Subgroups — #35322994

Order 21248.

366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909219	C₂	1	1	yes	maximal, characteristic
2	C₄×#14581728	C₂	1	1	yes	maximal, characteristic
3	#24905571	C₂	1	4	yes	maximal
4	C₂×#500×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₂²	1	2	yes	—
6	C₂×#14581728	C₂²	1	2	yes	—
7	C₄×#13050574	—	2	1	no	—
8	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
10	#19127466	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₈₃	—	2	1	no	—
12	#500×C₈₃	—	2	1	no	—
13	#500×C₈₃	—	2	1	no	—
14	#500×C₈₃	C₄	1	2	yes	—
15	#500×C₈₃	C₄	1	2	yes	—
16	C₂×#107×C₈₃	—	2	1	no	—
17	C₂×#107×C₈₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×#13050574	—	2	2	no	—
20	#14581732	—	2	4	no	—
21	C₂×C₈₃⋊C₁₆	—	2	2	no	—
22	#14581728	—	2	2	no	—
23	#14581728	Q₈	1	4	yes	complemented
24	#14581703	—	4	1	no	—
25	C₄×C₈₃⋊C₈	—	4	1	no	—
26	C₂²×Q₈×C₈₃	—	4	1	no	—
27	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₈₃	—	2	2	no	—
30	C₄⋊Q₈×C₈₃	—	2	2	no	—
31	C₄⋊Q₈×C₈₃	—	4	1	no	—
32	C₄⋊Q₈×C₈₃	—	4	1	no	—
33	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
35	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
36	#107×C₈₃	—	2	2	no	—
37	#107×C₈₃	—	2	2	no	—
38	#107×C₈₃	—	2	2	no	—
39	#107×C₈₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
44	#13050574	—	2	4	no	—
45	#13050574	—	4	2	no	—
46	C₂×C₈₃⋊C₈	—	4	2	no	—
47	C₈₃⋊C₁₆	—	4	2	no	—
48	C₂×Q₈×C₈₃	—	4	2	no	—
49	C₂×Q₈×C₈₃	—	4	2	no	—
50	C₂×Q₈×C₈₃	—	8	4	no	—
51	C₂²×C₄×C₈₃	—	4	2	no	—
52	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	—	2	2	no	—
56	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₈₃	#41	1	2	yes	—
62	C₄⋊C₄×C₈₃	—	2	4	no	—
63	C₄⋊C₄×C₈₃	—	4	2	no	—
64	C₄⋊C₄×C₈₃	—	4	2	no	—
65	C₄⋊C₄×C₈₃	—	4	2	no	—
66	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₈₃	D₈	1	1	yes	characteristic
69	C₄²×C₈₃	D₈	1	1	yes	characteristic
70	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
71	Q₈×C₈₃	—	8	4	no	—
72	Q₈×C₈₃	—	8	4	no	—
73	C₂×C₄×C₈₃	—	2	2	no	—
74	C₂×C₄×C₈₃	—	2	2	no	—
75	C₂×C₄×C₈₃	—	4	4	no	—
76	C₂×C₄×C₈₃	—	8	4	no	—
77	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#102	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₈₃	#137	1	2	yes	—
85	C₈×C₈₃	—	2	1	no	—
86	C₈×C₈₃	—	2	1	no	—
87	C₈×C₈₃	C₄×Q₈	1	2	yes	—
88	C₈×C₈₃	#124	1	1	yes	characteristic
89	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
90	C₈₃⋊C₈	—	4	4	no	—
91	C₂²×C₈₃	—	2	1	no	—
92	C₂²×C₈₃	—	2	1	no	—
93	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
94	C₂²×C₈₃	#475	1	2	yes	—
95	C₄×C₈₃	—	2	1	no	—
96	C₄×C₈₃	—	2	1	no	—
97	C₄×C₈₃	—	8	4	no	—
98	C₄×C₈₃	#361	1	2	yes	—
99	C₄×C₈₃	#362	1	2	yes	—
100	C₄×C₈₃	#391	1	1	yes	characteristic
101	C₄×C₈₃	#419	1	1	yes	characteristic
102	C₄×C₈₃	#474	1	2	yes	—
103	#13279	—	83	1	no	maximal, Sylow
104	C₂×C₈₃	—	2	2	no	—
105	C₂×C₈₃	#1988	1	1	yes	characteristic
106	C₂×C₈₃	C₂×#361	1	1	yes	characteristic
107	C₂×C₈₃	C₂×#362	1	1	yes	characteristic
108	C₂×#500	D₈₃	1	1	yes	characteristic
109	#2150	—	83	1	no	—
110	C₄×#108	—	83	1	no	—
111	#1387	—	83	4	no	—
112	C₈₃	#13279	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂×M₃₂	—	83	2	no	—
115	#500	—	2	1	no	—
116	#500	—	2	1	no	—
117	#500	Dic₈₃	1	2	yes	—
118	#500	Dic₈₃	1	2	yes	—
119	C₂×#108	—	83	2	no	—
120	C₂×#107	—	2	1	no	—
121	C₂×#107	C₂×D₈₃	1	2	yes	—
122	C₄×M₁₆	—	166	1	no	—
123	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
124	C₄⋊C₁₆	—	166	1	no	—
125	C₄⋊C₁₆	—	166	1	no	—
126	#341	—	166	1	no	—
127	C₂²×Q₈	—	4	1	no	—
128	C₂×M₁₆	—	166	2	no	—
129	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
138	M₃₂	—	166	4	no	—
139	C₂×C₁₆	—	166	2	no	—
140	#108	—	83	4	no	—
141	#108	—	166	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	C₂×Dic₈₃	1	4	yes	—
146	#97	—	332	1	no	—
147	C₄×C₈	—	332	1	no	—
148	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
149	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	#13050578	1	1	yes	characteristic
157	C₂²×C₄	#13050578	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
159	M₁₆	—	332	2	no	—
160	M₁₆	—	166	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	332	2	no	—
163	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
164	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
167	C₂×C₈	Q₈×D₈₃	1	2	yes	—
168	C₂×C₈	#13050591	1	2	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
174	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
175	C₄²	#13050575	1	1	yes	characteristic
176	C₄²	#13050575	1	1	yes	characteristic
177	C₁₆	—	332	2	no	—
178	C₂³	C₂×#13050578	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	#14581713	1	1	yes	characteristic
186	C₂×C₄	#14581713	1	1	yes	characteristic
187	C₂×C₄	#14581713	1	1	yes	characteristic
188	C₂×C₄	#14581713	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
190	C₂×C₄	#14581768	1	2	yes	—
191	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
192	C₂×C₄	#14581823	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	332	4	no	—
196	C₈	Q₈×Dic₈₃	1	2	yes	—
197	C₈	#14581781	1	1	yes	characteristic
198	C₈	#14581790	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×#14581713	1	1	yes	characteristic
202	C₂²	#19127983	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	#19127514	1	2	yes	—
207	C₄	#19127517	1	2	yes	—
208	C₄	#19127608	1	1	yes	characteristic
209	C₄	#19127710	1	1	yes	characteristic
210	C₄	#19127978	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	#24908416	1	1	yes	characteristic
213	C₂	C₂×#19127514	1	1	yes	characteristic
214	C₂	C₂×#19127517	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14581713
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#500×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#102
prime radical	C₂³×C₈₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#500	D₈₃
hypocenter	C₈₃	#13279
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
23. #14581728	Q₈	semidirect
112. C₈₃	#13279	semidirect

Maximal subgroups

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

subgroup	index
#24909219	2
C₄×#14581728	2
#24905571	2
C₂×#500×C₈₃	2
#13279	83