Subgroups — C₂×#24909640 #35559141

Order 21248.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128103	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19127819	C₂	1	1	yes	maximal, characteristic
3	#24909640	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19127524	C₂	1	2	yes	maximal, complemented
5	C₂×#19127512	C₂	1	2	yes	maximal
6	C₂×#364×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14581814	C₂²	1	1	yes	characteristic
8	C₂×#14581798	—	2	1	no	—
9	C₂×#14581784	—	2	1	no	—
10	#19128103	—	2	2	no	—
11	#19128103	C₂²	1	4	yes	complemented
12	#19127819	—	2	2	no	—
13	#19127819	C₂²	1	4	yes	—
14	C₂×#14581728	C₂²	1	1	yes	characteristic
15	C₂×#14581726	—	2	1	no	—
16	C₂×#14581724	C₂²	1	2	yes	—
17	C₂×#14581720	—	2	1	no	—
18	#19127524	C₂²	1	8	yes	complemented
19	#19127512	C₂²	1	8	yes	—
20	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	#364×C₈₃	C₂²	1	4	yes	complemented
23	C₂×#13050590	—	2	1	no	—
24	#14581814	—	2	1	no	—
25	#14581814	—	2	1	no	—
26	#14581814	Q₈	1	2	yes	—
27	#14581814	C₂³	1	2	yes	—
28	C₂×Dic₃₃₂	Q₈	1	1	yes	characteristic
29	C₂×#13050579	—	2	1	no	—
30	C₂×D₆₆₄	Q₈	1	1	yes	characteristic
31	#14581798	—	2	4	no	—
32	#14581798	—	4	2	no	—
33	C₂×#13050574	—	2	1	no	—
34	C₂×#13050573	—	4	1	no	—
35	C₂×C₈×D₈₃	—	4	1	no	—
36	#14581784	—	2	4	no	—
37	#14581784	—	4	2	no	—
38	C₂×C₄⋊Dic₈₃	—	2	2	no	—
39	C₂×Dic₈₃⋊C₄	—	4	2	no	—
40	#14581728	—	2	1	no	—
41	#14581728	C₂³	1	2	yes	—
42	#14581726	—	2	4	no	—
43	#14581724	—	2	2	no	—
44	#14581724	C₂³	1	4	yes	—
45	#14581720	—	2	4	no	—
46	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
49	M₃₂×C₈₃	—	2	2	no	—
50	M₃₂×C₈₃	C₂³	1	4	yes	—
51	C₂×C₁₆×C₈₃	—	2	2	no	—
52	C₄×C₈×C₈₃	D₄	1	2	yes	—
53	C₄×C₈×C₈₃	C₂³	1	2	yes	—
54	#13050590	—	2	2	no	—
55	#13050590	—	2	2	no	—
56	#13050590	—	4	1	no	—
57	#13050590	—	4	1	no	—
58	C₂×Dic₁₆₆	—	2	1	no	—
59	C₂×D₃₃₂	—	2	1	no	—
60	C₂×C₂²⋊D₈₃	—	4	1	no	—
61	C₂²×Dic₈₃	—	4	2	no	—
62	C₂×C₄×D₈₃	—	4	1	no	—
63	Dic₃₃₂	—	2	1	no	—
64	Dic₃₃₂	C₂×Q₈	1	2	yes	—
65	#13050579	—	2	2	no	—
66	#13050579	—	2	2	no	—
67	D₆₆₄	—	2	1	no	—
68	D₆₆₄	C₂×Q₈	1	2	yes	—
69	#13050574	—	2	2	no	—
70	#13050574	—	4	1	no	—
71	#13050573	—	4	4	no	—
72	C₂×C₈₃⋊C₈	—	4	1	no	—
73	C₈×D₈₃	—	4	4	no	—
74	C₄⋊Dic₈₃	—	2	4	no	—
75	C₄⋊Dic₈₃	—	4	2	no	—
76	Dic₈₃⋊C₄	—	4	8	no	—
77	C₂²×C₄×C₈₃	—	2	1	no	—
78	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
80	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
81	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
82	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
89	C₄²×C₈₃	—	2	1	no	—
90	C₄²×C₈₃	C₂×D₄	1	2	yes	—
91	C₁₆×C₈₃	—	2	4	no	—
92	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂²×D₈₃	—	4	1	no	—
94	C₂×C₄×C₈₃	—	2	2	no	—
95	C₂×C₄×C₈₃	—	4	2	no	—
96	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
102	C₂²⋊D₈₃	—	4	2	no	—
103	C₂²⋊D₈₃	—	4	2	no	—
104	C₂×Dic₈₃	—	4	1	no	—
105	C₂×Dic₈₃	—	4	4	no	—
106	C₂×Dic₈₃	—	8	4	no	—
107	D₃₃₂	—	2	2	no	—
108	D₃₃₂	—	4	1	no	—
109	C₄×D₈₃	—	4	2	no	—
110	C₄×D₈₃	—	4	2	no	—
