Subgroups — C₂×#22603153 #32911352

Order 18176.

59186 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22603153	C₂	1	64	yes	maximal, direct product factor
2	C₂×#17986835	C₂	1	15	yes	maximal, complemented
3	C₂×#17986834	C₂	1	20	yes	maximal, complemented
4	C₂²×#14556175	C₂	1	15	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×D₇₁	C₂	1	12	yes	maximal, complemented
6	C₂×#585×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986835	C₂²	1	480	yes	complemented
8	#17986834	C₂²	1	640	yes	complemented
9	C₂×#14556176	C₂²	1	30	yes	complemented
10	C₂×#14556175	C₂²	1	120	yes	complemented
11	C₂×#14556175	C₂²	1	60	yes	complemented
12	C₂×#14556175	C₂²	1	480	yes	complemented
13	2₋¹⁺⁴×D₇₁	C₂²	1	384	yes	complemented
14	C₂×#14556174	C₂²	1	20	yes	complemented
15	C₂×#14556173	C₂²	1	90	yes	complemented
16	C₂²×#11765865	C₂²	1	45	yes	complemented
17	C₂²×#11765864	C₂²	1	90	yes	complemented
18	C₂×#41×D₇₁	C₂²	1	120	yes	complemented
19	C₂³×Dic₁₄₂	C₂²	1	15	yes	complemented
20	C₂²×Q₈×D₇₁	C₂²	1	30	yes	complemented
21	#585×C₇₁	C₂²	1	32	yes	complemented
22	C₂×2₋¹⁺⁴×C₇₁	C₂²	1	6	yes	complemented
23	C₂×H(5,𝔽₂)×C₇₁	C₂²	1	10	yes	complemented
24	C₂²×#41×C₇₁	C₂²	1	15	yes	complemented
25	#14556176	C₂³	1	480	yes	complemented
26	#14556175	C₂³	1	1920	yes	complemented
27	#14556175	C₂³	1	960	yes	complemented
28	#14556174	C₂³	1	320	yes	complemented
29	#14556173	C₂³	1	1440	yes	complemented
30	C₂×#11765866	C₂³	1	40	yes	—
31	C₂×#11765865	C₂³	1	180	yes	—
32	C₂×#11765865	C₂³	1	60	yes	—
33	C₂×#11765865	C₂³	1	720	yes	complemented
34	C₂×#11765864	C₂³	1	360	yes	—
35	C₂×#11765864	C₂³	1	1440	yes	complemented
36	#41×D₇₁	C₂³	1	1920	yes	complemented
37	C₂²×Dic₁₄₂	C₂³	1	90	yes	—
38	C₂²×Dic₁₄₂	C₂³	1	45	yes	—
39	C₂²×Dic₁₄₂	C₂³	1	240	yes	complemented
40	C₂×Q₈×D₇₁	C₂³	1	120	yes	—
41	C₂×Q₈×D₇₁	C₂³	1	480	yes	complemented
42	C₂²×D₂₈₄	C₂³	1	15	yes	—
43	C₂²×C₂²⋊D₇₁	C₂³	1	90	yes	—
44	C₂×D₄×D₇₁	C₂³	1	120	yes	—
45	C₂³×Dic₇₁	C₂³	1	30	yes	—
46	C₂²×C₄×D₇₁	C₂³	1	90	yes	—
47	2₋¹⁺⁴×C₇₁	C₂³	1	96	yes	complemented
48	H(5,𝔽₂)×C₇₁	C₂³	1	160	yes	complemented
49	C₂×#41×C₇₁	C₂³	1	60	yes	—
50	C₂×#41×C₇₁	C₂³	1	20	yes	—
51	C₂×#41×C₇₁	C₂³	1	240	yes	complemented
52	C₂²×Q₈×C₇₁	C₂³	1	15	yes	—
53	C₂²×D₄×C₇₁	C₂³	1	45	yes	—
54	C₂³×C₄×C₇₁	C₂³	1	15	yes	—
55	C₂³×D₇₁	C₂⁴	1	30	yes	—
56	#11765866	C₂⁴	1	320	yes	—
57	#11765865	C₂⁴	1	1440	yes	—
58	#11765865	C₂⁴	1	480	yes	—
