Subgroups — C₂×#14521851 #16860461

Order 3968.

1512 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758566	C₂	1	2	yes	maximal, complemented
2	#14521851	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11758491	C₂	1	4	yes	maximal, complemented
4	C₂×#11758526	C₂	1	2	yes	maximal, complemented
5	C₂×#11758496	C₂	1	2	yes	maximal, complemented
6	C₂²×D₂₄₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈×D₃₁	C₂	1	2	yes	maximal, complemented
8	C₂×#11758478	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
11	#11758566	—	2	8	no	—
12	#11758566	C₂²	1	16	yes	complemented
13	C₂×#92499	C₂²	1	4	yes	complemented
14	C₂×D₄×D₃₁	—	2	4	no	—
15	C₂×D₄×D₃₁	C₂²	1	4	yes	complemented
16	C₂×#92495	—	2	2	no	—
17	C₂×#92495	C₂²	1	1	yes	characteristic
18	C₂²×D₁₂₄	—	2	2	no	—
19	C₂²×D₁₂₄	C₂²	1	2	yes	complemented
20	C₃₁×#135	C₂²	1	8	yes	complemented
21	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
22	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
23	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
25	#11758526	C₂²	1	16	yes	complemented
26	C₂×#92481	C₂²	1	4	yes	complemented
27	C₂×#92479	C₂²	1	4	yes	complemented
28	C₂×#92474	C₂²	1	1	yes	characteristic, complemented
29	#11758496	C₂²	1	16	yes	complemented
30	#11758491	C₂²	1	32	yes	complemented
31	D₈×D₃₁	C₂²	1	16	yes	complemented
32	C₂×D₂₄₈	C₂²	1	2	yes	complemented
33	C₂×D₂₄₈	C₂²	1	2	yes	complemented
34	C₂×D₂₄₈	C₂²	1	8	yes	complemented
35	#11758478	C₂²	1	8	yes	complemented
36	C₂×#92469	C₂²	1	2	yes	complemented
37	C₂×C₈×D₃₁	C₂²	1	2	yes	complemented
38	C₂³×D₃₁	—	2	2	no	—
39	C₂³×D₃₁	—	2	4	no	—
40	#41×C₃₁	—	2	4	no	—
41	#41×C₃₁	C₂³	1	8	yes	complemented
42	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
43	C₂×D₄×C₃₁	—	2	2	no	—
44	C₂×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₃₁	—	2	2	no	—
46	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₃₁	—	2	4	no	—
48	C₂×C₂²⋊D₃₁	D₄	1	2	yes	—
49	#92499	—	2	8	no	—
50	#92499	C₂³	1	16	yes	complemented
51	D₄×D₃₁	—	2	32	no	—
52	D₄×D₃₁	—	2	8	no	—
53	D₄×D₃₁	C₂³	1	16	yes	complemented
54	#92495	—	2	16	no	—
55	#92495	D₄	1	4	yes	—
56	#92495	C₂³	1	4	yes	—
57	C₂×D₁₂₄	—	2	4	no	—
58	C₂×D₁₂₄	—	2	2	no	—
59	C₂×D₁₂₄	—	2	4	no	—
60	C₂×D₁₂₄	—	2	16	no	—
61	C₂×D₁₂₄	D₄	1	1	yes	characteristic
62	C₂×D₁₂₄	C₂³	1	4	yes	—
63	C₂×D₁₂₄	C₂³	1	8	yes	complemented
64	C₂×C₄×D₃₁	—	2	4	no	—
65	C₂×C₄×D₃₁	C₂³	1	2	yes	—
66	C₂×Dic₆₂	D₄	1	1	yes	characteristic
67	Q₁₆×C₃₁	C₂³	1	4	yes	complemented
68	SD₁₆×C₃₁	C₂³	1	8	yes	complemented
69	D₈×C₃₁	C₂³	1	4	yes	complemented
70	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂³	1	4	yes	complemented
