Subgroups — #35310108

Order 21248.

626 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907202	C₂	1	2	yes	maximal
2	C₂×#19127466	C₂	1	2	yes	maximal
3	C₂×C₈₃⋊#338	C₂	1	2	yes	maximal
4	C₂×#405×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13050574	C₂²	1	4	yes	—
6	#19127470	—	2	4	no	—
7	#19127466	—	2	4	no	—
8	C₈₃⋊#338	C₄	1	8	yes	complemented
9	C₂²×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
11	C₂×#117×C₈₃	C₂²	1	1	yes	characteristic
12	#405×C₈₃	—	2	4	no	—
13	C₂×#97×C₈₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₄	1	2	yes	—
15	C₂×#13050574	—	2	4	no	—
16	C₂×#13050574	—	2	4	no	—
17	C₂×#13050574	—	2	4	no	—
18	C₂×#13050574	—	2	4	no	—
19	C₂×#13050574	—	2	4	no	—
20	C₂×#13050574	C₂×C₄	1	8	yes	—
21	C₂²×C₈₃⋊C₈	—	2	4	no	—
22	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₈₃	—	2	2	no	—
24	C₂×M₁₆×C₈₃	—	2	2	no	—
25	C₂×M₁₆×C₈₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₈₃	—	2	2	no	—
27	C₂²×C₈×C₈₃	—	2	2	no	—
28	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	D₄	1	4	yes	—
32	#117×C₈₃	—	2	2	no	—
33	#117×C₈₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₃	D₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₈₃	Q₈	1	2	yes	—
36	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
37	#97×C₈₃	—	2	4	no	—
38	C₄×C₈×C₈₃	—	2	4	no	—
39	#13050574	—	2	8	no	—
40	#13050574	—	2	8	no	—
41	#13050574	—	4	4	no	—
42	#13050574	—	4	4	no	—
43	C₂×C₈₃⋊C₈	—	2	8	no	—
44	C₂×C₈₃⋊C₈	—	4	4	no	—
45	C₂×C₈₃⋊C₈	—	4	4	no	—
46	C₂⁴×C₈₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	#41	1	2	yes	—
53	C₂²×C₄×C₈₃	#41	1	2	yes	—
54	C₂²×C₄×C₈₃	#41	1	2	yes	—
55	C₂²×C₄×C₈₃	#41	1	2	yes	—
56	M₁₆×C₈₃	—	2	8	no	—
57	C₂×C₈×C₈₃	—	2	2	no	—
58	C₂×C₈×C₈₃	—	2	1	no	—
59	C₂×C₈×C₈₃	—	2	1	no	—
60	C₂×C₈×C₈₃	—	2	1	no	—
61	C₂×C₈×C₈₃	—	2	1	no	—
62	C₂×C₈×C₈₃	—	2	2	no	—
63	C₂×C₈×C₈₃	—	2	8	no	—
64	C₂×C₈×C₈₃	C₄²	1	4	yes	—
65	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₈₃	—	2	4	no	—
68	C₂²⋊C₄×C₈₃	—	2	4	no	—
69	C₄²×C₈₃	—	2	2	no	—
70	C₄²×C₈₃	—	2	2	no	—
71	C₂³×C₈₃	—	4	2	no	—
72	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₈₃	C₄×Q₈	1	2	yes	—
74	C₂³×C₈₃	C₂²⋊Q₈	1	2	yes	—
75	C₂³×C₈₃	#125	1	2	yes	—
76	C₂×C₄×C₈₃	—	2	4	no	—
77	C₂×C₄×C₈₃	—	4	4	no	—
78	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₃	#117	1	4	yes	—
81	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₈₃	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₈₃	#121	1	2	yes	—
88	C₂×C₄×C₈₃	#123	1	4	yes	—
89	C₂×C₄×C₈₃	#124	1	2	yes	—
90	C₂×C₄×C₈₃	#126	1	4	yes	—
91	C₈×C₈₃	—	2	4	no	—
92	C₈×C₈₃	—	2	2	no	—
93	C₈×C₈₃	—	2	2	no	—
94	C₈₃⋊C₈	—	4	8	no	—
95	C₂²×C₈₃	—	2	2	no	—
96	C₂²×C₈₃	—	2	4	no	—
97	C₂²×C₈₃	—	4	2	no	—
98	C₂²×C₈₃	—	4	2	no	—
99	C₂²×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₈₃	#380	1	1	yes	characteristic
101	C₂²×C₈₃	#382	1	1	yes	characteristic
102	C₂²×C₈₃	#387	1	4	yes	—
103	C₄×C₈₃	—	4	4	no	—
104	C₄×C₈₃	#376	1	1	yes	characteristic
105	C₄×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₈₃	#381	1	1	yes	characteristic
107	C₄×C₈₃	#383	1	1	yes	characteristic
108	C₄×C₈₃	#388	1	4	yes	—
109	#10815	—	83	1	no	maximal, Sylow
110	C₂×C₈₃	—	2	1	no	—
111	C₂×C₈₃	—	2	1	no	—
112	C₂×C₈₃	—	2	4	no	—
113	C₂×C₈₃	#1434	1	1	yes	characteristic
114	C₂×C₈₃	#1813	1	2	yes	—
115	C₂×#405	D₈₃	1	1	yes	characteristic
116	#1742	—	83	2	no	—
117	C₂×#341	—	83	2	no	—
118	C₂×#338	—	83	2	no	—
119	C₈₃	#10815	1	1	yes	characteristic, complemented, Sylow