59	#11765864	C₂⁴	1	2880	yes	—
60	C₂×Dic₁₄₂	C₂⁴	1	90	yes	—
61	C₂×Dic₁₄₂	C₂⁴	1	720	yes	—
62	C₂×Dic₁₄₂	C₂⁴	1	10	yes	—
63	C₂×Dic₁₄₂	C₂⁴	1	360	yes	—
64	Q₈×D₇₁	C₂⁴	1	960	yes	—
65	C₂×D₂₈₄	C₂⁴	1	30	yes	—
66	C₂×D₂₈₄	C₂⁴	1	6	yes	—
67	C₂×D₂₈₄	C₂⁴	1	120	yes	—
68	C₂×C₂²⋊D₇₁	C₂⁴	1	120	yes	—
69	C₂×C₂²⋊D₇₁	C₂⁴	1	720	yes	—
70	D₄×D₇₁	C₂⁴	1	960	yes	—
71	C₂²×Dic₇₁	C₂⁴	1	90	yes	—
72	C₂²×Dic₇₁	C₂⁴	1	240	yes	—
73	C₂×C₄×D₇₁	C₂⁴	1	120	yes	—
74	C₂×C₄×D₇₁	C₂⁴	1	720	yes	—
75	C₂⁴×C₇₁	C₂⁴	1	15	yes	—
76	#41×C₇₁	C₂⁴	1	480	yes	—
77	#41×C₇₁	C₂⁴	1	160	yes	—
78	C₂×Q₈×C₇₁	C₂⁴	1	20	yes	—
79	C₂×Q₈×C₇₁	C₂⁴	1	120	yes	—
80	C₂×D₄×C₇₁	C₂⁴	1	60	yes	—
81	C₂×D₄×C₇₁	C₂⁴	1	360	yes	—
82	C₂²×C₄×C₇₁	C₂⁴	1	45	yes	—
83	C₂²×C₄×C₇₁	C₂⁴	1	15	yes	—
84	C₂²×C₄×C₇₁	C₂⁴	1	120	yes	—
85	C₂³×C₇₁	—	4	30	no	—
86	C₂³×C₇₁	C₂⁵	1	15	yes	—
87	C₂³×C₇₁	C₂⁵	1	60	yes	—
88	C₂²×D₇₁	—	4	60	no	—
89	C₂²×D₇₁	C₂⁵	1	12	yes	—
90	C₂²×D₇₁	C₂⁵	1	120	yes	—
91	Q₈×C₇₁	C₂⁵	1	80	yes	—
92	D₄×C₇₁	C₂⁵	1	240	yes	—
93	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂⁵	1	15	yes	—
95	C₂×C₄×C₇₁	C₂⁵	1	60	yes	—
96	C₂×C₄×C₇₁	C₂⁵	1	180	yes	—
97	C₂²⋊D₇₁	C₂⁵	1	480	yes	—
98	C₂×Dic₇₁	C₂⁵	1	20	yes	—
99	C₂×Dic₇₁	C₂⁵	1	360	yes	—
100	D₂₈₄	C₂⁵	1	120	yes	—
101	D₂₈₄	C₂⁵	1	24	yes	—
102	C₄×D₇₁	C₂⁵	1	480	yes	—
103	Dic₁₄₂	C₂⁵	1	360	yes	—
104	Dic₁₄₂	C₂⁵	1	40	yes	—
105	C₂²×C₇₁	—	2	30	no	—
106	C₂²×C₇₁	—	4	60	no	—
107	C₂²×C₇₁	C₂⁶	1	30	yes	—
108	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
109	C₂×D₇₁	—	2	24	no	—
110	C₂×D₇₁	—	4	120	no	—
111	C₂×D₇₁	C₂⁶	1	24	yes	—
112	C₄×C₇₁	C₂⁶	1	2	yes	—
113	C₄×C₇₁	C₂⁶	1	30	yes	—
114	Dic₇₁	C₂⁶	1	40	yes	—
115	C₂×2₋¹⁺⁶	—	71	1	no	maximal, Sylow
116	C₂×C₇₁	—	2	30	no	—
117	C₂×C₇₁	2₋¹⁺⁶	1	2	yes	complemented
118	C₂×C₇₁	C₂⁷	1	1	yes	characteristic
119	D₇₁	—	2	24	no	—
120	2₋¹⁺⁶	—	71	64	no	—
121	C₂×#585	—	71	20	no	—
122	C₂×#585	—	71	15	no	—
123	C₂×#585	D₇₁	1	1	yes	characteristic, complemented
124	C₂²×2₋¹⁺⁴	—	71	12	no	—
125	C₂²×2₋¹⁺⁴	—	71	15	no	—
126	C₇₁	C₂×2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
127	#585	—	71	640	no	—
128	#585	—	71	480	no	—
129	#585	C₂×D₇₁	1	32	yes	complemented
130	C₂×2₋¹⁺⁴	—	71	384	no	—
131	C₂×2₋¹⁺⁴	—	71	120	no	—
132	C₂×2₋¹⁺⁴	—	71	60	no	—
133	C₂×2₋¹⁺⁴	—	71	480	no	—
134	C₂×2₋¹⁺⁴	—	71	30	no	—
135	C₂×2₋¹⁺⁴	C₂×D₇₁	1	6	yes	complemented
136	C₂×H(5,𝔽₂)	—	71	20	no	—
137	C₂×H(5,𝔽₂)	—	71	90	no	—
138	C₂×H(5,𝔽₂)	C₂×D₇₁	1	10	yes	complemented
