Subgroups — C₂×D₄²×C₂₃₉ #44482232

Order 30592.

2536 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₂₃₉	C₂	1	4	yes	maximal, complemented
2	D₄²×C₂₃₉	C₂	1	16	yes	maximal, direct product factor
3	C₂×#127×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#121×C₂₃₉	C₂	1	4	yes	maximal, complemented
5	C₂×#120×C₂₃₉	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×D₄×C₂₃₉	C₂	1	2	yes	maximal, complemented
7	C₂⁵×C₂₃₉	C₂²	1	4	yes	complemented
8	C₂²×D₄×C₂₃₉	—	2	32	no	—
9	C₂²×D₄×C₂₃₉	C₂²	1	2	yes	—
10	C₂²×D₄×C₂₃₉	C₂²	1	8	yes	complemented
11	C₂²×D₄×C₂₃₉	C₂²	1	4	yes	complemented
12	C₂²×D₄×C₂₃₉	C₂²	1	2	yes	complemented
13	C₂²×D₄×C₂₃₉	C₂²	1	32	yes	complemented
14	C₂³×C₄×C₂₃₉	C₂²	1	4	yes	complemented
15	#127×C₂₃₉	C₂²	1	8	yes	complemented
16	#121×C₂₃₉	C₂²	1	32	yes	complemented
17	#120×C₂₃₉	C₂²	1	32	yes	complemented
18	C₄×D₄×C₂₃₉	C₂²	1	16	yes	complemented
19	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	complemented
20	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	4	yes	complemented
21	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	4	yes	complemented
22	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic, complemented
23	C₂⁴×C₂₃₉	—	2	32	no	—
24	C₂⁴×C₂₃₉	—	4	8	no	—
25	C₂⁴×C₂₃₉	C₂³	1	4	yes	—
26	C₂⁴×C₂₃₉	C₂³	1	4	yes	—
27	C₂⁴×C₂₃₉	C₂³	1	16	yes	complemented
28	C₂×D₄×C₂₃₉	—	2	32	no	—
29	C₂×D₄×C₂₃₉	—	2	128	no	—
30	C₂×D₄×C₂₃₉	—	2	16	no	—
31	C₂×D₄×C₂₃₉	—	4	4	no	—
32	C₂×D₄×C₂₃₉	D₄	1	16	yes	direct product factor
33	C₂×D₄×C₂₃₉	C₂³	1	8	yes	—
34	C₂×D₄×C₂₃₉	C₂³	1	32	yes	complemented
35	C₂×D₄×C₂₃₉	C₂³	1	16	yes	complemented
36	C₂×D₄×C₂₃₉	C₂³	1	8	yes	complemented
37	C₂²×C₄×C₂₃₉	—	2	16	no	—
38	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃₉	C₂³	1	16	yes	complemented
42	C₄⋊C₄×C₂₃₉	C₂³	1	8	yes	complemented
43	C₂²⋊C₄×C₂₃₉	C₂³	1	16	yes	complemented
44	C₂²⋊C₄×C₂₃₉	C₂³	1	16	yes	complemented
45	C₄²×C₂₃₉	C₂³	1	4	yes	complemented
46	C₂³×C₂₃₉	—	2	8	no	—
47	C₂³×C₂₃₉	—	2	8	no	—
48	C₂³×C₂₃₉	—	2	16	no	—
49	C₂³×C₂₃₉	—	2	64	no	—
50	C₂³×C₂₃₉	—	2	8	no	—
51	C₂³×C₂₃₉	—	4	16	no	—
52	C₂³×C₂₃₉	—	4	64	no	—
53	C₂³×C₂₃₉	C₂×D₄	1	16	yes	complemented
54	C₂³×C₂₃₉	C₂⁴	1	8	yes	—
55	C₂³×C₂₃₉	C₂⁴	1	8	yes	—
56	C₂³×C₂₃₉	C₂⁴	1	1	yes	characteristic
