Subgroups — #35375264

Order 21248.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918049	C₂	1	1	yes	maximal, characteristic
2	#24916560	C₂	1	2	yes	maximal
3	#24906333	C₂	1	4	yes	maximal, complemented
4	#24906312	C₂	1	4	yes	maximal, complemented
5	#24905935	C₂	1	2	yes	maximal
6	#24905755	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#1568	C₂	1	1	yes	maximal, characteristic
8	#19128258	C₂²	1	2	yes	—
9	#19128106	C₄	1	1	yes	characteristic
10	#19128081	C₄	1	1	yes	characteristic
11	#19128052	C₄	1	2	yes	—
12	#19127942	C₄	1	2	yes	—
13	#19127942	C₄	1	2	yes	—
14	#19127829	—	2	4	no	—
15	#19127828	C₂²	1	4	yes	—
16	#19127825	C₂²	1	4	yes	—
17	C₄×Dic₃₃₂	C₂²	1	2	yes	—
18	C₄×Dic₃₃₂	C₂²	1	2	yes	—
19	#19127759	—	2	4	no	—
20	#19127754	C₂²	1	1	yes	characteristic
21	#19127754	C₂²	1	1	yes	characteristic
22	#19127431	C₂²	1	8	yes	—
23	#19127423	C₂²	1	2	yes	—
24	#19127423	C₂²	1	2	yes	—
25	#432×C₈₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
27	#335×C₈₃	C₂²	1	2	yes	—
28	#334×C₈₃	C₂²	1	2	yes	—
29	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
30	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
31	#14581792	C₂×C₄	1	2	yes	—
32	#14581792	C₂×C₄	1	2	yes	—
33	#14581784	—	2	2	no	—
34	#14581771	—	2	2	no	—
35	#14581771	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₆₆	—	2	2	no	—
37	C₄×Dic₁₆₆	C₂³	1	4	yes	—
38	#14581751	—	2	2	no	—
39	#14581726	C₂×C₄	1	2	yes	—
40	#14581726	C₂×C₄	1	2	yes	—
41	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
42	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
43	Dic₈₃⋊C₈	—	2	4	no	—
44	Dic₈₃⋊C₈	—	2	4	no	—
45	#14581717	C₂×C₄	1	4	yes	—
46	#14581717	C₂×C₄	1	4	yes	—
47	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
48	C₂²⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
49	#14581702	—	2	4	no	—
50	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
51	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
52	C₈×Dic₈₃	—	2	4	no	—
53	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂³	1	4	yes	—
57	C₂²⋊C₈×C₈₃	—	2	2	no	—
58	#97×C₈₃	—	2	2	no	—
59	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₆₆	—	2	2	no	—
64	C₂×Dic₁₆₆	C₂²×C₄	1	4	yes	—
65	Dic₃₃₂	—	4	2	no	—
66	Dic₃₃₂	—	4	2	no	—
67	C₂×C₈₃⋊C₈	—	2	2	no	—
68	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
69	C₄⋊Dic₈₃	—	2	2	no	—
70	C₄⋊Dic₈₃	—	2	2	no	—
71	C₄⋊Dic₈₃	—	2	2	no	—
72	C₄⋊Dic₈₃	C₂²×C₄	1	4	yes	—
73	Dic₈₃⋊C₄	—	2	4	no	—
74	Dic₈₃⋊C₄	—	2	4	no	—
75	C₂²⋊Dic₈₃	—	2	4	no	—
76	C₄×Dic₈₃	—	2	4	no	—
77	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
80	C₂×C₈×C₈₃	—	2	4	no	—
81	C₂×C₈×C₈₃	—	4	1	no	—
82	C₂×C₈×C₈₃	—	4	1	no	—
83	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₃	#41	1	1	yes	characteristic
90	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
91	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₈₃	—	2	1	no	—
93	C₂×C₄×C₈₃	—	2	1	no	—
94	C₂×C₄×C₈₃	—	2	2	no	—
95	C₂×C₄×C₈₃	—	4	1	no	—
96	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₈₃	—	2	4	no	—
103	C₂×Dic₈₃	—	2	4	no	—
104	Dic₁₆₆	—	2	4	no	—
105	Dic₁₆₆	—	2	4	no	—
106	Dic₁₆₆	—	8	2	no	—
107	C₈×C₈₃	—	2	1	no	—
108	C₈×C₈₃	—	2	1	no	—
109	C₈×C₈₃	—	2	1	no	—
110	C₈×C₈₃	—	2	1	no	—
111	C₈×C₈₃	—	4	4	no	—
112	C₈₃⋊C₈	—	2	4	no	—
113	C₈₃⋊C₈	—	4	2	no	—
114	C₂²×C₈₃	—	2	1	no	—
115	C₂²×C₈₃	—	2	1	no	—
116	C₂²×C₈₃	—	2	1	no	—
117	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₈₃	—	2	1	no	—
119	C₄×C₈₃	—	2	1	no	—
120	C₄×C₈₃	—	2	1	no	—
121	C₄×C₈₃	—	2	1	no	—
122	C₄×C₈₃	#568	1	1	yes	characteristic
123	C₄×C₈₃	#568	1	1	yes	characteristic
124	C₄×C₈₃	#578	1	2	yes	—
125	Dic₈₃	—	4	4	no	—
126	Dic₈₃	—	4	4	no	—
127	#17991	—	83	1	no	maximal, Sylow
128	C₂×C₈₃	—	4	1	no	—
129	C₂×C₈₃	#2932	1	1	yes	characteristic
