Subgroups — C₂×#24909580 #35558958

Order 21248.

678 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19127968	C₂	1	2	yes	maximal, complemented
3	#24909580	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19127531	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19127514	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14581818	—	2	2	no	—
8	C₂×#14581798	C₂²	1	1	yes	characteristic
9	#19128117	—	2	2	no	—
10	#19128117	C₂²	1	4	yes	complemented
11	C₂×#14581732	C₂²	1	1	yes	characteristic
12	C₂×#14581731	—	2	1	no	—
13	C₂×C₁₆×D₈₃	—	2	1	no	—
14	#19127968	—	2	4	no	—
15	#19127968	C₂²	1	8	yes	complemented
16	C₂×#14581728	C₂²	1	2	yes	—
17	C₂×#14581709	—	2	2	no	—
18	#19127531	C₂²	1	4	yes	complemented
19	#19127514	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₈₃	C₂²	1	2	yes	—
21	C₂×D₁₆×C₈₃	—	2	1	no	—
22	C₂×D₁₆×C₈₃	—	2	1	no	—
23	C₂×D₁₆×C₈₃	—	2	2	no	—
24	C₂×D₁₆×C₈₃	C₂²	1	4	yes	complemented
25	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
26	C₂×#13050590	D₄	1	1	yes	characteristic
27	#14581818	—	2	8	no	—
28	#14581818	—	4	4	no	—
29	C₂×#13050578	—	4	2	no	—
30	C₂×#13050575	—	4	2	no	—
31	#14581798	—	2	1	no	—
32	#14581798	—	2	1	no	—
33	#14581798	D₄	1	2	yes	—
34	#14581798	C₂³	1	2	yes	—
35	C₂×#13050574	—	2	2	no	—
36	C₂×#13050574	D₄	1	1	yes	characteristic
37	C₂×#13050573	—	2	1	no	—
38	C₂×C₈×D₈₃	—	2	1	no	—
39	#14581732	—	2	1	no	—
40	#14581732	C₂³	1	2	yes	—
41	#14581731	—	2	4	no	—
42	C₂×C₈₃⋊C₁₆	—	2	1	no	—
43	C₁₆×D₈₃	—	2	4	no	—
44	#14581728	—	2	2	no	—
45	#14581728	C₂³	1	4	yes	—
46	#14581709	—	2	8	no	—
47	C₂²×D₄×C₈₃	—	2	2	no	—
48	C₂×D₈×C₈₃	—	2	2	no	—
49	C₂×D₈×C₈₃	—	2	2	no	—
50	C₂×D₈×C₈₃	—	2	2	no	—
51	C₂×D₈×C₈₃	—	2	2	no	—
52	C₂×D₈×C₈₃	—	2	2	no	—
53	C₂×D₈×C₈₃	C₂³	1	4	yes	—
54	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
55	D₁₆×C₈₃	—	2	4	no	—
56	D₁₆×C₈₃	—	2	4	no	—
57	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
60	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
61	#13050590	—	2	1	no	—
62	#13050590	—	2	1	no	—
63	#13050590	D₈	1	2	yes	—
64	#13050590	C₂×D₄	1	2	yes	—
65	C₂×Dic₁₆₆	D₈	1	1	yes	characteristic
66	C₂×D₃₃₂	D₈	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	#13050578	—	4	8	no	—
70	#13050575	—	4	8	no	—
71	#13050574	—	2	1	no	—
72	#13050574	—	2	4	no	—
73	#13050574	—	4	2	no	—
74	#13050574	C₂×D₄	1	2	yes	—
75	#13050573	—	2	2	no	—
76	#13050573	—	2	2	no	—
77	C₂×C₈₃⋊C₈	—	2	1	no	—
78	C₂×C₈₃⋊C₈	—	4	2	no	—
79	C₈×D₈₃	—	2	2	no	—
80	C₈×D₈₃	—	2	2	no	—
81	C₈₃⋊C₁₆	—	2	2	no	—
82	C₂⁴×C₈₃	—	4	2	no	—
83	C₂×D₄×C₈₃	—	2	4	no	—
84	C₂×D₄×C₈₃	—	4	2	no	—
85	C₂×D₄×C₈₃	—	4	2	no	—
86	C₂×D₄×C₈₃	—	4	2	no	—
87	C₂×D₄×C₈₃	—	4	2	no	—
88	C₂×D₄×C₈₃	—	4	2	no	—
89	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
90	D₈×C₈₃	—	2	4	no	—
91	D₈×C₈₃	—	2	4	no	—
92	D₈×C₈₃	—	4	2	no	—
93	D₈×C₈₃	—	4	2	no	—
94	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
100	C₁₆×C₈₃	C₂×D₄	1	2	yes	—
101	C₁₆×C₈₃	C₂×D₄	1	2	yes	—
102	C₂³×C₈₃	—	4	4	no	—
103	C₂³×C₈₃	—	8	2	no	—
104	C₂³×C₈₃	—	8	2	no	—
105	C₂³×C₈₃	—	16	4	no	—
106	C₂³×C₈₃	#135	1	1	yes	characteristic
107	C₂²×D₈₃	—	2	1	no	—
108	D₄×C₈₃	—	4	4	no	—
109	D₄×C₈₃	—	4	4	no	—