57	D₄×C₂₃₉	—	2	64	no	—
58	D₄×C₂₃₉	—	2	32	no	—
59	D₄×C₂₃₉	—	4	8	no	—
60	D₄×C₂₃₉	C₂×D₄	1	32	yes	direct product factor
61	C₂×C₄×C₂₃₉	—	2	8	no	—
62	C₂×C₄×C₂₃₉	—	2	2	no	—
63	C₂×C₄×C₂₃₉	—	2	32	no	—
64	C₂×C₄×C₂₃₉	C₂×D₄	1	8	yes	complemented
65	C₂×C₄×C₂₃₉	C₂⁴	1	2	yes	—
66	C₂×C₄×C₂₃₉	C₂⁴	1	8	yes	—
67	C₂×C₄×C₂₃₉	C₂⁴	1	4	yes	—
68	C₂²×C₂₃₉	—	2	32	no	—
69	C₂²×C₂₃₉	—	2	8	no	—
70	C₂²×C₂₃₉	—	2	8	no	—
71	C₂²×C₂₃₉	—	2	16	no	—
72	C₂²×C₂₃₉	—	2	8	no	—
73	C₂²×C₂₃₉	—	4	64	no	—
74	C₂²×C₂₃₉	—	4	16	no	—
75	C₂²×C₂₃₉	C₂²×D₄	1	4	yes	—
76	C₂²×C₂₃₉	C₂²×D₄	1	16	yes	complemented
77	C₂²×C₂₃₉	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₂₃₉	C₂⁵	1	1	yes	characteristic
79	C₄×C₂₃₉	—	2	8	no	—
80	C₄×C₂₃₉	—	2	2	no	—
81	C₄×C₂₃₉	C₂²×D₄	1	8	yes	complemented
82	C₂×C₂₃₉	—	2	16	no	—
83	C₂×C₂₃₉	—	4	8	no	—
84	C₂×C₂₃₉	D₄²	1	4	yes	direct product factor
85	C₂×C₂₃₉	C₂³×D₄	1	2	yes	—
86	C₂×C₂₃₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
87	C₂₃₉	C₂×D₄²	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×D₄²	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×D₄	C₂×C₂₃₉	1	4	yes	complemented
90	D₄²	C₂×C₂₃₉	1	16	yes	direct product factor
91	C₂×#127	C₂×C₂₃₉	1	1	yes	characteristic, complemented
92	C₂×#121	C₂×C₂₃₉	1	4	yes	complemented
93	C₂×#120	C₂×C₂₃₉	1	4	yes	complemented
94	C₂×C₄×D₄	C₂×C₂₃₉	1	2	yes	complemented
95	C₂⁵	C₂²×C₂₃₉	1	4	yes	complemented
96	C₂²×D₄	—	2	32	no	—
97	C₂²×D₄	C₂²×C₂₃₉	1	2	yes	—
98	C₂²×D₄	C₂²×C₂₃₉	1	8	yes	complemented
99	C₂²×D₄	C₂²×C₂₃₉	1	4	yes	complemented
100	C₂²×D₄	C₂²×C₂₃₉	1	2	yes	complemented
101	C₂²×D₄	C₂²×C₂₃₉	1	32	yes	complemented
102	C₂³×C₄	C₂²×C₂₃₉	1	4	yes	complemented
103	#127	C₂²×C₂₃₉	1	8	yes	complemented
104	#121	C₂²×C₂₃₉	1	32	yes	complemented
105	#120	C₂²×C₂₃₉	1	32	yes	complemented
106	C₄×D₄	C₂²×C₂₃₉	1	16	yes	complemented
107	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	complemented
108	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	4	yes	complemented
109	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	4	yes	complemented
110	C₂×C₄²	C₂²×C₂₃₉	1	1	yes	characteristic, complemented
111	C₂⁴	—	2	32	no	—
112	C₂⁴	—	4	8	no	—
113	C₂⁴	C₂³×C₂₃₉	1	4	yes	—
114	C₂⁴	C₂³×C₂₃₉	1	4	yes	—
