Subgroups — C₂×#24907080 #35392098

Order 21248.

878 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	C₂×#19128104	C₂	1	2	yes	maximal
3	C₂×#19128080	C₂	1	2	yes	maximal
4	#24907080	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#500×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13050580	—	2	4	no	—
8	C₂×#14581792	C₂²	1	2	yes	—
9	#19128106	—	2	2	no	—
10	#19128106	C₂²	1	4	yes	—
11	#19128104	—	2	4	no	—
12	#19128104	C₂²	1	8	yes	—
13	#19128080	—	2	4	no	—
14	#19128080	C₂²	1	8	yes	—
15	C₂×#14581726	—	2	2	no	—
16	C₂×#14581725	—	2	4	no	—
17	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×C₈₃⋊C₈	C₂²	1	4	yes	—
20	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
21	#500×C₈₃	—	2	2	no	—
22	#500×C₈₃	C₂²	1	4	yes	—
23	C₂×#107×C₈₃	—	2	2	no	—
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₆₆	—	2	2	no	—
26	C₂×#13050580	—	2	16	no	—
27	C₂×#13050580	—	4	16	no	—
28	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
29	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
30	#14581792	—	2	2	no	—
31	#14581792	—	2	2	no	—
32	#14581792	—	2	2	no	—
33	#14581792	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₈₃	—	2	2	no	—
35	C₂×C₄⋊Dic₈₃	—	2	2	no	—
36	#14581726	—	2	8	no	—
37	#14581725	—	2	16	no	—
38	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
39	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
40	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
41	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
42	C₂²×Q₈×C₈₃	—	2	2	no	—
43	C₂²×C₈×C₈₃	D₄	1	2	yes	—
44	C₄⋊Q₈×C₈₃	—	2	2	no	—
45	C₄⋊Q₈×C₈₃	—	2	2	no	—
46	C₄⋊Q₈×C₈₃	—	2	2	no	—
47	C₄⋊Q₈×C₈₃	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
50	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
51	#107×C₈₃	—	2	8	no	—
52	C₄×C₈×C₈₃	D₄	1	2	yes	—
53	C₄×C₈×C₈₃	C₂³	1	2	yes	—
54	C₂×Dic₁₆₆	—	2	4	no	—
55	C₂×Dic₁₆₆	—	4	8	no	—
56	C₂×Dic₁₆₆	—	4	2	no	—
57	C₂²×Dic₈₃	—	4	2	no	—
58	#13050580	—	4	32	no	—
59	C₂×C₈₃⋊C₈	D₈	1	8	yes	—
60	C₂×C₈₃⋊C₈	Q₁₆	1	8	yes	complemented
61	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
62	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
63	C₄⋊Dic₈₃	—	2	4	no	—
64	C₄⋊Dic₈₃	—	2	4	no	—
65	C₄⋊Dic₈₃	—	4	2	no	—
66	C₄⋊Dic₈₃	—	4	2	no	—
67	C₂×Q₈×C₈₃	—	2	4	no	—
68	C₂×Q₈×C₈₃	—	4	2	no	—
69	C₂×Q₈×C₈₃	—	4	8	no	—
70	C₂²×C₄×C₈₃	—	4	2	no	—
71	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₃	Q₁₆	1	8	yes	complemented
75	C₂×C₈×C₈₃	C₂×D₄	1	4	yes	—
76	C₄⋊C₄×C₈₃	—	2	4	no	—
77	C₄⋊C₄×C₈₃	—	2	4	no	—
78	C₄⋊C₄×C₈₃	—	4	2	no	—
79	C₄⋊C₄×C₈₃	—	4	2	no	—
80	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
84	C₂³×C₈₃	#127	1	1	yes	characteristic
85	Q₈×C₈₃	—	4	8	no	—
86	Q₈×C₈₃	—	8	4	no	—
87	C₂×C₄×C₈₃	—	4	4	no	—
88	C₂×C₄×C₈₃	—	8	4	no	—
89	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
91	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂×D₈	1	4	yes	—
93	C₂×C₄×C₈₃	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₈₃	C₂×Q₁₆	1	4	yes	—
95	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₈₃	—	4	4	no	—
99	C₂×Dic₈₃	—	8	4	no	—
100	Dic₁₆₆	—	4	8	no	—
101	Dic₁₆₆	—	8	4	no	—
102	C₈×C₈₃	C₂×Q₁₆	1	8	yes	complemented
103	C₈₃⋊C₈	C₂×D₈	1	8	yes	—
104	C₈₃⋊C₈	C₂×Q₁₆	1	8	yes	complemented
105	C₂²×C₈₃	#493	1	2	yes	—
106	C₂²×C₈₃	#494	1	2	yes	—
107	C₂²×C₈₃	#494	1	2	yes	—
108	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
109	C₄×C₈₃	—	8	4	no	—
110	C₄×C₈₃	#493	1	2	yes	—
111	C₄×C₈₃	#494	1	2	yes	—
112	C₄×C₈₃	#494	1	2	yes	—
113	C₄×C₈₃	C₂²×D₈	1	2	yes	—
114	C₄×C₈₃	C₂²×Q₁₆	1	2	yes	—
115	C₄×C₈₃	C₂²×Q₁₆	1	2	yes	—
116	Dic₈₃	—	8	4	no	—
117	C₂×#1713	—	83	1	no	maximal, Sylow
118	C₂×C₈₃	#1713	1	4	yes	complemented
119	C₂×C₈₃	C₂×#493	1	1	yes	characteristic
120	C₂×C₈₃	C₂×#494	1	1	yes	characteristic
121	C₂×C₈₃	C₂×#494	1	1	yes	characteristic
122	C₂×#500	—	83	1	no	—
123	C₂×#500	—	83	2	no	—
124	C₂×#500	D₈₃	1	1	yes	characteristic
125	C₂×#494	—	83	2	no	—
126	#1713	—	83	8	no	—
127	C₂×C₈²	—	83	1	no	—
