Subgroups — C₂×#24905510 #35315641

Order 21248.

518 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128078	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127454	C₂	1	1	yes	maximal, characteristic, complemented
3	#24905510	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₆₆₄	—	2	1	no	—
6	C₂²×D₆₆₄	C₄	1	2	yes	—
7	C₂×#14581789	—	2	1	no	—
8	#19128078	—	2	1	no	—
9	#19128078	—	2	1	no	—
10	#19128078	C₄	1	2	yes	—
11	#19128078	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
13	#19127454	—	2	1	no	—
14	#19127454	C₂²	1	2	yes	complemented
15	C₂²×C₁₆×C₈₃	—	2	1	no	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₈₃	—	2	1	no	—
18	C₄⋊C₁₆×C₈₃	C₂²	1	2	yes	complemented
19	C₂²×D₃₃₂	—	2	2	no	—
20	C₂²×D₃₃₂	—	4	1	no	—
21	C₂²×D₃₃₂	—	4	1	no	—
22	C₂×D₆₆₄	—	2	2	no	—
23	C₂×D₆₆₄	—	2	2	no	—
24	C₂×D₆₆₄	—	2	2	no	—
25	C₂×D₆₆₄	—	4	4	no	—
26	C₂×D₆₆₄	—	4	2	no	—
27	C₂×D₆₆₄	—	4	2	no	—
28	C₂×D₆₆₄	C₂×C₄	1	4	yes	—
29	C₂²×C₈₃⋊C₈	—	4	1	no	—
30	#14581789	—	2	2	no	—
31	#14581789	—	2	2	no	—
32	#14581789	—	4	1	no	—
33	#14581789	—	4	1	no	—
34	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
35	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
36	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₈₃	—	2	2	no	—
40	C₂×C₁₆×C₈₃	—	4	2	no	—
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	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
45	C₂³×D₈₃	—	4	2	no	—
46	C₂×D₃₃₂	—	2	4	no	—
47	C₂×D₃₃₂	—	4	4	no	—
48	C₂×D₃₃₂	—	4	4	no	—
49	C₂×D₃₃₂	—	4	2	no	—
50	C₂×D₃₃₂	—	4	2	no	—
51	C₂×D₃₃₂	—	4	2	no	—
52	C₂×D₃₃₂	—	4	2	no	—
53	C₂×D₃₃₂	—	4	2	no	—
54	C₂×D₃₃₂	—	8	4	no	—
55	C₂×D₃₃₂	—	8	2	no	—
56	C₂×D₃₃₂	—	8	2	no	—
57	D₆₆₄	—	4	4	no	—
58	D₆₆₄	—	4	4	no	—
59	D₆₆₄	—	4	4	no	—
60	D₆₆₄	—	4	2	no	—
61	D₆₆₄	—	4	2	no	—
62	C₂×C₈₃⋊C₈	—	4	2	no	—
63	C₂×C₈₃⋊C₈	—	8	2	no	—
64	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
67	C₂×C₈×C₈₃	—	2	2	no	—
68	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₈₃	D₈	1	2	yes	—
71	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
72	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₈₃	D₈	1	1	yes	characteristic
76	C₄²×C₈₃	SD₁₆	1	1	yes	characteristic
77	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₈₃	—	4	2	no	—
79	C₂³×C₈₃	#102	1	1	yes	characteristic
80	C₂²×D₈₃	—	4	4	no	—
81	C₂²×D₈₃	—	8	4	no	—
82	C₂²×D₈₃	—	8	4	no	—
83	C₂²×D₈₃	—	8	4	no	—
84	C₂×C₄×C₈₃	—	2	2	no	—
85	C₂×C₄×C₈₃	#100	1	2	yes	—
86	C₂×C₄×C₈₃	#102	1	2	yes	—
87	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
88	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₈₃	D₁₆	1	2	yes	—
91	C₂×C₄×C₈₃	SD₃₂	1	2	yes	—
92	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
95	D₃₃₂	—	4	4	no	—
96	D₃₃₂	—	4	4	no	—
97	D₃₃₂	—	8	4	no	—
98	D₃₃₂	—	8	4	no	—
99	D₃₃₂	—	8	4	no	—
100	D₃₃₂	—	8	4	no	—
101	D₃₃₂	—	8	2	no	—