115	C₂⁴	C₂³×C₂₃₉	1	16	yes	complemented
116	C₂×D₄	—	2	32	no	—
117	C₂×D₄	—	2	128	no	—
118	C₂×D₄	—	2	16	no	—
119	C₂×D₄	—	4	4	no	—
120	C₂×D₄	D₄×C₂₃₉	1	16	yes	direct product factor
121	C₂×D₄	C₂³×C₂₃₉	1	8	yes	—
122	C₂×D₄	C₂³×C₂₃₉	1	32	yes	complemented
123	C₂×D₄	C₂³×C₂₃₉	1	16	yes	complemented
124	C₂×D₄	C₂³×C₂₃₉	1	8	yes	complemented
125	C₂²×C₄	—	2	16	no	—
126	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₂₃₉	1	4	yes	—
128	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
129	C₂²×C₄	C₂³×C₂₃₉	1	16	yes	complemented
130	C₄⋊C₄	C₂³×C₂₃₉	1	8	yes	complemented
131	C₂²⋊C₄	C₂³×C₂₃₉	1	16	yes	complemented
132	C₂²⋊C₄	C₂³×C₂₃₉	1	16	yes	complemented
133	C₄²	C₂³×C₂₃₉	1	4	yes	complemented
134	C₂³	—	2	8	no	—
135	C₂³	—	2	8	no	—
136	C₂³	—	2	16	no	—
137	C₂³	—	2	64	no	—
138	C₂³	—	2	8	no	—
139	C₂³	—	4	16	no	—
140	C₂³	—	4	64	no	—
141	C₂³	C₂×D₄×C₂₃₉	1	16	yes	complemented
142	C₂³	C₂⁴×C₂₃₉	1	8	yes	—
143	C₂³	C₂⁴×C₂₃₉	1	8	yes	—
144	C₂³	C₂⁴×C₂₃₉	1	1	yes	characteristic
145	D₄	—	2	64	no	—
146	D₄	—	2	32	no	—
147	D₄	—	4	8	no	—
148	D₄	C₂×D₄×C₂₃₉	1	32	yes	direct product factor
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	32	no	—
152	C₂×C₄	C₂×D₄×C₂₃₉	1	8	yes	complemented
153	C₂×C₄	C₂⁴×C₂₃₉	1	2	yes	—
154	C₂×C₄	C₂⁴×C₂₃₉	1	8	yes	—
155	C₂×C₄	C₂⁴×C₂₃₉	1	4	yes	—
156	C₂²	—	2	32	no	—
157	C₂²	—	2	8	no	—
158	C₂²	—	2	8	no	—
159	C₂²	—	2	16	no	—
160	C₂²	—	2	8	no	—
161	C₂²	—	4	64	no	—
162	C₂²	—	4	16	no	—
163	C₂²	C₂²×D₄×C₂₃₉	1	4	yes	—
164	C₂²	C₂²×D₄×C₂₃₉	1	16	yes	complemented
165	C₂²	H(5,𝔽₂)×C₂₃₉	1	2	yes	—
166	C₂²	C₂⁵×C₂₃₉	1	1	yes	characteristic
167	C₄	—	2	8	no	—
168	C₄	—	2	2	no	—
169	C₄	C₂²×D₄×C₂₃₉	1	8	yes	complemented
170	C₂	—	2	16	no	—
171	C₂	—	4	8	no	—
172	C₂	D₄²×C₂₃₉	1	4	yes	direct product factor
173	C₂	C₂³×D₄×C₂₃₉	1	2	yes	—
174	C₂	C₂×H(5,𝔽₂)×C₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×C₂₃₉	C₂	semidirect
2. D₄²×C₂₃₉	C₂	direct (N × H)
3. C₂×#127×C₂₃₉	C₂	semidirect
4. C₂×#121×C₂₃₉	C₂	semidirect
5. C₂×#120×C₂₃₉	C₂	semidirect
6. C₂×C₄×D₄×C₂₃₉	C₂	semidirect
7. C₂⁵×C₂₃₉	C₂²	semidirect
10. C₂²×D₄×C₂₃₉	C₂²	semidirect
11. C₂²×D₄×C₂₃₉	C₂²	semidirect
12. C₂²×D₄×C₂₃₉	C₂²	semidirect
13. C₂²×D₄×C₂₃₉	C₂²	semidirect
14. C₂³×C₄×C₂₃₉	C₂²	semidirect
15. #127×C₂₃₉	C₂²	semidirect
