Subgroups — #35375265

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	#24916558	C₂	1	2	yes	maximal
2	#24906335	C₂	1	4	yes	maximal, complemented
3	#24906314	C₂	1	4	yes	maximal, complemented
4	#24906226	C₂	1	1	yes	maximal, characteristic
5	#24905936	C₂	1	2	yes	maximal
6	#24905755	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#3170	C₂	1	1	yes	maximal, characteristic
8	#19127832	—	2	4	no	—
9	#19127832	C₂²	1	4	yes	—
10	Q₈×C₈₃⋊C₈	C₂²	1	4	yes	—
11	#19127762	—	2	4	no	—
12	#19127758	C₂²	1	2	yes	—
13	#19127754	C₂²	1	1	yes	characteristic
14	#19127754	C₂²	1	1	yes	characteristic
15	#19127454	C₂²	1	2	yes	—
16	#19127452	C₂²	1	2	yes	—
17	Q₈⋊(C₈₃⋊C₈)	C₂²	1	8	yes	—
18	#19127423	C₂²	1	2	yes	—
19	#19127423	C₂²	1	2	yes	—
20	#533×C₈₃	C₂²	1	2	yes	—
21	#500×C₈₃	C₄	1	1	yes	characteristic
22	#494×C₈₃	C₄	1	1	yes	characteristic
23	#487×C₈₃	C₄	1	2	yes	—
24	#467×C₈₃	C₄	1	2	yes	—
25	#467×C₈₃	C₄	1	2	yes	—
26	C₄×Q₁₆×C₈₃	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₈₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
29	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
30	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
31	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
32	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
33	C₄⋊(C₈₃⋊C₈)	C₂³	1	4	yes	—
34	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
35	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
36	C₂²⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
37	#14581703	—	2	4	no	—
38	#14581703	—	2	2	no	—
39	C₄×C₈₃⋊C₈	—	2	4	no	—
40	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
41	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
42	C₂×Q₁₆×C₈₃	C₂×C₄	1	2	yes	—
43	C₂×Q₁₆×C₈₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₈₃	C₂×C₄	1	2	yes	—
47	C₄⋊Q₈×C₈₃	C₂×C₄	1	2	yes	—
48	#125×C₈₃	—	2	2	no	—
49	C₂²⋊Q₈×C₈₃	—	2	2	no	—
50	C₂²⋊Q₈×C₈₃	C₂×C₄	1	4	yes	—
51	C₄×Q₈×C₈₃	—	2	2	no	—
52	C₄×Q₈×C₈₃	C₂³	1	4	yes	—
53	#117×C₈₃	—	2	2	no	—
54	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
55	#107×C₈₃	C₂×C₄	1	2	yes	—
56	#107×C₈₃	C₂×C₄	1	2	yes	—
57	#103×C₈₃	C₂×C₄	1	4	yes	—
58	#103×C₈₃	C₂×C₄	1	4	yes	—
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×C₈₃⋊C₈	—	2	4	no	—
64	C₂×C₈₃⋊C₈	—	2	2	no	—
65	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
66	C₂×Q₈×C₈₃	—	2	2	no	—
67	C₂×Q₈×C₈₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
71	Q₁₆×C₈₃	—	4	2	no	—
72	Q₁₆×C₈₃	—	4	2	no	—
73	C₂×C₈×C₈₃	—	4	1	no	—
74	C₂×C₈×C₈₃	—	4	1	no	—
75	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₈₃	—	2	2	no	—
80	C₄⋊C₄×C₈₃	—	2	2	no	—
81	C₄⋊C₄×C₈₃	—	2	2	no	—
82	C₄⋊C₄×C₈₃	—	2	4	no	—
83	C₄⋊C₄×C₈₃	—	2	4	no	—
84	C₄⋊C₄×C₈₃	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×C₈₃	—	2	4	no	—
86	C₄²×C₈₃	—	2	4	no	—
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	Q₈×C₈₃	—	2	4	no	—
93	Q₈×C₈₃	—	2	4	no	—
94	Q₈×C₈₃	—	8	2	no	—
95	C₂×C₄×C₈₃	—	2	1	no	—
96	C₂×C₄×C₈₃	—	2	1	no	—
97	C₂×C₄×C₈₃	—	2	2	no	—
98	C₂×C₄×C₈₃	—	2	4	no	—
99	C₂×C₄×C₈₃	—	2	4	no	—
100	C₂×C₄×C₈₃	—	4	1	no	—
101	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
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	C₈₃⋊C₈	—	2	4	no	—
112	C₈₃⋊C₈	—	4	2	no	—
113	C₈₃⋊C₈	—	4	4	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	—	4	4	no	—
123	C₄×C₈₃	—	4	4	no	—
124	C₄×C₈₃	#568	1	1	yes	characteristic
125	C₄×C₈₃	#568	1	1	yes	characteristic
126	C₄×C₈₃	#578	1	2	yes	—
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	D₈₃	1	1	yes	characteristic