139	C₂²×#41	—	71	120	no	—
140	C₂²×#41	—	71	90	no	—
141	C₂²×#41	—	71	45	no	—
142	C₂²×#41	C₂×D₇₁	1	15	yes	complemented
143	C₂³×Q₈	—	71	30	no	—
144	C₂³×Q₈	—	71	15	no	—
145	2₋¹⁺⁴	—	71	1920	no	—
146	2₋¹⁺⁴	—	71	960	no	—
147	2₋¹⁺⁴	—	71	480	no	—
148	2₋¹⁺⁴	C₂²×D₇₁	1	96	yes	complemented
149	H(5,𝔽₂)	—	71	320	no	—
150	H(5,𝔽₂)	—	71	1440	no	—
151	H(5,𝔽₂)	C₂²×D₇₁	1	160	yes	complemented
152	C₂×#41	—	71	1920	no	—
153	C₂×#41	—	71	40	no	—
154	C₂×#41	—	71	360	no	—
155	C₂×#41	—	71	1440	no	—
156	C₂×#41	—	71	180	no	—
157	C₂×#41	—	71	60	no	—
158	C₂×#41	—	71	720	no	—
159	C₂×#41	C₂²×D₇₁	1	60	yes	—
160	C₂×#41	C₂²×D₇₁	1	20	yes	—
161	C₂×#41	C₂²×D₇₁	1	240	yes	complemented
162	C₂²×Q₈	—	71	120	no	—
163	C₂²×Q₈	—	71	480	no	—
164	C₂²×Q₈	—	71	90	no	—
165	C₂²×Q₈	—	71	45	no	—
166	C₂²×Q₈	—	71	240	no	—
167	C₂²×Q₈	C₂²×D₇₁	1	15	yes	—
168	C₂²×D₄	—	71	90	no	—
169	C₂²×D₄	—	71	120	no	—
170	C₂²×D₄	—	71	15	no	—
171	C₂²×D₄	C₂²×D₇₁	1	45	yes	—
172	C₂³×C₄	—	71	30	no	—
173	C₂³×C₄	—	71	90	no	—
174	C₂³×C₄	C₂²×D₇₁	1	15	yes	—
175	C₂⁴	—	71	30	no	—
176	C₂⁴	C₂³×D₇₁	1	15	yes	—
177	#41	—	71	320	no	—
178	#41	—	71	2880	no	—
179	#41	—	71	1440	no	—
180	#41	—	71	480	no	—
181	#41	C₂³×D₇₁	1	480	yes	—
182	#41	C₂³×D₇₁	1	160	yes	—
183	C₂×Q₈	—	71	960	no	—
184	C₂×Q₈	—	71	90	no	—
185	C₂×Q₈	—	71	10	no	—
186	C₂×Q₈	—	71	720	no	—
187	C₂×Q₈	—	71	360	no	—
188	C₂×Q₈	C₂³×D₇₁	1	20	yes	—
189	C₂×Q₈	C₂³×D₇₁	1	120	yes	—
190	C₂×D₄	—	71	120	no	—
191	C₂×D₄	—	71	720	no	—
192	C₂×D₄	—	71	960	no	—
193	C₂×D₄	—	71	30	no	—
194	C₂×D₄	—	71	6	no	—
195	C₂×D₄	—	71	120	no	—
196	C₂×D₄	C₂³×D₇₁	1	60	yes	—
197	C₂×D₄	C₂³×D₇₁	1	360	yes	—
198	C₂²×C₄	—	71	90	no	—
199	C₂²×C₄	—	71	240	no	—
200	C₂²×C₄	—	71	120	no	—
201	C₂²×C₄	—	71	720	no	—
202	C₂²×C₄	C₂³×D₇₁	1	45	yes	—
203	C₂²×C₄	C₂³×D₇₁	1	15	yes	—
204	C₂²×C₄	C₂³×D₇₁	1	120	yes	—
205	C₂³	—	4	30	no	—
206	C₂³	—	284	60	no	—
207	C₂³	—	71	120	no	—
208	C₂³	—	71	12	no	—
209	C₂³	C₂⁴×D₇₁	1	15	yes	—
210	C₂³	C₂⁴×D₇₁	1	60	yes	—
211	Q₈	—	71	360	no	—
212	Q₈	—	71	40	no	—
213	Q₈	C₂⁴×D₇₁	1	80	yes	—
214	D₄	—	71	480	no	—
215	D₄	—	71	120	no	—
216	D₄	—	71	24	no	—
217	D₄	C₂⁴×D₇₁	1	240	yes	—
218	C₂×C₄	—	71	360	no	—
219	C₂×C₄	—	71	20	no	—
220	C₂×C₄	—	71	480	no	—
221	C₂×C₄	C₂⁴×D₇₁	1	180	yes	—
222	C₂×C₄	C₂⁴×D₇₁	1	60	yes	—
223	C₂×C₄	C₂⁴×D₇₁	1	15	yes	—
224	C₂×C₄	C₂⁴×D₇₁	1	1	yes	characteristic
225	C₂²	—	2	30	no	—
226	C₂²	—	4	60	no	—
227	C₂²	—	284	120	no	—
