Subgroups — #35559764

Order 21248.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128081	C₂	1	1	yes	maximal, characteristic
2	#24909710	C₂	1	4	yes	maximal, complemented
3	#24909692	C₂	1	4	yes	maximal, complemented
4	#24909182	C₂	1	4	yes	maximal
5	C₄×#14581732	C₂	1	1	yes	maximal, characteristic
6	C₈₃×#3160	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₃₂	—	2	1	no	—
8	C₂²×Dic₃₃₂	C₂²	1	2	yes	—
9	C₂×#14581792	—	2	1	no	—
10	C₂×#14581732	C₂²	1	2	yes	—
11	#19128081	—	2	1	no	—
12	#19128081	—	2	1	no	—
13	#19128081	C₂²	1	2	yes	—
14	#19128081	C₂²	1	2	yes	—
15	#19127504	C₂²	1	16	yes	—
16	#19127477	C₂²	1	2	yes	—
17	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
18	#533×C₈₃	—	2	1	no	—
19	#500×C₈₃	C₂²	1	2	yes	—
20	#499×C₈₃	—	2	1	no	—
21	#499×C₈₃	C₂²	1	2	yes	—
22	#498×C₈₃	—	2	1	no	—
23	#427×C₈₃	—	2	1	no	—
24	#427×C₈₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₆₆	—	2	2	no	—
27	C₂×Dic₃₃₂	—	2	2	no	—
28	C₂×Dic₃₃₂	—	2	2	no	—
29	C₂×Dic₃₃₂	—	2	2	no	—
30	C₂×Dic₃₃₂	—	2	8	no	—
31	C₂×Dic₃₃₂	—	4	4	no	—
32	C₂×Dic₃₃₂	C₂³	1	4	yes	—
33	#14581792	—	2	2	no	—
34	#14581792	—	2	2	no	—
35	#14581792	—	4	2	no	—
36	C₂×C₄⋊Dic₈₃	—	4	2	no	—
37	#14581732	—	2	4	no	—
38	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
39	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₈₃	—	2	2	no	—
42	#125×C₈₃	—	2	2	no	—
43	C₂²⋊Q₈×C₈₃	—	4	2	no	—
44	C₄×Q₈×C₈₃	—	4	2	no	—
45	#117×C₈₃	—	2	2	no	—
46	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
47	#107×C₈₃	—	2	2	no	—
48	#107×C₈₃	C₂³	1	4	yes	—
49	#106×C₈₃	—	2	2	no	—
50	#106×C₈₃	—	2	2	no	—
51	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₆₆	—	2	4	no	—
56	C₂×Dic₁₆₆	—	4	4	no	—
57	C₂×Dic₁₆₆	—	4	4	no	—
58	C₂×Dic₁₆₆	—	4	2	no	—
59	C₂²×Dic₈₃	—	4	2	no	—
60	Dic₃₃₂	—	2	8	no	—
61	Dic₃₃₂	—	4	4	no	—
62	Dic₃₃₂	—	4	4	no	—
63	Dic₃₃₂	—	4	4	no	—
64	C₄⋊Dic₈₃	—	4	4	no	—
65	C₄⋊Dic₈₃	—	4	4	no	—
66	C₈₃⋊C₁₆	—	2	4	no	—
67	C₂×Q₈×C₈₃	—	4	2	no	—
68	C₂²×C₄×C₈₃	D₈	1	2	yes	—
69	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
71	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₃	#41	1	4	yes	—
75	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₈₃	—	2	4	no	—
77	C₄⋊C₄×C₈₃	—	4	4	no	—
78	C₄⋊C₄×C₈₃	—	4	2	no	—
79	C₂²⋊C₄×C₈₃	—	4	2	no	—
80	C₄²×C₈₃	—	4	2	no	—
81	C₄²×C₈₃	D₈	1	2	yes	—
82	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₈₃	—	8	4	no	—
86	C₂×C₄×C₈₃	—	2	4	no	—
87	C₂×C₄×C₈₃	—	4	4	no	—
88	C₂×C₄×C₈₃	#124	1	2	yes	—
89	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₈₃	—	4	4	no	—
95	C₂×Dic₈₃	—	8	4	no	—
96	Dic₁₆₆	—	4	4	no	—
97	Dic₁₆₆	—	4	4	no	—
98	Dic₁₆₆	—	8	4	no	—
99	C₈×C₈₃	#124	1	2	yes	—
100	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
101	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
102	C₈×C₈₃	C₂×#41	1	2	yes	—
103	C₂²×C₈₃	—	2	2	no	—
104	C₂²×C₈₃	#486	1	2	yes	—
105	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₈₃	—	2	2	no	—
107	C₄×C₈₃	—	8	4	no	—
108	C₄×C₈₃	#486	1	2	yes	—
109	C₄×C₈₃	#510	1	4	yes	—
110	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
111	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
112	Dic₈₃	—	8	4	no	—
113	#33705	—	83	1	no	maximal, Sylow
114	C₂×C₈₃	—	2	2	no	—
115	C₂×C₈₃	C₂×#486	1	1	yes	characteristic
116	C₂×C₈₃	C₂×#510	1	2	yes	—
117	#3160	D₈₃	1	1	yes	characteristic
118	C₂×#494	—	83	1	no	—