102	D₃₃₂	—	8	2	no	—
103	C₈×C₈₃	—	2	1	no	—
104	C₈×C₈₃	—	2	1	no	—
105	C₈×C₈₃	#102	1	2	yes	—
106	C₈×C₈₃	C₂×D₈	1	1	yes	characteristic
107	C₈×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₈₃⋊C₈	—	8	2	no	—
109	C₂²×C₈₃	#331	1	2	yes	—
110	C₂²×C₈₃	#357	1	2	yes	—
111	C₂²×C₈₃	#361	1	2	yes	—
112	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
113	C₂×D₈₃	—	8	4	no	—
114	C₂×D₈₃	—	8	4	no	—
115	C₂×D₈₃	—	8	4	no	—
116	C₂×D₈₃	—	16	8	no	—
117	C₄×C₈₃	—	2	1	no	—
118	C₄×C₈₃	—	2	1	no	—
119	C₄×C₈₃	#331	1	2	yes	—
120	C₄×C₈₃	#357	1	2	yes	—
121	C₄×C₈₃	C₂×#100	1	1	yes	characteristic
122	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
123	C₄×C₈₃	C₂×D₁₆	1	1	yes	characteristic
124	C₄×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
125	C₂×#1361	—	83	1	no	maximal, Sylow
126	C₂×C₈₃	#1361	1	4	yes	complemented
127	C₂×C₈₃	C₂×#331	1	1	yes	characteristic
128	C₂×C₈₃	C₂×#357	1	1	yes	characteristic
129	C₂×C₈₃	C₂×#361	1	1	yes	characteristic
130	D₈₃	—	16	4	no	—
131	C₂×#493	—	83	1	no	—
132	C₂×C₄⋊C₁₆	D₈₃	1	1	yes	characteristic, complemented
133	C₂×#335	—	83	1	no	—
134	#1361	—	83	4	no	—
135	C₈₃	C₂×#1361	1	1	yes	characteristic, complemented, Sylow
136	C₂²×D₈	—	83	2	no	—
137	C₂²×D₈	—	166	1	no	—
138	C₂×#127	—	166	1	no	—
139	C₂²×C₁₆	—	2	1	no	—
140	#493	—	83	2	no	—
141	#493	—	83	2	no	—
142	#493	—	166	1	no	—
143	#493	—	166	1	no	—
144	C₂×C₄⋊C₈	—	166	1	no	—
145	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
146	C₄⋊C₁₆	—	2	1	no	—
147	C₄⋊C₁₆	C₂×D₈₃	1	2	yes	complemented
148	#335	—	83	2	no	—
149	#335	—	166	1	no	—
150	C₂²×D₄	—	332	1	no	—
151	C₂²×D₄	—	332	1	no	—
152	C₂²×D₄	—	166	2	no	—
153	C₂×D₈	—	332	4	no	—
154	C₂×D₈	—	332	2	no	—
155	C₂×D₈	—	332	2	no	—
156	C₂×D₈	—	83	4	no	—
157	C₂×D₈	—	166	2	no	—
158	C₂×D₈	—	166	2	no	—
159	C₂×D₈	—	166	2	no	—
160	C₂²×C₈	—	332	1	no	—
161	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
163	#127	—	332	1	no	—
164	#127	—	332	1	no	—
165	#127	—	166	2	no	—
166	#127	—	166	2	no	—
167	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	4	2	no	—
170	C₄⋊C₈	—	332	1	no	—
171	C₄⋊C₈	—	166	2	no	—
172	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
173	C₄×C₈	D₃₃₂	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
176	C₂⁴	—	332	2	no	—
177	C₂×D₄	—	332	2	no	—
178	C₂×D₄	—	332	2	no	—
179	C₂×D₄	—	332	4	no	—
180	C₂×D₄	—	332	4	no	—
181	C₂×D₄	—	332	2	no	—
182	C₂×D₄	—	332	2	no	—
183	C₂×D₄	—	332	2	no	—
184	C₂×D₄	—	664	4	no	—
185	C₂×D₄	—	664	2	no	—
186	C₂×D₄	—	664	2	no	—
187	C₂×D₄	—	166	4	no	—
188	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
190	C₂²×C₄	#13050579	1	1	yes	characteristic
191	D₈	—	332	4	no	—
192	D₈	—	332	4	no	—
193	D₈	—	332	2	no	—
194	D₈	—	332	2	no	—
195	D₈	—	332	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	332	2	no	—
198	C₂×C₈	—	664	2	no	—
