Subgroups — C₂×#22581789 #31821212

Order 17792.

5956 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581789	C₂	1	32	yes	maximal, direct product factor
2	C₂×#17984890	C₂	1	2	yes	maximal, complemented
3	C₂×#17984889	C₂	1	4	yes	maximal, complemented
4	C₂×#17984888	C₂	1	4	yes	maximal, complemented
5	C₂×#17984887	C₂	1	6	yes	maximal, complemented
6	C₂²×#14555785	C₂	1	6	yes	maximal, complemented
7	C₂×#41×D₁₃₉	C₂	1	8	yes	maximal, complemented
8	C₂²×#41×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#17984890	C₂²	1	32	yes	complemented
10	#17984889	C₂²	1	64	yes	complemented
11	#17984888	C₂²	1	64	yes	complemented
12	#17984887	C₂²	1	96	yes	complemented
13	C₂×#14555786	C₂²	1	8	yes	complemented
14	C₂×#14555785	C₂²	1	12	yes	complemented
15	C₂×#14555785	C₂²	1	4	yes	complemented
16	C₂×#14555785	C₂²	1	12	yes	complemented
17	C₂×#14555785	C₂²	1	12	yes	complemented
18	C₂×#14555785	C₂²	1	96	yes	complemented
19	C₂×#14555784	C₂²	1	24	yes	complemented
20	#41×D₁₃₉	C₂²	1	128	yes	complemented
21	C₂²×Dic₂₇₈	C₂²	1	6	yes	complemented
22	C₂×Q₈×D₁₃₉	C₂²	1	8	yes	complemented
23	C₂²×D₅₅₆	C₂²	1	6	yes	complemented
24	C₂²×C₂²⋊D₁₃₉	C₂²	1	12	yes	complemented
25	C₂×D₄×D₁₃₉	C₂²	1	24	yes	complemented
26	C₂²×C₄×D₁₃₉	C₂²	1	12	yes	complemented
27	C₂×#41×C₁₃₉	C₂²	1	4	yes	complemented
28	C₂×#41×C₁₃₉	C₂²	1	4	yes	complemented
29	C₂×#41×C₁₃₉	C₂²	1	16	yes	complemented
30	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
31	C₂²×D₄×C₁₃₉	C₂²	1	3	yes	complemented
32	C₂³×C₄×C₁₃₉	C₂²	1	3	yes	complemented
33	C₂³×D₁₃₉	C₂³	1	12	yes	—
34	#14555786	C₂³	1	64	yes	complemented
35	#14555785	C₂³	1	96	yes	complemented
36	#14555785	C₂³	1	32	yes	complemented
37	#14555785	C₂³	1	96	yes	complemented
38	#14555785	C₂³	1	96	yes	complemented
39	#14555784	C₂³	1	192	yes	complemented
40	C₂×Dic₂₇₈	C₂³	1	12	yes	—
41	C₂×Dic₂₇₈	C₂³	1	4	yes	—
42	C₂×Dic₂₇₈	C₂³	1	48	yes	complemented
43	Q₈×D₁₃₉	C₂³	1	64	yes	complemented
44	C₂×D₅₅₆	C₂³	1	12	yes	—
45	C₂×D₅₅₆	C₂³	1	4	yes	—
46	C₂×D₅₅₆	C₂³	1	48	yes	complemented
47	C₂×C₂²⋊D₁₃₉	C₂³	1	24	yes	—
48	C₂×C₂²⋊D₁₃₉	C₂³	1	8	yes	—
49	C₂×C₂²⋊D₁₃₉	C₂³	1	96	yes	complemented
50	D₄×D₁₃₉	C₂³	1	192	yes	complemented
51	C₂²×Dic₁₃₉	C₂³	1	12	yes	—
52	C₂×C₄×D₁₃₉	C₂³	1	8	yes	—
53	C₂×C₄×D₁₃₉	C₂³	1	24	yes	—
54	C₂×C₄×D₁₃₉	C₂³	1	96	yes	complemented
55	C₂⁴×C₁₃₉	C₂³	1	3	yes	—
56	#41×C₁₃₉	C₂³	1	32	yes	complemented
57	#41×C₁₃₉	C₂³	1	32	yes	complemented
58	C₂×Q₈×C₁₃₉	C₂³	1	4	yes	—
59	C₂×Q₈×C₁₃₉	C₂³	1	8	yes	complemented
60	C₂×D₄×C₁₃₉	C₂³	1	12	yes	—
