Subgroups — #35303628

Order 21248.

454 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128106	C₂	1	1	yes	maximal, characteristic
2	#24908411	C₂	1	2	yes	maximal
3	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
4	C₂×#19127449	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	C₂²	1	2	yes	—
7	#19128106	—	2	2	no	—
8	#19128106	C₄	1	4	yes	complemented
9	C₂×#14581726	—	2	1	no	—
10	C₂×#14581726	—	2	1	no	—
11	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
12	#19127463	—	2	4	no	—
13	#19127454	—	2	2	no	—
14	#19127449	—	2	4	no	—
15	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₃	—	2	1	no	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
19	#336×C₈₃	—	2	1	no	—
20	C₂²×Dic₁₆₆	—	2	2	no	—
21	C₂²×C₈₃⋊C₈	—	2	2	no	—
22	#14581792	—	2	2	no	—
23	#14581792	—	2	2	no	—
24	#14581792	—	2	2	no	—
25	#14581792	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₈₃	—	2	2	no	—
27	C₂×C₄⋊Dic₈₃	—	2	2	no	—
28	#14581726	—	2	4	no	—
29	#14581726	—	2	4	no	—
30	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
31	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
32	C₂²×C₈×C₈₃	—	2	1	no	—
33	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₈₃	D₄	1	2	yes	—
36	C₂×C₄²×C₈₃	—	2	1	no	—
37	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₈₃	—	2	1	no	—
39	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
40	C₄×C₈×C₈₃	—	4	2	no	—
41	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₈₃	D₄	1	2	yes	—
43	#95×C₈₃	—	2	1	no	—
44	C₂×Dic₁₆₆	—	2	4	no	—
45	C₂×Dic₁₆₆	—	4	2	no	—
46	C₂×Dic₁₆₆	—	8	4	no	—
47	C₂²×Dic₈₃	—	4	2	no	—
48	C₂×C₈₃⋊C₈	—	4	2	no	—
49	C₂×C₈₃⋊C₈	—	4	4	no	—
50	C₄⋊Dic₈₃	—	2	4	no	—
51	C₄⋊Dic₈₃	—	2	4	no	—
52	C₄⋊Dic₈₃	—	4	2	no	—
53	C₄⋊Dic₈₃	—	4	2	no	—
54	C₂²×C₄×C₈₃	—	2	2	no	—
55	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₈₃	—	2	1	no	—
59	C₂×C₈×C₈₃	—	2	1	no	—
60	C₂×C₈×C₈₃	—	2	2	no	—
61	C₂×C₈×C₈₃	—	4	1	no	—
62	C₂×C₈×C₈₃	—	4	1	no	—
63	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
66	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
67	C₄⋊C₄×C₈₃	—	2	2	no	—
68	C₄⋊C₄×C₈₃	Q₁₆	1	4	yes	—
69	C₄²×C₈₃	—	4	1	no	—
70	C₄²×C₈₃	—	4	1	no	—
71	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₈₃	#41	1	1	yes	characteristic
75	C₂³×C₈₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₈₃	—	2	2	no	—
77	C₂×C₄×C₈₃	—	2	4	no	—
78	C₂×C₄×C₈₃	—	4	2	no	—
79	C₂×C₄×C₈₃	—	4	2	no	—
80	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₈₃	#135	1	2	yes	—
90	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
91	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
92	C₂×Dic₈₃	—	4	4	no	—
93	C₂×Dic₈₃	—	8	4	no	—
94	Dic₁₆₆	—	8	4	no	—
95	Dic₁₆₆	—	8	4	no	—
96	C₈×C₈₃	—	2	2	no	—
97	C₈×C₈₃	—	4	2	no	—
98	C₈×C₈₃	#103	1	4	yes	—
99	C₈₃⋊C₈	—	8	2	no	—
100	C₂²×C₈₃	#390	1	1	yes	characteristic
101	C₂²×C₈₃	#461	1	1	yes	characteristic
102	C₂²×C₈₃	#462	1	1	yes	characteristic
103	C₂²×C₈₃	#467	1	2	yes	—
104	C₂²×C₈₃	#472	1	2	yes	—
105	C₄×C₈₃	—	2	1	no	—
106	C₄×C₈₃	—	2	1	no	—
107	C₄×C₈₃	—	4	4	no	—
108	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
109	C₄×C₈₃	#419	1	1	yes	characteristic
110	C₄×C₈₃	C₄×Q₁₆	1	2	yes	—
111	C₄×C₈₃	#461	1	1	yes	characteristic
112	C₄×C₈₃	#462	1	1	yes	characteristic
113	C₄×C₈₃	#494	1	1	yes	characteristic
114	C₄×C₈₃	#495	1	1	yes	characteristic
115	Dic₈₃	—	8	4	no	—
116	#9730	—	83	1	no	maximal, Sylow
117	C₂×C₈₃	#1675	1	2	yes	—
118	C₂×C₈₃	#1676	1	2	yes	—
119	C₂×C₈₃	#1969	1	1	yes	characteristic
120	C₂×C₈₃	#1978	1	1	yes	characteristic
121	C₂×C₈₃	#1981	1	1	yes	characteristic
122	C₂×#500	—	83	1	no	—
123	#1987	—	83	2	no	—
124	C₈×C₄⋊C₄	D₈₃	1	1	yes	characteristic
125	C₂×#335	—	83	1	no	—
126	C₂×#333	—	83	2	no	—
127	C₈₃	#9730	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₄⋊Q₈	—	83	2	no	—
129	#500	—	83	4	no	—
130	#500	—	166	2	no	—