110	D₄×C₈₃	—	8	2	no	—
111	D₄×C₈₃	—	8	2	no	—
112	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₈₃	—	2	2	no	—
119	C₂²⋊D₈₃	—	2	2	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	D₃₃₂	—	2	1	no	—
122	D₃₃₂	C₂×D₈	1	2	yes	—
123	C₄×D₈₃	—	2	2	no	—
124	C₄×D₈₃	—	2	2	no	—
125	Dic₁₆₆	—	2	1	no	—
126	Dic₁₆₆	C₂×D₈	1	2	yes	—
127	C₈×C₈₃	#121	1	1	yes	characteristic
128	C₈×C₈₃	#121	1	1	yes	characteristic
129	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
131	C₈₃⋊C₈	—	2	2	no	—
132	C₈₃⋊C₈	—	4	4	no	—
133	C₂²×C₈₃	—	2	2	no	—
134	C₂²×C₈₃	—	8	4	no	—
135	C₂²×C₈₃	—	16	4	no	—
136	C₂²×C₈₃	—	16	4	no	—
137	C₂²×C₈₃	—	16	4	no	—
138	C₂²×C₈₃	#466	1	1	yes	characteristic
139	C₂²×C₈₃	#466	1	1	yes	characteristic
140	C₂²×C₈₃	C₂×#135	1	1	yes	characteristic
141	C₂×D₈₃	—	2	2	no	—
142	C₂×D₈₃	—	4	2	no	—
143	C₄×C₈₃	#466	1	1	yes	characteristic
144	C₄×C₈₃	#466	1	1	yes	characteristic
145	C₄×C₈₃	C₂×#121	1	1	yes	characteristic
146	C₄×C₈₃	C₂²×D₈	1	1	yes	characteristic
147	Dic₈₃	—	2	2	no	—
148	C₂×#2221	—	83	1	no	maximal, Sylow
149	C₂×C₈₃	—	2	1	no	—
150	C₂×C₈₃	—	2	1	no	—
151	C₂×C₈₃	—	16	4	no	—
152	C₂×C₈₃	#2221	1	2	yes	complemented
153	C₂×C₈₃	C₂×#466	1	1	yes	characteristic
154	D₈₃	—	4	2	no	—
155	C₂²×D₁₆	D₈₃	1	1	yes	characteristic, complemented
156	C₂×#504	—	83	1	no	—
157	C₂×#472	—	83	2	no	—
158	#2221	—	83	8	no	—
159	C₂×#368	—	83	1	no	—
160	C₂×#361	—	83	2	no	—
161	C₈₃	C₂×#2221	1	1	yes	characteristic, complemented, Sylow
162	C₂×AGL(1,ℤ/8)	—	166	2	no	—
163	C₂²×D₈	C₂×D₈₃	1	2	yes	—
164	C₂×#131	—	83	1	no	—
165	C₂×D₁₆	—	2	1	no	—
166	C₂×D₁₆	—	2	1	no	—
167	C₂×D₁₆	—	2	2	no	—
168	C₂×D₁₆	C₂×D₈₃	1	4	yes	complemented
169	#504	—	83	4	no	—
170	#504	—	166	2	no	—
171	C₂×M₃₂	—	83	1	no	—
172	C₂×M₃₂	—	166	1	no	—
173	C₂²×C₁₆	—	166	1	no	—
174	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
175	#472	—	83	8	no	—
176	#472	—	166	4	no	—
177	C₂×#108	—	83	2	no	—
178	C₂×#100	—	166	2	no	—
179	#368	—	83	4	no	—
180	#361	—	83	8	no	—
181	C₂×#41	—	83	1	no	—
182	C₂²×D₄	—	2	2	no	—
183	AGL(1,ℤ/8)	—	332	4	no	—
184	AGL(1,ℤ/8)	—	166	8	no	—
185	C₂×SD₁₆	—	332	2	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	332	2	no	—
192	C₂×D₈	C₂²×D₈₃	1	4	yes	—
193	#131	—	83	2	no	—
194	#131	—	83	2	no	—
195	#131	—	166	1	no	—
196	#131	—	166	1	no	—
197	C₂×M₁₆	—	83	1	no	—
198	C₂×M₁₆	—	166	2	no	—
199	C₂×M₁₆	—	166	1	no	—
200	C₂²×C₈	—	166	1	no	—
201	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
202	D₁₆	—	2	4	no	—
203	D₁₆	—	2	4	no	—
204	M₃₂	—	83	2	no	—
205	M₃₂	—	166	1	no	—
206	M₃₂	—	166	4	no	—
207	C₂×C₁₆	—	166	1	no	—
208	C₂×C₁₆	—	166	4	no	—
209	C₂×C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
210	C₂×C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
211	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
212	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
213	#108	—	83	4	no	—
214	#108	—	166	2	no	—
215	#100	—	166	8	no	—
216	C₂⁴	—	4	2	no	—
217	#41	—	83	2	no	—
218	#41	—	83	2	no	—
219	#41	—	166	1	no	—
220	#41	—	166	1	no	—
221	C₂×Q₈	—	83	1	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	83	1	no	—
229	C₂×D₄	—	166	1	no	—
230	C₂²×C₄	—	166	1	no	—