73	#92481	C₂³	1	16	yes	complemented
74	#92479	C₂³	1	16	yes	complemented
75	#92474	C₂³	1	4	yes	complemented
76	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
77	D₂₄₈	C₂³	1	8	yes	complemented
78	D₂₄₈	C₂³	1	8	yes	complemented
79	#92469	C₂³	1	8	yes	complemented
80	C₈×D₃₁	C₂³	1	8	yes	complemented
81	C₂³×C₃₁	—	2	2	no	—
82	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×D₃₁	—	2	4	no	—
84	C₂²×D₃₁	—	2	8	no	—
85	C₂²×D₃₁	—	2	16	no	—
86	C₂²×D₃₁	—	8	8	no	—
87	C₂²×D₃₁	—	8	4	no	—
88	C₂²×D₃₁	C₂×D₄	1	2	yes	—
89	Q₈×C₃₁	—	2	2	no	—
90	Q₈×C₃₁	C₂⁴	1	4	yes	—
91	D₄×C₃₁	—	2	8	no	—
92	D₄×C₃₁	—	2	2	no	—
93	D₄×C₃₁	C₂⁴	1	4	yes	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	8	no	—
96	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
100	C₂²⋊D₃₁	—	2	16	no	—
101	C₂²⋊D₃₁	C₂×D₄	1	8	yes	—
102	C₂×Dic₃₁	C₂×D₄	1	2	yes	—
103	D₁₂₄	—	2	16	no	—
104	D₁₂₄	—	2	8	no	—
105	D₁₂₄	—	2	4	no	—
106	D₁₂₄	C₂×D₄	1	4	yes	—
107	D₁₂₄	C₂⁴	1	8	yes	—
108	C₄×D₃₁	—	2	16	no	—
109	C₄×D₃₁	C₂×D₄	1	4	yes	—
110	C₄×D₃₁	C₂⁴	1	4	yes	—
111	Dic₆₂	C₂×D₄	1	4	yes	—
112	C₈×C₃₁	C₂⁴	1	2	yes	—
113	C₈×C₃₁	C₂⁴	1	2	yes	—
114	C₃₁⋊C₈	C₂⁴	1	4	yes	—
115	C₂×#576	—	31	1	no	maximal, Sylow
116	C₂²×C₃₁	—	2	2	no	—
117	C₂²×C₃₁	—	2	4	no	—
118	C₂²×C₃₁	—	4	4	no	—
119	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
120	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₃₁	—	2	4	no	—
122	C₂×D₃₁	—	2	8	no	—
123	C₂×D₃₁	—	4	8	no	—
124	C₂×D₃₁	—	8	16	no	—
125	C₂×D₃₁	—	8	8	no	—
126	C₂×D₃₁	C₂²×D₄	1	4	yes	—
127	C₄×C₃₁	—	2	4	no	—
128	C₄×C₃₁	C₂²×D₄	1	2	yes	—
129	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
131	Dic₃₁	C₂²×D₄	1	4	yes	—
132	C₂×H(5,𝔽₂)	—	31	2	no	—
133	#576	—	31	16	no	—
134	C₂×AGL(1,ℤ/8)	—	31	2	no	—
135	C₂×AGL(1,ℤ/8)	—	31	4	no	—
136	C₂×#135	—	31	2	no	—
137	C₂×#135	D₃₁	1	1	yes	characteristic, complemented
138	C₂²×D₈	—	31	2	no	—
139	C₂²×D₈	—	31	1	no	—
140	C₂×#131	—	31	1	no	—
141	C₂×C₃₁	—	2	2	no	—
142	C₂×C₃₁	—	4	4	no	—
143	C₂×C₃₁	#576	1	2	yes	complemented
144	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
145	D₃₁	—	2	4	no	—
146	D₃₁	—	4	8	no	—
147	H(5,𝔽₂)	—	31	16	no	—
148	H(5,𝔽₂)	—	62	8	no	—
149	C₂×#41	—	31	1	no	—
150	C₂×#41	—	31	4	no	—
151	C₂×#41	—	62	2	no	—
152	C₂×#41	C₂×D₃₁	1	2	yes	complemented
153	C₂²×D₄	—	31	4	no	—
154	C₂²×D₄	—	31	2	no	—
155	C₂²×D₄	—	62	2	no	—
156	C₂²×D₄	—	62	4	no	—
157	AGL(1,ℤ/8)	—	31	16	no	—
158	AGL(1,ℤ/8)	—	31	32	no	—
159	#135	—	31	16	no	—
160	#135	C₂×D₃₁	1	8	yes	complemented
161	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic, complemented
162	C₂×SD₁₆	—	31	4	no	—
163	C₂×SD₁₆	C₂×D₃₁	1	2	yes	complemented
164	C₂×D₈	—	31	4	no	—
165	C₂×D₈	—	31	16	no	—