120	C₂²×M₁₆	—	83	4	no	—
121	C₂²×M₁₆	C₂×D₈₃	1	1	yes	characteristic
122	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
123	C₂×#117	C₂×D₈₃	1	1	yes	characteristic
124	#405	—	2	4	no	—
125	C₂×#97	Dic₈₃	1	2	yes	—
126	C₂×C₄×C₈	Dic₈₃	1	2	yes	—
127	#343	—	166	4	no	—
128	#341	—	166	4	no	—
129	#338	—	83	8	no	—
130	C₂³×C₄	C₂²×D₈₃	1	1	yes	characteristic
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	83	8	no	—
134	C₂×M₁₆	—	166	4	no	—
135	C₂×M₁₆	—	166	4	no	—
136	C₂×M₁₆	—	166	4	no	—
137	C₂×M₁₆	—	166	4	no	—
138	C₂×M₁₆	—	166	4	no	—
139	C₂×M₁₆	C₄×D₈₃	1	4	yes	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	166	4	no	—
143	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
144	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₈₃	1	4	yes	—
147	#117	—	2	2	no	—
148	#117	—	2	2	no	—
149	C₂×C₄⋊C₄	D₃₃₂	1	2	yes	—
150	C₂×C₂²⋊C₄	Dic₁₆₆	1	2	yes	—
151	C₂×C₄²	C₂×Dic₈₃	1	2	yes	—
152	#97	—	2	4	no	—
153	C₄×C₈	—	2	4	no	—
154	C₂⁴	C₂×Dic₁₆₆	1	1	yes	characteristic
155	C₂²×C₄	C₄⋊Dic₈₃	1	4	yes	—
156	C₂²×C₄	C₂×C₄×D₈₃	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
159	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
160	C₂²×C₄	#13050590	1	2	yes	—
161	C₂²×C₄	#13050590	1	2	yes	—
162	C₂²×C₄	#13050590	1	2	yes	—
163	C₂²×C₄	#13050590	1	2	yes	—
164	M₁₆	—	2	8	no	—
165	M₁₆	—	332	4	no	—
166	M₁₆	—	332	4	no	—
167	M₁₆	—	166	8	no	—
168	M₁₆	—	166	8	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	332	4	no	—
177	C₂×C₈	—	332	4	no	—
178	C₂×C₈	—	166	8	no	—
179	C₂×C₈	C₄×Dic₈₃	1	4	yes	—
180	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
181	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₂³	—	4	2	no	—
187	C₂³	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
188	C₂³	C₄×Dic₁₆₆	1	2	yes	—
189	C₂³	#14581771	1	2	yes	—
190	C₂³	#14581784	1	2	yes	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
195	C₂×C₄	#14581749	1	4	yes	—
196	C₂×C₄	#14581751	1	1	yes	characteristic
197	C₂×C₄	#14581751	1	1	yes	characteristic
198	C₂×C₄	#14581751	1	1	yes	characteristic
199	C₂×C₄	#14581751	1	1	yes	characteristic
200	C₂×C₄	C₄×C₂²⋊D₈₃	1	4	yes	—
201	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
202	C₂×C₄	#14581766	1	2	yes	—
203	C₂×C₄	#14581775	1	4	yes	—
204	C₂×C₄	#14581780	1	2	yes	—
205	C₂×C₄	#14581786	1	4	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	4	no	—
209	C₈	—	332	8	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
215	C₂²	#19127561	1	1	yes	characteristic
216	C₂²	#19127567	1	1	yes	characteristic
217	C₂²	#19127585	1	4	yes	—
218	C₄	—	4	4	no	—
219	C₄	#19127547	1	1	yes	characteristic
220	C₄	C₄×C₂²⋊Dic₈₃	1	1	yes	characteristic
221	C₄	#19127564	1	1	yes	characteristic
222	C₄	#19127569	1	1	yes	characteristic
223	C₄	#19127592	1	4	yes	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	—	2	4	no	—
227	C₂	#24905690	1	1	yes	characteristic
228	C₂	#24907513	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14581751
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₃
Fitting	C₂×#405×C₈₃	C₂
socle	C₂²×C₈₃	#380
prime radical	C₂⁴×C₈₃	C₂×Q₈
cosocle	C₂³×C₄×C₈₃	C₂³
hypercenter	C₂×#405	D₈₃
hypocenter	C₈₃	#10815
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. C₈₃⋊#338	C₄	semidirect
119. C₈₃	#10815	semidirect

Maximal subgroups

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

subgroup	index
#24907202	2
C₂×#19127466	2
C₂×C₈₃⋊#338	2
C₂×#405×C₈₃	2
#10815	83