231	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
232	SD₁₆	—	332	8	no	—
233	D₈	—	2	4	no	—
234	D₈	—	2	4	no	—
235	D₈	—	4	2	no	—
236	D₈	—	4	2	no	—
237	D₈	—	332	8	no	—
238	M₁₆	—	332	2	no	—
239	M₁₆	—	83	2	no	—
240	M₁₆	—	166	4	no	—
241	M₁₆	—	166	1	no	—
242	M₁₆	—	166	2	no	—
243	M₁₆	—	166	2	no	—
244	C₂×C₈	—	332	2	no	—
245	C₂×C₈	—	166	1	no	—
246	C₂×C₈	—	166	2	no	—
247	C₂×C₈	—	166	2	no	—
248	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
251	C₂×C₈	#13050589	1	1	yes	characteristic
252	C₂×C₈	#13050589	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
254	C₁₆	—	166	2	no	—
255	C₁₆	C₂×C₂²⋊D₈₃	1	2	yes	—
256	C₁₆	C₂×C₂²⋊D₈₃	1	2	yes	—
257	C₂³	—	4	4	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	16	4	no	—
261	C₂³	—	166	1	no	—
262	C₂³	#14581811	1	1	yes	characteristic
263	Q₈	—	83	2	no	—
264	Q₈	—	166	1	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	2	no	—
268	D₄	—	8	2	no	—
269	D₄	—	83	2	no	—
270	D₄	—	166	2	no	—
271	D₄	—	166	2	no	—
272	D₄	—	166	1	no	—
273	C₂×C₄	—	166	1	no	—
274	C₂×C₄	—	166	2	no	—
275	C₂×C₄	—	166	2	no	—
276	C₂×C₄	#14581763	1	1	yes	characteristic
277	C₂×C₄	#14581763	1	1	yes	characteristic
278	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
279	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
280	C₂×C₄	#14581811	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
282	C₈	—	332	4	no	—
283	C₈	—	166	2	no	—
284	C₈	#14581763	1	1	yes	characteristic
285	C₈	#14581763	1	1	yes	characteristic
286	C₈	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
287	C₈	C₂×#13050589	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	16	4	no	—
291	C₂²	—	16	4	no	—
292	C₂²	—	16	4	no	—
293	C₂²	—	332	2	no	—
294	C₂²	—	166	2	no	—
295	C₂²	#19127934	1	1	yes	characteristic
296	C₂²	#19127934	1	1	yes	characteristic
297	C₂²	C₂×#14581811	1	1	yes	characteristic
298	C₄	—	166	2	no	—
299	C₄	#19127934	1	1	yes	characteristic
300	C₄	#19127934	1	1	yes	characteristic
301	C₄	C₂×#14581763	1	1	yes	characteristic
302	C₄	C₂×D₈×D₈₃	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	16	4	no	—
306	C₂	—	332	2	no	—
307	C₂	#24909580	1	2	yes	direct product factor
308	C₂	C₂×#19127934	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19127934
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂²×D₁₆×C₈₃	C₂
socle	C₂²×C₈₃	#466
hypercenter	C₂²×D₁₆	D₈₃
hypocenter	C₈₃	C₂×#2221
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19128117	C₂	semidirect
2. C₂×#19127968	C₂	semidirect
3. #24909580	C₂	direct (N × H)
4. C₂×#19127531	C₂	semidirect
5. C₂×#19127514	C₂	semidirect
6. C₂²×D₁₆×C₈₃	C₂	semidirect
10. #19128117	C₂²	semidirect
15. #19127968	C₂²	semidirect
18. #19127531	C₂²	semidirect
19. #19127514	C₂²	semidirect
24. C₂×D₁₆×C₈₃	C₂²	semidirect
152. C₂×C₈₃	#2221	semidirect
155. C₂²×D₁₆	D₈₃	semidirect
161. C₈₃	C₂×#2221	semidirect
168. C₂×D₁₆	C₂×D₈₃	semidirect
307. C₂	#24909580	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×#19128117	2
C₂×#19127968	2
#24909580	2
C₂×#19127531	2
C₂×#19127514	2
C₂²×D₁₆×C₈₃	2
C₂×#2221	83

Subgroups — C2×#24909580 #35558958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909580 #35558958