111	Dic₁₆₆	—	2	2	no	—
112	Dic₁₆₆	—	4	1	no	—
113	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
114	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
116	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
117	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
118	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
119	C₈₃⋊C₈	—	4	2	no	—
120	C₂²×C₈₃	—	2	2	no	—
121	C₂²×C₈₃	#476	1	1	yes	characteristic
122	C₂²×C₈₃	#476	1	1	yes	characteristic
123	C₂²×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×D₈₃	—	4	2	no	—
125	C₂×D₈₃	—	8	2	no	—
126	C₄×C₈₃	—	4	2	no	—
127	C₄×C₈₃	#476	1	1	yes	characteristic
128	C₄×C₈₃	#476	1	1	yes	characteristic
129	C₄×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
131	Dic₈₃	—	4	2	no	—
132	Dic₈₃	—	8	4	no	—
133	C₂×#2231	—	83	1	no	maximal, Sylow
134	C₂×C₈₃	—	2	1	no	—
135	C₂×C₈₃	—	2	1	no	—
136	C₂×C₈₃	#2231	1	2	yes	complemented
137	C₂×C₈₃	C₂×#476	1	1	yes	characteristic
138	D₈₃	—	8	2	no	—
139	C₂×#499	—	83	1	no	—
140	C₂×#443	—	83	1	no	—
141	#2231	—	83	8	no	—
142	C₂×#365	—	83	2	no	—
143	C₂×#364	D₈₃	1	1	yes	characteristic, complemented
144	C₂×#360	—	83	2	no	—
145	C₈₃	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
146	C₂×#135	—	83	1	no	—
147	C₂×#131	—	166	1	no	—
148	C₂×#125	—	166	1	no	—
149	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
150	#499	—	83	4	no	—
151	#499	—	166	2	no	—
152	#443	—	83	4	no	—
153	#443	—	166	2	no	—
154	C₂×#108	—	83	1	no	—
155	C₂×#107	—	166	1	no	—
156	C₂×#106	—	83	2	no	—
157	C₂×#104	—	166	1	no	—
158	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
159	#365	—	83	8	no	—
160	#364	C₂×D₈₃	1	4	yes	complemented
161	#360	—	83	8	no	—
162	C₂×#41	—	166	1	no	—
163	#135	—	83	2	no	—
164	#135	—	83	2	no	—
165	#135	—	166	1	no	—
166	#135	—	166	1	no	—
167	C₂×Q₁₆	—	83	1	no	—
168	C₂×SD₁₆	—	166	1	no	—
169	C₂×D₈	—	83	1	no	—
170	#131	—	332	2	no	—
171	#131	—	166	4	no	—
172	C₂×M₁₆	—	332	1	no	—
173	C₂×M₁₆	—	166	1	no	—
174	C₂²×C₈	—	332	1	no	—
175	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
177	#125	—	332	2	no	—
178	#125	—	166	4	no	—
179	C₂×C₄⋊C₄	—	332	2	no	—
180	C₂×C₄⋊C₄	—	166	2	no	—
181	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
182	M₃₂	—	2	2	no	—
183	M₃₂	C₂²×D₈₃	1	4	yes	—
184	C₂×C₁₆	—	2	2	no	—
185	#108	—	83	2	no	—
186	#108	—	166	1	no	—
187	#107	—	166	4	no	—
188	#106	—	83	4	no	—
189	#106	—	166	2	no	—
190	#104	—	166	4	no	—
191	C₄×C₈	D₃₃₂	1	2	yes	—
192	C₄×C₈	C₂²×D₈₃	1	2	yes	—
193	#41	—	332	1	no	—
194	#41	—	332	1	no	—
195	#41	—	166	2	no	—
196	#41	—	166	2	no	—
197	C₂×Q₈	—	166	1	no	—
198	C₂×D₄	—	332	1	no	—
199	C₂×D₄	—	166	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	332	1	no	—
202	C₂²×C₄	—	332	2	no	—
203	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
204	Q₁₆	—	83	2	no	—
205	Q₁₆	—	166	1	no	—
206	SD₁₆	—	166	2	no	—
207	SD₁₆	—	166	2	no	—
208	D₈	—	83	2	no	—
209	D₈	—	166	1	no	—
210	M₁₆	—	332	1	no	—
211	M₁₆	—	332	4	no	—
212	M₁₆	—	166	2	no	—
213	C₂×C₈	—	332	1	no	—
214	C₂×C₈	—	332	4	no	—
