Subgroups — #35290995

Order 21248.

292 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907615	C₂	1	1	yes	maximal, characteristic
2	#24907580	C₂	1	1	yes	maximal, characteristic
3	C₈₃×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
5	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
8	#19127456	—	2	1	no	—
9	#19127456	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₈₃	—	2	1	no	—
11	C₄³×C₈₃	C₂²	1	1	yes	characteristic
12	#336×C₈₃	—	2	1	no	—
13	#336×C₈₃	C₄	1	2	yes	—
14	C₂²×C₈₃⋊C₈	—	2	2	no	—
15	C₂²×C₈₃⋊C₈	—	2	2	no	—
16	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
17	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
18	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
19	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
20	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
21	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
22	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₈₃	—	2	2	no	—
24	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₃	Q₈	1	1	yes	characteristic
31	C₄⋊C₈×C₈₃	—	4	1	no	—
32	C₄⋊C₈×C₈₃	—	4	1	no	—
33	C₂×C₈₃⋊C₈	—	4	2	no	—
34	C₂×C₈₃⋊C₈	—	4	2	no	—
35	C₂×C₈₃⋊C₈	—	4	2	no	—
36	C₂×C₈₃⋊C₈	—	4	2	no	—
37	C₂×C₈₃⋊C₈	—	4	2	no	—
38	C₂×C₈₃⋊C₈	—	4	2	no	—
39	C₂²×C₄×C₈₃	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	—	4	2	no	—
47	C₂×C₈×C₈₃	—	4	2	no	—
48	C₂×C₈×C₈₃	—	4	2	no	—
49	C₄²×C₈₃	—	2	1	no	—
50	C₄²×C₈₃	—	2	1	no	—
51	C₄²×C₈₃	—	2	2	no	—
52	C₄²×C₈₃	—	4	1	no	—
53	C₄²×C₈₃	—	4	1	no	—
54	C₄²×C₈₃	—	4	1	no	—
55	C₄²×C₈₃	D₈	1	1	yes	characteristic
56	C₄²×C₈₃	D₈	1	1	yes	characteristic
57	C₄²×C₈₃	SD₁₆	1	1	yes	characteristic
58	C₄²×C₈₃	SD₁₆	1	1	yes	characteristic
59	C₄²×C₈₃	SD₁₆	1	1	yes	characteristic
60	C₄²×C₈₃	SD₁₆	1	1	yes	characteristic
61	C₄²×C₈₃	Q₁₆	1	1	yes	characteristic
62	C₄²×C₈₃	Q₁₆	1	1	yes	characteristic
63	C₂³×C₈₃	#95	1	1	yes	characteristic
64	C₂×C₄×C₈₃	—	2	1	no	—
65	C₂×C₄×C₈₃	—	2	1	no	—
66	C₂×C₄×C₈₃	—	2	1	no	—
67	C₂×C₄×C₈₃	—	2	1	no	—
68	C₂×C₄×C₈₃	—	2	1	no	—
69	C₂×C₄×C₈₃	—	2	1	no	—
70	C₂×C₄×C₈₃	—	2	1	no	—
71	C₂×C₄×C₈₃	—	2	1	no	—
72	C₂×C₄×C₈₃	—	2	1	no	—
73	C₂×C₄×C₈₃	—	2	1	no	—
74	C₂×C₄×C₈₃	—	2	1	no	—
75	C₂×C₄×C₈₃	—	2	1	no	—
76	C₂×C₄×C₈₃	—	2	1	no	—
77	C₂×C₄×C₈₃	#100	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#101	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
84	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
85	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
86	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
87	C₂×C₄×C₈₃	#106	1	1	yes	characteristic
88	C₂×C₄×C₈₃	#106	1	1	yes	characteristic
89	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
90	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
91	C₂×C₄×C₈₃	#108	1	2	yes	—
92	C₈×C₈₃	—	8	2	no	—
93	C₈₃⋊C₈	—	8	2	no	—
94	C₈₃⋊C₈	—	8	2	no	—
95	C₂²×C₈₃	H(3,ℤ/4)	1	1	yes	characteristic
96	C₂²×C₈₃	#340	1	1	yes	characteristic
97	C₂²×C₈₃	#340	1	1	yes	characteristic
98	C₂²×C₈₃	#341	1	1	yes	characteristic
99	C₂²×C₈₃	#343	1	1	yes	characteristic
100	C₂²×C₈₃	#344	1	1	yes	characteristic
101	C₂²×C₈₃	#344	1	1	yes	characteristic
102	C₄×C₈₃	—	2	1	no	—
103	C₄×C₈₃	—	2	1	no	—
104	C₄×C₈₃	—	2	1	no	—
105	C₄×C₈₃	—	2	1	no	—
106	C₄×C₈₃	—	2	2	no	—
107	C₄×C₈₃	—	2	1	no	—
108	C₄×C₈₃	—	2	1	no	—
109	C₄×C₈₃	—	2	1	no	—
110	C₄×C₈₃	—	2	1	no	—
111	C₄×C₈₃	—	4	1	no	—
112	C₄×C₈₃	#331	1	1	yes	characteristic
113	C₄×C₈₃	#332	1	1	yes	characteristic
114	C₄×C₈₃	#332	1	1	yes	characteristic
115	C₄×C₈₃	#333	1	1	yes	characteristic
116	#7040	—	83	1	no	maximal, Sylow
117	C₂×C₈₃	#1286	1	1	yes	characteristic
118	C₂×C₈₃	#1286	1	1	yes	characteristic
119	C₂×C₈₃	#1290	1	1	yes	characteristic
120	C₂×C₈₃	#1296	1	1	yes	characteristic
121	C₂×C₈₃	#1301	1	1	yes	characteristic