228	C₂²	—	71	24	no	—
229	C₂²	—	142	24	no	—
230	C₂²	C₂⁵×D₇₁	1	30	yes	—
231	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
232	C₄	—	71	40	no	—
233	C₄	C₂⁵×D₇₁	1	30	yes	—
234	C₄	C₂⁵×D₇₁	1	2	yes	—
235	C₂	—	2	30	no	—
236	C₂	—	142	24	no	—
237	C₂	#22603153	1	2	yes	direct product factor
238	C₂	C₂⁶×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁵×D₇₁
commutator, cosocle	C₂×C₇₁	C₂⁷
Frattini	C₂	C₂⁶×D₇₁
Fitting	C₂×#585×C₇₁	C₂
socle	C₂²×C₇₁	C₂⁶
hypercenter	C₂×#585	D₇₁
hypocenter	C₇₁	C₂×2₋¹⁺⁶
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. #22603153	C₂	direct (N × H)
2. C₂×#17986835	C₂	semidirect
3. C₂×#17986834	C₂	semidirect
4. C₂²×#14556175	C₂	semidirect
5. C₂×2₋¹⁺⁴×D₇₁	C₂	semidirect
6. C₂×#585×C₇₁	C₂	semidirect
7. #17986835	C₂²	semidirect
8. #17986834	C₂²	semidirect
9. C₂×#14556176	C₂²	semidirect
10. C₂×#14556175	C₂²	semidirect
11. C₂×#14556175	C₂²	semidirect
12. C₂×#14556175	C₂²	semidirect
13. 2₋¹⁺⁴×D₇₁	C₂²	semidirect
14. C₂×#14556174	C₂²	semidirect
15. C₂×#14556173	C₂²	semidirect
16. C₂²×#11765865	C₂²	semidirect
17. C₂²×#11765864	C₂²	semidirect
18. C₂×#41×D₇₁	C₂²	semidirect
19. C₂³×Dic₁₄₂	C₂²	semidirect
20. C₂²×Q₈×D₇₁	C₂²	semidirect
21. #585×C₇₁	C₂²	semidirect
22. C₂×2₋¹⁺⁴×C₇₁	C₂²	semidirect
23. C₂×H(5,𝔽₂)×C₇₁	C₂²	semidirect
24. C₂²×#41×C₇₁	C₂²	semidirect
25. #14556176	C₂³	semidirect
26. #14556175	C₂³	semidirect
27. #14556175	C₂³	semidirect
28. #14556174	C₂³	semidirect
29. #14556173	C₂³	semidirect
33. C₂×#11765865	C₂³	semidirect
35. C₂×#11765864	C₂³	semidirect
36. #41×D₇₁	C₂³	semidirect
39. C₂²×Dic₁₄₂	C₂³	semidirect
41. C₂×Q₈×D₇₁	C₂³	semidirect
47. 2₋¹⁺⁴×C₇₁	C₂³	semidirect
48. H(5,𝔽₂)×C₇₁	C₂³	semidirect
51. C₂×#41×C₇₁	C₂³	semidirect
117. C₂×C₇₁	2₋¹⁺⁶	semidirect
123. C₂×#585	D₇₁	semidirect
126. C₇₁	C₂×2₋¹⁺⁶	semidirect
129. #585	C₂×D₇₁	semidirect
135. C₂×2₋¹⁺⁴	C₂×D₇₁	semidirect
138. C₂×H(5,𝔽₂)	C₂×D₇₁	semidirect
142. C₂²×#41	C₂×D₇₁	semidirect
148. 2₋¹⁺⁴	C₂²×D₇₁	semidirect
151. H(5,𝔽₂)	C₂²×D₇₁	semidirect
161. C₂×#41	C₂²×D₇₁	semidirect
237. C₂	#22603153	direct (N × H)

Maximal subgroups

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

subgroup	index
#22603153	2
C₂×#17986835	2
C₂×#17986834	2
C₂²×#14556175	2
C₂×2₋¹⁺⁴×D₇₁	2
C₂×#585×C₇₁	2
C₂×2₋¹⁺⁶	71

Subgroups — C2×#22603153 #32911352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22603153 #32911352