Subgroups — #35302962

Order 21248.

378 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908182	C₂	1	2	yes	maximal
2	C₂×#19127452	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127423	C₂	1	1	yes	maximal, characteristic
4	#24905304	C₂	1	2	yes	maximal
5	C₄×#106×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	#19127456	—	2	2	no	—
11	#19127452	—	2	2	no	—
12	#19127423	—	2	2	no	—
13	C₂×#106×C₈₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
16	#340×C₈₃	C₄	1	1	yes	characteristic
17	#340×C₈₃	C₄	1	1	yes	characteristic
18	C₂²×C₈₃⋊C₈	—	2	2	no	—
19	C₂²×C₈₃⋊C₈	—	2	1	no	—
20	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	—	4	4	no	—
24	C₄×C₈₃⋊C₈	—	2	1	no	—
25	C₄×C₈₃⋊C₈	—	2	1	no	—
26	C₄×C₈₃⋊C₈	—	2	1	no	—
27	C₄×C₈₃⋊C₈	Q₈	1	2	yes	—
28	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₈₃	—	2	2	no	—
33	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
34	#106×C₈₃	—	2	4	no	—
35	C₄×C₈×C₈₃	D₄	1	2	yes	—
36	C₄×C₈×C₈₃	Q₈	1	2	yes	—
37	#95×C₈₃	—	2	2	no	—
38	C₂×C₈₃⋊C₈	—	2	2	no	—
39	C₂×C₈₃⋊C₈	—	2	2	no	—
40	C₂×C₈₃⋊C₈	—	2	2	no	—
41	C₂×C₈₃⋊C₈	—	4	1	no	—
42	C₂×C₈₃⋊C₈	—	4	1	no	—
43	C₂×C₈₃⋊C₈	—	4	2	no	—
44	C₂×C₈₃⋊C₈	—	4	2	no	—
45	C₂×C₈₃⋊C₈	—	4	2	no	—
46	C₂×C₈₃⋊C₈	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₈₃	—	2	2	no	—
48	C₂²×C₄×C₈₃	—	2	2	no	—
49	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₈₃	—	2	2	no	—
53	C₂×C₈×C₈₃	—	2	2	no	—
54	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
56	C₄⋊C₄×C₈₃	—	2	2	no	—
57	C₄⋊C₄×C₈₃	—	2	2	no	—
58	C₄⋊C₄×C₈₃	—	2	2	no	—
59	C₄⋊C₄×C₈₃	M₁₆	1	4	yes	—
60	C₄²×C₈₃	—	4	2	no	—
61	C₄²×C₈₃	—	4	2	no	—
62	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₈₃	#41	1	1	yes	characteristic
66	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	1	no	—
68	C₂×C₄×C₈₃	—	2	1	no	—
69	C₂×C₄×C₈₃	—	2	4	no	—
70	C₂×C₄×C₈₃	—	4	2	no	—
71	C₂×C₄×C₈₃	—	4	2	no	—
72	C₂×C₄×C₈₃	—	4	2	no	—
73	C₂×C₄×C₈₃	—	4	2	no	—
74	C₂×C₄×C₈₃	—	4	2	no	—
75	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
80	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₈₃	#131	1	2	yes	—
83	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
84	C₂×C₄×C₈₃	#135	1	2	yes	—
85	C₈×C₈₃	—	2	2	no	—
86	C₈×C₈₃	—	2	2	no	—
87	C₈₃⋊C₈	—	2	4	no	—
88	C₈₃⋊C₈	—	4	2	no	—
89	C₈₃⋊C₈	—	8	2	no	—
90	C₂²×C₈₃	#389	1	1	yes	characteristic
91	C₂²×C₈₃	#435	1	2	yes	—
92	C₂²×C₈₃	C₄×SD₁₆	1	2	yes	—
93	C₂²×C₈₃	#444	1	1	yes	characteristic
94	C₂²×C₈₃	#444	1	1	yes	characteristic
95	C₄×C₈₃	—	2	1	no	—
96	C₄×C₈₃	—	2	1	no	—
97	C₄×C₈₃	—	4	4	no	—
98	C₄×C₈₃	—	8	2	no	—
99	C₄×C₈₃	#423	1	1	yes	characteristic
100	C₄×C₈₃	#424	1	1	yes	characteristic
101	C₄×C₈₃	#446	1	2	yes	—
102	C₄×C₈₃	#465	1	2	yes	—
103	C₄×C₈₃	#498	1	1	yes	characteristic
104	C₄×C₈₃	#499	1	1	yes	characteristic
105	#9597	—	83	1	no	maximal, Sylow
106	C₂×C₈₃	#1582	1	2	yes	—
107	C₂×C₈₃	#1583	1	2	yes	—
108	C₂×C₈₃	#1940	1	1	yes	characteristic
109	C₂×C₈₃	#1941	1	1	yes	characteristic
110	C₂×C₈₃	#1946	1	1	yes	characteristic
111	#1939	—	83	2	no	—
112	C₄×#106	D₈₃	1	1	yes	characteristic
113	C₂×#334	—	83	1	no	—
114	C₂×#322	—	83	1	no	—
115	#1279	—	83	2	no	—
116	C₈₃	#9597	1	1	yes	characteristic, complemented, Sylow
117	C₂×#106	Dic₈₃	1	2	yes	—
118	C₂×C₄⋊C₈	—	83	2	no	—