122	C₂×C₈₃	#1301	1	1	yes	characteristic
123	C₂×C₈₃	#1307	1	1	yes	characteristic
124	#1842	—	83	1	no	—
125	#1831	—	83	1	no	—
126	#1831	D₈₃	1	1	yes	characteristic
127	C₈₃	#7040	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	—	83	2	no	—
130	C₂×C₄⋊C₈	—	166	1	no	—
131	C₂×C₄⋊C₈	—	166	1	no	—
132	C₂×C₄⋊C₈	—	166	1	no	—
133	C₄³	C₂×D₈₃	1	1	yes	characteristic
134	#336	—	2	1	no	—
135	#336	—	83	2	no	—
136	#336	—	166	1	no	—
137	#336	Dic₈₃	1	2	yes	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	166	2	no	—
140	C₂²×C₈	—	166	2	no	—
141	C₂×C₄²	Dic₁₆₆	1	1	yes	characteristic
142	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
143	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
144	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
148	C₄⋊C₈	—	4	1	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄⋊C₈	—	332	2	no	—
151	C₄⋊C₈	—	332	1	no	—
152	C₄⋊C₈	—	332	1	no	—
153	C₄⋊C₈	—	332	1	no	—
154	C₄⋊C₈	—	332	1	no	—
155	C₄⋊C₈	—	332	1	no	—
156	C₄⋊C₈	—	332	1	no	—
157	C₂²×C₄	C₄×Dic₈₃	1	1	yes	characteristic
158	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
161	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	332	2	no	—
168	C₂×C₈	—	332	2	no	—
169	C₂×C₈	—	332	2	no	—
170	C₂×C₈	—	332	2	no	—
171	C₂×C₈	—	332	2	no	—
172	C₂×C₈	—	332	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	4	1	no	—
177	C₄²	—	4	1	no	—
178	C₄²	—	4	1	no	—
179	C₄²	#13050575	1	1	yes	characteristic
180	C₄²	D₆₆₄	1	1	yes	characteristic
181	C₄²	#13050577	1	1	yes	characteristic
182	C₄²	#13050578	1	1	yes	characteristic
183	C₄²	#13050579	1	1	yes	characteristic
184	C₄²	#13050579	1	1	yes	characteristic
185	C₄²	#13050580	1	1	yes	characteristic
186	C₄²	Dic₃₃₂	1	1	yes	characteristic
187	C₂³	C₈₃⋊#95	1	1	yes	characteristic
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	#14581708	1	1	yes	characteristic
202	C₂×C₄	#14581710	1	1	yes	characteristic
203	C₂×C₄	#14581712	1	1	yes	characteristic
204	C₂×C₄	#14581713	1	1	yes	characteristic
205	C₂×C₄	#14581714	1	1	yes	characteristic
206	C₂×C₄	#14581714	1	1	yes	characteristic
207	C₂×C₄	#14581715	1	1	yes	characteristic
208	C₂×C₄	#14581716	1	1	yes	characteristic
209	C₂×C₄	#14581717	1	1	yes	characteristic
210	C₂×C₄	#14581717	1	1	yes	characteristic
211	C₂×C₄	#14581723	1	1	yes	characteristic
212	C₂×C₄	#14581724	1	1	yes	characteristic
213	C₂×C₄	#14581725	1	1	yes	characteristic
214	C₂×C₄	#14581726	1	1	yes	characteristic
215	C₂×C₄	#14581727	1	2	yes	—
216	C₈	—	8	2	no	—
217	C₈	—	664	2	no	—
218	C₈	—	664	2	no	—
219	C₂²	#19127457	1	1	yes	characteristic
220	C₂²	#19127463	1	1	yes	characteristic
221	C₂²	#19127464	1	1	yes	characteristic
222	C₂²	#19127465	1	1	yes	characteristic
223	C₂²	#19127469	1	1	yes	characteristic
224	C₂²	#19127471	1	1	yes	characteristic
225	C₂²	#19127472	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	1	no	—
236	C₄	#19127444	1	1	yes	characteristic
237	C₄	#19127446	1	1	yes	characteristic
238	C₄	#19127448	1	1	yes	characteristic
239	C₄	#19127449	1	1	yes	characteristic
240	C₂	#24905321	1	1	yes	characteristic
241	C₂	#24905322	1	1	yes	characteristic
242	C₂	#24905332	1	1	yes	characteristic
243	C₂	#24905347	1	1	yes	characteristic
244	C₂	#24905358	1	1	yes	characteristic
245	C₂	#24905359	1	1	yes	characteristic
246	C₂	#24905373	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₈₃⋊#95
commutator	C₂²×C₄×C₈₃	C₄²
Frattini	C₄³	C₂×D₈₃
Fitting	C₈₃×#1831	C₂
socle, prime radical	C₂³×C₈₃	#95
cosocle	C₄³×C₈₃	C₂²
hypercenter	#1831	D₈₃
hypocenter	C₈₃	#7040
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
127. C₈₃	#7040	semidirect

Maximal subgroups

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

subgroup	index
#24907615	2
#24907580	2
C₈₃×#1831	2
#7040	83