166	C₂×D₈	—	31	2	no	—
167	C₂×D₈	—	31	2	no	—
168	C₂×D₈	—	31	8	no	—
169	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic, complemented
170	#131	—	31	8	no	—
171	C₂×M₁₆	—	31	1	no	—
172	C₂×M₁₆	—	31	2	no	—
173	C₂²×C₈	—	31	2	no	—
174	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
175	C₃₁	C₂×#576	1	1	yes	characteristic, complemented, Sylow
176	C₂⁴	—	62	4	no	—
177	C₂⁴	—	62	2	no	—
178	#41	—	2	4	no	—
179	#41	—	31	4	no	—
180	#41	—	31	4	no	—
181	#41	—	31	16	no	—
182	#41	—	62	16	no	—
183	#41	—	62	8	no	—
184	#41	C₂²×D₃₁	1	8	yes	complemented
185	C₂×Q₈	—	31	1	no	—
186	C₂×Q₈	C₂²×D₃₁	1	2	yes	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	31	2	no	—
189	C₂×D₄	—	31	16	no	—
190	C₂×D₄	—	31	1	no	—
191	C₂×D₄	—	31	4	no	—
192	C₂×D₄	—	31	8	no	—
193	C₂×D₄	—	62	16	no	—
194	C₂×D₄	—	62	4	no	—
195	C₂×D₄	—	62	8	no	—
196	C₂×D₄	—	62	4	no	—
197	C₂×D₄	—	62	32	no	—
198	C₂×D₄	—	62	2	no	—
199	C₂×D₄	—	62	4	no	—
200	C₂×D₄	C₂²×D₃₁	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	31	2	no	—
203	C₂²×C₄	—	62	4	no	—
204	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
205	Q₁₆	C₂²×D₃₁	1	4	yes	complemented
206	SD₁₆	—	31	16	no	—
207	SD₁₆	C₂²×D₃₁	1	8	yes	complemented
208	D₈	—	31	16	no	—
209	D₈	—	31	8	no	—
210	D₈	—	31	8	no	—
211	D₈	C₂²×D₃₁	1	4	yes	complemented
212	M₁₆	—	31	4	no	—
213	M₁₆	—	31	8	no	—
214	C₂×C₈	—	31	2	no	—
215	C₂×C₈	—	31	8	no	—
216	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
217	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
218	C₂×C₈	C₂²×D₃₁	1	4	yes	complemented
219	C₂³	—	2	2	no	—
220	C₂³	—	31	2	no	—
221	C₂³	—	62	16	no	—
222	C₂³	—	62	4	no	—
223	C₂³	—	62	8	no	—
224	C₂³	—	248	4	no	—
225	C₂³	—	248	8	no	—
226	C₂³	D₄×D₃₁	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	31	4	no	—
229	Q₈	C₂³×D₃₁	1	4	yes	—
230	D₄	—	2	8	no	—
231	D₄	—	2	2	no	—
232	D₄	—	31	8	no	—
233	D₄	—	31	4	no	—
234	D₄	—	31	8	no	—
235	D₄	—	62	16	no	—
236	D₄	—	62	16	no	—
237	D₄	—	62	8	no	—
238	D₄	—	62	4	no	—
239	D₄	C₂³×D₃₁	1	4	yes	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	31	2	no	—
243	C₂×C₄	—	31	4	no	—
244	C₂×C₄	—	31	4	no	—
245	C₂×C₄	—	62	16	no	—
246	C₂×C₄	D₄×D₃₁	1	2	yes	—
247	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂³×D₃₁	1	2	yes	—
249	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
250	C₈	—	31	4	no	—
251	C₈	C₂³×D₃₁	1	2	yes	—
252	C₈	C₂³×D₃₁	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	31	4	no	—
257	C₂²	—	62	4	no	—
258	C₂²	—	62	8	no	—
259	C₂²	—	124	8	no	—
260	C₂²	—	248	8	no	—
261	C₂²	—	248	16	no	—
262	C₂²	C₂×D₄×D₃₁	1	2	yes	—
263	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	31	4	no	—
