Subgroups — #35319439

Order 21248.

514 conjugacy classes of subgroups.

The subgroup lattice has 266 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	2	yes	maximal
2	C₂×#19127477	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#24905541	C₂	1	2	yes	maximal, complemented
5	C₂×#405×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
7	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
8	C₂²⋊(C₈₃⋊C₁₆)	C₄	1	8	yes	complemented
9	#19127477	—	2	2	no	—
10	C₄×C₈₃⋊C₁₆	—	2	2	no	—
11	C₂²×M₁₆×C₈₃	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₈₃	C₄	1	1	yes	characteristic
14	#405×C₈₃	—	2	2	no	—
15	#405×C₈₃	—	2	2	no	—
16	C₂×#97×C₈₃	C₄	1	1	yes	characteristic
17	C₂×#97×C₈₃	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₈₃⋊C₁₆	—	2	2	no	—
21	C₂×C₈₃⋊C₁₆	—	2	2	no	—
22	C₂×C₈₃⋊C₁₆	—	2	2	no	—
23	C₂×C₈₃⋊C₁₆	—	2	2	no	—
24	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
25	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₈₃	—	2	2	no	—
28	C₂×M₁₆×C₈₃	—	2	2	no	—
29	C₂×M₁₆×C₈₃	C₈	1	4	yes	—
30	C₂²×C₈×C₈₃	—	2	1	no	—
31	C₂²×C₈×C₈₃	—	2	1	no	—
32	C₂²×C₈×C₈₃	—	2	2	no	—
33	C₂²×C₈×C₈₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
38	#117×C₈₃	—	2	2	no	—
39	#117×C₈₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₈₃	C₈	1	2	yes	—
41	C₂×C₂²⋊C₄×C₈₃	C₈	1	2	yes	—
42	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
44	#97×C₈₃	—	2	2	no	—
45	#97×C₈₃	—	2	2	no	—
46	C₄×C₈×C₈₃	—	2	1	no	—
47	C₄×C₈×C₈₃	—	2	1	no	—
48	C₄×C₈×C₈₃	D₄	1	2	yes	—
49	C₄×C₈×C₈₃	D₄	1	2	yes	—
50	C₈₃⋊C₁₆	—	2	4	no	—
51	C₈₃⋊C₁₆	—	2	4	no	—
52	C₂⁴×C₈₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	—	2	1	no	—
54	C₂²×C₄×C₈₃	—	2	1	no	—
55	C₂²×C₄×C₈₃	—	2	1	no	—
56	C₂²×C₄×C₈₃	—	2	1	no	—
57	C₂²×C₄×C₈₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
63	M₁₆×C₈₃	—	2	8	no	—
64	C₂×C₈×C₈₃	—	2	1	no	—
65	C₂×C₈×C₈₃	—	2	1	no	—
66	C₂×C₈×C₈₃	—	2	1	no	—
67	C₂×C₈×C₈₃	—	2	1	no	—
68	C₂×C₈×C₈₃	—	2	4	no	—
69	C₂×C₈×C₈₃	—	2	4	no	—
70	C₂×C₈×C₈₃	C₄²	1	2	yes	—
71	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
75	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₃	#41	1	2	yes	—
79	C₄⋊C₄×C₈₃	—	2	4	no	—
80	C₂²⋊C₄×C₈₃	—	2	4	no	—
81	C₄²×C₈₃	—	2	1	no	—
82	C₄²×C₈₃	—	2	1	no	—
83	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₈₃	M₁₆	1	2	yes	—
85	C₂³×C₈₃	—	2	1	no	—
86	C₂³×C₈₃	—	2	1	no	—
87	C₂³×C₈₃	—	2	4	no	—
88	C₂³×C₈₃	C₄×C₈	1	2	yes	—
89	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₈₃	—	2	1	no	—
91	C₂×C₄×C₈₃	—	2	1	no	—
92	C₂×C₄×C₈₃	—	2	4	no	—
93	C₂×C₄×C₈₃	—	2	2	no	—
94	C₂×C₄×C₈₃	—	2	2	no	—
95	C₂×C₄×C₈₃	—	2	2	no	—
96	C₂×C₄×C₈₃	—	2	2	no	—
97	C₂×C₄×C₈₃	—	2	2	no	—
98	C₂×C₄×C₈₃	—	2	4	no	—
99	C₂×C₄×C₈₃	C₄×C₈	1	2	yes	—
100	C₂×C₄×C₈₃	C₂²⋊C₈	1	4	yes	—
101	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
107	C₈×C₈₃	—	2	2	no	—
108	C₈×C₈₃	—	2	2	no	—
109	C₈×C₈₃	C₂²⋊C₈	1	4	yes	—
110	C₈×C₈₃	C₄×D₄	1	2	yes	—
111	C₈×C₈₃	C₄×D₄	1	2	yes	—
112	C₂²×C₈₃	—	2	4	no	—
113	C₂²×C₈₃	—	2	2	no	—
114	C₂²×C₈₃	—	4	2	no	—
115	C₂²×C₈₃	—	4	2	no	—
116	C₂²×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₈₃	#431	1	1	yes	characteristic
119	C₂²×C₈₃	#504	1	4	yes	—
120	C₄×C₈₃	—	2	4	no	—
121	C₄×C₈₃	—	2	2	no	—
122	C₄×C₈₃	—	4	2	no	—
123	C₄×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₈₃	C₄×M₁₆	1	1	yes	characteristic
125	C₄×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₈₃	#431	1	1	yes	characteristic
127	#12600	—	83	1	no	maximal, Sylow
128	C₂×C₈₃	—	4	2	no	—
129	C₂×C₈₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₈₃	#2108	1	2	yes	—
131	C₂×C₈₃	#2113	1	2	yes	—
132	C₂×C₈₃	#2164	1	2	yes	—
133	C₂×#405	D₈₃	1	1	yes	characteristic
134	C₂×C₂²⋊C₁₆	—	83	2	no	—
135	C₂×#346	—	83	1	no	—
136	C₂×C₄×C₁₆	—	83	1	no	—
137	#1374	—	83	2	no	—