130	C₂×C₈₃	#2950	1	1	yes	characteristic
131	C₂×C₈₃	#2957	1	1	yes	characteristic
132	#3170	—	83	1	no	—
133	#2985	—	83	2	no	—
134	#1587	—	83	4	no	—
135	#1584	—	83	4	no	—
136	#1568	D₈₃	1	1	yes	characteristic
137	#1497	—	83	2	no	—
138	#1452	—	83	1	no	—
139	C₈₃	#17991	1	1	yes	characteristic, complemented, Sylow
140	#533	—	83	2	no	—
141	#500	—	83	1	no	—
142	#494	—	83	1	no	—
143	#487	—	83	2	no	—
144	#467	—	83	2	no	—
145	#467	—	83	2	no	—
146	#446	—	83	4	no	—
147	#446	—	166	4	no	—
148	C₈×Q₈	—	83	4	no	—
149	C₄×Q₁₆	—	83	2	no	—
150	C₄×Q₁₆	—	83	2	no	—
151	#433	—	166	4	no	—
152	#432	C₂×D₈₃	1	2	yes	—
153	#431	—	83	1	no	—
154	#431	—	83	1	no	—
155	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
156	#335	C₂×D₈₃	1	2	yes	—
157	#334	C₂×D₈₃	1	2	yes	—
158	Q₈⋊C₈	—	83	8	no	—
159	#322	—	83	2	no	—
160	#322	—	83	2	no	—
161	C₂×Q₁₆	—	83	2	no	—
162	C₂×Q₁₆	—	83	2	no	—
163	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
165	C₄⋊Q₈	—	83	2	no	—
166	C₄⋊Q₈	—	83	2	no	—
167	#125	—	166	2	no	—
168	C₂²⋊Q₈	—	83	4	no	—
169	C₂²⋊Q₈	—	166	2	no	—
170	C₄×Q₈	—	83	4	no	—
171	C₄×Q₈	—	166	2	no	—
172	#117	—	166	2	no	—
173	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
174	#107	—	83	2	no	—
175	#107	—	83	2	no	—
176	C₄⋊C₈	—	83	2	no	—
177	C₄⋊C₈	—	166	4	no	—
178	C₄⋊C₈	—	166	4	no	—
179	C₄⋊C₈	—	166	1	no	—
180	C₄⋊C₈	C₂²×D₈₃	1	4	yes	—
181	#103	—	83	4	no	—
182	#103	—	83	4	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₂²⋊C₈	—	83	2	no	—
185	C₂²⋊C₈	—	166	1	no	—
186	#97	—	2	2	no	—
187	#97	—	166	4	no	—
188	C₄×C₈	—	83	2	no	—
189	C₄×C₈	—	83	2	no	—
190	C₄×C₈	—	166	4	no	—
191	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
192	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
195	C₂×Q₈	—	83	4	no	—
196	C₂×Q₈	—	166	2	no	—
197	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
198	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
199	C₂²×C₄	#13050591	1	1	yes	characteristic
200	Q₁₆	—	332	2	no	—
201	Q₁₆	—	332	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	83	4	no	—
206	C₂×C₈	—	166	2	no	—
207	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
211	C₄⋊C₄	—	83	4	no	—
212	C₄⋊C₄	—	166	4	no	—
213	C₄⋊C₄	—	166	2	no	—
214	C₄⋊C₄	—	166	2	no	—
215	C₄⋊C₄	—	166	4	no	—
216	C₄⋊C₄	—	166	2	no	—
217	C₂²⋊C₄	—	166	4	no	—
218	C₄²	—	166	4	no	—
219	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
220	C₄²	D₄×D₈₃	1	1	yes	characteristic
221	C₄²	#13050591	1	1	yes	characteristic
222	C₄²	C₂³×D₈₃	1	1	yes	characteristic
223	C₂³	#14581754	1	1	yes	characteristic
224	Q₈	—	664	2	no	—
225	Q₈	—	166	4	no	—
226	Q₈	—	166	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	166	4	no	—
232	C₂×C₄	—	166	4	no	—
233	C₂×C₄	#14581754	1	1	yes	characteristic
234	C₂×C₄	#14581754	1	1	yes	characteristic
235	C₂×C₄	#14581754	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	4	4	no	—
244	C₈	—	332	2	no	—
245	C₈	—	166	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×#14581754	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	332	4	no	—
255	C₄	—	332	4	no	—
256	C₄	#19128467	1	1	yes	characteristic
257	C₄	#19128467	1	1	yes	characteristic
258	C₄	#19128515	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#24916180	1	1	yes	characteristic
261	C₂	#24916330	1	1	yes	characteristic
262	C₂	#24916379	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1568	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#1568	D₈₃
hypocenter	C₈₃	#17991
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #24906333	C₂	semidirect
4. #24906312	C₂	semidirect
139. C₈₃	#17991	semidirect

Maximal subgroups

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

subgroup	index
#24918049	2
#24916560	2
#24906333	2
#24906312	2
#24905935	2
#24905755	2
C₈₃×#1568	2
#17991	83