Subgroups — #35320346

Order 21248.

538 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127522	C₂	1	2	yes	maximal, complemented
2	#24909074	C₂	1	2	yes	maximal
3	#24905545	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₂²	1	4	yes	—
6	#19127522	—	2	4	no	—
7	C₂²⋊(C₈₃⋊C₁₆)	—	2	4	no	—
8	C₂³×C₈×C₈₃	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₈₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₈₃	—	2	4	no	—
12	C₂×C₄×C₈×C₈₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#14581732	—	2	4	no	—
17	#14581732	—	2	4	no	—
18	C₂×C₈₃⋊C₁₆	—	2	4	no	—
19	C₂³×C₄×C₈₃	—	2	1	no	—
20	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₃	—	2	2	no	—
22	C₂²×C₈×C₈₃	—	2	2	no	—
23	C₂²×C₈×C₈₃	—	2	2	no	—
24	C₂²×C₈×C₈₃	—	2	2	no	—
25	C₂²×C₈×C₈₃	C₈	1	4	yes	—
26	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	D₄	1	2	yes	—
30	C₂²×C₈×C₈₃	Q₈	1	2	yes	—
31	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	—	2	1	no	—
33	C₂×C₄²×C₈₃	—	2	2	no	—
34	C₂×C₄²×C₈₃	—	2	2	no	—
35	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₃	—	2	2	no	—
38	C₄×C₈×C₈₃	—	2	2	no	—
39	C₄×C₈×C₈₃	—	2	2	no	—
40	C₄×C₈×C₈₃	D₄	1	4	yes	—
41	C₈₃⋊C₁₆	—	4	4	no	—
42	C₂⁴×C₈₃	M₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₈₃	—	2	4	no	—
44	C₂²×C₄×C₈₃	—	2	8	no	—
45	C₂²×C₄×C₈₃	—	2	1	no	—
46	C₂²×C₄×C₈₃	—	2	1	no	—
47	C₂²×C₄×C₈₃	—	2	1	no	—
48	C₂²×C₄×C₈₃	—	2	1	no	—
49	C₂²×C₄×C₈₃	—	2	1	no	—
50	C₂²×C₄×C₈₃	—	2	1	no	—
51	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	—	2	1	no	—
56	C₂×C₈×C₈₃	—	2	1	no	—
57	C₂×C₈×C₈₃	—	2	1	no	—
58	C₂×C₈×C₈₃	—	2	1	no	—
59	C₂×C₈×C₈₃	—	2	8	no	—
60	C₂×C₈×C₈₃	—	2	4	no	—
61	C₂×C₈×C₈₃	—	2	4	no	—
62	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
67	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₈₃	#41	1	2	yes	—
71	C₄²×C₈₃	—	2	4	no	—
72	C₄²×C₈₃	—	2	1	no	—
73	C₄²×C₈₃	—	2	1	no	—
74	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₈₃	M₁₆	1	2	yes	—
76	C₂³×C₈₃	—	2	4	no	—
77	C₂³×C₈₃	—	2	1	no	—
78	C₂³×C₈₃	—	2	1	no	—
79	C₂³×C₈₃	C₄⋊C₈	1	2	yes	—
80	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₈₃	—	2	16	no	—
82	C₂×C₄×C₈₃	—	2	4	no	—
83	C₂×C₄×C₈₃	—	2	4	no	—
84	C₂×C₄×C₈₃	—	2	1	no	—
85	C₂×C₄×C₈₃	—	2	2	no	—
86	C₂×C₄×C₈₃	—	2	2	no	—
87	C₂×C₄×C₈₃	—	2	1	no	—
88	C₂×C₄×C₈₃	—	2	2	no	—
89	C₂×C₄×C₈₃	—	2	2	no	—
90	C₂×C₄×C₈₃	—	2	2	no	—
91	C₂×C₄×C₈₃	—	2	2	no	—
92	C₂×C₄×C₈₃	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
99	C₈×C₈₃	—	2	2	no	—
100	C₈×C₈₃	—	2	2	no	—
101	C₈×C₈₃	C₂²⋊C₈	1	4	yes	—
102	C₈×C₈₃	#121	1	2	yes	—
103	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
104	C₂²×C₈₃	—	2	4	no	—
105	C₂²×C₈₃	—	2	4	no	—
106	C₂²×C₈₃	—	2	2	no	—
107	C₂²×C₈₃	—	2	2	no	—
108	C₂²×C₈₃	—	2	2	no	—
109	C₂²×C₈₃	#364	1	4	yes	complemented
110	C₂²×C₈₃	#407	1	1	yes	characteristic
111	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₈₃	#432	1	1	yes	characteristic
113	C₄×C₈₃	—	2	4	no	—
114	C₄×C₈₃	—	2	4	no	—
115	C₄×C₈₃	—	2	2	no	—
116	C₄×C₈₃	#380	1	1	yes	characteristic
117	C₄×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₈₃	#423	1	1	yes	characteristic
119	C₄×C₈₃	#431	1	1	yes	characteristic
120	#12788	—	83	1	no	maximal, Sylow
121	C₂×C₈₃	—	2	1	no	—
122	C₂×C₈₃	—	2	4	no	—
123	C₂×C₈₃	—	2	1	no	—
124	C₂×C₈₃	#1797	1	1	yes	characteristic
125	C₂×C₈₃	C₂×#364	1	2	yes	—
126	C₂²×C₄×C₈	D₈₃	1	1	yes	characteristic
127	C₂×#364	—	83	2	no	—