199	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
200	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
201	C₂×C₈	#13050575	1	2	yes	—
202	C₂×C₈	#13050577	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
204	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
205	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
206	C₄²	D₆₆₄	1	1	yes	characteristic
207	C₄²	#13050579	1	1	yes	characteristic
208	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
209	C₁₆	—	4	2	no	—
210	C₂³	—	332	4	no	—
211	C₂³	—	664	4	no	—
212	C₂³	—	664	4	no	—
213	C₂³	—	664	4	no	—
214	C₂³	#14581714	1	1	yes	characteristic
215	D₄	—	332	4	no	—
216	D₄	—	332	4	no	—
217	D₄	—	664	4	no	—
218	D₄	—	664	4	no	—
219	D₄	—	664	2	no	—
220	D₄	—	664	2	no	—
221	D₄	—	664	4	no	—
222	D₄	—	664	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	#14581708	1	2	yes	—
225	C₂×C₄	#14581712	1	2	yes	—
226	C₂×C₄	#14581714	1	1	yes	characteristic
227	C₂×C₄	#14581714	1	1	yes	characteristic
228	C₂×C₄	#14581714	1	1	yes	characteristic
229	C₂×C₄	D₁₃₂₈	1	2	yes	—
230	C₂×C₄	#14581737	1	2	yes	—
231	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
233	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	664	2	no	—
237	C₈	#14581712	1	2	yes	—
238	C₈	C₂×#13050575	1	1	yes	characteristic
239	C₈	C₂×#13050577	1	1	yes	characteristic
240	C₂²	—	1328	8	no	—
241	C₂²	—	664	4	no	—
242	C₂²	—	664	4	no	—
243	C₂²	—	664	4	no	—
244	C₂²	#19127444	1	2	yes	—
245	C₂²	#19127503	1	2	yes	—
246	C₂²	#19127515	1	2	yes	—
247	C₂²	C₂×#14581714	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	#19127444	1	2	yes	—
251	C₄	#19127503	1	2	yes	—
252	C₄	C₂×#14581708	1	1	yes	characteristic
253	C₄	C₂×#14581712	1	1	yes	characteristic
254	C₄	C₂×D₁₃₂₈	1	1	yes	characteristic
255	C₄	C₂×#14581737	1	1	yes	characteristic
256	C₂	—	1328	4	no	—
257	C₂	#24905510	1	4	yes	direct product factor
258	C₂	C₂×#19127444	1	1	yes	characteristic
259	C₂	C₂×#19127503	1	1	yes	characteristic
260	C₂	C₂×#19127515	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581714
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₂×C₄⋊C₁₆×C₈₃	C₂
socle	C₂³×C₈₃	#102
prime radical	C₂×#19128078	C₂
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₃
hypocenter	C₈₃	C₂×#1361
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19127454	C₂	semidirect
3. #24905510	C₂	direct (N × H)
4. C₂×C₄⋊C₁₆×C₈₃	C₂	semidirect
14. #19127454	C₂²	semidirect
18. C₄⋊C₁₆×C₈₃	C₂²	semidirect
126. C₂×C₈₃	#1361	semidirect
132. C₂×C₄⋊C₁₆	D₈₃	semidirect
135. C₈₃	C₂×#1361	semidirect
147. C₄⋊C₁₆	C₂×D₈₃	semidirect
257. C₂	#24905510	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×#19128078	2
C₂×#19127454	2
#24905510	2
C₂×C₄⋊C₁₆×C₈₃	2
C₂×#1361	83

Subgroups — C2×#24905510 #35315641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24905510 #35315641