138	C₈₃	#12600	1	1	yes	characteristic, complemented, Sylow
139	C₂²×M₁₆	Dic₈₃	1	1	yes	characteristic
140	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
141	C₂×#117	Dic₈₃	1	1	yes	characteristic
142	C₂²×C₁₆	—	83	2	no	—
143	C₂²×C₁₆	—	83	2	no	—
144	#405	—	2	2	no	—
145	#405	—	2	2	no	—
146	C₂×#97	Dic₈₃	1	1	yes	characteristic
147	C₂×#97	Dic₈₃	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
150	C₂²⋊C₁₆	—	83	8	no	—
151	#346	—	166	2	no	—
152	C₄×C₁₆	—	166	2	no	—
153	C₂³×C₄	C₂×Dic₈₃	1	1	yes	characteristic
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₈₃⋊C₈	1	4	yes	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₄×D₈₃	1	4	yes	—
161	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
165	#117	—	2	2	no	—
166	#117	—	2	2	no	—
167	C₂×C₄⋊C₄	C₈₃⋊C₈	1	2	yes	—
168	C₂×C₂²⋊C₄	C₈₃⋊C₈	1	2	yes	—
169	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
171	C₂×C₁₆	—	83	4	no	—
172	C₂×C₁₆	—	83	4	no	—
173	C₂×C₁₆	—	166	2	no	—
174	C₂×C₁₆	—	166	2	no	—
175	C₂×C₁₆	—	166	2	no	—
176	C₂×C₁₆	—	166	2	no	—
177	#97	—	2	2	no	—
178	#97	—	2	2	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
182	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
183	C₂⁴	C₂×C₈₃⋊C₈	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₄×Dic₈₃	1	2	yes	—
189	C₂²×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
190	C₂²×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
191	C₂²×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
194	M₁₆	—	2	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
203	C₂×C₈	C₂×C₈₃⋊C₈	1	2	yes	—
204	C₂×C₈	C₂×C₈₃⋊C₈	1	2	yes	—
205	C₂×C₈	#13050574	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
207	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
209	C₂×C₈	#13050590	1	2	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	C₂²⋊Dic₈₃	1	2	yes	—
215	C₄²	#13050574	1	2	yes	—
216	C₁₆	—	166	4	no	—
217	C₁₆	—	166	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	C₄×C₈₃⋊C₈	1	2	yes	—
222	C₂³	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
223	C₂×C₄	—	2	4	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	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	C₄×C₈₃⋊C₈	1	2	yes	—
233	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	4	yes	—
234	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
236	C₂×C₄	#14581751	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
238	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
239	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	C₂²⋊(C₈₃⋊C₈)	1	4	yes	—
243	C₈	C₄×C₂²⋊D₈₃	1	2	yes	—
244	C₈	C₄×C₂²⋊D₈₃	1	2	yes	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	C₂×C₄×C₈₃⋊C₈	1	1	yes	characteristic
250	C₂²	C₂×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
251	C₂²	#19127754	1	1	yes	characteristic
252	C₂²	#19128118	1	4	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	2	no	—
256	C₄	C₄×C₂²⋊Dic₈₃	1	1	yes	characteristic
257	C₄	C₄×#13050574	1	1	yes	characteristic
258	C₄	C₂×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
259	C₄	#19127754	1	1	yes	characteristic
260	C₂	—	4	2	no	—
261	C₂	C₄×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
262	C₂	#24909060	1	2	yes	—
263	C₂	#24909077	1	2	yes	—
264	C₂	#24909270	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂×C₄×C₈
Fitting	C₂×#405×C₈₃	C₂
socle	C₂³×C₈₃	C₂²×C₈
prime radical	C₂⁴×C₈₃	C₂×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#405	D₈₃
hypocenter	C₈₃	#12600
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. C₂×#19127477	C₂	semidirect
3. C₂×C₄×C₈₃⋊C₁₆	C₂	semidirect
4. #24905541	C₂	semidirect
8. C₂²⋊(C₈₃⋊C₁₆)	C₄	semidirect
138. C₈₃	#12600	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
C₂×#19127477	2
C₂×C₄×C₈₃⋊C₁₆	2
#24905541	2
C₂×#405×C₈₃	2
#12600	83