266	C₄	C₂×D₄×D₃₁	1	2	yes	—
267	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
268	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	4	no	—
271	C₂	—	62	4	no	—
272	C₂	—	124	8	no	—
273	C₂	#14521851	1	2	yes	direct product factor
274	C₂	C₂²×D₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₄×C₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₃₁
Fitting	C₂×C₃₁×#135	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	C₂×#135	D₃₁
hypocenter	C₃₁	C₂×#576
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#11758566	C₂	semidirect
2. #14521851	C₂	direct (N × H)
3. C₂×#11758491	C₂	semidirect
4. C₂×#11758526	C₂	semidirect
5. C₂×#11758496	C₂	semidirect
6. C₂²×D₂₄₈	C₂	semidirect
7. C₂×D₈×D₃₁	C₂	semidirect
8. C₂×#11758478	C₂	semidirect
9. C₂×C₃₁×#135	C₂	semidirect
10. C₂×#41×C₃₁	C₂²	semidirect
12. #11758566	C₂²	semidirect
13. C₂×#92499	C₂²	semidirect
15. C₂×D₄×D₃₁	C₂²	semidirect
19. C₂²×D₁₂₄	C₂²	semidirect
20. C₃₁×#135	C₂²	semidirect
21. C₂×Q₁₆×C₃₁	C₂²	semidirect
22. C₂×SD₁₆×C₃₁	C₂²	semidirect
23. C₂×D₈×C₃₁	C₂²	semidirect
24. C₂²×C₈×C₃₁	C₂²	semidirect
25. #11758526	C₂²	semidirect
26. C₂×#92481	C₂²	semidirect
27. C₂×#92479	C₂²	semidirect
28. C₂×#92474	C₂²	semidirect
29. #11758496	C₂²	semidirect
30. #11758491	C₂²	semidirect
31. D₈×D₃₁	C₂²	semidirect
32. C₂×D₂₄₈	C₂²	semidirect
33. C₂×D₂₄₈	C₂²	semidirect
34. C₂×D₂₄₈	C₂²	semidirect
35. #11758478	C₂²	semidirect
36. C₂×#92469	C₂²	semidirect
37. C₂×C₈×D₃₁	C₂²	semidirect
41. #41×C₃₁	C₂³	semidirect
50. #92499	C₂³	semidirect
53. D₄×D₃₁	C₂³	semidirect
63. C₂×D₁₂₄	C₂³	semidirect
67. Q₁₆×C₃₁	C₂³	semidirect
68. SD₁₆×C₃₁	C₂³	semidirect
69. D₈×C₃₁	C₂³	semidirect
72. C₂×C₈×C₃₁	C₂³	semidirect
73. #92481	C₂³	semidirect
74. #92479	C₂³	semidirect
75. #92474	C₂³	semidirect
77. D₂₄₈	C₂³	semidirect
78. D₂₄₈	C₂³	semidirect
79. #92469	C₂³	semidirect
80. C₈×D₃₁	C₂³	semidirect
137. C₂×#135	D₃₁	semidirect
143. C₂×C₃₁	#576	semidirect
152. C₂×#41	C₂×D₃₁	semidirect
160. #135	C₂×D₃₁	semidirect
161. C₂×Q₁₆	C₂×D₃₁	semidirect
163. C₂×SD₁₆	C₂×D₃₁	semidirect
169. C₂×D₈	C₂×D₃₁	semidirect
174. C₂²×C₈	C₂×D₃₁	semidirect
175. C₃₁	C₂×#576	semidirect
184. #41	C₂²×D₃₁	semidirect
205. Q₁₆	C₂²×D₃₁	semidirect
207. SD₁₆	C₂²×D₃₁	semidirect
211. D₈	C₂²×D₃₁	semidirect
218. C₂×C₈	C₂²×D₃₁	semidirect
273. C₂	#14521851	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×#11758566	2
#14521851	2
C₂×#11758491	2
C₂×#11758526	2
C₂×#11758496	2
C₂²×D₂₄₈	2
C₂×D₈×D₃₁	2
C₂×#11758478	2
C₂×C₃₁×#135	2
C₂×#576	31

Subgroups — C2×#14521851 #16860461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521851 #16860461