128	#2112	—	83	2	no	—
129	#1376	—	83	2	no	—
130	C₈₃	#12788	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₈	Dic₈₃	1	1	yes	characteristic
132	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
133	C₂²×C₄²	Dic₈₃	1	1	yes	characteristic
134	C₂×M₃₂	—	83	4	no	—
135	C₂×C₄×C₈	—	2	4	no	—
136	C₂×C₄×C₈	Dic₈₃	1	1	yes	characteristic
137	C₂×C₄×C₈	Dic₈₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
140	#364	—	166	4	no	—
141	C₂²⋊C₁₆	—	166	4	no	—
142	C₂³×C₄	—	2	1	no	—
143	C₂³×C₄	C₂×Dic₈₃	1	1	yes	characteristic
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	C₈₃⋊C₈	1	4	yes	—
149	C₂²×C₈	Dic₁₆₆	1	2	yes	—
150	C₂²×C₈	D₃₃₂	1	2	yes	—
151	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
152	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
153	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
159	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
160	M₃₂	—	166	4	no	—
161	M₃₂	—	166	4	no	—
162	C₂×C₁₆	—	166	4	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	C₂²⋊D₈₃	1	4	yes	—
167	C₂⁴	#13050574	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	C₄⋊Dic₈₃	1	2	yes	—
177	C₂²×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
178	C₂²×C₄	#13050574	1	1	yes	characteristic
179	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
188	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
189	C₂×C₈	C₂×C₈₃⋊C₈	1	2	yes	—
190	C₂×C₈	C₂×C₈₃⋊C₈	1	2	yes	—
191	C₂×C₈	#13050574	1	2	yes	—
192	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
193	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
194	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
195	C₂×C₈	#13050590	1	2	yes	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	C₂²⋊Dic₈₃	1	2	yes	—
200	C₄²	#13050574	1	2	yes	—
201	C₁₆	—	332	4	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	4	no	—
205	C₂³	C₄⋊(C₈₃⋊C₈)	1	2	yes	—
206	C₂³	C₂×#13050574	1	1	yes	characteristic
207	C₂×C₄	—	2	16	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
221	C₂×C₄	#14581751	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
223	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
224	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	C₂²⋊(C₈₃⋊C₈)	1	4	yes	—
228	C₈	#14581766	1	2	yes	—
229	C₈	#14581771	1	2	yes	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	#19127522	1	4	yes	complemented
236	C₂²	#19127665	1	1	yes	characteristic
237	C₂²	C₂×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
238	C₂²	#19127758	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	#19127561	1	1	yes	characteristic
243	C₄	C₂×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
244	C₄	#19127725	1	1	yes	characteristic
245	C₄	#19127754	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	—	2	4	no	—
249	C₂	#24907443	1	1	yes	characteristic
250	C₂	C₂×#19127522	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂²×C₄×C₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₈₃	M₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂²×C₄×C₈	D₈₃
hypocenter	C₈₃	#12788
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19127522	C₂	semidirect
3. #24905545	C₂	semidirect
109. C₂²×C₈₃	#364	semidirect
130. C₈₃	#12788	semidirect
235. C₂²	#19127522	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127522	2
#24909074	2
#24905545	2
C₂²×C₄×C₈×C₈₃	2
#12788	83