61	C₂×D₄×C₁₃₉	C₂³	1	24	yes	complemented
62	C₂²×C₄×C₁₃₉	C₂³	1	6	yes	—
63	C₂²×C₄×C₁₃₉	C₂³	1	6	yes	—
64	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
66	C₂²×C₄×C₁₃₉	C₂³	1	24	yes	complemented
67	C₂²×D₁₃₉	—	4	24	no	—
68	C₂²×D₁₃₉	C₂⁴	1	8	yes	—
69	C₂²×D₁₃₉	C₂⁴	1	48	yes	—
70	Dic₂₇₈	C₂⁴	1	48	yes	—
71	Dic₂₇₈	C₂⁴	1	16	yes	—
72	D₅₅₆	C₂⁴	1	48	yes	—
73	D₅₅₆	C₂⁴	1	16	yes	—
74	C₂²⋊D₁₃₉	C₂⁴	1	96	yes	—
75	C₂²⋊D₁₃₉	C₂⁴	1	32	yes	—
76	C₂×Dic₁₃₉	C₂⁴	1	8	yes	—
77	C₂×Dic₁₃₉	C₂⁴	1	48	yes	—
78	C₄×D₁₃₉	C₂⁴	1	32	yes	—
79	C₄×D₁₃₉	C₂⁴	1	96	yes	—
80	C₂³×C₁₃₉	—	4	6	no	—
81	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
82	C₂³×C₁₃₉	C₂⁴	1	6	yes	—
83	C₂³×C₁₃₉	C₂⁴	1	12	yes	—
84	Q₈×C₁₃₉	C₂⁴	1	16	yes	—
85	D₄×C₁₃₉	C₂⁴	1	48	yes	—
86	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
87	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
88	C₂×C₄×C₁₃₉	C₂⁴	1	4	yes	—
89	C₂×C₄×C₁₃₉	C₂⁴	1	12	yes	—
90	C₂×C₄×C₁₃₉	C₂⁴	1	24	yes	—
91	C₂×C₄×C₁₃₉	C₂⁴	1	24	yes	—
92	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
93	C₂²×C₁₃₉	—	2	2	no	—
94	C₂²×C₁₃₉	—	2	12	no	—
95	C₂²×C₁₃₉	—	4	12	no	—
96	C₂²×C₁₃₉	C₂⁵	1	2	yes	—
97	C₂²×C₁₃₉	C₂⁵	1	12	yes	—
98	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
99	C₂×D₁₃₉	—	2	16	no	—
100	C₂×D₁₃₉	—	4	48	no	—
101	C₂×D₁₃₉	C₂⁵	1	16	yes	—
102	C₄×C₁₃₉	C₂⁵	1	2	yes	—
103	C₄×C₁₃₉	C₂⁵	1	2	yes	—
104	C₄×C₁₃₉	C₂⁵	1	12	yes	—
105	Dic₁₃₉	C₂⁵	1	16	yes	—
106	C₂×C₁₃₉	—	2	12	no	—
107	C₂×C₁₃₉	—	2	2	no	—
108	C₂×C₁₃₉	#585	1	2	yes	complemented
109	C₂×C₁₃₉	C₂⁶	1	1	yes	characteristic
110	D₁₃₉	—	2	16	no	—
111	C₁₃₉	C₂×#585	1	1	yes	characteristic, complemented, Sylow
112	C₂×#585	—	139	1	no	maximal, Sylow
113	#585	—	139	32	no	—
114	C₂×2₋¹⁺⁴	—	139	4	no	—
115	C₂×2₋¹⁺⁴	—	139	2	no	—
116	C₂×H(5,𝔽₂)	—	139	4	no	—
117	C₂×H(5,𝔽₂)	—	139	6	no	—
118	C₂²×#41	—	139	8	no	—
119	C₂²×#41	—	139	6	no	—
120	C₂²×#41	D₁₃₉	1	1	yes	characteristic, complemented
121	2₋¹⁺⁴	—	139	64	no	—
122	2₋¹⁺⁴	—	139	32	no	—
123	H(5,𝔽₂)	—	139	64	no	—
124	H(5,𝔽₂)	—	139	96	no	—
125	C₂×#41	—	139	128	no	—
126	C₂×#41	—	139	8	no	—
127	C₂×#41	—	139	24	no	—
128	C₂×#41	—	139	12	no	—
129	C₂×#41	—	139	4	no	—
130	C₂×#41	—	139	12	no	—
131	C₂×#41	—	139	12	no	—
132	C₂×#41	—	139	96	no	—
133	C₂×#41	C₂×D₁₃₉	1	4	yes	complemented
134	C₂×#41	C₂×D₁₃₉	1	4	yes	complemented
135	C₂×#41	C₂×D₁₃₉	1	16	yes	complemented
136	C₂²×Q₈	—	139	8	no	—
137	C₂²×Q₈	—	139	6	no	—
138	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic, complemented
139	C₂²×D₄	—	139	12	no	—