16. #121×C₂₃₉	C₂²	semidirect
17. #120×C₂₃₉	C₂²	semidirect
18. C₄×D₄×C₂₃₉	C₂²	semidirect
19. C₂×C₄⋊C₄×C₂₃₉	C₂²	semidirect
20. C₂×C₂²⋊C₄×C₂₃₉	C₂²	semidirect
21. C₂×C₂²⋊C₄×C₂₃₉	C₂²	semidirect
22. C₂×C₄²×C₂₃₉	C₂²	semidirect
27. C₂⁴×C₂₃₉	C₂³	semidirect
32. C₂×D₄×C₂₃₉	D₄	direct (N × H)
34. C₂×D₄×C₂₃₉	C₂³	semidirect
35. C₂×D₄×C₂₃₉	C₂³	semidirect
36. C₂×D₄×C₂₃₉	C₂³	semidirect
41. C₂²×C₄×C₂₃₉	C₂³	semidirect
42. C₄⋊C₄×C₂₃₉	C₂³	semidirect
43. C₂²⋊C₄×C₂₃₉	C₂³	semidirect
44. C₂²⋊C₄×C₂₃₉	C₂³	semidirect
45. C₄²×C₂₃₉	C₂³	semidirect
53. C₂³×C₂₃₉	C₂×D₄	semidirect
60. D₄×C₂₃₉	C₂×D₄	direct (N × H)
64. C₂×C₄×C₂₃₉	C₂×D₄	semidirect
76. C₂²×C₂₃₉	C₂²×D₄	semidirect
81. C₄×C₂₃₉	C₂²×D₄	semidirect
84. C₂×C₂₃₉	D₄²	direct (N × H)
87. C₂₃₉	C₂×D₄²	direct (N × H)
88. C₂×D₄²	C₂₃₉	direct (N × H)
89. C₂³×D₄	C₂×C₂₃₉	semidirect
90. D₄²	C₂×C₂₃₉	direct (N × H)
91. C₂×#127	C₂×C₂₃₉	semidirect
92. C₂×#121	C₂×C₂₃₉	semidirect
93. C₂×#120	C₂×C₂₃₉	semidirect
94. C₂×C₄×D₄	C₂×C₂₃₉	semidirect
95. C₂⁵	C₂²×C₂₃₉	semidirect
98. C₂²×D₄	C₂²×C₂₃₉	semidirect
99. C₂²×D₄	C₂²×C₂₃₉	semidirect
100. C₂²×D₄	C₂²×C₂₃₉	semidirect
101. C₂²×D₄	C₂²×C₂₃₉	semidirect
102. C₂³×C₄	C₂²×C₂₃₉	semidirect
103. #127	C₂²×C₂₃₉	semidirect
104. #121	C₂²×C₂₃₉	semidirect
105. #120	C₂²×C₂₃₉	semidirect
106. C₄×D₄	C₂²×C₂₃₉	semidirect
107. C₂×C₄⋊C₄	C₂²×C₂₃₉	semidirect
108. C₂×C₂²⋊C₄	C₂²×C₂₃₉	semidirect
109. C₂×C₂²⋊C₄	C₂²×C₂₃₉	semidirect
110. C₂×C₄²	C₂²×C₂₃₉	semidirect
115. C₂⁴	C₂³×C₂₃₉	semidirect
120. C₂×D₄	D₄×C₂₃₉	direct (N × H)
122. C₂×D₄	C₂³×C₂₃₉	semidirect
123. C₂×D₄	C₂³×C₂₃₉	semidirect
124. C₂×D₄	C₂³×C₂₃₉	semidirect
129. C₂²×C₄	C₂³×C₂₃₉	semidirect
130. C₄⋊C₄	C₂³×C₂₃₉	semidirect
131. C₂²⋊C₄	C₂³×C₂₃₉	semidirect
132. C₂²⋊C₄	C₂³×C₂₃₉	semidirect
133. C₄²	C₂³×C₂₃₉	semidirect
141. C₂³	C₂×D₄×C₂₃₉	semidirect
148. D₄	C₂×D₄×C₂₃₉	direct (N × H)
152. C₂×C₄	C₂×D₄×C₂₃₉	semidirect
164. C₂²	C₂²×D₄×C₂₃₉	semidirect
169. C₄	C₂²×D₄×C₂₃₉	semidirect
172. C₂	D₄²×C₂₃₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₂₃₉	2
D₄²×C₂₃₉	2
C₂×#127×C₂₃₉	2
C₂×#121×C₂₃₉	2
C₂×#120×C₂₃₉	2
C₂×C₄×D₄×C₂₃₉	2
C₂×D₄²	239

Subgroups — C2×D42×C239 #44482232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄²×C₂₃₉ #44482232