119	C₂×C₄⋊C₈	—	166	2	no	—
120	C₂×C₄×C₈	—	83	1	no	—
121	C₂×C₄×C₈	—	83	1	no	—
122	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
123	C₄×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
124	#340	Dic₈₃	1	1	yes	characteristic
125	#340	Dic₈₃	1	1	yes	characteristic
126	#336	—	166	2	no	—
127	#334	—	166	2	no	—
128	#322	—	166	2	no	—
129	C₂²×C₈	—	83	2	no	—
130	C₂²×C₈	—	166	2	no	—
131	C₂²×C₈	—	166	1	no	—
132	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
133	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
135	C₂×C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
138	#106	—	2	4	no	—
139	C₄⋊C₈	—	332	4	no	—
140	C₄⋊C₈	—	166	2	no	—
141	C₄⋊C₈	—	166	2	no	—
142	C₄×C₈	—	83	2	no	—
143	C₄×C₈	—	166	1	no	—
144	C₄×C₈	—	166	1	no	—
145	C₄×C₈	—	166	1	no	—
146	C₄×C₈	Dic₁₆₆	1	2	yes	—
147	C₄×C₈	D₃₃₂	1	2	yes	—
148	#95	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
152	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
153	C₂²×C₄	#13050589	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	332	1	no	—
157	C₂×C₈	—	332	1	no	—
158	C₂×C₈	—	332	2	no	—
159	C₂×C₈	—	332	2	no	—
160	C₂×C₈	—	332	2	no	—
161	C₂×C₈	—	83	4	no	—
162	C₂×C₈	—	166	2	no	—
163	C₂×C₈	—	166	2	no	—
164	C₂×C₈	—	166	2	no	—
165	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
166	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	#13050574	1	4	yes	—
171	C₄²	—	4	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
174	C₄²	Q₈×D₈₃	1	1	yes	characteristic
175	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
176	C₄²	#13050591	1	1	yes	characteristic
177	C₂³	D₄×Dic₈₃	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
187	C₂×C₄	Q₈×Dic₈₃	1	1	yes	characteristic
188	C₂×C₄	#14581767	1	1	yes	characteristic
189	C₂×C₄	#14581772	1	1	yes	characteristic
190	C₂×C₄	#14581783	1	1	yes	characteristic
191	C₂×C₄	#14581791	1	1	yes	characteristic
192	C₂×C₄	C₂×#13050574	1	2	yes	—
193	C₂×C₄	#14581799	1	2	yes	—
194	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
195	C₂×C₄	#14581813	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	2	2	no	—
198	C₈	—	332	2	no	—
199	C₈	—	664	2	no	—
200	C₈	—	166	4	no	—
201	C₂²	#19127600	1	1	yes	characteristic
202	C₂²	#19127773	1	2	yes	—
203	C₂²	SD₁₆×Dic₈₃	1	2	yes	—
204	C₂²	#19127823	1	1	yes	characteristic
205	C₂²	#19127823	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	4	no	—
209	C₄	—	8	2	no	—
210	C₄	#19127725	1	1	yes	characteristic
211	C₄	#19127728	1	1	yes	characteristic
212	C₄	#19127832	1	2	yes	—
213	C₄	#19127931	1	2	yes	—
214	C₄	#19128098	1	1	yes	characteristic
215	C₄	#19128101	1	1	yes	characteristic
216	C₂	#24906300	1	2	yes	—
217	C₂	#24906307	1	2	yes	—
218	C₂	#24908187	1	1	yes	characteristic
219	C₂	#24908192	1	1	yes	characteristic
220	C₂	#24908217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₃
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₄×#106×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₄×#106	D₈₃
hypocenter	C₈₃	#9597
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
116. C₈₃	#9597	semidirect

Maximal subgroups

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

subgroup	index
#24908182	2
C₂×#19127452	2
C₂×#19127423	2
#24905304	2
C₄×#106×C₈₃	2
#9597	83