119	#2245	—	83	4	no	—
120	#2241	—	83	4	no	—
121	#2141	—	83	4	no	—
122	C₄×M₃₂	—	83	1	no	—
123	C₈₃	#33705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₁₆	—	83	2	no	—
125	C₂²×Q₁₆	—	166	1	no	—
126	#533	—	2	1	no	—
127	C₂×C₄⋊Q₈	—	166	1	no	—
128	C₂×M₃₂	—	83	2	no	—
129	#500	C₂×D₈₃	1	2	yes	—
130	#499	—	2	1	no	—
131	#499	C₂×D₈₃	1	2	yes	—
132	#498	—	2	1	no	—
133	#494	—	83	2	no	—
134	#494	—	83	2	no	—
135	#494	—	166	1	no	—
136	#494	—	166	1	no	—
137	#427	—	2	1	no	—
138	#427	C₂×D₈₃	1	2	yes	—
139	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
140	#358	—	83	16	no	—
141	#346	—	83	2	no	—
142	C₄×C₁₆	—	83	2	no	—
143	C₂²×Q₈	—	166	2	no	—
144	C₂×Q₁₆	—	332	4	no	—
145	C₂×Q₁₆	—	83	4	no	—
146	C₂×Q₁₆	—	166	2	no	—
147	C₂×Q₁₆	—	166	2	no	—
148	C₂×Q₁₆	—	166	2	no	—
149	C₂×Q₁₆	—	166	8	no	—
150	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	332	2	no	—
154	C₄⋊Q₈	—	166	2	no	—
155	C₄⋊Q₈	—	166	2	no	—
156	#125	—	2	2	no	—
157	C₂²⋊Q₈	—	4	2	no	—
158	C₄×Q₈	—	4	2	no	—
159	#117	—	2	2	no	—
160	C₂×C₄⋊C₄	—	332	2	no	—
161	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
162	M₃₂	—	166	4	no	—
163	C₂×C₁₆	—	83	4	no	—
164	#107	—	2	2	no	—
165	#107	C₂²×D₈₃	1	4	yes	—
166	#106	—	2	2	no	—
167	#106	—	2	2	no	—
168	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	332	2	no	—
174	C₂×Q₈	—	332	4	no	—
175	C₂×Q₈	—	332	4	no	—
176	C₂×Q₈	—	166	4	no	—
177	C₂²×C₄	—	332	2	no	—
178	C₂²×C₄	#13050575	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
180	Q₁₆	—	332	4	no	—
181	Q₁₆	—	332	4	no	—
182	Q₁₆	—	332	4	no	—
183	Q₁₆	—	166	8	no	—
184	C₂×C₈	#13050577	1	4	yes	—
185	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
188	C₂×C₈	#13050591	1	4	yes	—
189	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	332	4	no	—
194	C₄⋊C₄	—	332	4	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	—	4	2	no	—
197	C₄²	#13050575	1	2	yes	—
198	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
200	C₁₆	—	166	4	no	—
201	C₂³	C₂×#13050575	1	1	yes	characteristic
202	Q₈	—	8	4	no	—
203	Q₈	—	332	4	no	—
204	Q₈	—	332	4	no	—
205	Q₈	—	664	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	332	4	no	—
209	C₂×C₄	—	664	4	no	—
210	C₂×C₄	#14581781	1	2	yes	—
211	C₂×C₄	C₂×#13050575	1	2	yes	—
212	C₂×C₄	C₂×#13050575	1	2	yes	—
213	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
214	C₂×C₄	C₂×#13050577	1	2	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
216	C₈	#14581781	1	2	yes	—
217	C₈	C₂×#13050577	1	2	yes	—
218	C₈	C₂×#13050577	1	2	yes	—
219	C₈	C₂×#13050591	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#19128049	1	2	yes	—
222	C₂²	C₂²×#13050575	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	664	4	no	—
226	C₄	#19128049	1	2	yes	—
227	C₄	#19128141	1	4	yes	—
228	C₄	C₂×#14581781	1	1	yes	characteristic
229	C₄	C₂²×#13050577	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#19128049	1	1	yes	characteristic
232	C₂	C₂×#19128141	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#3160	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	#3160	D₈₃
hypocenter	C₈₃	#33705
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #24909710	C₂	semidirect
3. #24909692	C₂	semidirect
123. C₈₃	#33705	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128081	2
#24909710	2
#24909692	2
#24909182	2
C₄×#14581732	2
C₈₃×#3160	2
#33705	83