128	C₈₃	C₂×#1713	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₁₆	—	166	4	no	—
130	C₂×C₄⋊Q₈	—	83	2	no	—
131	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
132	#500	—	2	2	no	—
133	#500	—	83	8	no	—
134	#500	—	83	4	no	—
135	#500	—	166	4	no	—
136	#500	—	166	2	no	—
137	#500	C₂×D₈₃	1	4	yes	—
138	#494	—	83	8	no	—
139	#494	—	166	4	no	—
140	C₂×#107	—	2	2	no	—
141	C₂×#107	—	166	4	no	—
142	C₂×#107	—	166	2	no	—
143	C₂×C₄×C₈	—	83	1	no	—
144	C₂×C₄×C₈	—	83	1	no	—
145	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
146	C₈²	—	83	4	no	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	—	166	2	no	—
149	C₂×Q₁₆	—	332	16	no	—
150	C₂×Q₁₆	—	166	16	no	—
151	C₂²×C₈	—	83	2	no	—
152	C₂²×C₈	—	83	2	no	—
153	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	83	4	no	—
158	C₄⋊Q₈	—	166	2	no	—
159	C₄⋊Q₈	—	166	2	no	—
160	C₄⋊Q₈	—	166	2	no	—
161	C₄⋊Q₈	C₂²×D₈₃	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	166	2	no	—
165	C₂×C₄⋊C₄	—	166	2	no	—
166	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
167	#107	—	2	8	no	—
168	#107	—	166	16	no	—
169	#107	—	166	8	no	—
170	C₄×C₈	—	83	2	no	—
171	C₄×C₈	—	83	2	no	—
172	C₄×C₈	—	83	2	no	—
173	C₄×C₈	—	83	2	no	—
174	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
175	C₄×C₈	C₂²×D₈₃	1	2	yes	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	8	no	—
179	C₂×Q₈	—	332	2	no	—
180	C₂×Q₈	—	332	8	no	—
181	C₂×Q₈	—	166	4	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	332	2	no	—
184	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
187	Q₁₆	—	332	32	no	—
188	C₂×C₈	—	83	8	no	—
189	C₂×C₈	—	83	8	no	—
190	C₂×C₈	—	83	4	no	—
191	C₂×C₈	—	83	4	no	—
192	C₂×C₈	#13050580	1	8	yes	complemented
193	C₂×C₈	C₂×C₂²⋊D₈₃	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	332	2	no	—
199	C₄⋊C₄	—	332	2	no	—
200	C₄⋊C₄	—	166	4	no	—
201	C₄⋊C₄	—	166	4	no	—
202	C₄²	D₄×D₈₃	1	1	yes	characteristic
203	C₄²	D₄×D₈₃	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
205	C₄²	C₂³×D₈₃	1	1	yes	characteristic
206	C₂³	#14581788	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	Q₈	—	8	4	no	—
209	Q₈	—	332	8	no	—
210	Q₈	—	664	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	332	4	no	—
214	C₂×C₄	—	664	4	no	—
215	C₂×C₄	#14581788	1	1	yes	characteristic
216	C₂×C₄	#14581788	1	1	yes	characteristic
217	C₂×C₄	#14581788	1	1	yes	characteristic
218	C₂×C₄	D₈×D₈₃	1	4	yes	—
219	C₂×C₄	Q₁₆×D₈₃	1	4	yes	—
220	C₂×C₄	C₂×#13050580	1	4	yes	—
221	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
224	C₈	—	83	8	no	—
225	C₈	—	83	8	no	—
226	C₈	C₂×#13050580	1	8	yes	complemented
227	C₂²	#19128076	1	2	yes	—
228	C₂²	#19128079	1	2	yes	—
229	C₂²	#19128080	1	2	yes	—
230	C₂²	C₂×#14581788	1	1	yes	characteristic
231	C₄	—	8	4	no	—
232	C₄	—	664	4	no	—
233	C₄	#19128076	1	2	yes	—
234	C₄	#19128079	1	2	yes	—
235	C₄	#19128080	1	2	yes	—
236	C₄	C₂×D₈×D₈₃	1	2	yes	—
237	C₄	C₂×Q₁₆×D₈₃	1	2	yes	—
238	C₄	C₂²×#13050580	1	2	yes	—
239	C₂	#24907080	1	4	yes	direct product factor
240	C₂	C₂×#19128076	1	1	yes	characteristic
241	C₂	C₂×#19128079	1	1	yes	characteristic
242	C₂	C₂×#19128080	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581788
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×#500×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#127
hypercenter	C₂×#500	D₈₃
hypocenter	C₈₃	C₂×#1713
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #24907080	C₂	direct (N × H)
60. C₂×C₈₃⋊C₈	Q₁₆	semidirect
74. C₂×C₈×C₈₃	Q₁₆	semidirect
102. C₈×C₈₃	C₂×Q₁₆	semidirect
104. C₈₃⋊C₈	C₂×Q₁₆	semidirect
118. C₂×C₈₃	#1713	semidirect
128. C₈₃	C₂×#1713	semidirect
192. C₂×C₈	#13050580	semidirect
226. C₈	C₂×#13050580	semidirect
239. C₂	#24907080	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19128106	2
C₂×#19128104	2
C₂×#19128080	2
#24907080	2
C₂×C₈×C₈₃⋊C₈	2
C₂×#500×C₈₃	2
C₂×#1713	83

Subgroups — C2×#24907080 #35392098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24907080 #35392098