131	C₂×#107	—	166	1	no	—
132	C₂×#107	—	166	1	no	—
133	C₂×C₄⋊C₈	—	83	2	no	—
134	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
135	C₂×C₄×C₈	—	2	1	no	—
136	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
138	#340	—	166	4	no	—
139	#336	—	2	1	no	—
140	#335	—	166	2	no	—
141	#333	—	166	4	no	—
142	C₂²×Q₈	—	166	2	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	166	2	no	—
145	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
147	C₄⋊Q₈	—	83	4	no	—
148	C₄⋊Q₈	—	166	2	no	—
149	C₄⋊Q₈	—	166	2	no	—
150	C₄⋊Q₈	—	166	2	no	—
151	C₂×C₄⋊C₄	—	166	2	no	—
152	C₂×C₄⋊C₄	—	166	2	no	—
153	C₂×C₄⋊C₄	D₃₃₂	1	2	yes	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
156	#107	—	166	4	no	—
157	#107	—	166	4	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	166	2	no	—
160	C₄⋊C₈	—	166	2	no	—
161	C₄⋊C₈	D₃₃₂	1	2	yes	—
162	C₄×C₈	—	4	2	no	—
163	C₄×C₈	C₄×D₈₃	1	2	yes	—
164	C₄×C₈	D₃₃₂	1	2	yes	—
165	#95	—	2	1	no	—
166	C₂×Q₈	—	332	2	no	—
167	C₂×Q₈	—	664	4	no	—
168	C₂×Q₈	—	166	4	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	332	2	no	—
171	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
173	C₂²×C₄	#13050590	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	332	2	no	—
180	C₂×C₈	—	332	4	no	—
181	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
182	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
183	C₂×C₈	#13050578	1	2	yes	—
184	C₂×C₈	#13050580	1	2	yes	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	332	2	no	—
187	C₄⋊C₄	—	332	2	no	—
188	C₄⋊C₄	—	166	4	no	—
189	C₄⋊C₄	—	166	4	no	—
190	C₄⋊C₄	Dic₃₃₂	1	4	yes	—
191	C₄²	—	4	1	no	—
192	C₄²	—	4	1	no	—
193	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
194	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
195	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
196	C₄²	#13050590	1	1	yes	characteristic
197	C₂³	#14581766	1	1	yes	characteristic
198	Q₈	—	664	4	no	—
199	Q₈	—	664	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	332	4	no	—
205	C₂×C₄	—	664	4	no	—
206	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
208	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
209	C₂×C₄	#14581766	1	1	yes	characteristic
210	C₂×C₄	#14581780	1	1	yes	characteristic
211	C₂×C₄	#14581789	1	1	yes	characteristic
212	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
213	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
214	C₂×C₄	C₂×Dic₃₃₂	1	2	yes	—
215	C₂×C₄	#14581814	1	2	yes	—
216	C₂×C₄	#14581822	1	1	yes	characteristic
217	C₂×C₄	#14581822	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	4	2	no	—
220	C₈	—	664	2	no	—
221	C₈	#14581715	1	4	yes	—
222	C₂²	#19127603	1	1	yes	characteristic
223	C₂²	#19127906	1	1	yes	characteristic
224	C₂²	#19127911	1	1	yes	characteristic
225	C₂²	#19127942	1	2	yes	—
226	C₂²	#19127971	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	4	no	—
230	C₄	—	664	4	no	—
231	C₄	C₂×#14581715	1	1	yes	characteristic
232	C₄	#19127709	1	1	yes	characteristic
233	C₄	C₄×Dic₃₃₂	1	2	yes	—
234	C₄	#19127906	1	1	yes	characteristic
235	C₄	#19127911	1	1	yes	characteristic
236	C₄	#19128081	1	1	yes	characteristic
237	C₄	#19128085	1	1	yes	characteristic
238	C₂	#24906862	1	2	yes	—
239	C₂	#24906867	1	2	yes	—
240	C₂	#24908324	1	1	yes	characteristic
241	C₂	#24908373	1	1	yes	characteristic
242	C₂	#24908386	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581766
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈×C₄⋊C₄×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#121
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₈₃
hypocenter	C₈₃	#9730
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
8. #19128106	C₄	semidirect
127. C₈₃	#9730	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128106	2
#24908411	2
C₂×#19127454	2
C₂×#19127449	2
C₈×C₄⋊C₄×C₈₃	2
#9730	83