215	C₂×C₈	#13050579	1	1	yes	characteristic
216	C₂×C₈	#13050579	1	1	yes	characteristic
217	C₂×C₈	#13050579	1	2	yes	—
218	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
219	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
220	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
221	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
222	C₂×C₈	#13050589	1	1	yes	characteristic
223	C₂×C₈	#13050589	1	1	yes	characteristic
224	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
225	C₄⋊C₄	—	332	8	no	—
226	C₄⋊C₄	—	332	2	no	—
227	C₄⋊C₄	—	166	4	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×D₃₃₂	1	2	yes	—
230	C₁₆	—	2	4	no	—
231	C₂³	—	332	1	no	—
232	C₂³	#14581820	1	1	yes	characteristic
233	Q₈	—	332	1	no	—
234	Q₈	—	166	2	no	—
235	D₄	—	332	2	no	—
236	D₄	—	332	2	no	—
237	D₄	—	332	1	no	—
238	D₄	—	166	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	332	1	no	—
242	C₂×C₄	—	332	2	no	—
243	C₂×C₄	—	332	2	no	—
244	C₂×C₄	—	332	4	no	—
245	C₂×C₄	—	664	4	no	—
246	C₂×C₄	#14581772	1	1	yes	characteristic
247	C₂×C₄	#14581772	1	1	yes	characteristic
248	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
249	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
250	C₂×C₄	#14581820	1	1	yes	characteristic
251	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
252	C₈	—	332	2	no	—
253	C₈	#14581772	1	1	yes	characteristic
254	C₈	#14581772	1	1	yes	characteristic
255	C₈	C₂×#13050579	1	2	yes	—
256	C₈	C₂×#13050579	1	2	yes	—
257	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
258	C₈	C₂×#13050589	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	332	2	no	—
261	C₂²	—	664	2	no	—
262	C₂²	#19127992	1	1	yes	characteristic
263	C₂²	#19127992	1	1	yes	characteristic
264	C₂²	C₂×#14581820	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	332	2	no	—
267	C₄	—	664	4	no	—
268	C₄	#19127992	1	1	yes	characteristic
269	C₄	#19127992	1	1	yes	characteristic
270	C₄	C₂×#14581772	1	1	yes	characteristic
271	C₄	C₂²×#13050579	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	664	2	no	—
275	C₂	#24909640	1	2	yes	direct product factor
276	C₂	C₂×#19127992	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19127992
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#364×C₈₃	C₂
socle	C₂²×C₈₃	#476
prime radical	C₂×#14581814	C₂²
hypercenter	C₂×#364	D₈₃
hypocenter	C₈₃	C₂×#2231
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#19128103	C₂	semidirect
3. #24909640	C₂	direct (N × H)
4. C₂×#19127524	C₂	semidirect
6. C₂×#364×C₈₃	C₂	semidirect
11. #19128103	C₂²	semidirect
18. #19127524	C₂²	semidirect
22. #364×C₈₃	C₂²	semidirect
136. C₂×C₈₃	#2231	semidirect
143. C₂×#364	D₈₃	semidirect
145. C₈₃	C₂×#2231	semidirect
160. #364	C₂×D₈₃	semidirect
275. C₂	#24909640	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×#19128103	2
C₂×#19127819	2
#24909640	2
C₂×#19127524	2
C₂×#19127512	2
C₂×#364×C₈₃	2
C₂×#2231	83

Subgroups — C2×#24909640 #35559141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909640 #35559141