140	C₂²×D₄	—	139	24	no	—
141	C₂²×D₄	—	139	6	no	—
142	C₂²×D₄	C₂×D₁₃₉	1	3	yes	complemented
143	C₂³×C₄	—	139	12	no	—
144	C₂³×C₄	C₂×D₁₃₉	1	3	yes	complemented
145	C₂⁴	—	139	12	no	—
146	C₂⁴	C₂²×D₁₃₉	1	3	yes	—
147	#41	—	139	64	no	—
148	#41	—	139	192	no	—
149	#41	—	139	96	no	—
150	#41	—	139	32	no	—
151	#41	—	139	96	no	—
152	#41	—	139	96	no	—
153	#41	C₂²×D₁₃₉	1	32	yes	complemented
154	#41	C₂²×D₁₃₉	1	32	yes	complemented
155	C₂×Q₈	—	139	64	no	—
156	C₂×Q₈	—	139	12	no	—
157	C₂×Q₈	—	139	4	no	—
158	C₂×Q₈	—	139	48	no	—
159	C₂×Q₈	C₂²×D₁₃₉	1	4	yes	—
160	C₂×Q₈	C₂²×D₁₃₉	1	8	yes	complemented
161	C₂×D₄	—	139	24	no	—
162	C₂×D₄	—	139	8	no	—
163	C₂×D₄	—	139	96	no	—
164	C₂×D₄	—	139	192	no	—
165	C₂×D₄	—	139	12	no	—
166	C₂×D₄	—	139	4	no	—
167	C₂×D₄	—	139	48	no	—
168	C₂×D₄	C₂²×D₁₃₉	1	12	yes	—
169	C₂×D₄	C₂²×D₁₃₉	1	24	yes	complemented
170	C₂²×C₄	—	139	12	no	—
171	C₂²×C₄	—	139	24	no	—
172	C₂²×C₄	—	139	96	no	—
173	C₂²×C₄	—	139	8	no	—
174	C₂²×C₄	C₂²×D₁₃₉	1	6	yes	—
175	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
176	C₂²×C₄	C₂²×D₁₃₉	1	6	yes	—
177	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₃₉	1	24	yes	complemented
179	C₂³	—	4	6	no	—
180	C₂³	—	556	24	no	—
181	C₂³	—	139	8	no	—
182	C₂³	—	139	48	no	—
183	C₂³	C₂³×D₁₃₉	1	6	yes	—
184	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
185	C₂³	C₂³×D₁₃₉	1	12	yes	—
186	Q₈	—	139	48	no	—
187	Q₈	—	139	16	no	—
188	Q₈	C₂³×D₁₃₉	1	16	yes	—
189	D₄	—	139	96	no	—
190	D₄	—	139	32	no	—
191	D₄	—	139	48	no	—
192	D₄	—	139	16	no	—
193	D₄	C₂³×D₁₃₉	1	48	yes	—
194	C₂×C₄	—	139	8	no	—
195	C₂×C₄	—	139	48	no	—
196	C₂×C₄	—	139	32	no	—
197	C₂×C₄	—	139	96	no	—
198	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
199	C₂×C₄	C₂³×D₁₃₉	1	24	yes	—
200	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₃₉	1	4	yes	—
202	C₂×C₄	C₂³×D₁₃₉	1	24	yes	—
203	C₂×C₄	C₂³×D₁₃₉	1	12	yes	—
204	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
205	C₂²	—	2	2	no	—
206	C₂²	—	2	12	no	—
207	C₂²	—	4	12	no	—
208	C₂²	—	278	16	no	—
209	C₂²	—	556	48	no	—
210	C₂²	—	139	16	no	—
211	C₂²	C₂⁴×D₁₃₉	1	12	yes	—
212	C₂²	C₂⁴×D₁₃₉	1	2	yes	—
213	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
214	C₄	—	139	16	no	—
215	C₄	C₂⁴×D₁₃₉	1	2	yes	—
216	C₄	C₂⁴×D₁₃₉	1	12	yes	—
217	C₄	C₂⁴×D₁₃₉	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	2	12	no	—
220	C₂	—	278	16	no	—
221	C₂	#22581789	1	2	yes	direct product factor
222	C₂	C₂⁵×D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃₉
commutator, cosocle	C₂×C₁₃₉	C₂⁶
Frattini	C₂	C₂⁵×D₁₃₉