133	#2985	—	83	2	no	—
134	#1587	—	83	4	no	—
135	#1584	—	83	4	no	—
136	#1568	—	83	1	no	—
137	#1497	—	83	2	no	—
138	#1452	—	83	1	no	—
139	C₈₃	#17991	1	1	yes	characteristic, complemented, Sylow
140	#533	C₂×D₈₃	1	2	yes	—
141	#500	Dic₈₃	1	1	yes	characteristic
142	#494	Dic₈₃	1	1	yes	characteristic
143	#487	Dic₈₃	1	2	yes	—
144	#467	Dic₈₃	1	2	yes	—
145	#467	Dic₈₃	1	2	yes	—
146	#446	—	83	4	no	—
147	#446	—	166	4	no	—
148	C₈×Q₈	—	83	4	no	—
149	C₄×Q₁₆	C₂×D₈₃	1	2	yes	—
150	C₄×Q₁₆	C₂×D₈₃	1	2	yes	—
151	#433	—	166	4	no	—
152	#432	—	83	2	no	—
153	#431	—	83	1	no	—
154	#431	—	83	1	no	—
155	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
156	#335	—	83	2	no	—
157	#334	—	83	2	no	—
158	Q₈⋊C₈	—	83	8	no	—
159	#322	—	83	2	no	—
160	#322	—	83	2	no	—
161	C₂×Q₁₆	C₂×Dic₈₃	1	2	yes	—
162	C₂×Q₁₆	C₂×Dic₈₃	1	2	yes	—
163	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
165	C₄⋊Q₈	C₂×Dic₈₃	1	2	yes	—
166	C₄⋊Q₈	C₂×Dic₈₃	1	2	yes	—
167	#125	—	2	2	no	—
168	C₂²⋊Q₈	—	2	2	no	—
169	C₂²⋊Q₈	C₂×Dic₈₃	1	4	yes	—
170	C₄×Q₈	—	2	2	no	—
171	C₄×Q₈	C₂²×D₈₃	1	4	yes	—
172	#117	—	2	2	no	—
173	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
174	#107	C₂×Dic₈₃	1	2	yes	—
175	#107	C₂×Dic₈₃	1	2	yes	—
176	C₄⋊C₈	—	83	2	no	—
177	C₄⋊C₈	—	83	4	no	—
178	C₄⋊C₈	—	166	4	no	—
179	C₄⋊C₈	—	166	4	no	—
180	C₄⋊C₈	—	166	1	no	—
181	#103	C₂×Dic₈₃	1	4	yes	—
182	#103	C₂×Dic₈₃	1	4	yes	—
183	C₂²⋊C₈	—	83	2	no	—
184	C₂²⋊C₈	—	166	2	no	—
185	C₂²⋊C₈	—	166	1	no	—
186	#97	—	166	4	no	—
187	#97	—	166	2	no	—
188	C₄×C₈	—	83	2	no	—
189	C₄×C₈	—	83	2	no	—
190	C₄×C₈	—	166	4	no	—
191	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
192	C₄×C₈	C₂×Dic₈₃	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	—	2	2	no	—
196	C₂×Q₈	C₂²×Dic₈₃	1	4	yes	—
197	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
198	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
199	C₂²×C₄	#13050589	1	1	yes	characteristic
200	Q₁₆	—	4	2	no	—
201	Q₁₆	—	4	2	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	83	4	no	—
205	C₂×C₈	—	166	4	no	—
206	C₂×C₈	—	166	2	no	—
207	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	C₂²×Dic₈₃	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
220	C₄²	D₄×D₈₃	1	1	yes	characteristic
221	C₄²	#13050589	1	1	yes	characteristic
222	C₄²	C₂³×D₈₃	1	1	yes	characteristic
223	C₂³	D₄×Dic₈₃	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	2	4	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
235	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
236	C₂×C₄	C₂³×Dic₈₃	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#13050589	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	—	332	2	no	—
244	C₈	—	332	4	no	—
245	C₈	—	166	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×D₄×Dic₈₃	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	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	#19128468	1	1	yes	characteristic
257	C₄	#19128468	1	1	yes	characteristic
258	C₄	#19128514	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#24916181	1	1	yes	characteristic
261	C₂	#24916331	1	1	yes	characteristic
262	C₂	#24916380	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×#3170	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#3170	D₈₃
hypocenter	C₈₃	#17991
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #24906335	C₂	semidirect
3. #24906314	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
#24916558	2
#24906335	2
#24906314	2
#24906226	2
#24905936	2
#24905755	2
C₈₃×#3170	2
#17991	83