteristic
132	#423	—	166	1	no	—
133	C₂×#103	—	2	1	no	—
134	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
135	C₄⋊C₁₆	—	166	1	no	—
136	C₂²⋊C₁₆	—	166	1	no	—
137	C₄×C₁₆	—	83	2	no	—
138	#335	—	83	2	no	—
139	#335	—	166	1	no	—
140	C₂²×Q₈	—	2	1	no	—
141	C₂×Q₁₆	—	2	4	no	—
142	C₂×Q₁₆	—	2	1	no	—
143	C₂×Q₁₆	C₂×Dic₈₃	1	2	yes	—
144	C₂×M₁₆	—	332	1	no	—
145	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	4	1	no	—
150	C₄×Q₈	—	4	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	4	1	no	—
153	C₂×C₄²	—	4	1	no	—
154	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
155	C₂×C₁₆	—	166	2	no	—
156	#107	—	2	1	no	—
157	#107	C₂×Dic₈₃	1	2	yes	—
158	C₄⋊C₈	—	332	1	no	—
159	C₄⋊C₈	—	166	2	no	—
160	#103	—	2	2	no	—
161	#103	—	2	2	no	—
162	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
173	C₂²×C₄	#13050575	1	1	yes	characteristic
174	C₂²×C₄	#13050578	1	1	yes	characteristic
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	2	2	no	—
177	Q₁₆	#13050574	1	4	yes	complemented
178	M₁₆	—	664	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	332	2	no	—
182	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
183	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
184	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
185	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	—	4	2	no	—
192	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
193	C₄²	#13050575	1	1	yes	characteristic
194	C₄²	#13050578	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
196	C₁₆	—	332	2	no	—
197	C₂³	#14581713	1	1	yes	characteristic
198	Q₈	—	2	4	no	—
199	Q₈	—	4	4	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	#14581713	1	1	yes	characteristic
209	C₂×C₄	#14581713	1	1	yes	characteristic
210	C₂×C₄	#14581713	1	1	yes	characteristic
211	C₂×C₄	#14581735	1	2	yes	—
212	C₂×C₄	#14581738	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
214	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
215	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	664	2	no	—
220	C₈	C₂×#13050574	1	2	yes	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	#19127505	1	2	yes	—
224	C₂²	C₂×#14581713	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	4	no	—
229	C₄	—	8	2	no	—
230	C₄	#19127505	1	2	yes	—
231	C₄	#19127665	1	1	yes	characteristic
232	C₄	#19127718	1	1	yes	characteristic
233	C₄	C₂×#14581735	1	1	yes	characteristic
234	C₄	C₂×#14581738	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	#24905888	1	1	yes	characteristic
237	C₂	C₂×#19127505	1	1	yes	characteristic
238	C₂	#24909191	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₂×C₄×Q₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#102
prime radical	C₂³×C₈₃	#102
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₈₃
hypocenter	C₈₃	#11658
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #24905427	C₂	semidirect
60. Q₁₆×C₈₃	M₁₆	semidirect
124. C₈₃	#11658	semidirect
177. Q₁₆	#13050574	semidirect

Maximal subgroups

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

subgroup	index
#24909262	2
#24906222	2
#24905427	2
C₂×C₄×Q₁₆×C₈₃	2
#11658	83