Fitting	C₂²×#41×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂⁵
hypercenter	C₂²×#41	D₁₃₉
hypocenter	C₁₃₉	C₂×#585
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #22581789	C₂	direct (N × H)
2. C₂×#17984890	C₂	semidirect
3. C₂×#17984889	C₂	semidirect
4. C₂×#17984888	C₂	semidirect
5. C₂×#17984887	C₂	semidirect
6. C₂²×#14555785	C₂	semidirect
7. C₂×#41×D₁₃₉	C₂	semidirect
8. C₂²×#41×C₁₃₉	C₂	semidirect
9. #17984890	C₂²	semidirect
10. #17984889	C₂²	semidirect
11. #17984888	C₂²	semidirect
12. #17984887	C₂²	semidirect
13. C₂×#14555786	C₂²	semidirect
14. C₂×#14555785	C₂²	semidirect
15. C₂×#14555785	C₂²	semidirect
16. C₂×#14555785	C₂²	semidirect
17. C₂×#14555785	C₂²	semidirect
18. C₂×#14555785	C₂²	semidirect
19. C₂×#14555784	C₂²	semidirect
20. #41×D₁₃₉	C₂²	semidirect
21. C₂²×Dic₂₇₈	C₂²	semidirect
22. C₂×Q₈×D₁₃₉	C₂²	semidirect
23. C₂²×D₅₅₆	C₂²	semidirect
24. C₂²×C₂²⋊D₁₃₉	C₂²	semidirect
25. C₂×D₄×D₁₃₉	C₂²	semidirect
26. C₂²×C₄×D₁₃₉	C₂²	semidirect
27. C₂×#41×C₁₃₉	C₂²	semidirect
28. C₂×#41×C₁₃₉	C₂²	semidirect
29. C₂×#41×C₁₃₉	C₂²	semidirect
30. C₂²×Q₈×C₁₃₉	C₂²	semidirect
31. C₂²×D₄×C₁₃₉	C₂²	semidirect
32. C₂³×C₄×C₁₃₉	C₂²	semidirect
34. #14555786	C₂³	semidirect
35. #14555785	C₂³	semidirect
36. #14555785	C₂³	semidirect
37. #14555785	C₂³	semidirect
38. #14555785	C₂³	semidirect
39. #14555784	C₂³	semidirect
42. C₂×Dic₂₇₈	C₂³	semidirect
43. Q₈×D₁₃₉	C₂³	semidirect
46. C₂×D₅₅₆	C₂³	semidirect
49. C₂×C₂²⋊D₁₃₉	C₂³	semidirect
50. D₄×D₁₃₉	C₂³	semidirect
54. C₂×C₄×D₁₃₉	C₂³	semidirect
56. #41×C₁₃₉	C₂³	semidirect
57. #41×C₁₃₉	C₂³	semidirect
59. C₂×Q₈×C₁₃₉	C₂³	semidirect
61. C₂×D₄×C₁₃₉	C₂³	semidirect
66. C₂²×C₄×C₁₃₉	C₂³	semidirect
108. C₂×C₁₃₉	#585	semidirect
111. C₁₃₉	C₂×#585	semidirect
120. C₂²×#41	D₁₃₉	semidirect
133. C₂×#41	C₂×D₁₃₉	semidirect
134. C₂×#41	C₂×D₁₃₉	semidirect
135. C₂×#41	C₂×D₁₃₉	semidirect
138. C₂²×Q₈	C₂×D₁₃₉	semidirect
142. C₂²×D₄	C₂×D₁₃₉	semidirect
144. C₂³×C₄	C₂×D₁₃₉	semidirect
153. #41	C₂²×D₁₃₉	semidirect
154. #41	C₂²×D₁₃₉	semidirect
160. C₂×Q₈	C₂²×D₁₃₉	semidirect
169. C₂×D₄	C₂²×D₁₃₉	semidirect
178. C₂²×C₄	C₂²×D₁₃₉	semidirect
221. C₂	#22581789	direct (N × H)

Maximal subgroups

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

subgroup	index
#22581789	2
C₂×#17984890	2
C₂×#17984889	2
C₂×#17984888	2
C₂×#17984887	2
C₂²×#14555785	2
C₂×#41×D₁₃₉	2
C₂²×#41×C₁₃₉	2
C₂×#585	139

Subgroups — C2×#22581789 #31821212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581789 #31821212